ON THE USE OF MLDS IN THE STUDY OF DEPTH
AND LIGHTNESS PERCEPTION
v orgelegt v on
Guiller mo Andres Aguilar Cornejo, M.S c.
aus Santiago, Chile
v on der Fakultät IV - Elektrotechnik und Informatik
der T echnischen Univ ersität Berlin
zur Erlangung des akademischen Grades
Doktor der Natur wissenschaften
- Dr . rer . nat. -
genehmigte Dissertation
Promotionsausschuss:
V orsitzender: Prof. Dr . Klaus Oberma y er
Gutachterin: Frau Dr . Marianne Maertens
Gutachter: Prof. Dr . Marc Alexa
Gutachter: Prof. Dr . Felix W ichmann
Gutachter: Prof. Dr . Kenneth Knoblauch
T ag der wissenschaftlichen A ussprache: 4 . S eptember 2017
Berlin 2017

T o my grandmother ,
who w as alw a ys there.
1923 – 2015

A B S T R A C T
An open question in vision research is ho w to measure the perceptual dimension
ev oked b y the stimulus in a reliable w a y . Although a v ariety of psy chophysical
procedures ar e a v ailable, it is still a challenge to find methods that are ef ficient
and a v oid critical confounds, such as strategies triggered b y dif ficult and un-
natural tasks used b y discrimination methods. In this doctoral thesis I propose
the use of Maximum Likelihood Dif ference S caling (MLDS, Maloney & Y ang,
2003 ) as a reliable tool for measuring perception. MLDS is a method based on
judgments of appearance of clearly visible stimulus dif ferences in an easy and
intuitiv e task, and it allo ws the estimation of perceptual scales in an ef ficient
w a y .
Here I first use numerical simulations to test the accuracy and precision of the
scales deriv ed with MLDS, and I also tested the ef fect of violations of the model
assumptions. The results of these simulations establish the v alidity of MLDS as
a method for measuring appearance. Then, w e ev aluated MLDS experimentally
in the domain of lightness perception. W e measured perceptual lightness scales
under dif ferent viewing conditions and w e v alidate the deriv ed scales empiri-
cally b y predicting lightness matches that w ere deriv ed in a classical asymmetric
matching task. A large practical benefit of MLDS is that it r enders the task easy
for the subject and thus minimizing the potential influence of strategies. At the
same time the perceptual scales pr o vide a more direct estimate of internal v ari-
ables against which theoretical models of appearance can be tested.
In a third part I study the r elationship betw een MLDS and discrimination
methods as suggested b y Devinck & Knoblauch ( 2012 ). In simulations MLDS
w as more efficient than the traditional 2 -AFC discrimination method while at
the same time pro viding analogous sensitivity estimates. I also tested this equiv-
alence experimentally in a slant-from-texture task, for which sensitivity has been
pre viously studied in the literature. Here I found v ar ying degrees of equiv alence
and it remains to be tested in the future whether these dif ferences are due to
v

true dif ferences in the perceptual repr esentation, or to violations of the model
assumptions.
T ogether with the use of realistic stimuli, MLDS offers a r eliable method to
measure the perceptual dimension, and in that w a y enabling the testing of theo-
retical models of perceptual infer ence.
ZUSAMMENF ASSUNG
Eine of fene Frage in der visuellen W ahrnehmungsforschung ist, wie sich eine
durch einen Stimulus e v ozierte W ahr nehmungsdimension reliabel messen lässt.
T rotz einer A usw ahl existierender psy chophysischer Methoden, bleibt es eine
Herausforderung ef fiziente Methoden zu finden, die kritische Konfundierungen
v er hinder n, wie es zum Beispiel bei Diskriminationsaufgaben mit schwierigen
und unnatürlichen A ufgaben der Fall sein kann. In dieser Dissertation stelle ich
Maximum Likelihood Differ ence Scaling (MLDS, Maloney & Y ang, 2003 ) als eine re-
liable Methode zur Messung v on W ahr nehmungseindrücken v or . MLDS basiert
auf der Bew ertung deutlich sichtbarer Stimulusunterschiede in einer intuitiv en
und einfachen A ufgabe und er möglicht die effiziente und reliable Schätzung
perzeptueller Skalen.
In einem ersten S chritt wird MLDS als eine zuv erlässig Methode zur Messung
v on W ahr nehmunseindrücken etabliert, indem ihre Genauigkeit und Präzision
bestimmt wird so wie V erletzungen v on Modellannahmen numerisch simuliert
w erden. In einem nächsten Schritt wird MLDS im Bereich der Helligkeitsw ahr-
nehmung experimentell ev aluiert, indem gezeigt wird, dass MLDS erfolgr eich
W ahr nehmungsskalen für v erschiedene visuelle Kontextbedingungen bestimmt.
Die gemessenen Skalen w eisen w eitgehend Helligkeitskonstanz auf. MLDS er -
forderte dafür den V ergleich v on Stimuli innerhalb eines visuellen Kontexts, w as
zu einer V ereinfachung der A ufgabe für die V ersuchsperson sowie zur V er mei-
dung v on Problemen geführt hat, w elche mit V ergleichen über visuelle Kontext-
bedingungen hinw eg in V erbindung gebracht w er den. Zusätzlich hierzu, schien
MLDS der Methode des asymmetrischen V ergleichs für die Bestimmung v on Hel-
vi

ligkeitskonstanz überlegen, da diese im Gegensatz zu MLDS nur ein indirektes
Maß liefer n kann und W ahr nehmungsskalen nicht direkt misst.
Der Zusammenhang zwischen MLDS und Diskriminanzv erfahren wur de auch
im Rahmen der Signalentdeckungstheorie untersucht, wie zuletzt v on Devinck &
Knoblauch ( 2012 ) v orgeschlagen. Simulationen haben gezeigt, dass MLDS ef fizi-
enter ist und in der S ensitivitätsschätzung quantitativ ähnlich zu traditionellen
2 -AFC Diskriminanzv erfahren, jedoch nur w enn alle Modellannahmen erfüllt
w aren. Diese Äquiv alenz wur de zudem experimentell in einer slant-from-textur e
A ufgabe getestet, für die S ensitivitätsmaße bereits untersucht wurden. Ich fand
unterschiedliche Abstufungen v on Übereinstimmung, die entw eder tatsächliche
Unterschiede in der perzeptuellen Repräsentation oder eine V erletzung der Mo-
dellannahmen darstellen können.
Zusammen mit der V er w endung realistischer Stimuli bietet MLDS eine relia-
blere Methode zur Messung perzeptueller Dimensionen und ermöglicht so die
T estung theoretischer Modelle perzeptueller Inferenz.
vii

P U B L I C A T I O N S
Parts of this thesis ha v e been published in:
Aguilar , G., W ichmann, F . A., & Maertens, M. ( 2017 ). Comparing sensitivity esti-
mates from MLDS and for ced-choice methods in a slant-from-texture experiment.
Journal of V ision, 17 ( 1 ): 37 , 1 - 18 . doi: 10 . 1167 / 17 . 1 . 37
W iebel C.B.*, Aguilar G.*, Maertens M. ( 2017 ). Maximum likelihood difference
scales represent per ceptual magnitudes and predict appearance matches.
Journal of V ision, 17 ( 4 ): 1 , 1 - 14 . doi: 10 . 1167 / 17 . 4 . 1
*: equal contribution
W ichmann F . A., Janssen D. H. J., Geirhos R., Aguilar G., Schütt H. H., Maertens
M., & Bethge M. ( 2017 ). Methods and measurements to compare men against
machines. Electr onic Imaging, 2017 ( 14 ): 36 - 45 .
doi: 10 . 2352 /ISSN. 2470 - 1173 . 2017 . 14 .HVEI- 113
ix

W ahrlich es ist nicht das W issen, sondern das Lernen,
nicht das Besitzen sondern das Erwerben,
nicht das Da-Seyn, sondern das Hinkommen,
was den gr össten Genuss gewährt
It is not knowledge, but the act of learning,
not possession but the act of getting ther e,
which grants the gr eatest enjoyment.
— Carl Fr iedr ich Gauss
xi

ACKNOWLEDGMENTS
First I w ant to thank my advisor Marianne Maertens. I’m really thankful for all
the time, patience, ideas and feedback that she pro vided, making the y ears on
my Ph.D. the best I’v e had. She has taught me ho w good psy chophysics is done,
and ho w to a v oid the traps of the hypes. Her contagious critical thinking made
all this w ork possible.
I w ant to also thank Felix W ichmann, for the stimulating discussions and his
advice on both projects here exposed; in particular for his critical feedback r e-
garding the w ork on MLDS and signal detection theor y . I also thank my col-
leagues Christiane W iebel and Da vid Higgins. W ith Christiane w e had a fruitful
collaboration that lead us to publish our study in co-authorship. And with Chris-
tiane and Da vid w e shared, more than an office, memorable times of w ork and
laughs together .
I also thank the numerous re view ers of the manuscripts, who pro vided insight-
ful comments on them and consequently also helped impro v e this thesis: Michael
Landy , Kenneth Knoblauch, Bart Anderson, Richard Murra y , Frank Jäkel, Da vid
Brainard, Michael Kubo vy , Laurence Malone y and one other anonymous re-
view er . Also I thank the funding resources that allo w ed these y ears of w ork: the
Graduate Research T raining ’S ensor y Computation and Neural Systems’ (GRK
1589 / 1 - 2 ), and Marianne Maertens’s project (DFG MA 5127 / 1 - 1 ), both fr om the
Ger man Research Foundation (DFG).
Finally , I w ould like to thank my family back in Chile, and to Ba, my partner
in crime, for his company , patience, and encouragement in these last y ears. And
to the rest of my ne w family in Ger many: Karin Ludwig, Dirk W ar nick, Rafaela
W ahl, and the Berlin Bruisers’ family . They inspir e me to continue in this life
jour ne y of lear ning and disco v er y .
xiii

C O N T E N T S
I b a c k g r o u n d 1
1 measuring per ception 3
2 scaling methods 1 3
3 maximum likelihood difference scaling 2 1
4 s i m u l a t i o n s 35
II using mlds to measure appearance 47
5 i n t r o d u c t i o n 49
6 m e t h o d s 55
7 r e s u l t s 61
8 discussion 6 9
III using mlds to measure sensitivity 75
9 i n t r o d u c t i o n 77
10 s i m u l a t i o n s 83
11 e x p e r i m e n t s 93
12 d i s c u s s i o n 103
IV discussion and outlook 109
1 3 general discussion and outlook 1 1 1
a a p p e n d i x 127
r e f e r e n c e s 147
xv

A C R O N Y M S
MLDS Maximum Likelihood Dif ference S caling
JNDs Just-noticeable dif ferences
GLM Generalized linear model
2 -AFC tw o-alter nativ e forced-choice
2 -IFC tw o-inter v al forced-choice
xvi

Part I
BACKGROUND

1
MEASURING PERCEPTION
An open question in vision research is ho w the visual system constructs percep-
tion giv en the ambiguous infor mation arriving to the retina. W e nor mally ha v e
a stable and constant percept of the w orld’s attributes, despite the changes in
viewing contexts that pr oduce a radical change in the actual stimulation imping-
ing our sensory organs. This phenomenon is kno wn as perceptual constancy , Perceptual constancy
and it can be defined as the ability of a perceptual system to pr o vide constant
representation of stimulus attributes despite the significant change in vie wing
conditions or context. The human visual system is constant to object’s attributes
such as size and surface reflectance (i.e. color), among others.
Figure 1 . 1 illustrates the issue in the domain of lightness perception. W e con-
sider an illumination source, like the sun, that emits a flux of light in all dir ec-
tions. W e also consider an object, like the cube in Figure 1 . 1 where its surface
reflects part of the light it receiv es. The r eflectance of the object is the proportion
of light reflected and it is a physical quality of the object’s material. The flux
of reflected light or luminance , measured in cd/m 2 , arriv es to the retina and it
depends on a combination of both the illumination and the object reflectance.
Multiple combinations of illumination and reflectance can produce the same
luminance v alue; luminance is thus ambiguous with respect to both of its com-
ponents. Luminance is the pr oximal stimulus and the product of the image for -
mation process that occurs when the distal stimuli (i.e. illumination, reflectance,
and ev entually inter v ening media) are projected into the retina. Lightness is the
counterpart in the perceptual dimension of the distal attribute, in this case light-
ness is the perceiv ed surface reflectance, and it is the outcome of the process of
perceptual inference.
The ambiguity of the retinal input can be best illustrated with the ‘Adelson Adelson checkerboard
checker -shado w illusion’ sho wn in Figure 1 . 2 A. In this image tw o checks, one
inside and one outside the shado w (in ’plain vie w’) are identical in luminance
3

4 measuring per ception
Illumination
Surface
reflectance
Luminance
[cd/m²]
Intervening
Medium
Distal Proximal Perceptual
Image formation
perceptual
inference
Lightness
?

Figure 1 . 1 : Distal, proximal and per ceptual dimensions in the domain of lightness perception.
but dif ferent in reflectance. The retinal input (or gra yv alue in the picture) is
identical, ho w ev er lightness follo ws closely the actual reflectance of the checks,
the one in shado w perceiv ed as lighter than the one outside of it.
T o understand ho w the visual system can accomplish perceptual constancy it
is necessary the reliable measurement of perception, i.e. the perceptual dimen-
sion ev oked b y a stimulus. After more than a centur y of psy chophysical research
it is still a challenge to do measurements in a reliable w a y . This challenge is tw o-
fold: choosing the type of stimuli to probe the visual system, and choosing the
type of psy chophysical method (Figure 1 . 3 ).
Stimuli can be broadly classified accor ding to their complexity or naturalism, Stimulus naturalism
ranging from simple, w ell controlled but artificial stimuli, to more naturalis-
tic and complex stimuli such as the Adelson’s checker -shado w illusion. Simple
stimuli are usually flat tw o-dimensional arra ys, such as Gabor patches or sinu-
soidal gratings. In the case of lightness perception, examples ar e the simultane-
ous brightness contrast displa y , or disk-and-annulus stimuli (e.g. W allach, 1948 ;
Rudd & Zemach, 2007 ). Figure 1 . 2 B sho ws the simultaneous brightness contrast
displa y , where tw o equiluminant squares are embedded in surrounds of dif fer -

measuring per ception 5
A B

Figure 1 . 2 : A. Adelson’s checker -shado w illusion. The luminance of tw o checks (inside and outside
the shado w , arro ws) are identical, ho w ev er they ar e perceiv ed as ha ving different surface
reflectance, i.e the y differ in lightness . B. Simultaneous brightness contrast stimulus. T w o
equiluminant squares are surr ounded b y backgrounds of dif ferent luminance, appearing
dif ferent in lightness. The luminance v alues of the checks and their a v erage surround are
equiv alent betw een panels A and B. Adapted from Maertens et al. ( 2015 ).
ent luminance. The squares are equiluminant and appear dif ferent in lightness:
the square embedded in a dark surround appears lighter than the one embedded
in a light surround.
The direction of the ef fect in the simultaneous brightness contrast displa y is
similar to the Adelson checker -shado w illusion (panel A of same Figure). A crit-
ical dif ference, ho w ev er , is that the simultaneous contrast displa y is composed
b y a flat tw o-dimensional arra y of luminance v alues, and thus it has not a clear
distal source. Betw een panels A and B in Figure 1 . 2 the luminance v alues of the
equiluminant checks and squares ha v e been equated, as w ell as the surround
where the y are embedded. Maertens et al. ( 2015 ) used these stimuli and quan-
tified the magnitude of the illusion ef fect, i.e. the lightness differ ence betw een
the tw o equiluminant checks (or squares). They found that the ef fect is bigger
for the Adelson’s checker -shado w illusion than for the simultaneous brightness
displa y , despite that the tar gets w ere equiluminant and they w ere surrounded b y
similar luminance configuration. They concluded that the dif ference in the effect

6 measuring per ception
Stimuli
Measuring perception
Method
Simple Naturalistic Performance-
based
Appearance-
based
well controlled
artificial
less controlled objective subjective
Part III:
MLDS to measure
sensitivity
Part II:
MLDS to measure
appearance

Figure 1 . 3 : The challenges of measuring per ception are tw o-fold: the choice of stimulus and the type
of psy chophysical method. Part II and III refer to the sections on this thesis.
magnitude is likely due to the fact that the Adelson’s checker-shado w illusion is
more realistic than the simultaneous brightness contrast, pr obing the visual sys-
tem in a more appropriate w a y and this being reflected in a higher perceptual
constancy .
In this thesis w e chose stimuli of inter mediate complexity and realism in the
domain of lightness perception. Stimulus realism is not the main topic of this
dissertation, but I will further explore and discuss the issue of ho w to incor porate
more realism to stimuli at the end of this thesis (Chapter 13 ).
1 . 1 t w o t y p e s o f m e t h o d s
The second challenge in the measurement of per ception is the choice of psy-
chophysical method, and is the main focus of this thesis. Historically , there has
been a division betw een tw o major approaches (Figure 1 . 3 ): the measurement of
perfor mance on stimulus discriminability , and the measurement of stimulus ap-
pearance (Kingdom & Prins, 2010 ). It is said that perfor mance is measured when
obser v ers are asked to tell tw o stimuli apart, and appearance when obser v ers are
asked to report ho w stimuli ’look like’.

1 . 1 two types of methods 7
Perfor mance methods estimate sensitivity to stimulus dif ferences, or ’just no-
ticeable dif ferences’, that usually requires the discrimination of stimuli that are Performance
methods
v er y similar and close together in the stimulus continuum. In these type of tasks
judgments are correct or incorrect, as defined b y the actual physical stimulus
attribute, and therefore a performance measure can be calculated.
Follo wing a reductionist appr oach and dra wing from the methods in classical
Physics, perfor mance methods ha v e been extensiv ely used in the study of the
visual system. Perfor mance-based methods focus on the measurement of sensi-
tivity , i.e. thresholds, and use tasks that are usually difficult and unintuitiv e be-
cause they pr obe discrimination for small, near -threshold stimulus dif ferences.
Examples of these tasks are y es/no or forced-choice procedur es. Commonly ,
perfor mance-based methods use simple stimuli.
Although widely used, the reductionist approach in performance-based meth-
ods can be problematic. It is assumed that b y studying the visual system with
simple stimuli will ev entually lead to explain the function of the visual system
when confronted with naturally-occurring stimuli. In the natural environment
the visual system is ho w ev er alw a ys confronted with complex scenes that ha v e
clear distal sources and with a multitude of cues (as described abo v e).
Alter nativ ely , appearance methods, such as scaling or matching methods, aim
to measure ho w stimuli ’look like’ b y letting the obser v ers adjust a probe that Appearance
methods
matches their perception (matching), or b y asking them to judge set of stimuli
that v ar y in some dimension of interest (scaling). In appearance methods there is
no correct or incorrect r esponse, as the whole point of measurement is to estab-
lish the mapping betw een the stimulus attribute and its perceptual dimension, i.e.
a perceptual scale. An example of an appearance method is the method of pair ed
comparisons (in Thurstonian scaling), where pairs of stimuli are presented and
the obser v er judges their similarity along some perceptual dimension of inter -
est. By analyzing the similarity judgments a perceptual scale can be constructed
(T orgerson, 1958 )
T raditionally these tw o different kind of measur ement ha v e had distinct pro-
cedures, tasks and statistical analysis tools. The division has been a sour ce of
contro v ersy in the literature that expands until toda y (e.g. Luce & Krumhansl,
1988 ; Gescheider, 1997 ; Kingdom & Prins, 2010 ). Intuitiv ely one w ould expect

8 measuring per ception
that the ability to discriminate tw o stimuli has to depend on ho w they appear ,
because discrimination must rely on common mechanisms of perceptual r epre-
sentation from where also appearance judgments ar e dra wn. S ome researchers
agree to this equiv alency , while others ar gue that the tw o types of measurement
ev oke distinct perceptual mechanisms and therefore cannot be easily equated.
For some domains such as size perception, performance and appearance mea-
surements dif fer significantly (Ross, 1997 ). In lightness perception, Whittle ( 1994 )
sho w ed that sensitivity to luminance increments (or decrements) follo w closely
the judgments of appearance in a partition scaling task (re view ed also b y King-
dom, 2016 ). The distinction betw een methods of appearance and discrimination
is critical because models of perceptual infer ence pro vide different predictions if
judgments are based on one or the other . T aking the results of Whittle ( 1994 ) into
account, discrimination thresholds in the Adelson’s checkerboar d for increments
(or decrements) measured on equiluminant checks – in shado w and plain view –
should be dif ferent if the y depend on perceiv ed lightness. Contrarily , thresholds
should be equal if they depend on luminance instead of lightness.
It is still an open question in the field which type of method, perfor mance-
based or appearance-based, is better at probing successfully the per ceptual di-
mension. As suggested abo v e, it w ould seem that appearance-based methods
could be better because they can pr o vide tasks that are not as dif ficult as dis-
crimination of near -threshold stimulus differ ences. In this thesis I focus on the
use of appearance-based methods for studying perception, as it will be discussed
in the follo wing.
1 . 2 measurement pr oblems
As pointed out correctly b y Runeson ( 1977 ) “the fact that subjects do judge a
certain v ariable does not pro v e that they possess per ceptual mechanisms of an
appropriate kind. When the task does not fit the per ceptual mechanisms w e
must expect the subject to try to compensate b y using intellectual abilities, and
such results will not be rele v ant to the study of perception”. In fact, there ha v e
been reports that call the v alidity of some standard psy chophysical measurement

1 . 3 m o t i va t i o n 9
methods into question. The follo wing tw o examples illustrate that the problem
that equally af fects perfor mance-based and appearance-based methods.
Ekroll and Faul ( 2013 ) asked observ ers to match a target color in an asymmet-
ric matching experiment using simultaneous color contrast stimuli. The stim-
uli w ere conceiv ed in a 3 -D color space, but in order to accomplish satisfy-
ing matches obser v ers resorted to a fourth transparency dimension. Thus, un-
intended b y the experimenters, the dimensionality of the perceptual space ex-
ceeded that of the stimulus space (see also Logvinenko & Maloney, 2006 ). If trans-
parency had not been included as an adjustable dimension, obser v ers w ould
ha v e set unsatisfactor y adjustments not matching in appearance, and the experi-
menter could ha v e inter preted these results as a failure in constancy .
T odd, Christensen, and Guckes ( 2010 ), emplo ying a discrimination procedure,
measured apparent slant for textured surfaces that w ere slanted at differ ent an-
gles. The textured surfaces inevitably contain tw o dimensional ( 2 -D) cues such
as foreshortening or change of texture density that v ar y systematically with slant.
It w as assumed that the 2 -D cues are used b y the visual system to compute per -
ceiv ed slant, and that obser v ers compare the stimuli with respect to per ceiv ed
slant. This w as how ev er not the case; instead, the results fr om T odd et al. ( 2010 )
re v ealed that “obser v ers’ judgments w ere completely unaffected b y whether or
not the displa ys actually appeared slanted” and that obser v ers w ere doing the
judgments b y directly comparing 2 -D cues. Thus, these studies call for a revision
of the current methods used to measure per ceptual phenomena.
1 . 3 m o t i va t i o n
It is still an open question which methods are better for a reliable measure-
ment of perception, methods that could a v oid the aforementioned problems. A
recently introduced appearance-based method seems mor e suitable and ma y pro-
vide a better choice: Maximum Likelihood Dif ference S caling ( MLDS ).
MLDS has been recently introduced b y Maloney and Y ang ( 2003 ) and is a scal-
ing method aims to produce reliable and ef fi cient estimates of perceptual scales.
It can be used with stimuli that are more complex and with stimulus dif ferences

1 0 measuring perception
that are clearly visible (supra-threshold). In this thesis I pr opose the use of MLDS
for the reliable measurement of per ception.
First, I re view the a v ailable scaling methods with an emphasis on their kno wn
shortcomings, which are rele v ant for the ev aluation of MLDS (Chapter 2 ). Then, I
present MLDS in detail with its mathematical formulation and statistical methods
(Chapter 3 ).
The accuracy and precision of MLDS as a statistical tool has not y et been stud-
ied in detail, and this w as needed before any experimental application could be
done with MLDS . This thesis pro vides analyses of accuracy , precision and the ef-
fect of violations in MLDS model assumptions (Chapter 4 ), which w as dev eloped
simultaneously to experimental testing 1 .
Then, the v alidity of MLDS w as tested experimentally in a study of lightness
perception, presented in its entirety in Part II (W iebel et al., 2017 ). This study
sho ws ho w MLDS can be used to estimate perceptual scales in a scenario of per-
ceptual constancy . S o far MLDS has not been widely adopted b y the visual per-
ception community , likely because of the reluctance to commit to the v arious as-
sumptions that are required b y MLDS in order to statistically estimate per ceptual
scales. In this study w e sho w , ho w ev er , that other appearance-based procedures
such as asymmetric matching also assume the presence of inter nal scales which
are hidden and their shape can not be inferred fr om obser v ers’ matches.
It has recently been proposed that MLDS could be also used to deriv e measures
of discriminability , as an alter nativ e to perfor mance-based methods (Devinck &
Knoblauch, 2012 ). This possibility is inv estigated in-depth in the study presented
in its entirety in Part III (Aguilar et al., 2017 ). This study pro vides the theoretical
framew ork for relating MLDS with perfor mance-based (discrimination) methods,
b y for mulating MLDS in a framew ork of classical signal detection theor y (Green
& Sw ets, 1966 ). It tests whether MLDS could be used for measuring sensitivity ,
first using simulations and later experimentally using a slant-from-textur e task.
Although slant-from-texture stimuli can be classified as simple and artificial, it
1 These analyses ha v e been published mostly as Appendices and Supplementary Material in the tw o
studies presented in this thesis, Aguilar , W ichmann, and Maertens ( 2017 ) and W iebel, Aguilar , and
Maertens ( 2017 ), as w ell as in W ichmann et al. ( 2017 ). Here they are pr esented first for clarity .

1 . 3 m o t i va t i o n 11
w as used in this study in order to compare the results with pre vious w ork (Rosas,
W ichmann, & W agemans, 2004 ).
Finally , Part IV pro vides a general discussion of the findings and an outlook on
ho w to incorporate more realism to stimuli used in the study the visual system.

2
S C A L I N G M E T H O D S
The perception of a stimulus usually does not depend in a one-to-one relation-
ship on the physical stimulus. For example, the perceiv ed loudness of a sound
doubles when the physical intensity increases ten-fold, and not tw o-fold 1 . In or -
der to understand these type of relationships, w e need to measure the physical
stimulus as w ell as the perceiv ed sensation that it produces. In that w a y w e can
study ho w stimuli are mapped into sensations, and ultimately de v elop models
of ho w a perceptual system w orks.
The measurement of the physical stimulus is done b y the use of physical in-
struments, such as a photometer , a sound pressure meter , or a ruler . Ho w ev er ,
the measurement of the magnitude of sensation is not as straightforw ard, and
it has been the most dif ficult endea v or in perception r esearch. Ov er the last cen-
tury many procedur es of measuring magnitude sensation ha v e been devised,
collectiv ely called scaling methods . They aim to establish the specific r elationship Perceptual scale
betw een the physical stimulus ( x ) and the perceiv ed dimension ( Ψ ( x ) ), i.e. the es-
timation of a psychophysical magnitude function , per ceptual scale or transducer func-
tion .
2 . 1 fechner and earl y scaling methods
S caling dates back to the origin of psy chophysics. Fechner ( 1860 ) postulated that
there must be a relationship betw een the physical stimulus and the inter nal sen-
sation that produce in an observ er that can be measured quantitativ ely . W eber , a
predecessor of Fechner in the study of per ception, w orked on w eight perception
and disco v ered that the amount of stimulus intensity ( ∆x ) that must be added
to a baseline stimulus ( x ) to be noticeable is proportional to the baseline stim-
1 An example from the decibel scale.
13

1 4 scaling methods
ulus itself. Fechner for malized this relationship in the W eber ’s la w , that can be
expressed as
∆x = k · x W eber ’ s law ( 2 . 1 )
where k is a proportionality constant – the W eber ’s constant, or W eber ’s frac-
tion. W eber ’s la w has been found to hold in the mid intensity range for many
stimulus dimensions in vision and audition (Baird, 1978 ; Gescheider, 1997 ).
Fechner took W eber ’s la w and, b y postulating the existence of an inter nal di-
mension Ψ ( x ) , deriv ed mathematically a relationship betw een physical stimulus
x and its inter nal dimension. This relationship is not linear but logarithmic, and
it can be expressed as
Ψ ( x ) = C · ln ( x/x 0 ) Fechner ’ s law
where x 0 is the absolute threshold for the stimulus dimension of inter est and C
an arbitrary constant that depends on the experimental conditions (Baird, 1978 ).
Experimenters are only able to measure the physical stimulus dimension x di-
rectly , with appropriate physical instruments. Ho w can w e then measure the
inter nal, perception component Ψ ( x ) ?
Fechner proposed that a sensation scale can be constructed b y measuring and Fechnerian scaling
summing Just-noticeable dif ferences ( JNDs ) sequentially . A JND can be defined as
the dif ference in the stimulus dimension ∆x that can be minimally distinguished
for some perfor mance criterion, for example on 75 % of the time in a 2 -AFC
task. Fechnerian scaling (also called discrimination scaling) constructs a sensa-
tion scale b y measuring JNDs sequentially , starting at an absolute threshold, and
assigning equal steps in the sensation scale. The procedur e is repeated until the
whole discrimination scale is measured (Gescheider, 1997 ).
Fechnerian integration has been criticized, theoretically as w ell as for lack of
consistent experimental evidence to support it. A ke y assumption in Fechnerian
integration is that W eber ’s la w must hold, ho w ev er evidence for many dif ferent
stimulus domain sho w that W eber ’s la w does not hold in many experimental
scenarios such as extreme stimulus intensities (T orgerson, 1958 ; Bair d, 1978 ). An-
other criticism stands for the method itself. The measurement of sensation is

2 . 1 fechner and earl y scaling methods 1 5
Stimulus dimension
Perceptual dimension
Stimulus dimension
∆ x 1 ∆ x 2
∆Ψ 1
∆Ψ 2
∆ x 1 ∆ x 2
∆Ψ 1
∆Ψ 2
A B

Figure 2 . 1 : JNDs are not informativ e about the shape of the scale without assumptions of
the inter nal noise. T w o differ ent JNDs ( ∆x 1 and ∆x 2 ) can originate from a non-
linear perceptual scale with constant noise v ariance (additiv e noise, left), or
from a linear per ceptual scale with increasing noise v ariance (multiplicativ e
noise, right). Figure adapted from Kingdom and Prins ( 2010 ).
intrinsically noisy , and if done sequentially , measurement error could accumu-
late, giving less reliable estimates for increasing scale v alues.
Another criticism concer ns the assumed equality of JNDs in the sensation scale. JND dependency on
the noise distribution
In Fechnerian integration it is assumed that when JNDs are sequentially mea-
sured and summed, the same step is e v oked in the sensation scale, i.e. all subjec-
tiv e just-noticeable dif ferences ( ∆Ψ ) are equal. These assumptions has been chal-
lenged b y many authors (e.g. Gescheider, 1997 ; Kingdom & Prins, 2010 ), because
JNDs critically depend not only on the shape of the scale but on the distribution
of its noise. This criticism is illustrated in Figure 2 . 1 , where tw o dif ferent func-
tions are sho wn: one non-linear described b y Ψ ( x ) = x 0 . 5 in panel A, and one
strictly linear described b y Ψ ( x ) = x in panel B. For the non-linear function the
noise is constant along the sensation dimension, i.e. it is of equal-v ariance, or
’additiv e’ noise. For the linear function the noise increases along the sensation
dimension, i.e. it is of unequal-v ariance, or ’multiplicativ e’ noise. JNDs measured
at tw o lev els of the stimulus dimension ( ∆x 1 and ∆x 2 ) are ho w ev er identical for

1 6 scaling methods
both functions. It follo ws that JNDs are b y themselv es not infor mativ e for con-
structing a perceptual scale without considering the noise distribution, which b y
itself cannot be tested experimentally .
Due to these reasons the use of Fechnerian integration for deriving scales
has been a topic of strong contr o v ersy (Krueger, 1989 ; Gescheider, 1997 ) and
it is no w ada ys largely unused (Kingdom & Prins, 2010 ). Ho w ev er , the interest
of studying sensation magnitude (or appearance) and relating it with JND-style
discrimination methods (measuring sensitivity) has not decreased, because both
types of measurement pro vide insights into the underlying perceptual mecha-
nisms at w ork (e.g. Ross, 1997 ; Hillis & Brainard, 2005 , 2007 b; Maertens & W ich-
mann, 2013 ; Devinck & Knoblauch, 2012 ).
2 . 2 stevens and direct scaling
A completely dif ferent approach of measuring perception w as dev eloped b y
Stev ens ( 1957 , 1975 ). Unlike Fechner , he postulated that the sensation scale can
be probed directly b y coupling it with a numerical response scale, b y assuming
that there is a direct, one-to-one mapping. By asking observ ers to estimate the
magnitude of their sensations, the experimenter could construct a scale ’directly’.
He called these methods dir ect , as oppose to indirect methods relying on discrim-
ination, such as Fechnerian scaling. He devised se v eral methods, mostly impor-
tantly the method of magnitude estimation. In this method, obser v ers w ould be Magnitude
estimation presented with a initial stimulus which w ould be explicitly anchored to some
numerical v alue, sa y 50 . Then, obser v ers w ould be asked to rate the subsequent
stimuli according to that initial anchor . For example a stimulus that elicits the
’double’ sensation of the first w ould be assigned the number 100 , a stimulus half
the intensity w ould be assigned 25 , etc. Thus, it is assumed that obser v ers ha v e di-
rect access to the sensation scale, that the y can judge their sensation magnitude
and that they can assign numbers to them. It is ho w ev er unclear if obser v ers
can reliable do this. Ratio or magnitude estimation can ha v e strong unw anted
confounds, such as cognitiv e strategies or bias intrinsic to the use of numbers
(e.g. tendency to giv e rounded numbers) to name a fe w (re view ed in Gescheider,
1988 ; Marks & Algom, 1998 ).

2 . 3 thurstonian scaling 1 7
More fundamentally , direct scaling methods has been criticized as not pro vid-
ing more insights into the perceptual dimension than Fechnerian scaling. Instead,
it is said that experimental data from direct scaling only is used to confirm ad-
hoc definitions of the psy chophysical magnitude function, i.e. Stev ens’ po w er la w .
Other definition of functions that map stimulus with sensation and responses are
also possible, and direct scaling does not resolv e among the dif ferent possibilities
of mapping (re view ed in detail in Ch. 12 in Gescheider, 1997 ).
T o deal with these difficulties, Ste v ens dev eloped cross-modality matching, a Cr oss-modality matching
method that seeks to a v oid these problems. In this method the obser v er w ould
be presented tw o stimulus, one for a dif ferent sensor y modality , e.g. a bright spot
and a tone. The task is to adjust one of the stimuli (the light spot) in brightness
until it matches in sensation magnitude with the loudness of the tone. As both
sensation scales are assumed to map into the same numerical response scale, and
obser v ers are using this scale internally to match one modality with the other , it
is thought that potential biases are b ypassed. Ho w ev er , cross-modality matching
across sensory modalities is at least odd and unnatural. Humans w ould not
naturally adjust the brightness of a lamp with the loudness of a tone. Obser v ers
left to ambiguous or odd tasks can rely on countless dif ferent strategies, and the
result of such experiment does not impro v e our kno wledge of the perceptual
system.
Direct scaling methods – magnitude estimation, cr oss-modality matching, and
others – ha v e been found to be subject of strong ’contextual ef fects’ and a high
inter -individual differ ences. These effects can be true dif ferences in experimental
conditions and obser v ers, but they could also be confounds (T reisman, 1964 a,
1964 b; Gescheider, 1988 ; Marks & Algom, 1998 ). Although direct scaling w as
important for the establishment of the basic relationships of some modalities,
such as pitch and loudness perception, direct scaling has fallen into disuse in
the later decades (Kingdom & Prins, 2010 ).
2 . 3 thurstonian scaling
Thurstone ( 1927 a, 1927 b) proposed to study the per ceptual dimension without
ha ving (necessarily) a clear mapping with the stimulus dimension. He aimed

1 8 scaling methods
to study abstract sensations such as artw ork beauty , for which it is dif ficult or
ma ybe impossible to define the physical stimulus dimension. For this aim he
proposed the ’La w of Comparativ e Judgment’ and sev eral methods that allow
the estimation of a perceptual scale without a stimulus definition.
Thurstonian scaling w orks b y making observ ers judge the similarity of stim-
ulus pairs along some perceptual dimension of inter est (T orgerson, 1958 ). Im- Law of
Comparative
Judgment portantly in Thurstonian scaling is the notion that perceptual r epresentation is
noisy , that is, a stimulus produces not a fixed but a v ariable representation in
the perceptual dimension that is go v er ned b y random fluctuation. This notion is
analogous and preceded signal detection theory (Green & Sw ets, 1966 ). By ha v-
ing obser v ers judge pairs of stimuli repeatedly , and which are close to each other
in the perceptual dimension, a measure of performance (e.g. percentage correct)
can be deriv ed. By assuming a nor med dimension, e.g. equal-v ariance, Gaussian
noise, perfor mance for dif ferent stimulus pairs can be transformed into distances
in that dimension using a z-score calculation and thus stimuli can be located in
a perceptual scale. The specific methods and procedur es are re view ed in detail
in the scaling literature (T orgerson, 1958 ; McNicol, 1972 ; Marks & Gescheider,
2002 ).
The major dra wback of Thurstonian scaling is the requirement of judgments of
stimulus close to each other and on many repetitions. Needing a high amount of
data can be problematic when multiple conditions and stimulus dimensions are
of interest, especially when deriving scales on an individual basis. More funda-
mentally , Thurstonian scaling needs judgments of v er y closed spaced stimulus,
which can be considered near -threshold performance. S caling done from near -
threshold performance has been problematic, because it has been historically
argued that discrimination and appearance judgments belong to and can e v oke
dif ferent perceptual mechanisms (see Chapter 1 ). Ho w ev er , Thurstonian scaling
and its notion of noisy representation, together with signal detection theory , ha v e
pro vided the theoretical basis of the current ef forts in moder n psy chophysical
scaling (Maloney & Y ang, 2003 ).

2 . 4 p artition scaling 1 9
2 . 4 p artition scaling
Partition scaling results when the perceptual dimension is pr obed using the
method of adjustment (Fechner, 1860 ). The simplest procedure in partition scal-
ing is the bisection task, in which obser v ers are giv en tw o stimulus anchors, a
maximum and a minimum, and they ar e asked to adjust a middle stimulus so
that it bisects the inter v al perceptually . The task can be repeated with many
adjustment lev els, either sequentially or simultaneously , in order to obtain a
finer resolution of a scale (Gescheider, 1997 ). In the core, partition scaling – like
Thurstonian scaling and MLDS – relies on the judgment of interv als differ ences,
i.e. perceptual distances. The dif ference in partition scaling is that interv als are
not fixed and presented to the observ er , as in Thurstonian scaling, but rather is
the obser v er who adjust the inter v als to be perceptually equal.
Partition scaling has been used in a v ariety of stimulus domains, importantly Munsell scale
b y Munsell, Sloan, and Godlov e ( 1933 ) to deriv e the classical Munsell neutral
v alue scale. This scale is used as a standard for equal steps in lightness perception
for neutral gra y v alues. More recently , Whittle ( 1994 ) also used partition scaling
in the study of lightness and its relationship with discrimination thresholds.
Partition scales are not explored in detail in this thesis, ho w ev er they ha v e
potential to be used in conjunction with other scaling methods.
2 . 5 s u m m a r y
Ideally a method should pro vide a reliable and efficient estimation of the per -
ceptual scale. It should not rely on JND measurement because of the r easons ex-
posed in S ection 2 . 1 , and it should use a task that is intuitiv e and easy , with com-
parisons of clearly visible stimuli (supra-threshold), thus a v oiding confounds
due to task dif ficulty (Chapter 1 ). Under these criteria T able 2 . 1 pro vides a com-
parison o v er view of scaling methods, including MLDS which is r evie w ed in detail
in the next chapter .
Thurstonian and Fechnerian scaling rely on comparison of near -threshold
stimulus dif ferences, making the task dif ficult and inefficient. Magnitude estima-
tion can ha v e strong confounding ef fects due to the nature of its task of v erbal

2 0 scaling methods
relies on JNDs task type task easiness ef ficiency
Fechnerian scaling y es near -threshold comparison + lo w
Magnitude estimation no v erbal estimation +++ –
Thurstonian scaling no near -threshold comparison + lo w
Partition scaling no adjustment ++ –
MLDS no supra-threshold comparison +++ high
T able 2 . 1 : Comparison of scaling methods.
estimation, and it is thus no w ada ys largely unused. Partition scaling and MLDS
pro vide better alter nativ es than all of the abo v e, and MLDS seems suitable for
the estimation of scales that is ef ficient, using an easy task of supra-threshold
comparisons.
MLDS dra ws notions from Thurstonian scaling b y assuming noise in the ob-
ser v er judgments. Judgments are made for the comparison of three or four stim-
ulus exemplars, and it requires the definition of a stimulus dimension (unlike
Thurstonian scaling) which reduces the amount of comparisons needed to con-
struct a scale. It has been sho wn to be robust against differ ent noise distributions,
unlike Fechnerian integration, and it does not depend on the estimation of JNDs .
The next chapter presents MLDS , its mathematical definitions, estimation pro-
cedures, and assumptions. Then, Chapter 4 presents the results of simulations
aimed to measure the accuracy and precision of MLDS in pr eparation for ex-
perimental testing, as w ell as the ef fect of model violations on the estimation
results. These analysis w ere needed in or der to deter mine MLDS perfor mance
and v alidity before any experimental application. The experiments using MLDS
are presented in the tw o studies in Parts II and III of this thesis.

3
M A X I M U M L I K E L I H O O D D I F F E R E N C E S C A L I N G
MLDS is a scaling method dev eloped b y Maloney and Y ang ( 2003 ) that consist of
the construction of inter v al scales based on the judgment of inter v al dif ferences.
Contrary for other scaling methods, MLDS recognizes that obser v ers’ responses
are stochastic, and so it takes into account that judgments of interv al dif ferences
are noisy .
As all scaling methods, MLDS aimed to measure the psy chophysical magnitude
function, which maps the physical stimulus v ariable ( x ) with an inter nal sensa-
tion v ariable Ψ ( x ) (also noted Ψ x for simplicity). It is common to assume a po w er
function (after Stev ens, see Chapter 2 ) although other functions are allo w ed. A
po w er function can be described b y the for mula
Ψ ( x ) = x e Power function ( 3 . 1 )
where e is the exponent parameter and controls the curv ature of the function.
In MLDS the obser v er is presented with three (or four) stimulus exemplars ( x i )
that elicit discrete per ceptual responses ( Ψ ( x i ) ). An ensemble of three stimulus
exemplars is called a triad ( x 1 , x 2 , x 3 ), and an ensemble of four exemplars is
called a quadruple ( x 1 , x 2 , x 3 , x 4 ).
In the method of triads, the task of the observ er is to judge which pair of
stimulus elicits a bigger dif ference, either the pair ( x 1 , x 2 ) or the pair ( x 2 , x 3 ). The
obser v er is thus comparing the inter v al [ Ψ x 3 − Ψ x 2 ] with the inter v al [ Ψ x 2 − Ψ x 1 ] .
A decision v ariable can be written as the difference betw een these tw o interv als
∆ = | Ψ x 3 − Ψ x 2 | − | Ψ x 2 − Ψ x 1 | + ϵ Decision variable ( 3 . 2 )
The content of this chapter has been partly published in W ichmann et al. ( 2017 ), and as Appen-
dices and Supplementary Material in Aguilar et al. ( 2017 ) and in W iebel et al. ( 2017 ).
21

2 2 maximum likelihood difference scaling
MLDS includes stochasticity in the decision v ariable with the ter m ϵ , which
is Gaussian distributed noise, with zero-mean and v ariance σ 2 , i.e. ϵ ∼ N ( 0 , σ 2 ) .
Using this decision v ariable, the model selects the stimulus pair ( x 2 , x 3 ) as larger
when ∆ > 0 , other wise it selects the pair ( x 1 , x 2 ) .
Alter nativ ely , MLDS can be used with the method of quadruples, in which case
the decision v ariable is
∆ = | Ψ x 4 − Ψ x 3 | − | Ψ x 2 − Ψ x 1 | + ϵ ( 3 . 3 )
with same noise distribution and decision rule than for the method of triads.
This thesis uses only the method of triads.
Giving obser v er judgments to triads (or quadruples), the goal of MLDS is to
obtain an inter v al scale that reflects the underlying inter nal sensor y function
Ψ ( x ) and an estimate of the v ariance of the noise parameter in the model ˆ σ .
3 . 1 construction of triads
During the design of an MLDS experiment the number of stimulus exemplars ( x i )
needs to be decided in or der to construct the triads. First, the range of the stim-
ulus dimension under study must be decided. For example, if w e are interested
in the lightness domain, the stimulus dimension is physical reflectance, an unit-
less dimension that can range from 0 (no reflectance, per ceiv ed as black) and 1
(full reflectance, per ceiv ed as white). On this range, p differ ent stimuli can be
placed unifor mly . The triads are then constructed b y selecting all possible non-
o v erlapping inter v als using p discrete stimulus v alues. As an example, if p = 4 ,
the complete set of triads are ( 1 , 2 , 3 ) , ( 1 , 2 , 4 ) , ( 1 , 3 , 4 ) , ( 2 , 3 , 4 ) . The triplet ( 3 , 2 , 4 )
is not a v alid triad for MLDS because its inter v als o v erlap. The total number of
possible triads can be calculated as
n =  p
3  = p !
( p − 3 ) ! × 3 !
These complete set of triads must be repeated many times ( r ) to obtain reliable
estimates. Thus, a full MLDS experiment consist of an obser v er judging N = n × r

3 . 2 estima tion using a generalized linear model 2 3
p = 7 8 9 10 11 12 13 14 15 20
n = 35 56 84 120 165 220 286 364 455 1140
T able 3 . 1 : The number of possible triads ( N ) gro ws rapidly with increasing stimulus
number ( p ).
trials, which can be done in a blocked design, with r blocks containing n trials
each 1 . The order of the stimulus in the triad, with either ascending or descending
v alues, must be randomized so to a v oid systematic spatial or temporal arrange-
ment of the stimuli that could lead to confounds.
3 . 2 e s t i m a t i o n u s i n g a g e n e r a l i z e d l i n e a r m o d e l
Once stimulus triads are presented to the observ er and his/her answ ers collected,
w e can proceed to estimate the scale. MLDS has been implemented with tw o
dif ferent algorithms: using direct optimization (Malone y & Y ang, 2003 ) or as a
Generalized linear model ( GLM ) for logistic regression (Knoblauch & Malone y,
2008 , 2012 ). I focus on the GLM implementation as it is the new est and most used
one.
For the GLM implementation w e must further assume that the sensor y func-
tion is monotonically increasing, thus a v oiding the absolute v alue operation in
Equation 3 . 2 and leading to a decision v ariable that can be rewritten as a linear
combination of the sensory v ariables
∆ = ( Ψ x 3 − Ψ x 2 ) − ( Ψ x 2 − Ψ x 1 ) + ϵ ( 3 . 4 )
= Ψ x 3 − 2Ψ x 2 + Ψ x 1 + ϵ
1 In this w ork w e chose to use full repetitions of the complete set of triads. Pre vious w ork with
MLDS ha v e instead use a procedure that randomly dra ws a triad (from the complete set) until the
final number of desired trials is reached.

2 4 maximum likelihood difference scaling
A
Ψ( x )
x
x 2 x 5 x 8
Ψ( x 2 )
Ψ( x 5 )
Ψ( x 8 )
Stimulus
B C
Dif ference Scale
Stimulus
β 1
β 10
x
...
...
g (E[ Y ]) = *
β 1
β 2
β 3
β 4
β 5
β 6
β 7
β 8
β 9
β 10

Figure 3 . 1 : Estimation of scales using MLDS with triads. An example triad ( x 2 , x 5 , x 8 ) ev okes perceptual expe-
riences Ψ ( x ) on an hypothetical sensory function (A). The decision model for this example triad is
∆ = Ψ ( x 8 ) − 2Ψ ( x 5 ) + Ψ ( x 2 ) . (B) GLM construction. A design matrix (X) is constructed b y setting the
entries in the corresponding columns as the w eights of each Ψ ( x ) from the decision model. Shaded
ro ws indicate tw o repetitions of the same triad. Y is the binomial response v ariable. (C) The esti-
mated scale is obtained b y solving the GLM and finding the coefficients β , which correspond to the
scale v alues at different le v els of the physical v ariable. After W iebel et al. ( 2017 ).

3 . 2 estima tion using a generalized linear model 2 5
Under this assumption, MLDS can be reformulated as a GLM (Figure 3 . 1 ). The
GLM is set up b y taking the responses of an obser v er and assigning them to the
response v ariable ( Y ), and b y constructing a design matrix ( X ) that represents the
w eights of each Ψ ( x i ) component on the decision v ariable in a single-trial basis.
The goal is to find the coef ficients β that best account for the data ( Y ) giv en X .
For mally , the GLM is described b y
g ( E [ Y ]) = Xβ GLM ( 3 . 5 )
where Y is a v ector of length n with entries 0 or 1 , indicating the obser v er ’s
response (for first vs. second pair , respectiv ely). X is the design matrix of size
n × p , whereb y n is the total number of triads and p is the number of stimulus
lev els sampled as w ell as the number of estimated points on the perceptual
scale. Each ro w in matrix X contains non-zero entries ( 1 ,- 2 , 1 ) in the columns Design matrix
corresponding to the stimulus v alues for the presented triad v alues ( x 1 , x 2 , x 3 ),
and zero entries in the remaining p − 3 columns. The coef ficient v ector ˆ
β is of
length p and it contains the scale estimates (Figure 3 . 1 C).
The link function g () is required to establish the relationship betw een the lin-
ear pr edictors Xβ and the mean response v ariable E [ Y ] , where Y is binomially
distributed with n= 1 (also kno wn as a Bernoulli process). The default link func-
tion for MLDS is the inv erse of the Gaussian cumulativ e distribution function
( Φ − 1 ), as it has been sho wn to be robust against dif ferent noise distributions
(Maloney & Y ang, 2003 ), and it w as used throughout this w ork.
T o make the model solv able, the first ro w of the design matrix is dropped,
which ef fectiv ely anchors the scale at a minimum of zero, β 1 = 0 . The rest of
the coef ficients ˆ
β 2 ··· ˆ
β p are estimated b y maximum likelihood using standard
GLM solv ers. The coefficient v ector ˆ
β is the estimated dif ference scale, and it is a
inter v al scale (Figure 3 . 1 C). The estimation using GLM produces scales where its
maximum is related inv ersely to the estimated noise parameter of the model ˆ σ
ˆ
β p = 1
ˆ σ ( 3 . 6 )

2 6 maximum likelihood difference scaling
These type of sales are called ’unconstrained’ scales in MLDS literature (Knoblauch
& Maloney, 2012 ).
3 . 3 mlds and sign al detection theory
The decision model underlying MLDS can be also framed in ter ms of signal de-
tection theory . Figure 3 . 2 sho ws the equiv alence betw een MLDS on its original
for mulation and forced-choice pr ocedures used for signal detection theory . The
equiv alence has been suggested b y Devinck and Knoblauch ( 2012 ) and is ex-
plained in the follo wing.
An ’unconstrained scale’ in MLDS can be conv erted to a nor med scale defined
in units of d ′ , when the follo wing assumptions are met. First, it is assumed that
the decision process is not stochastic but deterministic. This w ould attribute all
of the obser v ed noise to the sensory representation, ψ ( x ) , which is a Gaussian
random v ariable with mean Ψ ( x ) . S econd, it is assumed that the noise is constant,
i.e. independent of the stimulus lev el. Finally , it is assumed that the sensor y
representations are independent of each other . It follows fr om these assumptions
that the ψ ( x ) are independent Gaussian random v ariables with equal v ariance 2 .
Then, the noise parameters can be ‘carried’ to the sensor y representation, b y
re writing the decision model (Equation 3 . 4 ) in this w a y
ψ x i ∼ N  Ψ x i , σ 2
4 
MLDS in
signal detection
theory formulation
( 3 . 7 )
∆ = ( ψ x 3 − ψ x 2 ) − ( ψ x 2 − ψ x 1 ) ( 3 . 8 )
The v ariance of the decision v ariable ∆ is σ 2 (Equation 3 . 4 ). When rewriting
the model equations, the v ariance of each sensor y representation ψ x must be
2 This restricted case can be deriv ed from the MLDS model only because: (i) the decision v ariable
after the differ encing is assumed to be Gaussian, for which the simplest case is when it is produced
b y underlying Gaussian distributed representations; and (ii) equal-v ariance is the simplest case to
relate the v ariance of each representation with the v ariance of the decision v ariable. Other models
(e.g. unequal-v ariance) cannot be deriv ed easily as they w ould be underconstrained.

3 . 3 mlds and sign al detection theory 2 7
Ψ x
Ψ x 3
Ψ x 2
Ψ x 1
∆ = ψ x 2 − ψ x 1
ψ x i ∼ N (Ψ x i , σ 2 )
∆= ( ψ x 3 − ψ x 2 ) − ( ψ x 2 − ψ x 1 )
∆ = | Ψ x 3 − Ψ x 2 |− | Ψ x 2 − Ψ x 1 | + 
B) MLDS in SDT formulation
criterion
Sensory
representation
Decision
choose pair
-1 1 0
C) Forced-choice
criterion
choose
equivalence
√ 2
Sensory representation function
S timulus
A) MLDS model
original formulation
 ∼ N (0 , σ 2 )
Signal
detection
theory
asumptions
equal variance, Gaussian
,
1 0
1
0
x 3 x 2
x 3
x 2
x 1
x
x 2
-1 1 0
ψ x 3
ψ x 2
ψ x 1
ψ x 2
ψ x 1

Figure 3 . 2 : MLDS in the signal detection frame w ork. (A) In its original for mulation the decision v ariable ( ∆ ) in MLDS
is defined as the differ ence betw een inter v als ( | Ψ s 3 , Ψ s 2 | and | Ψ s 2 , Ψ s 1 | ) and this differ ence is corrupted
b y Gaussian noise ( ϵ ). (B) In the signal detection formulation of MLDS the noise originates only from the
sensory representations ( ψ s ) which are assumed to be independent Gaussian random v ariables with equal
v ariance. In the signal detection v ersion of MLDS the model is equiv alent to forced-choice methods (C) at
the lev el of the sensory representation. S ee text for details. Adapted from Aguilar et al. ( 2017 ).

2 8 maximum likelihood difference scaling
adjusted so that Equation 3 . 4 still holds. Because the decision v ariable ∆ is com-
puted as a linear combination of four independent, Gaussian random v ariables,
its v ariance is four times the v ariance of each individual v ariable ψ x i . Therefore
each individual v ariance in the sensor y representation needs to be ‘corrected’ b y
a factor of 1 / 4 .
Y et, MLDS pro vides the noise estimate ˆ σ (Equation 3 . 6 ) as an estimate of param-
eter σ of the decision v ariable and not of the sensor y representation directly . By
kno wing the abo v e explained relationship betw een the v ariance in the sensor y
representation and in the decision v ariable, the dif ference scale can be adjusted
so as to represent the v ariance in the sensory representation. The ˆ σ estimated b y d’ scales
MLDS corresponds to four times the v ariance present in the sensory representa-
tion ψ x (Equation 3 . 7 ). Thus, the conv ersion is accomplished b y multiplying the
original scale b y a factor of tw o (as done also b y Devinck & Knoblauch, 2012 ).
For mally , the new transfor med scale maximum is tw o times the maximum of the
original scale
ˆ
β ′
p = 1
ˆ σ
2
= 2 1
ˆ σ = 2 ˆ
β p ( 3 . 9 )
This new scale ˆ
β ′ is in “ d ′ units”, i.e. an inter v al difference of one in the scale
dimension should represent a performance of d ′ of one, when all assumptions
are met. The parametrization in units of d ′ can be used to deriv e sensitivity (see
Chapter 10 ).
3 . 4 e s t i m a t i o n e r r o r
The v ariability of the scale estimation is deter mined using bootstrapping tech-
niques (Knoblauch & Maloney, 2008 , 2012 ). The goal is to estimate the v ariabil-
ity of the coef ficients ˆ
β . For that purpose, Equation 3 . 5 is rearranged in order to
compute the mean response probability for each triad fr om the fitted data
E [ Y ] = g − 1 ( X ˆ
β ) ( 3 . 10 )

3 . 4 estima tion error 2 9
The obtained v ector E [ Y ] contains the expected probability of a Bernoulli v ari-
able ( Y ) for each triad, in other w ords, the mean pr obability of binar y responses
giv en the presented stimulus v alues in each triad. These probabilities are used
to simulate a Ber noulli response in each triad, which is in turn used to estimate
a new set of coef ficients ˆ
β ∗
j , j = 1 .. p using the same GLM procedure. The coef-
ficients ˆ
β ∗
ij , j = 1 .. p are the i-th bootstrap sample, and many bootstrap samples
are dra wn b y repeating the simulation pr ocedure many times ( N s = 10000 ). A
matrix S with all the samples can be constructed with N s × p entries.
S i , j = 



ˆ
β ∗
1 , 1 ··· ˆ
β ∗
1 , p
.
.
. . . . .
.
.
ˆ
β ∗
N s , 1 ··· ˆ
β ∗
N s , p




( 3 . 11 )
From this matrix the confidence interv als for each scale estimate can be ob-
tained from the distribution of bootstrap samples at a confidence of ( 1 − 2α ) (e.g.
α = 0 . 025 for 95 % CIs).
There are multiple w a ys to obtain confidence inter v als from bootstrap samples. Boostrap
confidence
intervals
The simplest and straightfor w ard method is the ’percentile’ method, in which the
confidence inter v als are dra wn directly from the ( α ) and ( 1 − α ) percentiles of
the bootstrap samples distribution. The percentile method often pr oduce confi-
dence inter v als that are too small and do not represent the underlying v ariability ,
specially when the sample distribution violates the normality assumption (Efron
& T ibshirani, 1993 ).
T o a v oid these problems, Efr on and T ibshirani ( 1993 , pp. 184 - 188 ) proposed the
calculation of ‘bias-corrected and accelerated’ (BCa) confidence interv als. This
method is robust against ske w ed distributions and is recommended for stan-
dard use. It is also the method of choice in estimation of psy chometric functions
using bootstrap (W ichmann & Hill, 2001 ). BCa confidence inter v als are used
throughout this thesis.

3 0 maximum likelihood difference scaling
Figure 3 . 3 : Goodness of fit diagnostics for MLDS (as pro vided by the R package) for a simulated experiment.
(A) Cumulativ e distribution function of the linear predictors Xβ . The observ ed data is depicted
in the markers, and its 95 % confidence inter v al obtained by bootstrap as lines. (B) T est for
independence of deviance residuals. Histogram of ’number of runs’ obtained b y bootstrap, and
the position of the obser v ed number or runs (v ertical line). S ee text for details.
3 . 5 g o o d n e s s o f f i t
Unlike for linear models, the goodness of fit of a binomial GLM can be dif ficult
to ev aluate. T w o generic measures of GLM can be used. The first is the Akaike
infor mation criterion (AIC), a general metric of the likelihood w eighted b y the
number of parameters, and that is commonly used for model comparison. The
second is the ‘deviance accounted for ’ (DAF), which is the reduction of deviance
b y the model fit with respect to the deviance of the null model, i.e. a model
with only an intercept (W ood, 2006 ; Knoblauch & Maloney, 2012 ). The DAF is
conceptually similar to the R 2 measure in linear regression, interpreted as the
percentage of v ariance explained.
Ho w ev er , the ev aluation of the residuals of these type of GLM can be problem-
atic, giv en that they distribute binomially (Knoblauch & Maloney, 2008 , 2012 ).
For this reason W ood ( 2006 ) recommends to test binomial GLM using Monte Goodness of fit
diagnostics Carlo methods. The procedure inv olv es the simulation of binomial responses ac-
cording to the expected probabilities E [ Y ] (Equation 3 . 10 ). For each simulation,

3 . 6 measurement assumptions 3 1
the deviance r esiduals are obtained and sorted in a cumulativ e distribution func-
tion (cdf). Figure 3 . 3 A sho ws the experimental distribution of deviance residuals
(markers), and the ( 1 − α ) % confidence env elope obtained from all ( N s = 10000 )
simulated cdfs (blue lines). If the decision model in MLDS is appropriate, the
experimental distribution should be inside the confidence env elope, which is in-
deed the case in Figure 3 . 3 A. Additionally , the independence of the deviance
residuals can also be tested using the same Monte Carlo method. For each simu-
lation described abo v e, the deviance r esiduals are sorted according to the linear
predictors, and the number of times the sign in the deviance r esidual changes
is counted (‘number of runs’ in Figure 3 . 3 B). An independent distribution of de-
viance residuals should ha v e random distribution of sign and no evident tr end.
The v alue obtained experimentally (v ertical line in Figure 3 . 3 B) is compared with
the distribution obtained from the Monte Carlo simulations b y calculating a p-
v alue. This procedure produces a p-v alue that is not significant if the deviance
residuals sho w independence.
3 . 6 measurement assumptions
MLDS bases its dif ference model in classical measurement theory . Krantz, Luce,
Suppes, and Tv ersky ( 1971 , Ch. 4 , pp. 247 ) defines a series of axioms that must be
fulfilled to ensure the existence of a scale that can be deriv ed from the judgment
of stimulus dif ferences, or the judgment of inter v als. Importantly for MLDS are
the transitivity axiom and the ’w eak monotoncity’ axiom ( Axioms 3 and 4 in
Krantz et al. ( 1971 )).
The transitivity axiom states that if tw o inter v als ( x 1 , x 2 ) , ( x 2 , x 3 ) exist in the T ransitivity
axiom
set of all possible inter v als ( X ∗ ), then the inter v al ( x 1 , x 3 ) must be bigger than
each of them alone
if ( x 1 , x 2 ) , ( x 2 , x 3 ) ∈ X ∗
then ( x 1 , x 3 ) > ( x 1 , x 2 ) and ( x 1 , x 3 ) > ( x 2 , x 3 )
where > stands for the binary comparison operation ‘bigger than’. In other
w ords, this axiom states that the stimuli x i belong to an or dered dimension, for

3 2 maximum likelihood difference scaling
example length, luminance, brightness, etc. It translates experimentally into an
obser v er that can judge x 3 > x 2 > x 1 , and thus who is able to order the stimulus
increasingly .
The ’w eak monotonicity’ axiom states that giv en tw o pairs of contiguous inter- W eak
monotonicity
axiom v als, ( x 1 , x 2 ) , ( x 2 , x 3 ) and ( x 4 , x 5 ) , ( x 5 , x 6 ) ,
if ( x 1 , x 2 ) > ( x 4 , x 5 )
and if ( x 2 , x 3 ) > ( x 5 , x 6 )
then ( x 1 , x 3 ) > ( x 4 , x 6 )
This axiom implies that in the measured dimension the interv als increase as
the distance betw een the stimulus gets larger , in other w ords, the function is
monotonically increasing. Clear examples are lightness, depth and distance 3 .
This assumption can also be tested experimentally in MLDS with the method
of quadruples, b y analyzing directly if the obser v er ’s judgments comply with
the inequality stated in the axiom. For the method of triads this is not possible
due to ho w the triads themselv es are constructed. Thus, when using the method
of triads the monotonicity assumption is a requisite that must be assumed and
cannot be tested experimentally .
3 . 7 s u m m a r y o f m l d s
T o summarize, the goal of MLDS is to obtain an estimation of the inter nal scales
for a stimulus dimension of interest, b y asking obser v ers to judge stimulus dif-
ferences in triads or quadruples. It does so b y setting up a statistical model (a
generalized linear model) that estimates the scale v alues. The v ariability on the
estimation and the goodness of fit is ev aluated using bootstrap methods.
MLDS assumes:
• a function that maps the physical stimulus v ariable x to an inter nal percep-
tual scale Ψ ( x ) .
3 A counterexample w ould be a function with a shape of an inv erted U, with a peak in the middle
of the stimulus range.

3 . 7 summary of mlds 3 3
• a function Ψ ( x ) that is monotonically increasing. This allo ws the GLM im-
plementation and the compliance with axioms of measurement theory .
• the obser v er judges the dif ference betw een the stimulus exemplars using a
dif ference model (Equation 3 . 2 ).
• this judgment is corrupted b y independent, Gaussian distributed noise.
• the obser v er is able to order the stimulus incr easingly .
MLDS can be also parametrized in a signal detection framew ork, for which the
obtained dif ference scales are in units of d ′ .
In the next chapter I addressed some of these assumptions using simulations,
as w ell as deter mining the accuracy and precision of MLDS as a statistical tool. T o
anticipate, I found that MLDS could reliably estimate the generativ e perceptual
function with lo w bias, pro viding at the same time a reliable estimate of the noise
in the decision model when the equal-v ariance assumption holds. Additionally ,
MLDS pro vided confidence inter v als that w ere stable in precision and adequate
in co v erage. These results pro vided the basis for the follo wing experiments that
tested the perfor mance of MLDS experimentally (Part II, starting at Chapter 5 ).

4
S I M U L A T I O N S
Since its introduction b y Maloney and Y ang ( 2003 ), the bias and v ariability of
the MLDS estimation itself has not been studied in detail, as w ell as the effect
of violations of its model assumptions. In this chapter I explore these issues
using numerical simulations. These simulations also pro vide the upper limit of
precision when a limited amount of data is a v ailable, for the practical use case
in psy chophysical experiments.
4 . 1 specifica tion
The sensory functions (or transducer functions) w ere po w er functions with three
dif ferent exponents ( e = 0 . 5 , 1 . 0 and 2 . 0 ) that giv es the function a dif ferent cur v a-
ture. The stimulus dimension x w as set to the nor malized range [ 0 , 1 ] , and p = 8
stimulus v alues w ere linearly spaced on this range. As detailed in S ection 3 . 1
this spacing giv es n = 56 unique triads to be presented.
The sensory functions w ere subjected to Gaussian-distributed noise of either (i)
equal v ariance (‘additiv e noise’), or (ii) unequal v ariance (‘multiplicativ e’ noise).
For the equal-v ariance case, the dra w of the function’s response to the stimulus
x i can be expressed as
ψ x i ∼ N  Ψ x i , σ 2  Equal-variance
noise
( 4 . 1 )
where σ is the fixed le v el of the Gaussian-distributed noise, and it could ha v e
v alues of σ = { 0 . 035 , 0 . 07 , 0 . 14 } . The choice of these v alues w as infor med b y
the study reported in Aguilar et al. ( 2017 ) (and presented in part III of this
The content of this chapter has been partly published in W ichmann et al. ( 2017 ), and as Appen-
dices and Supplementary Material in Aguilar et al. ( 2017 ) and in W iebel et al. ( 2017 ).
35

3 6 simula tions
dissertation): ˆ σ = 0 . 07 w as the a v erage v alue of estimated noise found in eight
obser v ers, and 0 . 035 and 0 . 14 w ere the respectiv e half and double of that a v erage.
The range [ 0 . 035 , 0 . 14 ] co v ered all v alues obser v ed experimentally .
For the unequal v ariance case, the noise increased constantly with the stimulus
dimension, from σ min = 0 . 035 at x = 0 to σ max = 0 . 14 at x = 1 . Formally this
case can be expressed as
ψ x i ∼ N  Ψ x i , σ 2 ( x ) 
Unequal-variance
noise
( 4 . 2 )
where the standar d deviation of the Gaussian-distributed noise increase lin-
early with the stimulus v alue
σ 2 ( x ) = [ ( σ max − σ min ) · x + σ min ] 2
The unequal-v ariance noise case correspond to the most studied case of unequal-
v ariance: the noise increases with the stimulus dimension as obeying W eber ’s
la w .
The decision v ariable w as as assumed b y MLDS for the method of triads (Equa-
tion 3 . 8 ), re written here for clarity:
∆ = ( ψ x 3 − ψ x 2 ) − ( ψ x 2 − ψ x 1 ) ( 4 . 3 )
As described in S ection 3 . 1 , the set of unique triads needs to be repeated
many times, thus v ar ying the total number of trials in the experiment. In these
simulations the total number of trials w as v aried systematically with v alues of
N ∈ { 280 , 560 , 840 , 1680 , 2520 , 3360 } , which is the result of ha ving the set of n = 56
triads repeated r ∈ { 5 , 10 , 15 , 30 , 45 , 60 } times.
Each simulation w as fed into the MLDS analysis routine (S ection 3 . 2 ) that in-
cluded the estimation error using bootstrap (S ection 3 . 4 ) and the goodness of fit
analysis (S ection 3 . 5 ). The results are presented in the follo wing sections.

4 . 2 bias of scale and noise estima tes 3 7
σ = 0 . 035 σ = 0 . 14 σ = 0 . 035 0 . 14
x 0 . 5
x 1
x 2

Figure 4 . 1 : S cale estimation for three noise conditions and three dif ferent exponents for the sensory
function Ψ ( x ) ( M ± 1S . D . across N = 560 simulated trials, scales normalized to their
maximum.
4 . 2 b i a s o f s c a l e a n d n o i s e e s t i m a t e s
Figure 4 . 1 sho ws an example of the scales estimated b y MLDS for differ ent ex-
ponents and dif ferent noise conditions. S cale v alues are normalized to the range
[ 0 , 1 ] and depicted as errorbars sho wing the M ± 1S . D . across simulations. Ground
truth functions are sho wn as continuous lines. The estimated v alues follo w ed
closely the ground truth functions for all the noise conditions studied, indicat-
ing lo w bias. These results replicate pre vious w ork for the equal-v ariance case
(Maloney & Y ang, 2003 ), and additionally the y sho w ho w MLDS seems to be
robust against the presence of unequal-v ariance noise (or ’multiplicativ e’ noise,
Figure 4 . 1 right). In addition, lo w bias w as already present at relativ ely fe w
number of trials: Figure 4 . 1 sho ws an example with N = 560 trials, which is
equiv alent to repeat r = 10 times the total number of unique triads. Thus, MLDS
can estimate the underlying functions with a lo w bias, also in the presence of
multiplicativ e noise.
The scales estimated in MLDS using the ’unconstrained’ parametrization also
pro vide an estimate of the noise parameter ( ˆ σ ), because the scale maximum is

3 8 simula tions
Number of trials
Estimated noise
x 0 . 5 x 1 x 2
σ
. 035
. 07
. 035 → 0 . 14

Figure 4 . 2 : Noise estimation for all simulated conditions. Horizontal lines sho w the expected v alues, errorbars
indicate M ± 1S . D . across simulations. The dashed v ertical lines indicate the number of trials used
in the experiments in Part III (rightmost, for Ψ ( x ) = x 2 . 0 ) and in Part II (leftmost, for Ψ ( x ) = x 0 . 5 ).
inv ersely related to the noise estimated b y the model (Equation 3 . 6 ). Figure 4 . 2
sho ws the noise estimates as a function of the number of trials for all conditions.
The estimated noise (errorbars) w as in accordance with the expected v alues (hor -
izontal lines) for all conditions when the equal-v ariance assumption held. For
the unequal v ariance case (" 0 . 035 → 0 . 14 ", continuous gra y errobars) the esti-
mated noise had v alues betw een the minimum and maximum as expected for
the equal-v ariance case. For all conditions the estimated noise w as stable around
the expected v alue, showing lo w bias and lo w v ariability when more than ap-
prox. N = 1000 trials w ere used. Thus, MLDS can also reliably reco v er the noise
in the decision model when the equal-v ariance assumption holds.
4 . 3 va r i a b i l i t y a n d c o v e r a g e o f c o n f i d e n c e i n t e r v a l s
As detailed in S ection 3 . 4 , the error estimation in MLDS is calculated using boot-
strap techniques. Bootstrap allo ws to calculate confidence inter v als for the scale
v alues, and the width of the confidence inter v als are an indication of the v ari-
ability in the estimation procedure. Figur e 4 . 3 sho ws the width of the confidence

4 . 3 v ariability and co verage of confidence inter v als 3 9
Number of trials
CI width
x 0 . 5 x 1 x 2
σ
. 035
. 07
. 035 → 0 . 14

Figure 4 . 3 : W idth of the confidence inter v als (nor malized b y the scale v alue itself) as a function of
number of trials, for all simulated conditions. Example at a stimulus v alue x = 0 . 51 .
V ertical lines as in Figure 4 . 2
inter v als (nor malized b y the scale v alue itself) as a function of the number of tri-
als. It is expected that the estimation procedure gets more pr ecise (less v ariable)
as more trials are simulated and included in the analysis; this w as indeed the
case for all simulated conditions. The width of the confidence interv als decrease
in an exponential w a y , with the biggest gain in precision on the first N = 1500
trials.
Bootstrap techniques are kno wn to produce confidence inter v als that are too
narro w , i.e. with lo w coverage (W ichmann & Hill, 2001 ). Co v erage can be defined Coverage
as the number of times the true v alue (as defined in the ground truth function)
is included in the confidence inter v al across multiple simulations. Credible con-
fidence inter v als must ha v e a co v erage that is equal to the statistical confidence
that is used to calculate them. In this w a y , co v erage must be 95 % for confidence
inter v als calculated with 95 % statistical confidence lev el, i.e. the true v alue must
be included in 95 % of the simulations. As MLDS uses bootstrap to calculate
confidence inter v als, it is relev ant to check whether MLDS pro vides confidence
inter v als with adequate cov erage. Figure 4 . 4 sho ws the co v erage for all simu-
lated conditions. Co v erage approximated the expected 95 % as the number of
trials is increased, and it w as adequate for all conditions studied with N = 1000

4 0 simula tions
Number of trials
x 0 . 5 x 1 x 2
σ
. 035
. 07
Percentage %

Figure 4 . 4 : Cov erage of confidence inter v als as a function of number of trials. Horizontal dashed line
depicts 95 % co v erage; v ertical lines as in Figure 4 . 3 .
trials or more. Adequate co v erage w as largely independent of the underlying
noise lev el.
4 . 4 goodness of fit and viola tion of the equ al - v ariance assump -
t i o n
V iolations of some model assumptions are likely to occur in psy chophysical ex-
periments. It has been sho wn that some discrimination data is better fit when an
unequal-v ariance model is assumed (e.g. Kingdom & Prins, 2010 ; Goris, Putzeys,
W agemans, & W ichmann, 2013 ). Thus, it is also desirable to study the ef fect of
model violations in the outcome of MLDS , and deter mine whether the goodness
of fit procedure in MLDS can detect these cases.
Figure 4 . 5 sho ws the percentage of simulations in which the goodness of fit
w as tagged as acceptable according to the p-v alue described in S ection 3 . 5 . For
mostly all conditions the goodness of fit w as acceptable in more than 95 % of
the cases; the only exception w as the unequal-v ariance noise condition with ex-
ponent of tw o at large number of trials (gra y lines in Figure 4 . 5 right).

4 . 4 goodness of fit and viola tion of the equ al - v ariance assumption 4 1
Number of trials
x 0 . 5 x 1 x 2
σ
. 035
. 07
. 035 → 0 . 14
Percentage %

Figure 4 . 5 : Percentage of simulations with acceptable goodness of fit, as described in Section 3 . 5 . Horizontal
dashed line indicates 95 %.
These results indicate that the goodness of fit procedur e in MLDS mostly did
not detect the violation of the equal-v ariance noise assumption. MLDS masked the
underlying noise distribution when unkno wn, and therefore the outcome of an
MLDS experiment when unequal-v ariance noise is present will be indistinguish-
able from an equal-v ariance case. This is a clear disadv antage in comparison
to classical signal detection theory procedures, in which the unequal-v ariance
case can be estimated from discrimination data (McNicol, 1972 ; Knoblauch &
Maloney, 2008 ).
Ho w ev er , MLDS is robust in estimating the underlying function –for what it
w as designed for– in the presence of unequal-v ariance noise (Figure 4 . 1 ). In this
respect MLDS o v ercomes the shortcomings of Fechnerian integration, reco v ering
a scale that is independent of the noise distribution. As revie w ed in Equation 2 . 1 ,
scales deriv ed b y Fechnerian integration of JNDs cannot be guaranteed to reco v er
the true function, as JNDs depend on the noise distribution and the distribution
itself cannot be deter mined experimentally . MLDS a v oids this issue b y estimated
the scale v alues independently of the noise distribution.

4 2 simula tions
4 . 5 v i o l a t i o n o f o t h e r m o d e l a s s u m p t i o n s
Other assumptions in MLDS that could also be violated are re view ed here. It is
rele v ant to kno w the ef fect of the violations on estimation, as these assumptions
cannot be tested experimentally .
4 . 5 . 1 Corr elation
MLDS in the signal detection theory for m assumes that each dra w from the sen-
sory function is independent of each other (S ection 3 . 3 ). In the method of triads
the stimulus arra y is presented simultaneously , and therefore it is possible that
correlation ma y exist among the sensor y responses for a triad. In this case the
independence assumption w ould be violated, and the estimates of the noise re-
tur ned b y MLDS ma y dif fer . Using simulations w e can analyze quantitativ ely the
dif ferences that occur in the presence of correlated noise.
T o simulate correlated noise the obser v er model in Equation 4 . 1 must be mod- Correlated observer
ified b y expressing it in a v ectorized for m. In this w a y , each triad is a v ector
 x = ( x 1 , x 2 , x 3 ) that is presented to the observ er and ev okes a v ector of sensor y
responses ( ψ  x ) that is dra wn from a multiv ariate Gaussian process
ψ  x ∼ N  Ψ (  x ) , 
Σ 
with co v ariance matrix

Σ = 




σ 2 σ 2
c σ 2
c
σ 2
c σ 2 σ 2
c
σ 2
c σ 2
c σ 2





The co v ariance matrix ( 
Σ ) has diagonal components σ 2 and non-diagonal com-
ponents σ 2
c . The diagonal v alues σ 2 are the assumed v ariance v alues for the non-
correlated noise, and it w as set to σ 2 = 0 . 07 2 as in previous simulations. The
non-diagonal v alues σ 2
c represent the co v ariance betw een the sensor y v ariables

4 . 5 viola tion of other model assumptions 4 3
A B
Estimated noise
Percentage %

Figure 4 . 6 : Results of simulated obser v ers with added corr elated noise ( σ c ), violating the independence
assumption. (A) Estimated noise lev el ˆ σ as a function of the correlated noise σ c added to
the simulation, for a fixed non-correlated noise lev el ( σ = 0 . 07 ). Errorbars indicate M ±
S . D . across N= 100 simulations. (B) Per centage of simulations with acceptable goodness of
fit from panel (A). Horizontal dashed line indicates 95 %.
in that triad, and correlation is introduced b y setting σ 2
c > 0 . In principle each
non-diagonal entry in the cov ariance matrix could ha v e a different v alue, but for
simplicity they w ere all set to a fixed v alue ( σ 2
c ). This “correlated observ er" w as
simulated using the same procedure as pr eviously described.
Figure 4 . 6 A sho ws the simulation results b y plotting the noise v alue estimated
b y MLDS ( ˆ σ , y-axis) as a function of increasing correlated noise ( σ c , x-axis). When
no correlation is present ( σ c = 0 ) the estimated noise v alue matches the expected
v alue ( ˆ σ = 0 . 14 for a simulated σ = 0 . 07 , see Equation 3 . 9 ). When correlation is
added, MLDS returns a noise estimate that is lower than the expected under the
independence assumption. In addition, adding correlated noise w as not detected
b y the goodness of fit procedure, as sho wn in Figure 4 . 6 B. Thus, MLDS underesti-
mate the noise when the independence assumption is violated, and this violation
appears unnoticed to the goodness of fit diagnostics.

4 4 simula tions
A B
Percentage %
Estimated noise

Figure 4 . 7 : Results of simulated obser v ers with an absolute value decision rule . (A) Esti-
mated noise lev el ˆ σ as a function of the simulated noise σ . Errorbars indi-
cate M ± S . D . across N= 100 simulations. The gra y line indicates the expected
noise v alues. (B) Percentage of simulations with acceptable goodness of fit
from panel (A). Horizontal dashed line indicates 95 %.
4 . 5 . 2 Decision rule
The decision rule for MLDS in its original for mulation comprises the comparison
of tw o perceptual interv als with an absolute v alue operation, and corrupted b y
Gaussian-noise (Equation 3 . 2 ). This rule is further simplified to a simple dou-
ble dif ference rule b y eliminating the absolute v alue operation (Equation 3 . 8 ),
and thus allo wing the scale estimation using a GLM (S ection 3 . 2 ). Ho w ev er , in
the presence of sensory representations that are noisy , the original decision rule
could still be v alid, representing the direct comparison of interv als (as comparing
their absolute length)
∆ = | ψ x 3 − ψ x 2 | − | ψ x 2 − ψ x 1 | Absolute value
decision rule This rule is mathematically equiv alent to Equation 3 . 8 when the function Ψ ( x )
is deter ministic and monotonically increasing, and thus dif ferences inside the
absolute v alue operation are alw a ys positiv e. Ho w ev er , since w e modeled sen-
sory functions that are dra ws from a random (Gaussian) distribution, the use

4 . 6 summary 4 5
of the absolute v alue operation could ha v e a differ ent outcome. Differ ences can
occur when the noise is large enough to pr oduce inter v als that are negativ e, i.e.
when ψ x 2 − ψ x 1 < 0 and thus ψ x 2 − ψ x 1  = | ψ x 2 − ψ x 1 | . Using simulations w e
can quantify the ef fect of the change on the decision rule in MLDS estimation.
The results of these simulations are sho wn in Figure 4 . 7 A. The estimated noise
b y MLDS ( ˆ σ , y-axis) is plotted against the noise introduced in the simulation ( σ ,
x-axis) when the absolute v alue rule is applied. The estimated noise b y MLDS
(errorbars) w as consistently higher than the expected v alues (gra y line) at high
noise lev els. Thus, MLDS ov erestimate the noise in the presence of the absolute
v alue rule, but only when at large noise v alues (approx. larger than ˆ σ = 0 . 4 ).
This upper limit is much higher than noise v alues obser v ed experimentally in
human obser v ers. The goodness of fit diagnostics of MLDS also did not detect the
change in the decision rule (Figure 4 . 7 B).
4 . 6 s u m m a r y
Before the testing of MLDS in experiments w as carried out, the accuracy (bias)
and precision (v ariability) of the MLDS estimation w as quantified using simula-
tions. MLDS could reliably estimate the generativ e perceptual function with lo w
bias, pro viding at the same time a reliable estimate of the noise in the decision
model when the equal-v ariance assumption holds. In addition, MLDS pro vided
confidence inter v als that are stable in precision after approx. N = 1000 trials, and
with an adequate co v erage that w as independent of the simulated noise lev el.
In the case of unequal-v ariance noise distribution, MLDS could reliably reco v er
the shape of the function. The distribution of the noise, e.g. equal- or unequal-
v ariance, cannot be measured experimentally , therefore it is critical for MLDS to
be robust against dif ferent noise distributions. The goodness of fit procedur e did
not detect this model violation, and the estimated noise is non-indicativ e of the
underlying unequal-v ariance distribution.
V iolations of other model assumptions w ere also studied, b y introducing cor -
related noise, violating the independence assumption, and changing the decision
rule operation from a simple dif ference to an absolute v alue operation. These vi-

4 6 simula tions
olations also cannot be tested independently in experiments, and therefore it is
rele v ant to sho w that MLDS could be robust against them.
The results of these simulations established the v alidity of MLDS for estimat-
ing perceptual scales. The y also infor med some practical decisions that must be
taken during preparation of experiments, such as the stimulus spacing or num-
ber of trials, b y considering the accuracy and precision of the method obtained
from these simulations. In the part that follo ws (Part II) I present a study that
sho ws ho w MLDS w as used in the lightness domain, b y measuring perceptual
scales in a scenario of expected lightness constancy . In the study presented in
Part III the possibility of using MLDS to deriv e sensitivity , traditionally done with
perfor mance-based methods, is explored in depth using simulations as w ell as
experimental testing in a slant-from-texture task. Chapter 13 pr o vides a general
discussion of the results of both studies, and an o v erall ev aluation of MLDS .

Part II
USING MLDS T O MEASURE APPEARANCE
This part has been published in:
W iebel C.B.*, Aguilar G.*, Maertens M. ( 2017 ). Maximum likelihood differ -
ence scales represent per ceptual magnitudes and predict appearance matches.
Journal of V ision, 17 ( 4 ): 1 , 1 - 14 . doi: 10 . 1167 / 17 . 4 . 1 (postprint)
*: equal contribution

5
I N T R O D U C T I O N
One major objectiv e in the scientific study of perception is to understand ho w
psy chological experiences are linked to physical v ariables in the w orld (Fechner,
1860 ). Devising pr oper methods to quantify this relationship has tur ned out to be
challenging, because psy chological v ariables, contrar y to physical ones, cannot
be obser v ed directly , but must be inferred from obser v ers’ responses to properly
chosen stimuli (e.g. Gescheider, 1988 ). In the absence of a w ell-established mea-
surement theory (Krantz et al., 1971 ), Fechner ’s simple method of adjustment
(matching) is hard to beat and r emains widely used (Koenderink, 2013 ).
T o illustrate the problem let’s sa y w e are interested in the perceiv ed light-
ness of the target check (Fig. 5 . 1 A, r ed outline) presented behind a transparent
medium. Introducing a transparent medium betw een a surface and the obser v er
(Fig. 5 . 1 B) changes the mapping betw een surface reflectance and retinal lumi-
nance in a characteristic w a y (Fig. 5 . 1 C). The luminance range of surfaces seen
through a transparent medium is substantially r educed and potentially shifted
relativ e to the luminance range for surfaces seen in plain view . T o be inv ariant
against such changes the visual system has to “undo” these changes b y appro-
priate computations (e.g. Singh & Anderson, 2002 ; Singh, 2004 ; W iebel, Singh, &
Maertens, 2016 ). This approximate inv ariance of perceiv ed lightness across v ary-
ing luminance is kno wn as lightness constancy . W e kno w from experience and
from empirical studies that human observ ers are indeed largely inv ariant against
such fluctuations in retinal luminance. Ho w ev er , w e still lack a theoretical model
of ho w the visual system accomplishes lightness constancy . T o dev elop such a
model, w e must be able to measure the relationship betw een retinal luminance
and perceiv ed lightness in a reliable and comprehensiv e w a y . T o that end, w e ide-
ally w ant to estimate the functions describing this relationship, which are kno wn
as transducer functions or perceptual scales (e.g. Kingdom & Prins, 2010 ).
49

5 0 introd uction
high transparent dark
high transparent light
low transparent dark
low transparent light
B A

I2

matching experiment
atmospheric transfer
functions
Reflectance
Luminance
C

Figure 5 . 1 : Experimental stimuli. A. The basic stimulus is a 10 x 10 checkerboard composed of checks with 13
possible reflectance v alues. In an asymmetric matching task obser v ers adjust the luminance of an
exter nal test field so that is matches the perceiv ed lightness of a specified target check (here I 2 ).
Obser v ers are said to be lightness constant when their matches indicate the inv ersion of the v arious
reflectance-to-luminance mappings that are intr oduced b y differ ent transparent media (see B and C).
B. Checkerboards w ere also presented behind differ ent transparent media that v aried in reflectance
(dark and light) and in transmittance (high and lo w). C. Atmospheric transfer functions (A TFs)
relate target r eflectance (x-axis) to target luminance (y-axis) (Adelson, 2000 ). The color scheme
corresponds to the images in B. In the transparency conditions the luminance range is compressed
and/or shifted with respect to plain vie w . This is reflected in corresponding slope and intercept
changes of the respectiv e A TFs.

intr oduction 5 1
x M
x T
Ψ( x M ) == Ψ( x T )
x T
x M
B A
Ψ( x M
x T )
T arget luminance
Match luminance

Figure 5 . 2 : Perceptual processes underlying matching pr ocedures. (A) At each position, match
and target, ther e is a transducer function that relates retinal luminance ( x M , x T ) to
perceiv ed lightness ( Ψ ( x M ) , Ψ ( x T ) , insets on the stimulus). (B) What is measured
in a matching procedure ar e the luminances x M and x T that correspond to equal
perceiv ed lightness at both positions ( Ψ ( x M ) == Ψ ( x T ) ). After Maertens and W ich-
mann ( 2013 ).
The most commonly used method for measuring this relationship is the method
of adjustment, ev en though it does not pro vide a direct estimate of the transducer
functions and it presumes a number of operations on the part of the obser v er .
Figure 5 . 2 depicts the processes inv olv ed in adjustment or matching procedures
for perceiv ed lightness. An obser v er adjusts the intensity of a test stimulus so
that it looks identical to a giv en standard. It is assumed that the observ er inter-
nally compares magnitudes of perceiv ed lightness for the target ( Ψ ( x T ) ) and the
match ( Ψ ( x M ) ). What is being measured though, are not the transducer func-
tions relating the tw o, but the corresponding luminances of the tar get and the
match ( x T and x M , Fig. 5 . 2 B).
Another problem with the method arises when, as in the abo v e case, test and
match are presented in dif ferent contexts (asymmetric matching). In most cases
researchers ar e interested in such asymmetric comparisons because they allo w

5 2 introd uction
one to quantify the degree of per ceptual constancy . Such asymmetric compar -
isons become problematic, ho w ev er , when the differ ence in context causes ap-
pearance dif ferences that cannot be compensated along the dimension of the
adjustment (Brainard, Brunt, & Speigle, 1997 ; Ekr oll & Faul, 2013 ; Foster, 2003 ;
Logvinenko & Maloney, 2006 ; Logvinenko, Petrini, & Malone y, 2008 ). The con-
sequence w ould be an inaccurate or ev en inv alid measurement that does not
capture the perceptual r epresentation of the stimulus.
Recently , there ha v e been attempts to tackle the problems associated with
matching (Logvinenko & Malone y, 2006 ; Logvinenko et al., 2008 ; Radonjic, Cot-
taris, & Brainard, 2015 b; Radonji ´ c & Brainard, 2016 ; Umbach, 2013 ). While it is
widely accepted that obser v ers are relativ ely lightness constant under natural
viewing conditions, many experiments still find v arying amounts of constancy
for dif ferent viewing conditions, stimuli, task types or e v en instructions (Foster,
2011 ; Gilchrist et al., 1999 ). Such deviations might either be a consequence of
methodological problems like the ones just outlined, or a meaningful de viation
from constancy which then w ould need to be explained b y any successful light-
ness model. Progress in r ev ealing the underlying mechanisms for lightness per-
ception is therefore tightly coupled with choosing appr opriate and robust exper -
imental methods that allo w the comprehensiv e testing of theoretical models.
As an ef fort in this direction w e address the limitations of matching proce-
dures b y adopting the follo wing approach. W e measure the transducer functions
directly using Maximum-Likelihood Dif ference S caling ( MLDS , Maloney & Y ang,
2003 ). MLDS is a scaling method that allo ws the ef ficient estimation of percep-
tual scales, i.e. the transducer functions relating retinal luminance and per ceiv ed
lightness (Fig. 5 . 1 ). It has been used to study v arious perceptual dimensions
(e.g. Obein, Knoblauch, & V iénot, 2004 ; Fleming, Jäkel, & Maloney, 2011 ). Fur -
ther more, it is based on a signal detection model which potentially allo ws one
to relate measurements of appearance with measurements of discriminability
(Devinck & Knoblauch, 2012 ; Aguilar et al., 2017 ). Here w e used MLDS to measure
perceptual scales in dif ferent contexts using only within-context comparisons in
order to a v oid the procedural pr oblems of asymmetric matching. The estimated
scales are constructed from the judgment of per ceiv ed stimulus dif ferences and
not from the adjustment of a refer ence as in other scaling methods such as mag-

intr oduction 5 3
nitude estimation (Gescheider, 1988 ) or partition scaling (Whittle, 1994 ). MLDS
requires a straightforw ard per ceptual judgment and is thus less susceptible to
strategic influences.
T o scrutinize whether MLDS pro vides reliable per ceptual scales of lightness w e
v alidate the scales empirically and theoretically . First, w e use the estimated scales
to predict perceptual matches and compar e them to matches gathered in an inde-
pendent asymmetric matching experiment. S econd, w e compare the predictiv e
po w er of a contrast-based lightness model (Zeiner & Maertens, 2014 ; W iebel et
al., 2016 ) for scaling and matching data. T o anticipate, w e found that (a) the
empirical perceptual scales for dif ferent contexts w ere consistent with lightness
constancy , (b) matching data w ere w ell predicted b y the perceptual scales, and
(c) human lightness perception follo w ed a differ ence scale that corresponds to
a nor malized contrast metric. The predictiv e po w er of the contrast-based light-
ness model w as higher for the scaling than for the matching data, suggesting
that estimating perceptual scales has the adv antage of probing more dir ectly the
inter nal dimension under study .

6
M E T H O D S
6 . 1 o b s e r v e r s
T en naïv e obser v ers participated in the study , fiv e of them w ere female. Ob-
ser v ers’ age ranged from 19 to 32 y ears. All obser v ers had nor mal or corrected
to nor mal visual ability and w ere reimbursed for participation. Informed written
consent w as giv en b y all obser v ers prior to the experiment.
6 . 2 s t i m u l i a n d a p p a r a t u s
Stimuli w ere presented on a linearized 21 -inch Siemens SMM 2106 LS monitor
( 400 x 300 mm, 1024 x 768 px, 130 Hz). Presentation w as controlled b y a DataPixx
toolbox (Vpixx T echnologies, Inc., Saint-Bruno, QC, Canada) and custom presen-
tation softw are ( http://github.com/TUBvision/hrl ). Obser v ers w ere seated 110
cm a w a y from the screen in a dark experimental cabin. Observ ers’ responses
w ere registered with a ResponsePixx button-box (VPixxT echnologies, Inc., Saint-
Bruno, QC, Canada).
The stimuli w ere images of customized checkerboar ds composed of 10 x 10
checks (Fig. 5 . 1 ). The images w ere rendered using Po vra y (Persistence of V i-
sion Ra ytracer Pty . Ltd., W illiamsto wn, V ictoria, A ustralia, 2004 ). The position of
the checkerboard, the light sour ce and the camera w ere kept constant across all
images. Checks w ere assigned one out of thirteen surface reflectance v alues ac-
cording to the experimental design (see belo w). In the transparency conditions,
a transparent la y er w as placed betw een the checkerboard and the camera (Fig.
5 . 1 B). It w as positioned so as to co v er all target and their surrounding checks
in both the MLDS and the matching experiment. The transparency w as created
using alpha blending (Metelli’s episcotister model). The image luminances of the
background B and the foregr ound F are combined accor ding to some w eighting
55

5 6 methods
factor α so as to result in a new image luminance at the position of transpar ency
T = α × B + ( 1 − α ) × F . An α v alue of 0 corresponds to an opaque foreground
T = F , α of 1 corresponds to a fully transparent for eground T = B . The trans-
parent la y er v aried in transmittance and reflectance. The dark transparency had
a v alue of 0 . 35 in povray reflectance units ( 19cd/m 2 ) and the light transparency
of 2 ( 110cd/m 2 ). A v alue of α = 0 . 4 and 0 . 2 w as used in the high and lo w
transmittance condition respectiv ely . The rendered images w ere conv erted to
gra yscale images. The background luminance w as 141cd/m 2 . Detailed v alues of
luminance for each transparent medium can be found in Appendix T able A. 4 ).
In the matching experiment, an adjustable test field w as presented abo v e the
checkerboard to assess observ ers’ lightness matches (Fig. 5 . 1 A). The test field
w as embedded in a coplanar surround checkerboard that w as composed of 5 x 5
checks. The size of the test field w as 1 . 2 x 1 . 2 degrees visual angle and that of
the surround checkerboar d w as 3 x 3 degrees. The luminances of the checks in
the surround checkerboar d w ere fixed throughout the experiment and the lumi-
nances w ere chosen so that tw o adjacent checks did not ha v e the same luminance.
The mean luminance of the surround checks w as 178cd/m 2 , which is identical
to the mean luminance of the 13 checks in the main checkerboar d in plain view .
The surround checkerboar d w as presented in four different spatial arrangements
resulting from clockwise r otation of the original in steps of 90 degree. A config-
uration w as assigned randomly to each trial.
6 . 3 d e s i g n a n d p r o c e d u r e
Perceptual scales and asymmetric matching functions w ere measured for fiv e
dif ferent viewing conditions, a plain vie w condition and four transparency con-
ditions (Fig. 5 . 1 ).
6 . 3 . 1 MLDS experiment
W e used MLDS with the methods of triads (Figure 6 . 1 A, Knoblauch & Maloney,
2008 , 2012 ). W e used ten out of the 13 reflectance v alues to construct the triads.

6 . 3 design and pr ocedure 5 7
The lo w est and the tw o highest reflectance v alues w ere omitted to achiev e a fea-
sible number of trials. W ith p = 10 reflectance v alues the total number of unique
triads w as n = p ! / (( p − 3 ) ! × 3 ! ) = 10 ! / ( 7 ! × 3 ! ) = 120 . Each triad contained
three v alues that w ere selected so as to enclose non-o v erlapping inter v als. They
w ere presented in ascending ( x 1 < x 2 < x 3 ) or descending ( x 1 > x 2 > x 3 ) or der
(Knoblauch & Maloney, 2008 ). The r eference, x 2 (check I 2 in Fig. 6 . 1 A), w as lo-
cated betw een the tw o comparisons, x 1 and x 3 (checks B 2 and I 9 in Fig. 6 . 1 A). In
each trial obser v ers judged which comparison check x 1 or x 3 w as more different
in lightness from the refer ence. Obser v ers used a left or right response button to
indicate their choices. No time limit w as imposed.
T o keep the local context comparable for the elements of a triad w e controlled
the luminances of the eight checks surrounding each triad element. The same
eight luminance v alues w ere used for each triad element but they dif fered in
spatial arrangement. Their mean luminance w as 17 8cd/m 2 which w as identical
to the mean luminance of all checks seen in plain vie w . The remaining 73 checks
w ere dra wn randomly without replacement from a set consisting of six r epeats
of the 13 dif ferent reflectance v alues. This resulted in a slight v ariation of the
mean luminance of those checks betw een trials (up to 6cd/m 2 ). The checks w ere
positioned so that tw o neighboring checks did not ha v e the same reflectance.
Each triad w as repeated 10 times resulting in 1200 trials per viewing condition
and 6000 trials in total. T rials w ere randomized across viewing condition, triad
and target r eflectance. The experiment w as divided into sev eral sessions. A new
image w as created for each trial.
6 . 3 . 2 Matching experiment
T arget reflectances and vie wing conditions w ere identical to those in the MLDS ex-
periment. The target check w as presented at the position of the refer ence (check
I 2 in Figure 5 . 1 ) in the MLDS experiment. Obser v ers adjusted the luminance of
the exter nal test field to match the perceiv ed lightness of the target check. The
luminance w as adjusted b y pressing one of four buttons, tw o of them for coarse
adjustments ( ± 10cd/m 2 ) and the other tw o for fine adjustments ( ± 1cd/m 2 ). The
maximum luminance of the monitor w as 550cd/m 2 . Satisfactor y matches w ere

5 8 methods
confir med with a fifth button which initiated the next trial. No time limit w as
imposed on the adjustment procedure.
The eight checks surrounding the tar get w ere assigned in the same w a y as in
the MLDS experiment. The remaining 91 check reflectances w ere dra wn randomly
without replacement from a set consisting of eight r epeats of all 13 reflectance
v alues. Thus the mean luminance across trials w as comparable to that in the
MLDS experiment. Again, neighboring checks had to ha v e different r eflectances.
Each combination of target reflectance and vie wing condition w as repeated 10
times resulting in a total of 500 trials. A ne w image w as created for each trial
and trials w ere randomized across experimental conditions. 1
6 . 4 simula tion of observer models
W e used an ideal obser v er analysis to test whether MLDS could distinguish be-
tw een dif ferent generativ e models. In particular , w e tested a lightness-constant
against a luminance-based obser v er , tw o extremes of beha vioral judgments. The
model comparison is done as follo ws. W e define inter nal scales for each of
the tw o models (Fig. 6 . 1 B). For a luminance-based obser v er the luminance-to-
lightness mappings in dif ferent contexts coincide on a single function and dif fer
only in the range of luminance v alues (Fig. 6 . 1 B low er left panel). For mally , the
sensory representation function w as defined as
Ψ lum ( x ) = a · x + b
where x is luminance, and a , b are linear coef ficients calculated to map the range
of luminance in plain view [ L min , L max ] to the range [ 0 , 1 ] .
For a lightness-constant obser v er the mapping functions in dif ferent contexts
should ‘undo’ the transfor mations of image formation in which equal surface
reflectances are mapped onto dif ferent luminance ranges (Fig 5 . 1 C). Thus, w e
model this obser v er b y using inter nal mapping functions that are the inv erse
1 S ection “MLDS analysis" from the published paper w as omitted here to a v oid redundancy with
Chapter 3 of this thesis.

6 . 4 simula tion of observer models 5 9
functions of the A TFs sho wn in Figure 5 . 1 C. For mally , the sensor y representation
function w as defined as
Ψ light ( x ) = a i · x + b i i ∈ 1 ... 5
where x is luminance, and a i , b i are linear coef ficients calculated to map the
range of luminance for each viewing condition to the range [ 0 , 1 ] (for simplicity
w e used linear functions, but po w er functions could be used as w ell and w ould
not change our ideal obser v er results).
Each of the tw o obser v er models is used to generate responses in a ‘mock’
MLDS experiment that has the same number of triads and repetitions as the actual
experiment. For each triad and repetition, the decision v ariable w as calculated
as
∆ = [ Ψ ∗ ( x 3 ) − Ψ ∗ ( x 2 )] − [ Ψ ∗ ( x 2 ) − Ψ ∗ ( x 1 )] + ϵ ( 6 . 1 )
with ϵ ∼ N ( 0 , σ 2 ) , and Ψ ∗ is either Ψ lum or Ψ light . Simulated responses w ere
generated choosing the triad ( x 2 , x 3 ) when ∆ > 0 and ( x 1 , x 2 ) other wise. Finally ,
the simulated data w ere subjected to the MLDS analysis to obtain the coef ficients
β that constitute the scale v alues. Figure 6 . 1 B sho ws the model perceptual scales
(left) and the estimated scales (right), and it is evident that for the chosen noise
lev el ( σ = 0 . 15 ) the method reco v ers the underlying scale.
W e repeated the ideal obser v er analysis for a range of differ ent noise lev els ( σ ,
minimum = 0 . 01 and maximum 1 . 2 , see Appendix A. 1 ). The tw o obser v er mod-
els w ere distinguishable for a broad range of noise le v els up to approximately
0 . 4 . This upper -bound v alue w as much higher than the noise lev els that ha v e
been obser v ed in previous experiments (Knoblauch & Malone y, 2008 ; Devinck
& Knoblauch, 2012 ). W e therefore concluded that MLDS could be used to deriv e
meaningful scales because the y w ould allo w us to distinguish betw een these tw o
dif ferent obser v er models.

6 0 methods
Lightness constant observer
Luminance-based observer
Luminance

B2

I2

I9

A B

B2

I2

I9

Difference scale
Luminance
Difference scale
0
0
MLDS experiment Ψ( x )
Ψ( x 1 )
Ψ( x 1 )
Ψ( x 1 )
x 1
x 1
Ψ( x )

Figure 6 . 1 : Method of triad procedure and observ er models. A. In the triad comparison obser v ers
compared the lightness of three specified checks ( B 2 , I 2 and I 9 , marked with a red outline).
The upper panel sho ws a triad comparison in plain view , the lo w er panel a comparison
behind one of the transparent media. B. Simulation for a lightness constant (upper panels)
and a luminance-based obser v er (lo w er panels). For the lightness constant obser v er the
perceptual scales (upper left panel) correspond to an inv erse mapping of the atmospheric
transfer functions (Fig. 5 . 1 C). For the luminance-based obser v er (lo w er left panel) the
luminance-to-lightness mappings in dif ferent contexts coincide on a single function. W e
generated data for each of the models in simulations, and the estimated perceptual scales
are sho wn on the right panels. S ee text for details.

7
R E S U L T S
Figure 7 . 1 sho ws the perceptual scales measured in dif ferent viewing conditions
aggregated across all observ ers. The scales are inter v al scales with the minimum
anchored at zero and the maximum being inv ersely proportional to the estimated
noise (in MLDS ter minology referred to as ‘unconstrained scales’; Knoblauch &
Maloney, 2012 ).
The empirical scales are consistent with a lightness-constant observ er and not
with a luminance-based obser v er . This is evident from a comparison betw een
the model predictions (Fig. 6 . 1 B) and the observ ed result patter n (Fig. 7 . 1 A). Al-
though the estimated scales are not linear the y share crucial features with the
hypothetical scales. First, there is a dif ference in ‘intercept’ betw een perceptual
scales in the light and dark transparency conditions (blue vs. green lines in Fig.
7 . 1 A). S econd, the scales are steeper for transparent media with lo w er transmit-
tance than with higher transmittance (light vs dark colored lines in Fig. 7 . 1 A).
Figure 7 . 1 A also plots the Munsell neutral v alue scale (Munsell et al., 1933 ) that
w ould be predicted for our choice of luminances (dashed black line in Fig. 7 . 1 A).
The Munsell scale represents the expected scale that r elates equal steps in per-
ceiv ed lightness to luminance (Whittle, 1994 ). It w as calculated b y setting the
highest luminance in the plain view stimulus as the white refer ence, i.e. to the
maximum of one (Pauli, 1976 ). It is evident fr om Figure 7 . 1 A that the Munsell
scale is consistent with the perceptual scale estimated in our plain vie w con-
dition. The typical nonlinear shape indicates higher sensitivity for differ ences
betw een checks of lo w reflectances than for checks with high reflectances. This
has indeed been reported in pre vious w ork (e.g. Chubb, Landy , & Econopouly,
2004 ). T o aggregate scales across observ ers w e nor malized the scales of each indi-
vidual obser v er relativ e to the maximum scale v alue in the plain view condition.
The ranges of the scales dif fered betw een obser v ers because differ ent obser v ers
61

6 2 resul ts
ha v e different noise le v els. The data for individual obser v ers are pro vided in
Appendix Figure A. 1 .
Luminance [cd/m²]
Difference scale
Viewing context high transparent dark
high transparent light
low transparent dark
low transparent light
plain view
Reflectance [ povray a.u.]
0.0 0.5 1.0 1.5 2.0 0 100 200 300
0.0
0.2
0.4
0.6
0.8
1.0
1.2
B A

Figure 7 . 1 : MLDS dif ference scales in dif ferent viewing conditions. A. Dif ference scales
as a function of luminance. The functions depict the aggregated scales across
obser v ers (n= 10 ). For each obser v er the scales w ere nor malized with respect
to plain view , and then aggregated. The dashed black line depicts the Munsell
scale in plain view (see main text for a description of the Munsell scale). Err or
bars indicate M ± S . D . B. Same as in A. but scales are plotted as a function
of reflectance. S cale v alues (markers, M ± S . D .) w ere fitted with a po w er
function (lines) individually for each viewing condition.
7 . 1 scales as a function of reflect ance
T o better illustrate the degree of lightness constancy across conditions w e re-
placed luminance b y reflectance at the x-axis of the perceptual scales. In such a
perceiv ed lightness vs. reflectance plot the scales of a lightness-constant obser v er

7 . 1 scales as a function of reflect ance 6 3
should coincide on a single function. Figure 7 . 1 B sho ws that this w as indeed that
case.
T o assess the agreement betw een scales in differ ent conditions quantitativ ely
w e compared the functions that w ere fit in each condition against what w e call a
global fit in which the data from all conditions are fitted b y a single function. If
the data in dif ferent viewing conditions can be explained b y one inter nal model
then the global fit should account for the data as w ell as the individual fits for
each viewing condition. W e fitted the scale parameters in each condition and the
global scale with a po w er function Ψ ( x ) = ax e + b using a nonlinear least squares
method (Ritz & Streibig, 2008 ). T o ev aluate the goodness of fit w e computed R 2
v alues for linear fits to the data. The a v erage R 2 w as already reasonably high
( 0 . 86 ). W e then perfor med F-tests on nested models (po w er function vs. its linear
submodel with e = 1 ) which re v ealed that the po w er functions fitted the data
significantly better than the linear ones ( F min ( 1 , 97 ) = 15 . 6 , p < 0 . 001 ). From this
w e conclude that the po w er functions captured the data suf ficiently w ell.
W e used a general non-linear model to test whether applying single models to
the data in the fiv e dif ferent viewing conditions w ould result in better fits than
applying a global model to all data. W e compared the respectiv e sum of squares
for the global model with three parameters ( a , b , e ) and for the separate models
with fiv e times three parameters. There w as a benefit for the separate model fits
relativ e to the global model ( F ( 12 , 497 ) = 18 . 57 , p < 0 . 001 ). T o explore the cause
for this dif ference w e computed one-w a y repeated measures ANOV As for each
of the three parameters of the po w er functions. W e found a significant differ ence
betw een scales for the exponent parameter , e ( F ( 4 , 36 ) = 16 . 6 , p < 0 . 001 ) which
deter mines the cur v ature of the function. Posthoc tests on the exponents rev ealed
significant dif ferences betw een each of the light transparency conditions and the
plain view and the dark transpar ency with high transmittance (Bonferroni cor -
rected p < 0 . 05 ). The main dif ference betw een the light transparency conditions
and the plain view and the dark transpar ency (high transmittance) conditions is
the dif ference in cur v ature betw een these functions (Fig. 7 . 1 B).
The light transparency conditions are special insofar as during image forma-
tion the reflectance-to-luminance mapping undergoes a range r eduction and a
range shift (see Fig. 5 . 1 ). This means that checks seen through a light transparent

6 4 resul ts
medium undergo the gr eatest compression in its contrast range. The Michelson
contrast for targets in plain vie w range from - 0 . 84 to 0 . 4 , whereas in the lo w
transparent light condition range from - 0 . 16 to 0 . 16 (the contrast is computed r el-
ativ e to the mean luminance in the region of transparency). Ther efore, sensitivity
might be lo w er for this range of the stimuli.
7 . 2 per ceptu al scales and ma tching functions
W e illustrated in Figure 5 . 2 ho w the data recorded in matching pr ocedures are
related to per ceptual scales. Here w e sho w to what extent the theoretical relation-
ship can be corroborated b y experimental data. T o predict matching data from
perceptual scales, one needs to first find the scale v alue Ψ ( x T ) that corresponds
to a particular target luminance x T in one of the transpar ency conditions. In the
next step w e need to find the luminance v alue x M that corresponds to the scale
v alue at the match position ( Ψ ( x M ) ) assuming that obser v ers match the lightness
of the match region to that of the tar get region according to Ψ ( x M ) == Ψ ( x T ) .
W e did not measure a perceptual scale at the match position but instead adopt
the plain view scale to r epresent the scale for the matches. In order to be able to
read out x -v alues corresponding to any possible Ψ -v alue and vice versa w e fitted
the scales with po w er functions ( ψ ( x ) = ax e + b ) using a non-linear least squares
method. W e deriv ed the predicted matching data from the ‘unconstrained’ scales
individually for each obser v er , and w e then aggregated them in the same w a y as
the empirical data obtained from the matching experiment.
In Figure 7 . 2 empirical and predicted matches are plotted next to each other
(panels A and B, respectiv ely) and it can be seen that the y share some character -
istic features. The matching functions, like the scales (Fig. 7 . 1 A), dif fer in slope
and intercept betw een the different transparency conditions. Dif ferences in trans-
mittance are accompanied b y differences in slope and dif ferences in reflectance
are accompanied b y differences in inter cept. Unlike the scales the matching func-
tions are linear .
For a quantitativ e ev aluation of the degree of similarity betw een empirical and
predicted matching data w e computed linear regressions for each of the vie wing
conditions. W e used within-subject t-tests to compare slopes and intercepts be-

7 . 2 per ceptu al scales and ma tching functions 6 5
Viewing context high transparent dark
high transparent light
low transparent dark
low transparent light
plain view
B A
T arget Luminance [cd/m²] T arget Luminance [cd/m²]
Match Luminance [cd/m²]
Matching experiment Prediction derived from scales
0 10 0 20 0 30 0 40 0 50 0
0
10 0
20 0
30 0
40 0
50 0
0 10 0 20 0 30 0 40 0 50 0

Figure 7 . 2 : Empirical and predicted matching data. A. Results of the matching experi-
ment. The luminance adjusted in the matching field (y-axis) is plotted as a
function of target luminance (x-axis) in each vie wing conditions. Data w ere
aggregated across observ ers (n= 10 ). Error bars indicate mean ± S.D. (B) Same
as in (A) but for matches predicted from the estimated MLDS scales.

6 6 resul ts
tw een predicted and empirical functions. The a v erage slope and intercept v alues
are listed in Appendix T able A. 3 together with the relev ant test statistics. W e
found significant dif ferences betw een the predicted and the empirical functions
only for the dark transparent medium with a high transmittance.
7 . 3 predictive power of a contrast - based model
The estimated perceptual scales ar e an interesting test case for lightness mod-
els because they represent a more dir ect measurement of perceiv ed lightness
than the matching data. In particular w e compared ho w w ell our previously sug-
gested nor malized-contrast model (Zeiner & Maertens, 2014 ) could account for
both the scaling and the matching data.
The nor malized contrast model w as initially motiv ated b y the obser v ation that
the introduction of a transparent medium leads to a systematic change in con-
trast range of the respectiv e image region. It w as suggested that this change in
contrast range might ser v e as a cue to segregate the region fr om regions seen
in plain view (Anderson, 1999 ; Singh & Anderson, 2002 ; Singh, 2004 ). It has
been subsequently sho wn that the accompanying contrast statistics can be used
to accurately predict perceiv ed lightness (Singh & Anderson, 2002 ; Singh, 2004 ;
Zeiner & Maertens, 2014 ; W iebel et al., 2016 ). The nor malized contrast model
engages tw o processing steps: first the tar get intensity is nor malized relativ e to
its local surround b y computing the Michelson contrast betw een target and sur -
round. S econd this tar get contrast is nor malized relativ e to the contrast range
in the region of the transparency which is subsequently mapped to the contrast
range in plain view (for details of the normalized contrast model calculation
see Appendix). The so deriv ed nor malized contrast predicts observ ers’ lightness
matches in contrast units.
Figure 7 . 3 sho ws the aggregated data of both experiments as a function of the
model predictions. If the computed normalized contrast accounts w ell for dif fer-
ences in appearance then the functions should line up on top of each other and
they should become more linear (see Knoblauch & Maloney, 2012 for a similar
rationale underlying correlation perception). T ransforming the x-axis into units
of nor malized Michelson contrast did indeed linearize the perceptual scales. T o

7 . 3 predictive power of a contrast - based model 6 7
test ho w w ell the nor malized contrast model accounts for the v ariability betw een
the dif ferent context conditions, w e computed a global R 2 v alue. As described
before w e treat all data as if the y w ere coming fr om one underlying model. The
nor malized contrast measure accounts for 98 % of the v ariance in the scaling data
and for 88 % of the v ariance in the matching data. This indicates that the nor mal-
ized contrast measure is a better predictor for the scales than to the matching
data b y explaining more v ariance.
Viewing context high transparent dark
high transparent light
low transparent dark
low transparent light
plain view
B A
Normalized Michelson Contrast
Difference scale
MLDS Matching
− 1.0 0 .0 1.0
0.0
− 1.0 0 .0 1. 0
− 1.0
0.0
1.0
Normalized Michelson Contrast
Match contrast
0.4
0.8
1.2

Figure 7 . 3 : Perceptual scales (Fig. 7 . 1 A) and matching data (Fig. 7 . 2 A) plotted as a func-
tion of the Nor malized Michelson contrast. Dashed lines indicate a linear
fit to the data for all viewing contexts ( R 2 = 0 . 98 for MLDS , R 2 = 0 . 88 for
matching). Error bars indicate mean ± S.D. acr oss obser v ers.

8
D I S C U S S I O N
The goal of this w ork w as to better understand ho w psy chological experiences
are linked to physical v ariables. W e studied the question in the domain of light-
ness perception but the observ ed principles equally apply to other domains of
perceptual appearance. T o make progr ess to w ar ds that goal w e measured percep-
tual scales that link perceiv ed lightness to image luminance using MLDS . Our re-
sults sho w that the estimated perceptual scales (a) are consistent with a lightness-
constant obser v er model in all viewing contexts, (b) predict per ceptual equality
across dif ferent vie wing contexts, (c)indicate that human lightness perception
follo ws a dif ference scale that corresponds to a normalized contrast metric. The
nor malized contrast model accounted for more of the v ariance in the scaling
( 98 %) than in the matching data ( 88 %), suggesting that estimating perceptual
scales has the adv antage of probing more directly the internal lightness scale.
8 . 1 m l d s - based lightness scales
The estimated perceptual scales w ere in close correspondence with each other
(Fig. 7 . 1 B), i.e. perceiv ed lightness follow ed the actual check reflectances despite
substantial v ariations in check luminance across viewing conditions. This reflects
a high degree of lightness constancy . This w as corroborated b y the simulated
obser v er models, because the empirical scales w ere consistent with the lightness-
constant and not the luminance-based obser v er . The shape of the perceptual
scales follo w ed the shape of the classical Munsell scale. The perceptual scales
are an estimation of the transducer functions which cannot be unco v ered using
matching (see Fig. 5 . 2 ).
In addition to the MLDS experiment w e conducted a conv entional asymmet-
ric matching experiment. W e tested to what extent the postulated relationship
betw een perceptual scales and matching (Fig. 5 . 2 ) w ould be e vident in the data.
69

7 0 discussion
Predicted and empirical matching functions w ere consistent with each other (Fig.
7 . 2 ). The high degree of consistency is note w orthy because triad comparisons
and matching require dif ferent per ceptual judgments. Asymmetric matching can
be likened to measuring a rod of unkno wn length with a ruler whereas in triads
rods of dif ferent lengths w ould be compared among each other . The consistency
betw een both types of measurements indicates that the stimulus suitably con-
strains the perceptual response to judgments based on lightness, and not on
luminance. This cannot be taken for granted (Arend & Goldstein, 1987 ; Radon-
ji ´ c & Brainard, 2016 ) in particular since observ ers w ere not explicitly told what
dimension to judge.
A potential challenge when comparing perceptual scales measured in dif fer-
ent contexts is the necessary assumption of how scales are anchored. T w o per-
ceptual scales might ha v e the same shape but co v er a different range, implying
a dif ferent anchoring. Classical scaling experiments did not confront this pr ob-
lem because perceptual scales w ere measured in only one context, i.e. plain view .
The default of MLDS is to anchor the perceptual scales at zer o. This is an arbitrar y
choice and any linear transfor mation of the scale w ould be a v alid outcome of the
analysis. The good correspondence betw een the estimated scales and the match-
ing data in the present case suggests that there w as no substantial anchoring
problem.
As described b y Knoblauch and Maloney ( 2012 ), MLDS assumes that obser v ers
are stochastic in their judgments, with the noise originating at the decision le v el
(as sho wn in Eq. 6 . 1 ). This assumption implies that obser v ers are w orse at judg-
ing inter v al differences that ar e small, i.e. when [ Ψ ( x 3 ) − Ψ ( x 2 )] ∼ [ Ψ ( x 2 ) − Ψ ( x 1 )] .
This critical assumption in MLDS is dif ferent than other scaling methods, such as
Fechnerian scaling that uses integration of JNDs or other discrimination-based
scaling methods (Baird, 1978 ). These scaling methods assume a noise sour ce at
an early sensory representation lev el, and not at a late decision lev el. Here w e
compared perceptual lightness scales that w ere measured in dif ferent viewing
conditions, and hence could ha v e been associated with different amounts of de-
cision noise. This w as not what w e obser v ed. Although individual obser v ers
dif fered in their o v erall noise lev el, all scales measured for one obser v er had
comparable estimated noise lev els. How ev er , these assumptions must be consid-

8 . 2 al tern a tives to asymmetric ma tching 7 1
ered carefully , and ultimately their v alidity must be addressed experimentally
(Aguilar et al., 2017 ).
The estimated noise lev el is critical for the inter pretation of scales, also with
respect to the distinction betw een our tw o obser v er models (lightness constant
vs. luminance-based). This is possible only up to a limit at which obser v ers’ noise
is too large for the models to be distinguished. W e established in simulation that
this upper bound is at an estimated noise lev el ˆ σ = 0 . 4 (Appendix A. 1 ). In our
obser v ers the estimated ˆ σ v alues v aried from 0 . 13 to 0 . 21 for observ ers O 1 to
O 8 , i.e. v alues below the upper limit of model discriminability . For obser v er O 9
ˆ σ = 0 . 39 w as at the boundar y of discriminability , and for observ er O 10 ˆ σ = 0 . 71 ,
w as bey ond the upper limit. Thus, the noise lev el of observ er O 10 did not allow
a definite selection of either of the tw o models. The estimated noise lev el must be
also considered carefully when comparing scales against ideal observ er models.
8 . 2 al tern a tives to asymmetric ma tching
Asymmetric matching has been criticized in the past for mainly tw o reasons:
First, obser v ers’ matches reflect the underlying per ceptual magnitudes only indi-
rectly (Gescheider, 1988 ; Maertens & W ichmann, 2013 ). S econd, obser v ers’ matches
might not reflect perceptual identity but mer ely the best possible match (Brainard
et al., 1997 ; Ekroll & Faul, 2013 ; Foster, 2003 ; Logvinenko & Malone y, 2006 ).
In particular , the question whether lightness is represented b y more than one
dimension across dif ferent contexts has been tackled using dif ferent methods
(Logvinenko & Maloney, 2006 ; Logvinenko et al., 2008 ; Umbach, 2013 ). Be y ond
methodological shortcomings, asymmetric matching tasks ha v e also been crit-
icized for their lack of realism, because in real life w e rarely adjust the color
of an object but rather select objects based on their color . In tw o recent studies
Radonjic et al. ( 2015 b); Radonjic, Cottaris, and Brainard ( 2015 a) measur ed color
constancy in a color selection paradigm where the y asked obser v ers to select
which of tw o competitors w as more similar to a giv en target.
Their task w as analogous to the triad comparison used in MLDS , but the de-
sign w as different fr om the standard MLDS design. A limited number of tar gets
w as presented as anchor for a respectiv e set of competitors but these competitors

7 2 discussion
w ere not compared with each other . MLDS w ould inv olv e triad comparisons of
all possible combinations of targets and competitors. The data w ere analyzed
with a customized v ersion of MLDS . The crucial dif ference to our approach is
that in their critical condition target and competitors w ere presented in differ ent
illuminations. As a consequence, obser v ers’ judgments w ere subject to the same
comparison problem as in asymmetric matching. T o estimate a perceptual scale it
w as assumed that target and competitors are represented on a common underly-
ing dimension. In our w a y of thinking this means to skip the step of estimating
the dif ferent transducer functions (scales), which map luminance to perceiv ed
lightness in dif ferent contexts (Fig. 6 . 1 B), and to compare stimuli directly on
the inter nal axis. As w e ha v e outlined abov e this assumption is v alid only for
a lightness constant obser v er , i.e. for obser v ers whose perceptual scales in dif-
ferent vie wing situations ha v e comparable scale maxima. The authors reported
moderately high color constancy indices which w ere comparable to asymmetric
matches for the same type of stimuli (Radonjic et al., 2015 a, 2015 b). W e suggest
to include such cross-context comparisons only to v alidate predictions from the
MLDS -based scales as w e did here with the asymmetric matches.
8 . 3 models of lightness per ception
W e claim that perceptual scales are an important test case for models of lightness
perception because the y of fer a direct estimate of the transducer functions that
w e are interested in. A successful model should be able to explain both char -
acteristics of lightness appearance: perceptual equality acr oss contexts as w ell
as sensitivity dif ferences manifested in the shape of perceptual scales (Hillis &
Brainard, 2007 a).
If w e assume that the goal of the visual system is to accurately represent sur -
face reflectance, then reflectance w ould be the best predictor of per ceiv ed surface
lightness. Thus, for a perfectly lightness-constant obser v er the perceptual scales
measured in dif ferent contexts should perfectly o v erlap when plotted against re-
flectance. Our empirical scales are consistent with a lightness-constant observ er ,
ho w ev er they re v eal small deviations, especially for the tw o lighter transparent
media (Fig. 7 . 1 B). When w e plotted the scales as a function of nor malized con-

8 . 4 general conclusions 7 3
trast (W iebel et al., 2016 ; Zeiner & Maertens, 2014 ) instead of reflectance, the
dif ferences betw een scales w ere substantially reduced (Fig. 7 . 3 A). This means
that the nor malized contrast metric does not perfectly capture veridical surface
reflectances but is rather tightly correlated with them. One might be tempted to
conclude that the predictiv e po w er of the contrast-based model ‘exceeds’ that of
physical surface reflectances, because it accounts for the de viations from light-
ness constancy that w e obser v ed in the data.
This finding is consistent with the idea that the visual system, instead of do-
ing inv erse optics (e.g. Barro w & T enenbaum, 1978 ; D’Zmura & Iv erson, 1993 ),
might use a set of readily a v ailable but imperfect cues to infer stable proper -
ties of objects (e.g. Anderson, 2011 ; Fleming, 2014 ) . The inv olv ed computations
might not alw a ys lead to a veridical percept with respect to the physical w orld,
but to an o v erall reliable estimate of the appearance of objects (e.g. Marlo w , Kim,
& Anderson, 2012 ). The estimated scales w ere linearized b y the transfor mation
to contrast units, which implies that the model accounts for the sensitivity dif-
ferences betw een lo w and high reflectances (e.g. Lu & Sperling, 2012 ), a feature
which cannot be quantitativ ely captured with matching. The higher agreement
betw een the model and the perceptual scales (compared to matching) supports
the idea that the perceptual scales are a mor e direct and infor mativ e measure of
the inter nal v ariable of lightness and subject to few er sources of v ariability .
8 . 4 general conclusions
In this paper w e sho w that a scaling method is more po w erful than matching
in elucidating the perceptual repr esentation of surface lightness. MLDS pro vides
a direct estimate of the transducer functions that relate the physical dimension
of reflectance to the psy chological dimension of perceiv ed lightness. In addition,
MLDS a v oids the practical difficulties associated with asymmetric matching tasks
because all perceptual comparisons are made within the same vie wing context.
Obser v ers confir med that subjectiv ely the triad comparison required b y MLDS
w as a natural and straightfor w ard task.
S o why is it then that asymmetric matching remains the method of choice de-
spite the ob vious benefits of MLDS . W e suspect that experimenters feel slightly

7 4 discussion
uneasy about explicitly making and commiting to the v arious assumptions that
are required b y MLDS in order to statistically estimate the perceptual scales. Ho w-
ev er , as w e illustrate in Figure 5 . 2 , asymmetric matching procedur es also assume
the presence of internal scales but they are hidden and their shape can not be
inferred from observ ers’ matches. W e think that the present results ar e encour-
aging and adv ocate the estimation of scales, because they pr o vide a more direct
estimate of inter nal v ariables against which w e can test our theoretical models
of appearance.

Part III
USING MLDS T O MEASURE SENSITIVITY
This part has been published in:
Aguilar G., W ichmann F . A., Maertens M. ( 2017 ). Comparing sensitivity es-
timates from MLDS and for ced-choice methods in a slant-from-texture ex-
periment. Journal of V ision, 17 ( 1 ): 37 , 1 - 18 . doi: 10 . 1167 / 17 . 1 . 37 (postprint)

9
I N T R O D U C T I O N
Maximum likelihood dif ference scaling ( MLDS ) is a psy chophysical method that
allo ws the ef ficient characterization of perceptual scales (Malone y & Y ang, 2003 ;
Knoblauch & Maloney, 2012 ). Observ ers are asked to judge appearance dif fer-
ences for supra-threshold stimuli that v ar y along some dimension of interest,
and a scale is constructed based on the reported dif ferences in appearance. The
method has been used to study appearance in a v ariety of visual domains such as
color dif ferences (Maloney & Y ang, 2003 ), texture pr operties (Emrith, Chantler ,
Green, Malone y , & Clarke, 2010 ), surface glossiness (Obein et al., 2004 ), trans-
parency (Fleming et al., 2011 ) and material properties (Paulun, Ka w abe, Nishida,
& Fleming, 2015 ), as w ell as for the assessment of perceiv ed image quality in
compression-degraded images (Charrier , Maloney , Cherifi, & Knoblauch, 2007 ).
Recently , MLDS has been used to link stimulus appearance with stimulus
discriminability . Assuming an underlying signal detection model Devinck and
Knoblauch ( 2012 ) ha v e demonstrated a quantitativ e agreement betw een sensitiv-
ity estimates deriv ed from per ceptual scales ( MLDS ) and sensitivity estimates as-
sessed with a traditional forced-choice pr ocedure for the w atercolor ef fect. Their
finding is remarkable giv en the long ef fort in psy chophysical resear ch of relating
discrimination and appearance in a unified framew ork.
Relating stimulus appearance — the stimulus’ subjectiv e magnitude — to dis-
crimination — the ability to discriminate stimuli—dates back to the roots of
psy chophysical resear ch. Fechner ( 1860 ) proposed that b y summing equal sub-
jectiv e just-noticeable dif ferences (JND) and assuming W eber ’s la w , a function
could be constructed which relates stimulus’ subjectiv e magnitude and physical
magnitude (Baird, 1978 ). Soon Fechner ’s suggestion w as criticized, theoretically
as w ell as for lack of experimental evidence to support it (r evie w ed in detail in
Krueger, 1989 ).
77

7 8 introd uction
Stev ens ( 1957 , 1975 ) later proposed that subjectiv e magnitude could be directly
measured from observ er responses to supra-threshold stimuli. He devised ‘di-
rect’ methods to measure subjectiv e magnitude and deriv ed (pow er) functions
that w ould relate subjectiv e and physical magnitude (Stev ens, 1975 ; Gescheider,
1997 ; but c.f. T reisman, 1964 a, 1964 b). Ho w ev er , Stev ens’ proposal w as met with
equal criticism, partly because of the scales’ lack of predictiv e po w er for dis-
criminability and partly because of the methodological concer ns of asking ob-
ser v ers to numerically estimate or pro vide ratings of perceiv ed sensation (e.g.
Baird, 1989 ). Although a considerable amount of w ork has been done tr ying to
unify discrimination and appearance, so far the debate still continues and mixed
experimental evidence has been found (e.g. Krueger, 1989 ; Ross, 1997 ; Hillis &
Brainard, 2007 b). Thus, the finding of De vinck and Knoblauch ( 2012 ) that appear -
ance and sensitivity can be linked via MLDS is promising, because it suggests that
a supra-threshold method like MLDS could be used to predict sensitivity to near -
threshold stimulus dif ferences. Apart from potential theor etical implications,
Devinck and Knoblauch’s finding ma y be beneficial from a purely methodologi-
cal point of view , because MLDS requir es a considerably smaller amount of data
than traditional discrimination methods. Because of its ef ficiency MLDS could be
used to identify experimental settings in which appearance and discrimination
judgments are consistent, b y comparing sensitivity measured in discrimination
tasks (e.g. tw o-inter v al forced-choice) with sensitivity deriv ed from MLDS . The
goal of this w ork w as to further explore - theoretically and empirically - the p os-
sibility to use MLDS to predict near -threshold discrimination performance using
a slant-from-texture task.
9 . 1 s l a n t - f r o m - texture t asks
W e measure perceptual scales in a slant-from-textur e experiment. The percep-
tual scale that relates apparent and physical slant in slant-fr om-texture tasks has
a non-linear shape and it therefore pr o vides an interesting test case for predicting
sensitivity at dif ferent positions of the MLDS based scale. Slant-from-texture stim-
uli ha v e been used extensiv ely in the study of depth and surface perception (e.g.
Knill, 1998 ; Rosas et al., 2004 ; T odd, Thaler , & Dijkstra, 2005 ; V elisa vljevi ´ c & Elder,

9 . 2 mlds and the sign al detection model 7 9
Figure 9 . 1 : Example stimuli sho wing surfaces of differ ent slants cov ered with the ‘polka
dots’ texture. Here w e used the method of triads for MLDS where observ ers
judge which of the pairs exhibit a larger dif ference in perceiv ed slant, the
left-middle pair or the right-middle pair . Most obser v ers w ould report that
the right-middle pair ( 35 , 70 ) contains the larger slant dif ference, although
the physical slant dif ference is identical betw een the pairs: ( 0 , 35 ) vs. ( 35 , 70 ).
2006 ; Saunders & Backus, 2006 ), because texture gradients can e v oke a strong im-
pression of 3 -D slant in the absence of other cues (Saunders, 2003 ). Stimuli are
surfaces that are co v ered with a texture patter n such as randomly placed circu-
lar elements (or ‘polka dots’, Fig. 9 . 1 ). The surface is slanted at v ar ying degrees
relativ e to the fronto-parallel position r esulting in characteristic changes in the
polka dot patter ns. The slanted texture is vie w ed through an aperture to isolate
texture cues from other pictorial cues such as the shape and bor ders of the sur-
face (Knill, 1998 ; T odd et al., 2010 ). Using this type of stimuli it has been found
that sensitivity to slant is lo w er when the surface is close to the fronto-parallel
position than when the surface is slanted a w a y from it (Knill, 1998 ; Rosas et al.,
2004 ). The dif ference in sensitivity betw een 0 and 70 deg can be up to ten-fold
(Knill, 1998 ).
9 . 2 mlds and the sign al detection model
The decision model underlying the MLDS framew ork is depicted in Figure 3 . 2 A.
It is assumed that dif ferent stimulus lev els x i are associated with discrete per -
ceptual responses Ψ x i , and that observ ers compare different stimuli b y judging

8 0 introd uction
the dif ferences betw een the perceptual responses. The decision v ariable is as-
sumed to be corrupted b y decision noise, ϵ , which is assumed to be Gaussian
distributed with zero mean and v ariance σ 2 . MLDS estimates the perceptual scale
together with the noise associated with the judgments (Malone y & Y ang, 2003 ;
Knoblauch & Maloney, 2008 ).
The same perceptual pr ocess can be rephrased in a signal detection frame w ork
b y shifting the noise from the decision process to the sensory representation
(see Figure 3 . 2 B). In this w a y , the original MLDS scale can be transfor med to a
nor med scale in which the units on the perceptual axis r epresent differ ences in
units of d ′ . This transfor mation has been suggested b y Devinck and Knoblauch
( 2012 ) to compare supra- and near -threshold judgments in the w atercolor ef fect.
A detailed description of the transfor mation and the MLDS model is pro vided in
S ection 3 . 3 .
In order to apply this transfor mation the follo wing assumptions are made: 1 .
The sensory representations associated with each stimulus lev el are Gaussian
random v ariables with equal v ariance ( σ 2 ). 2 . They are independent. 3 . The deci-
sion process is deterministic. 4 . The sensor y representation function is monotoni-
cally increasing. This produces only positiv e v alues of sensor y response interv als
so that the absolute v alue operation can be remo v ed from the decision rule ( ∆
v ariable in Figs. 3 . 2 A and B). An MLDS decision model with the abov e assump-
tions is equiv alent to a signal detection model with equal-v ariance and Gaussian
distributed sensory representations, as depicted in Fig. 3 . 2 C 1 .
9 . 3 objectives
W e w ant to test whether and to what extent w e can assume the equiv alence of
MLDS and forced-choice pr ocedures for estimating sensitivity as it w as reported
b y Devinck and Knoblauch ( 2012 ) for the W atercolor ef fect. W e first examine the
theoretical equiv alence betw een both methods b y means of simulations. W e use
a kno wn observ er model to generate sensitivity estimates for both methods. In
1 Which, in turn, is analogous to Thurstone’s case V of the La w of Comparativ e Judgment
(Thurstone, 1927 b)

9 . 3 objectives 8 1
the present analysis w e e v aluate the adequacy of MLDS to predict sensitivity us-
ing the tw o-alter nativ e forced-choice ( 2 -AFC ) method as the standar d of reference
as the latter has pro v en its usefulness in the estimation of sensitivity o v er time.
W e quantify the amount of agreement betw een the tw o methods in the presence
of dif ferent violations in the assumptions underlying MLDS . W e then test the em-
pirical consistency betw een sensitivity estimates deriv ed with MLDS and forced-
choice procedures in tw o experiments with a slant-from-texture task. In Experi-
ment 1 obser v ers judge supra-threshold slant dif ferences and perceptual scales
are deriv ed from the judgments using MLDS . Fr om these scales w e deriv e sensi-
tivity estimates (thresholds) at dif ferent slant le v els. In Experiment 2 obser v ers
judge near -threshold slant dif ferences in a tw o-inter v al forced-choice ( 2 -IFC ) task.
S ensitivity estimates (thresholds) are deriv ed from psy chometric functions for
the same slant lev els as in Experiment 1 .
T o anticipate, the amount of agreement betw een sensitivity estimates from
the tw o methods v aried substantially across observ ers. The simulations sho w ed
that disagreement betw een the method might be due to violations of the model
assumptions underlying MLDS .

10
S I M U L A T I O N S
The sensory representation w as modelled as a po w er function, Ψ ( x ) = x e , with
exponent e = 2 . 0 (Fig. 3 . 2 A). W e used an exponent greater than one so that sen-
sitivity w ould increase with stimulus intensity , which is the case for slant-from-
texture (Knill, 1998 ). The sensory representation function w as used to simulate
responses of a model observ er for the MLDS and the 2 -IFC procedure. It w as as-
sumed to be a Gaussian random v ariable with the mean corresponding to Ψ ( x )
and unique v ariance σ 2 (Figs. 3 . 2 B-C). An example simulation is depicted in Fig-
ure 10 . 1 . Thresholds w ere deriv ed for a standard v alue of st = 0 . 6 from MLDS
scales (panel A) and from psy chometric functions in a 2 -IFC task (panel B).
10 . 1 mlds thresholds
W e perfor med the MLDS experiment with the method of triads (Maloney & Y ang,
2003 ; Knoblauch & Malone y, 2012 ). A triad consists of three stimuli, x 1 , x 2 and
x 3 . T o simulate a triad the generativ e model (Fig. 3 . 2 A) assigns perceptual re-
sponses, Ψ x i , to each of the three stimuli, x i . The simulated obser v er decides
which of the pairs, ( x 1 , x 2 ) or ( x 2 , x 3 ) , contains the bigger dif ference in perceiv ed
slant according to the decision model depicted in Figur e 3 . 2 B.
MLDS data (simulated and obser v ed) w ere analyzed with the R package MLDS ,
a v ailable in CRAN (Knoblauch & Maloney, 2008 ) and with p ython routines
based on numpy and scipy libraries. A p ython wrapper of the MLDS routines to-
gether with all subsequent analysis routines is a v ailable online ( http://github
.com/TUBvision/mlds ).
W e first estimated a perceptual scale from the simulated r esponses b y em-
plo ying the standard MLDS routines a v ailable in R (Knoblauch & Maloney, 2008 )
(see S ection 3 . 2 for a detailed description of the estimation procedure). W e then
deriv ed sensitivity estimates from the per ceptual scale follo wing the procedure
83

8 4 simula tions
suggested b y Devinck and Knoblauch ( 2012 ). T o do this w e re-parametrized the
original unconstrained scale so that the scale v alues are expressed in units of
d ′ . The details underlying the re-parametrization are explained in S ection 3 . 3 .
In the simulation w e deriv ed sensitivity estimates for eight standard v alues (ex-
periments w ere done with four standar d v alues). Due to the non-linear shape of
the perceptual scale the local slopes dif fered betw een different standar d v alues
and hence translated into dif ferent sensitivity lev els along the stimulus dimen-
sion. For each standard w e determined sensitivity at three perfor mance lev els
( d ′ = 0 . 5 , 1 and 2 ) abo v e and belo w the standard. T o deriv e the stimulus v alues
that corresponded to each d ′ dif ference for a giv en standar d, w e inter polated be-
tw een the sampled data points with a cubic spline fit ( ˆ
ϕ ( x ) , sho wn as solid dark
gra y line in Figure 10 . 1 A). The scale v alue ( ˆ
ϕ ( st ) in d ′ units) that corresponds
to a particular standard stimulus ( st ) and performance lev el ( d ′ ) w as read from
the fitted function. The readout can be described b y
ˆ
ϕ − 1  ˆ
ϕ ( st ) ± d ′  = ˆ
θ st
± d ′ | MLDS ( 10 . 1 )
in which the + ( − ) sign next to d ′ stands for comparison v alues abov e (belo w)
the standard, and ˆ
θ st
± d ′ | MLDS stands for a particular sensitivity v alue in stimulus
units as estimated b y MLDS .
10 . 2 2 - ifc thresholds
The same generativ e model is used to simulate responses in the 2 -IFC procedur e.
In each trial one response is generated for the standar d and one for the compar -
ison v alue. Perceptual responses are compared accor ding to the decision model
depicted in Figure 3 . 2 C. W e simulated the same number of trials that w e ran in
the beha vioral experiments (see sections 11 . 1 . 3 and 11 . 1 . 4 ).
T o allo w the comparison of thresholds across dif ferent standar d slants w e re-
port comparison v alues in ter ms of differences relativ e to each standard. W e
fitted separate psy chometric functions for positiv e and negativ e comparison v al-
ues (smaller and larger than the standar d). Psy chometric functions w ere W eibull
functions ( F ) with the guess rate ( γ ) set to 50 % chance lev el. The lapse rate

8 6 simula tions
( λ ), slope and position parameters of the psy chometric function w ere estimated
using Ba y es inference (Kuss, Jäkel, & W ichmann, 2005 ). W e used the psignifit 4 im-
plementation (S chütt, Har meling, Macke, & W ichmann, 2016 ) for function fitting,
estimation of confidence inter v als and analysis of goodness of fit. Each psy cho-
metric function w as estimated from a total of 320 trials ( 4 comparison v alues x
80 repeats) as in the experiments.
An example psy chometric function for one standar d slant is sho wn in Fig-
ure 10 . 1 B. Performance thresholds w ere obtained from each psy chometric func-
tion b y finding the stimulus v alue that produces a per centage correct correspond-
ing to a desired d ′ . Assuming the equal v ariance Gaussian case of a signal detec-
tion model (Green & Sw ets, 1966 ), d ′ can be conv erted to percentage correct and
vice v ersa, and the threshold can be read out b y
ˆ
F − 1 ( Fc ′ ) = ˆ
θ st
± d ′ | 2 IFC
where + ( − ) indicate comparisons abo v e (below) the standar d, Fc ′ = { 0 . 28 , 0 . 52 , 0 . 84 }
are the unscaled fractions correct (range betw een 0 and 1 ) that correspond to the
ra w fractions correct Fc = { 0 . 64 , 0 . 76 , 0 . 92 } (range betw een 0 . 5 and 1 . 0 ). These
fraction correct v alues Fc correspond to the perfor mance lev els of d ′ = 0 . 5 , 1 and
2 , respectiv ely , in a tw o-alter nativ e forced-choice task (Green & Sw ets, 1966 ).
10 . 3 threshold comp arison
In Figure 10 . 1 C the thresholds deriv ed with each method are plotted against
each other . They are expr essed as differ ences relativ e to the standard v alue. Per -
fect agreement betw een the tw o methods is indicated b y the main diagonal. T o
ev aluate the statistical significance of the dif ferences betw een thresholds w e esti-
mated the 95 % confidence inter v als for each of the thresholds using the bootstrap
technique (for details see S ection 10 . 5 ).
Thresholds w ere said to be in agreement when either one of the tw o confidence
inter v als of a data point (v ertical or horizontal corresponding to 2 -IFC and MLDS ,
respectiv ely) crossed the unity line. This criterion ensures that the point estimate
of one method is included in the 95 % confidence inter v al of the other method.

10 . 4 thresholds tha t could not be obt ained 8 7
In Figure 10 . 1 C all data points coincided with the unity line resulting in a 100 %
agreement.
W e used this measure to quantify the degree to which the consistency betw een
the thresholds. For eight dif ferent standar d v alues w e perfor med n= 1000 simu-
lations and Figure 10 . 2 A sho ws a summar y of the results for the a v erage of the
empirically obser v ed noise lev el, σ = 0 . 07 (green lines). Thresholds agreed in
more than 90 % of the cases, and the agreement w as also high across a range of
noise lev els that w e tested, from σ = 0 . 035 to σ = 0 . 14 , which includes all the
v alues of sensor y noise obser v ed in the experiments.
10 . 4 thresholds tha t could not be obt ained
The estimation procedure in either of the methods sometimes failed when sen-
sitivity w as low . When the stimulus w as in a range where the sensory function
is too shallo w , for example for v alues belo w 0 . 4 in the sensor y function in Fig-
ure 3 . 2 A, the interpolation of scale differences w as not possible. Similarly , the
psy chometric function w as sometimes so shallo w that it did not allo w the read
out of a threshold at a giv en performance lev el. These ‘failure’ cases pro vide an
additional test of consistency betw een the tw o methods, because when sensitiv-
ity is genuinely lo w both methods should fail to pro vide a threshold estimate.
W e counted the number of cases in which either one or both of the methods
did not pro vide a threshold estimate for a giv en perfor mance lev el. The results
are sho wn in Figure 10 . 2 A (gra y lines). It can be read from the Figure that both
methods did consistently fail to pro vide threshold estimates for standard v alues
near zero.
10 . 4 . 1 Model assumptions
T o test the effect of violations of some of the model assumptions on the agr ee-
ments betw een thresholds w e repeated the simulations with a modified genera-
tiv e model. W e introduced sensor y noise that w as not independent of the stimu-
lus lev el but instead increased with the stimulus v alue. W e also tested a model

8 8 simula tions
St anda rd va lue St anda rd va lue
B
0
20
40
60
80
10 0
Pe rc en t ag e
A
C D
Pe rc en t ag e
0
20
40
60
80
10 0
0.2 0 .4 0.6 0 .8 0.2 0 .4 0.6 0 .8
0.2 0 .4 0.6 0 .8 0.2 0 .4 0.6 0 .8
Total No t es tim at ed
Agreements in No t es tim at ed
Total Estimated
Agreements in E stim at ed
Pe rc en t ag e agreements

Figure 10 . 2 : MLDS and forced-choice thr esholds from simulations. At each standard le v el, thresholds for dif-
ferent performance lev els could be successfully estimated (dark green) and ha v e quantitativ e
agreement betw een them (light green). There w ere cases in which thr esholds could not be es-
timated (dark gra y), from which an agreement occurred when both methods w ere unable to
estimate it (light gra y). The sum of agreement cases for estimated and not estimated thresholds
is also sho wn (light blue). Percentage o v er 1000 simulations. (A) independent, equal-v ariance
case with noise lev el σ = 0 . 07 (B) independent, unequal-v ariance case with increasing noise
lev el from 0 . 035 to 0 . 14 (violation of equal-v ariance assumption), (C) same as (A) but with
added unifor m correlation in the sensory representation of ρ = 0 . 8 (violation of independence
assumption), and (D) same as (B) but with added unifor m correlation of ρ = 0 . 8 (both assump-
tions violated).

10 . 5 v ariability of threshold estima tes 8 9
that included unifor m correlations betw een the sensory representations (spe-
cific details in Chapter 4 ). These tw o modifications violate the assumptions of
equal v ariance (assumption 1 ) and independence (assumption 2 ). As illustrated
b y Kingdom ( 2016 ) and tested in simulations b y Maloney and Y ang ( 2003 ), the
scales themselv es are insensitiv e to a violation of the equal v ariance assumption.
Ho w ev er , violating the equal v ariance assumption did reduce the agreement be-
tw een thresholds (Figure 10 . 2 B) in particular for extreme standar d v alues where
the simulated noise w as respectiv ely lo w er or higher than in the equal-v ariance
model. The reason for this is illustrated in Figure 10 . 1 which sho ws ho w thresh-
old readout depends on noise on the sensory axis. Introducing correlations re-
duced the amount of agreement betw een thresholds independent of the standar d
v alue (Figure 10 . 2 C). W e obser v ed the smallest agreements when both assump-
tions are violated (Figure 10 . 2 D).
10 . 5 v ariability of threshold estima tes
T o study the v ariability of threshold estimates w e made use of the bootstrap sam-
ples that are already generated b y MLDS to calculate confidence inter v als (CIs) for
the scale v alues (error bars in Figure 10 . 1 A, Knoblauch & Maloney, 2012 ). Boot-
strap samples are generated from the r esponse probabilities that are observ ed
for each triad. Each bootstrap sample is a new per ceptual scale and b y default
MLDS generates 1000 of these bootstrapped scales. T o deriv e the bootstrap sam-
ples for a particular threshold v alue w e fitted a cubic spline to each bootstrap
scale and deter mined the slant v alue corresponding to the threshold v alue (see
Equation 10 . 1 ). From these bootstrap distributions w e obtained the 95 % confi-
dence inter v als for each threshold (a detailed description of the procedure can
be found in S ection 3 . 4 ).
T o compare the confidence inter v als associated with each method Figure 10 . 3 A
plots the widths of the respectiv e CIs against each other , for one example stan-
dard. The main diagonal indicates equal width in the confidence interv als, data
points abo v e the main diagonal indicate that the width of the CIs for thresholds
from MLDS w ere smaller than the width of the CIs for thresholds from 2 -IFC . For
all standard le v els (Figure 10 . 3 ) and for all tested noise le v els (see Appendix) the

9 0 simula tions
C .I. wi dth o f
thr es ho l d f ro m ML D S [ deg ]
C .I. wi dth o f
thr es ho l d f ro m 2- IFC [ deg ]
0.0 0 .5
0.5
B A
0.2 0 .4 0.6 0 .8
40
10 0
Percentage
Standard value
= 0. 03 5
= 0.0 7
= 0.1 4

Figure 10 . 3 : (A) Comparison of the v ariability in the threshold estimation. The width
of the confidence intervals ar e plotted against each other for multiple sim-
ulations at one standard stimulus v alue st = 0 . 4 as example. (B) Co v erage
of threshold estimated at dif ferent standard le v els, and for three differ ent
simulated noise lev els ( σ ). Expected co v erage of 95 % is sho wn as a black
dashed line.
majority of confidence inter v als ( 99 %) w as smaller in MLDS than in 2 -IFC . This is
curious because MLDS requires a smaller amount of data than 2 -IFC .
A smaller width in the confidence interv als could either be due to a truly
more precise estimate (less underlying v ariability), or alter nativ ely , it could re-
sult from an insuf ficient co v erage of the confidence inter v als. This is a common
problem in deriv ations using bootstrap techniques (W ichmann & Hill, 2001 ) and
w e tested this with an analysis of the co v erage of the scales and the deriv ed
thresholds. W e calculated co v erage b y counting ho w many times the ‘true’ v alue
(as defined in the generativ e model) w as contained in the estimated confidence
inter v al of a scale or threshold. For confidence inter v als to be credible, co v erage
across multiple simulations should reflect the confidence in the confidence inter -
v al, i.e. cov erage should be 95 % o v er multiple simulations for 95 % confidence
inter v als.
Co v erage of the scale estimates w as adequate for the range of noise lev els stud-
ied (Figures 4 . 4 and A. 7 ). MLDS thus pro vides credible confidence inter v als for
the scale estimates that it w as designed for . Ho w ev er , co v erage for the threshold

10 . 6 summary and discussion 9 1
estimates w as at best at 90 % for nominal v alues of 95 % (Figure 10 . 3 B), and for
stimulus v alues at shallow portions of the sensory function (e.g. smaller than
0 . 4 ) co v erage w as as lo w as 50 - 60 %. These results indicate that the confidence
inter v als for the thresholds deriv ed with MLDS w ere indeed too narro w . Thresh-
old v ariability might hence be underestimated when deriv ed from MLDS in the
w a y described abo v e. This is an important ca v eat when using MLDS to estimate
thresholds.
Upon suggestion of the re view ers w e perfor med a sanity check for the confi-
dence inter v als to test for their stability and bias when trial numbers are high.
W e repeated the scale and threshold estimation procedur e and calculated bias
and co v erage for increasingly lar ge trial numbers. For the scale estimation, the
patter n of results indicates that co v erage slightly impro v ed with an increasing
trial number (Figure 4 . 4 ). For the threshold estimation, co v erage did not impro v e
when the number of trials is tripled (Appendix Figure A. 11 ). This result suggest
that lo w co v erage w as not due to a small sample size.
10 . 6 summary and discussion
W e simulated an obser v er model with a kno wn sensor y representation function.
W e compared thresholds deriv ed from MLD scales with thresholds deriv ed from
a 2 -IFC procedure at dif ferent standard v alues and perfor mance lev els. W e found
a high degree of consistency betw een thresholds obtained with each method,
when all the assumptions are met (Fig. 10 . 2 A). The amount of agreement did
not depend on the sensor y noise lev el for the range of noise lev els that w as ob-
ser v ed experimentally . The estimation procedure fails to obtain thresholds when
sensitivity is lo w . In most of these cases both methods failed to estimate a thresh-
old which is a further indication of consistency betw een them. The v ariability
of threshold estimates, quantified as the width of their confidence interv als, is
smaller for MLDS than for 2 -IFC thresholds (Fig. 10 . 3 A). This finding needs to be
qualified b y the cov erage analysis which indicates that the bootstrapped confi-
dence inter v als for MLDS thresholds might be too small (Fig. 10 . 3 B).
The agreement betw een threshold estimates did not amount to the theoreti-
cally expected 100 %. This might be due to the rather small number of simulated

9 2 simula tions
trials. Ho w ev er , the simulations should capture actual psy chophysical experi-
ments where it is not practicable to collect lar ge numbers of trials. In addition,
they captur e the softw are pipeline of estimation and statistical inference, which
could be prone to dif ferent kind of pr oblems (e.g. numerical). Thus, the simula-
tion results establish an upper bound for the agreement that is expected for a
realistic amount of collected data and estimation procedur es.
Finally , w e found that violating the equal-v ariance assumption b y MLDS might
lead to disagreement betw een estimated thresholds. The disagreement is rele v ant
because unequal v ariance models might fit beha vioral data better when the equal
v ariance assumption is violated in real data (e.g. Goris et al., 2013 ).

11
EXPERIMENTS
11 . 1 m e t h o d s
11 . 1 . 1 Observers
Six naïv e (three male, age range betw een 23 and 29 ) and tw o experienced ob-
ser v ers (obser v ers “O 3 ” and “O 6 ”) participated in the study . All observ ers had
nor mal or corrected-to-normal visual acuity . The participation of naïv e obser v ers
w as v oluntary and financially compensated. Infor med written consent w as giv en
b y all obser v ers prior to the experiment.
11 . 1 . 2 Stimuli and Apparatus
Stimuli w ere planes textured with a ‘polka dot’ pattern and slanted about their
horizontal axis. They w ere generated in tw o steps. First, the textures containing
the ’polka dot’ patter n w ere generated as 2500 x 500 pixel images. The ‘polka dot‘
patter n is created using a hard cor e point pr ocess , which is a random spatial process
that a v oids dot super position b y applying an inhibition radius to each point.
Using the R package spatstat (Baddele y & T ur ner, 2005 ), w e generated fifteen
samples of this process follo wing specifications from pre vious w ork (Rosas et al.,
2004 ). The textures consisted of black dots ( 0 . 4 - 0 . 6 cd/m 2 , 12 pixels or 0 . 5 deg
visual angle in diameter in the fronto-parallel plane) on a gra y background area
( 48 - 52 cd/m 2 , Figure 9 . 1 ).
In a second step the textured planes w ere r endered in 3 -D using OpenGL
(Shreiner , W oo, Neider , & Da vis, 2005 ). The planes w ere slanted and perspec-
tiv ely projected into 2 -D. The so-generated planes w ere vie w ed through simu-
lated circular apertures that subtended 8 . 3 deg of visual angle and w ere added
at the depth of the screen distance.
93

9 4 experiments
Stimuli w ere displa y ed on a 24 . 1 -in. LCD monitor (Eizo CG 243 W 496 x 310 mm,
1920 x 1200 pixels, 60 Hz) located in a dark cabin. Obser v ers view ed the stimuli
monocularly with their dominant ey e at a distance of 60 cm. Ey e dominance
for each obser v er w as deter mined with the Miles test (Miles, 1930 ) prior to the
start of the experiment. The non-dominant e y e w as co v ered with an ey e-patch
and the head rested on a chin rest. Stimulus pr esentation w as controlled b y a
computer (Apple Mac Pro QuadCore 2 . 66 with a graphic car d Nvidia GeForce
7300 GT) that w as running custom-made softw are which w as based on python
and the visualization library pyglet . Obser v ers’ responses w ere registered via the
keyboar d.
11 . 1 . 3 Pr ocedure Experiment 1 : MLDS
In each trial, three stimulus exemplars that v aried in slant w ere presented next
to each other . Each of the slanted surfaces w as rendered independently and
view ed through a differ ent circular aperture (see Figure 9 . 1 ). Slant v alues ( s )
v aried betw een 0 (fronto-parallel) to 70 deg in steps of 10 deg. This spacing results
in p = 8 possible slant v alues and a total number of n = p ! / (( p − 3 ) ! × 3 ! ) = 56
unique triads.
By design each triad consists of stimuli that are slanted so that the tw o inter-
v als enclosed b y the three stimuli do not o v erlap. The stimuli in a triad w ere
presented in either ascending ( s 1 < s 2 < s 3 ) or descending ( s 1 > s 2 > s 3 ) or -
der , and the order w as randomized across trials. Obser v ers w ere asked to report
which of the pairs, ( s 1 , s 2 ) or ( s 2 , s 3 ), contained the bigger perceiv ed difference in
slant. Obser v ers view ed the stimulus configuration with no time limit for their
response. The y indicated their choice b y pressing a ke yboar d button and this
triggered the next trial after a dela y of one second. No feedback w as giv en as to
the correctness of the response.
The full set of unique triads w as presented in one experimental block and
15 such blocks w ere presented within one session. In total, each observ er judged
840 triads. This w as the same amount of trials used in the simulations. Obser v ers
could pause after each block. Before the experiment observ ers w ere sho wn tw o
to fiv e examples of extreme triads ( ( 0 , 10 , 70 ) and ( 0 , 60 , 70 ) deg), together with

11 . 1 m e t h o d s 95
the ‘correct’ answ ers and the corresponding ke yboar d presses. W e emplo y ed this
instruction method to ensure that observ ers understood the task. Comparing
stimulus inter v als is not an ob vious task and in previous experiments w e noted
that, instead of reporting the pair with the biggest perceiv ed difference , some
obser v ers reported the pair that included the most extreme slant.
11 . 1 . 4 Pr ocedure Experiment 2 : 2 -IFC
A standard 2 -IFC pr ocedure w as emplo y ed in Experiment 2 . A trial started with a
fixation cross that appeared for 1000 ms in the center of the scr een. Then the first
stimulus w as presented for 200 ms. Its contrast ramped on and off from zer o to
full contrast and back to zero within the first and last 50 ms of presentation so
that the stimulus w as seen at full contrast for 100 ms. After a blank inter-stimulus
inter v al of 500 ms the second stimulus w as presented with temporal parameters
identical to those of the first. After stimulus offset observ ers had to report which
of the tw o stimuli w as more slanted using a ke yboard button to indicate first
or second. Obser v ers did not receiv e feedback about their performance. Stan-
dard and comparison stimuli w ere randomly assigned to the first or the second
inter v al.
Discrimination perfor mance w as measured for the same four standard slant
v alues ( 26 , 37 , 53 and 64 degrees) for which MLDS thresholds w ere predicted.
Each standard slant w as compared with one of eight comparison slants (four
belo w and four abo v e the standar d slant) in a method of constant stimuli proce-
dure. In the first session the range of comparison stimuli for each standar d slant
w as selected based on the point estimates corresponding to perfor mance lev els
of d ′ = 0 . 5 , 1 , 2 and 3 that w ere deriv ed from the MLD scale (section 10 . 1 ). After
the first session the comparison v alues w ere adjusted so as to pro vide good cov er-
age of the psy chometric function (W ichmann & Hill, 2001 ). The full experimental
design contained 4 standards x 8 comparison v alues (four abo v e and four belo w
the standard) x 80 r epeats resulting in 2560 trials in total. This amount w as the
same as in the simulations. The presentation w as randomized and the total num-
ber of trials w as subdivided into 40 blocks of 64 trials each. Obser v ers completed

9 6 experiments
all trials in three to four sessions of maximum one hour duration. Experiment 2
w as run on a different da y than Experiment 1 and subsequent to it.
There are ob vious differ ences in stimulus spacing as w ell as in the number
of trials betw een both methods and both factors might af fect the shape of the
respectiv e fitted functions, scales or psy chometric functions. Ho w ev er , there is
no principled w a y to equate these aspects across the pr ocedures and w e w ould
argue that the y had little ef fect on the present results. W e perfor med goodness-of-
fit analyses for both procedures which sho w ed that the fitted functions captured
the data, and which also indicates that the stimulus choice w as reasonable.
11 . 2 r e s u l t s
The objectiv e of the experiments w as to compare sensitivity estimates from a
forced-choice and an MLDS pr ocedure at dif ferent positions along the perceptual
scale. Before w e report these results w e will show that the thr esholds from the
forced-choice task w ere comparable to those reported in earlier studies of slant-
from-texture discrimination (Rosas et al., 2004 ).
The procedure in the for ced-choice task (Experiment 2 ) w as identical to that
emplo y ed b y Rosas et al. ( 2004 ). T o capture sensitivity the y computed an “area”
measure, which w as defined as the region betw een the tw o psy chometric func-
tions fitted separately for smaller and larger comparison v alues enclosed b y the
60 % and 80 % percent performance lev els (see Fig. 10 . 1 B). This ‘area’ is small
when the psy chometric functions are steep, i.e. when sensitivity to slant dif fer -
ences is high, and conv ersely , it is lar ge when sensitivity is lo w . Thus, the calcu-
lated area measure is inv ersely related to the sensitivity at a particular standard
slant.
W e computed the area measure for each standard v alue and each obser v er . The
results are sho wn in Figure 11 . 1 . In order to a v erage across observ ers the area
measure w as nor malized relativ e to the highest v alue for each obser v er individ-
ually , because obser v ers had different o v erall sensitivity to slant (inter-observ er
v ariability). In all obser v ers the area measure w as maximal for a standard slant
of 26 deg indicating lo w est sensitivity . For comparison Figure 11 . 1 also sho ws
the mean nor malized area of the fiv e obser v ers reported in Rosas et al. ( 2004 ,

11 . 2 r e s u l t s 97
26 37 53 66
Standard S l an t [ deg ]
0.0
0.2
0.4
0.6
0.8
1.0 O bse rve rs
Ro sas e t a l . ( 2004 )
Area

Figure 11 . 1 : S ensitivity obtained from psy chometric functions in Experiment 2 . The ‘area’
enclosed b y the tw o psy chometric functions and the 60 % and 80 % per-
centage correct (y-axis, see Figure 10 . 1 B for a depiction) is plotted for the
dif ferent standards (x-axis). Areas w ere nor malized for each obser v er with
respect to the maximum and aggregated across observ ers. Data from Rosas
et al. ( 2004 ) is sho wn as reference (mean ± s.e.m.).
p. 1523 ). Apart from the v ariability betw een obser v ers, sensitivity increased with
slant which is in accordance with the data r eported b y Rosas et al. ( 2004 ).
11 . 2 . 1 Thr eshold comparison
Thresholds for MLDS and 2 -IFC w ere obtained in the same w a y as in the simula-
tions. Figure 11 . 2 sho ws the data of one single obser v er . Thresholds from both
methods are plotted against each other for performance lev els of d ′ = ± 0 . 5 , 1 , 2
and for the four standard v alues tested (panels). Data points lying on the main di-
agonal indicate a quantitativ e agreement betw een thresholds. This w as obser v ed
for thresholds obtained at standar d slants of 37 , 53 and 66 deg. For a standard
slant of 26 deg a correspondence betw een thresholds fr om both methods w as ob-
ser v ed for comparisons that w ere lar ger than the standard. For comparisons that
w ere belo w the standard MLDS thresholds w ere smaller than 2 -IFC thresholds. For

9 8 experiments
Th res ho l ds f rom M LDS [ deg ]
Th res ho l ds f rom 2-IFC [ deg ]
66 de g 26 de g 37 de g 53 de g
d’
- +
0.5
1.0
2.0
0.5
1.0
2.0
20 0 20
20
0
20
20 0 20 20 0 20 2 0 0 20

Figure 11 . 2 : Thr eshold comparison for one obser v er (O 1 ). Estimates of threshold from MLDS in Experiment
1 (x-axis) and from the psy chometric functions obtained in a 2 -IFC procedure in Experiment 2 (y-
axis) are sho wn for each standard (dif ferent panels) and d ′ performance lev el, for comparisons
abo v e (+, w arm colors) and below (-, cold colors) the standar d. Thresholds are expr essed as
relativ e v alues to the standar d. Error -bars denote the 95 % confidence inter v al of the point
estimate.
some combinations of perfor mance le v els and standard v alues thresholds from
either or both methods could not be calculated (see section 10 . 4 ).
As described for the simulations w e classified thresholds to be in agreement
when one of the confidence inter v als of either method crossed the identity line
(as in Fig. 10 . 1 C). Obser v ers differ ed substantially in their proportion of agree-
ment betw een thresholds. W e sorted them according to the amount of agr eement
in descending order (Fig. 11 . 3 ). Ther e w as agreement in 15 out of 16 data points
( 94 %) for obser v er O 1 in Figure 11 . 2 , 11 of 14 ( 79 %) for obser v er O 2 , 15 of 22 ( 68
%) for obser v er O 3 , 10 of 18 ( 56 %) for obser v er O 4 , 10 of 19 ( 53 %) for obser v er
O 5 , 10 of 21 ( 48 %) for obser v er O 6 , 5 of 20 ( 25 %) for obser v er O 7 , and 2 of
14 ( 14 %) for obser v er O 8 . For obser v ers O 7 and O 8 , the data points fell abo v e
the diagonal line for comparisons abo v e the standar d (Figure 11 . 3 , red markers),
and belo w the diagonal line for comparisons belo w the standard (blue markers).
This patter n of results indicates that for these tw o observ ers thresholds obtained
with MLDS w ere consistently smaller than thresholds obtained with 2 -IFC . In other
w ords, MLDS estimated a higher sensitivity than the 2 -IFC pr ocedure; the opposite

11 . 2 r e s u l t s 99
case did not occur . T aking all obser v ers and standard lev els together , 78 out of
144 ( 54 %) estimated thresholds agreed betw een the tw o methods.
11 . 2 . 2 Thr esholds that could not be obtained
Thresholds could not be obtained from either method for stimulus comparisons
that inv olv ed the lo w est standard v alue ( 26 deg) and/or comparison slant v alues
belo w 30 deg. For example, for the observ er depicted in Figure 11 . 2 thr esholds
from MLDS could not be obtained for performance lev els of d ′ = 1 , 2 for com-
parisons belo w the standar d slant of 26 deg. The reason for this discrepancy w as
a shallo w slope in the scale reflecting lo w sensitivity at that particular stimulus
lev el.
As in the simulations w e counted the number of cases in which either one or
both of the methods did not produce a threshold for our experimental results.
A total of 22 cases occurred in which either one of both thresholds could not
be obtained. Four out of the 22 cases w ere cases in which thresholds from MLDS
w ere missing (at standar d of 26 deg and 37 deg), elev en w ere cases in which
thresholds from 2 -IFC w ere missing (standard 26 deg and 37 deg) and se v en w ere
cases in which both thresholds w ere missing (all for a standar d of 26 deg). S o
the methods consistently estimated lo w sensitivity in 32 % of the cases for which
thresholds could not be obtained.
11 . 2 . 3 V ariability of threshold estimates
W e also deriv ed the v ariability of the threshold estimates for the experimental
data. W e found that the v ariability w as low er for MLDS than for 2 -IFC , consistent
with the simulations. Figure 11 . 4 sho ws the widths of the confidence inter v als for
the thresholds obtained with each method from all observ ers. As in Figure 10 . 3 A
confidence inter v als w ere smaller for thresholds from MLDS than for thresholds
from 2 -IFC . Ov erall for 142 of the 144 threshold comparisons ( 98 . 6 %) the width
of the confidence inter v al w as smaller for thresholds from MLDS (separate com-
parisons for each obser v er can be found in Appendix Figure A. 10 ).

1 0 0 experiments
Th res ho lds f rom M LD S [ deg ]
d’
- +
0.5
1.0
2.0
0.5
1.0
2.0
Th res ho lds f rom 2 -IF C [ deg ]
66 de g 26 de g 37 de g 53 de g
O2
O4
O3
O5
-20
0
20
-20
0
20
-20
0
20
-20
0
20

Figure 11 . 3 : Threshold comparison for sev en obser v ers (O 2 -O 8 ). Similar to Figure 11 . 2 , thresholds r elativ e
to the standard obtained fr om the tw o methods are compared b y obser v er (ro ws), standar d
(columns), and performance lev el ( d ′ ), for comparison v alues abo v e (+, w ar m colors) and
belo w (-, cold colors) the standar d. Obser v ers are sorted b y percentage of threshold agr eement
betw een the tw o methods in descending order .

11 . 2 r e s u l t s 101
Th res ho lds f rom M LD S [ deg ]
d’
- +
0.5
1.0
2.0
0.5
1.0
2.0
Th res ho lds f rom 2 -IF C [ deg ]
66 de g 26 de g 37 de g 53 de g
O6
O7
O8
-20
0
20
-20
0
20
-20
0
20
-20 0 20 -20 0 20 -20 0 20 -20 0 20

1 0 2 experiments
C .I. w idt h o f
thr es ho ld f rom M L DS [ deg ]
C .I. w idt h o f
thr es ho ld f rom 2 -IF C [ deg ]
0 5 10 15 2 0
0
5
10
15
20

Figure 11 . 4 : Comparison of the v ariability in the threshold estimation. The width of the
confidence inter v als from Figure 11 . 2 and Figure 11 . 3 are plotted against
each other , for all obser v ers and standar d v alues.

12
D I S C U S S I O N
The goal of the present study w as to test whether judgments of stimulus ap-
pearance and judgments of stimulus discriminability are mutually consistent
which w ould suggest that both types of judgments rely on a common percep-
tual representation of the stimulus dimension under study . The evidence on this
question is mixed (e.g. Krueger, 1989 ; Ross, 1997 ; Hillis & Brainard, 2007 b), but
comparing supra-threshold judgments in a MLDS pr ocedure and near-thr eshold
judgments in a forced-choice pr ocedure, Devinck and Knoblauch ( 2012 ) ha v e re-
ported that the tw o can be linked within a common signal detection framew ork.
Using slant-from-texture stimuli w e conducted tw o experiments that indepen-
dently measured the sensitivity to dif ferences in slant. In the first experiment ob-
ser v ers judged supra-threshold stimulus dif ferences and w e deriv ed thresholds
from per ceptual scales using the MLDS framew ork (Maloney & Y ang, 2003 ). In
the second experiment w e measured sensitivity in a conv entional tw o-alter nativ e
forced-choice pr ocedure and w e deriv ed thresholds from psy chometric functions.
For some obser v ers there w as agreement betw een thresholds obtained with both
methods but across observ ers the methods agreed in only 54 % of the cases. For
tw o obser v ers (O 7 and O 8 ) sensitivity estimates from the MLDS procedur e w ere
consistently higher than those from the for ced choice procedure.
The obser v ed lack of correspondence betw een the estimates could imply that
the tw o tasks do indeed probe dif ferent perceptual r epresentations of a stimu-
lus. Alter nativ ely , the lack of correspondence might result from violations of the
model assumptions, and hence w ould not be infor mativ e about the relationship
betw een appearance and discrimination tasks.
103

1 0 4 discussion
12 . 1 viola tions of model assumptions
The equiv alence betw een MLD scales and the 2 -IFC procedure used in the present
w ork relies on a number of theoretical assumptions concerning the sensor y rep-
resentation and the decision model. In the follo wing w e will describe the effect
of violations of one or more of these assumptions on the estimated scales and
the consequences for the estimation procedure.
12 . 1 . 1 Goodness of fit
The MLDS framew ork pro vides goodness of fit procedures that test the plausibil-
ity of the data being produced b y a differ ence scaling model (Knoblauch & Mal-
oney, 2008 , p. 219 - 222 ). In our data, the goodness of fit of the dif ference scales
w as insufficient for fiv e out of eight observ ers when the default parameter v al-
ues w ere used. W e follo w ed the refitting procedur e suggested b y Knoblauch and
Maloney ( 2008 , pp. 219 - 222 ) for these cases to modify the model specification.
The procedure includes the estimation of ‘guess’ and ‘lapse’ rates, and a split of
the ra w data into tw o parts that w ere e v aluated separately (detailed description
of the goodness of fit procedure is pr o vided in the Appendix A. 2 ). After the refit-
ting procedure w e obtained an appropriate goodness of fit for all obser v ers, and
w e deriv ed thresholds from the r efitted scales. The thresholds that w ere deriv ed
from the adjusted scales w ere not markedly dif ferent fr om the original ones. In
particular , the disagreement betw een thresholds that w e obser v ed in three ob-
ser v ers w as present with or without the goodness of fit adjustment. Thus, the
model violations that w ere detected b y the goodness of fit routines did not ha v e
much of an ef fect on the shape of the scale, at least for the present data.
12 . 1 . 2 Reconciling MLDS with 2 -IFC thr esholds.
The assumption of independence betw een dif ferent lev els of the sensor y repre-
sentation (assumption 2 ) is not and cannot be tested b y the goodness of fit rou-
tine. If this assumption is violated it w ould af fect the noise and it w ould require

12 . 1 viola tions of model assumptions 1 0 5
Th res ho l ds f rom M LDS [ deg ]
Th res ho l ds f rom 2-IFC [ deg ]
66 de g 26 de g 37 de g 53 de g
d’
- +
0.5
1.0
2.0
0.5
1.0
2.0
-20 0 20
-20
0
20
-20 0 20 -20 0 20 -2 0 0 20

Figure 12 . 1 : Threshold comparison for observ er O 8 when the dif ference scales are
rescaled b y a different factor to account for a possible dependence betw een
dif ferent realizations of the random v ariable that characterizes the sensory
representation (a factor of 0 . 6 corresponding to a correlation coef ficient of
0 . 9 ).
an adjustment of the scaling factor that transfor ms the original MLDS scale into a
scale in units of d ′ . The independence assumption w ould be violated when the
sensory r epresentations cannot be characterized as independent realizations of a
Gaussian random v ariable but are instead correlated with each other . W e tested
the ef fect of these kind of correlations in the sensor y representation in simula-
tions. Correlated sensory v ariables do indeed af fect the threshold estimates. T o
illustrate the ef fect w e sho w that the magnitude of the correlation can be chosen
so as to elicit a correspondence betw een thresholds deriv ed from MLDS scales and
from 2 -IFC . Figure 12 . 1 sho ws the thresholds for obser v er O 8 for a simulated case
in which the sensor y representations ar e highly correlated ( ρ = 0 . 9 ). As a conse-
quence of this correlation w e giv e up the independence assumption and w ould
ha v e to rescale the perceptual scale b y a factor of 0 . 6 (instead of the theoretical
factor of tw o). In this scenario the resulting thresholds from MLDS correspond
better with the thresholds from 2 -IFC . Thus, an alternativ e transfor mation that
accounts for a model violation can ‘produce’ a higher agreement betw een the
tw o types of thresholds. W e are not a w are of any method to test the assumption

1 0 6 discussion
of independence empirically and it is therefore not possible to e v aluate which of
the many possible transfor mations is closest to the true sensory representation.
A similar issue arises when w e scrutinize the ef fect of violating the assumed
decision rule (assumption 4 ). Based on the assumption that the sensory represen-
tation function is monotonically increasing the decision rule can be expressed as
a double dif ference operation ( ∆ v ariable in Figure 3 . 2 B) instead of an absolute
v alue operation ( ∆ v ariable in Figure 3 . 2 A). This change from an absolute to a
relativ e dif ference operation can ha v e noticeable effects when the sensory rep-
resentations are random v ariables (as assumed here) instead of fixed v alues. T o
explore the ef fect of the dif ferencing operation w e simulated an obser v er that
judged the triads b y using either one of the tw o decision rules. T o analyze the
ef fect on the estimated noise w e applied MLDS to each of the tw o types of sim-
ulated responses, and found that the absolute dif ference operation pr oduced
higher noise estimates than the double dif ference operation. This differ ence in-
creases progr essiv ely as the underlying sensor y noise increases. Thus, the tw o
decision rules can produce dif ferent results (see Section 4 . 5 . 1 and S ection 4 . 5 . 2
for simulations and details).
It is not possible to deter mine empirically whether or when obser v ers apply
an absolute or a relativ e dif ferencing rule, the y might ev en change the rule with
v ar ying difficulty of the judgment. One should be a w are that a deviation fr om the
assumed decision rule or a violation of the independence assumption ma y both
af fect the noise estimate although in opposite directions. Thus, the combined
contributions of both factors can produce v arious types of deviation from the
true scaling factor and this af fects the scale and the deriv ed sensitivity estimate.
12 . 2 v ariability of thresholds estima ted by mlds
As in the simulations w e obser v ed that the v ariability of thresholds deriv ed
from MLDS w as smaller than the v ariability of thresholds deriv ed from 2 -IFC (Fig-
ure 10 . 3 and Figure 11 . 4 ). Again, this is counterintuitiv e because MLDS requires
a smaller amount of data for the threshold deriv ation. Ho w ev er , the smaller v ari-
ability must be interpreted with care, because our simulations rev ealed that the
co v erage of MLDS deriv ed threshold might be insuf ficient and the width of the

12 . 3 conclusions 1 0 7
confidence inter v als might be underestimated. This should be considered for
hypotheses tests as it ma y lead to T ype-I errors.
Ho w ev er , apart from the co v erage problem associated with our threshold
deriv ation, MLDS pro vides an ef ficient method to acquire sensitivity estimates.
In the present Experiment 1 w e ran the MLDS procedur e with 840 trials which
took about 45 minutes per obser v er . In contrast, the 2 -IFC procedure in Experi-
ment 2 required 2560 trials and lasted three hours. Thus, the dif ference in both
the amount of data and the required acquisition time might be up to three to four
times more for 2 -IFC than for MLDS . MLDS thus pro vides an efficient alternativ e to
forced-choice pr ocedures to obtain a rough estimate of sensitivity .
12 . 3 conclusions
In the present experiment w e inv estigated the question of equiv alence of thresh-
olds deriv ed from an MLD scale and thresholds deriv ed from a forced-choice
procedure. Using simulations, w e established upper bounds for a possible agree-
ment considering the theoretical model assumptions, the finite amount of col-
lected data and the necessar y softw are pipeline. Experimentally , w e found v ar y-
ing degrees of correspondence betw een the methods for different observ ers. Out
of a total of 144 threshold estimates the methods’ sensitivity estimates dif fered
in 66 cases. W e discuss that the equiv alence of thresholds (or lack thereof) might
either indicate a corresponding equiv alence betw een the underlying perceptual
representations (or lack ther eof) as has been argued b y Devinck and Knoblauch
( 2012 ), or alter nativ ely , it might result from violations of the model assumptions.
An important point that has been made b y one of the revie w ers is that w e
ga v e the 2 -AFC method the benefit of histor y . In the present analysis w e used
the 2 -AFC method as a standard of r eference against which w e compared the sen-
sitivity estimates deriv ed with MLDS . Accordingly w e tested the effect of model
violations on threshold agreement only for the assumptions underlying MLDS .
Ho w ev er , considering the present data and the numerous benefits associated
with the experimental procedures of MLDS , it might be w arranted to try to elab-
orate the first principles case of which of the tw o methods w e w ould trust more
if w e started out de novo .

1 0 8 discussion
Our positiv e ev aluation of the MLDS method is corroborated b y recent results
from Kingdom ( 2016 ) who used MLDS to decide betw een competing theories of
inter nal noise in contrast transduction. In summar y , w e conclude that MLDS , as
state-of-the-art scaling method, seems to ha v e great potential to be used bey ond
the purpose that it w as originally designed for .

Part IV
DISCUSSION AND OUTLOOK

13
G E N E R A L D I S C U S S I O N A N D O U T L O O K
In this thesis I propose the use of MLDS as a tool for measuring per ception b y es-
timating perceptual scales in an ef ficient and reliable w a y . MLDS – an appearance
method – uses the judgment of clearly visible stimulus dif ferences which a v oids
the shortcomings of Fechnerian, Thurstonian, and direct scaling methods (Chap-
ter 2 ). Critically , MLDS uses the method of triads which pro vides an intuitiv e and
easy task for the obser v er , a v oiding the disadv antages of perfor mance-based (dis-
crimination) methods that use dif ficult and unintuitiv e tasks.
MLDS w as used to reliably estimate scales in cases of perceptual constancy .
MLDS can reco v er the underlying perceptual scales for dif ferent viewing contexts
but requiring only judgments inv olving within-context comparisons (Chapter 5 ).
In the case of lightness constancy , w e sho w ed how MLDS seems superior to asym-
metric matching procedures as these, unlike MLDS , do not estimate the internal
scales directly but rather their existence is assumed and cannot be infered fr om
obser v ers’ matches (Chapter 6 ). Additionally , matching data deriv ed from the
MLDS experiment follo w ed closely the data acquired in an asymmetric match-
ing procedure (Chapter 7 ), which suggests that MLDS indeed pr obes the inter nal
perceptual repr esentation, i.e. lightness.
MLDS can be also used for predicting sensitivity b y framing it in a signal detec-
tion theory for mulation (S ection 3 . 3 ). In simulations the estimation of sensitivity
deplo y ed with traditional perfor mance methods – from psy chometric functions
in a 2 -IFC procedure – w as equiv alent to the estimation of sensitivity from MLDS
when all model assumptions w ere met (Chapter 10 ). Critically , MLDS w as more
ef ficient, requiring less data than the traditional 2 -IFC procedur e. In an experi-
ment using a slant-from-texture task, w e found v ar ying degrees of agreement
betw een the methods for dif ferent obser v ers, which either indicates a lack of
equiv alence betw een the underlying perceptual representations, or alternativ ely ,
it ma y result from violations of the model assumptions (Chapter 11 ).
111

1 1 2 general discussion and outlook
13 . 1 the method of triads
Historically , the method of paired comparisons w as the first procedure used in
scaling that inv olv ed comparison of stimuli (Thurstone, 1927 a). The method of
pair comparisons commonly requires a significant amount of data for a scale to
be estimated. Later , the method of tetrads w as introduced b y T orgerson ( 1958 ) as
a direct extension of the method of paired comparisons, and it consisted of four
stimulus exemplars judged simultaneously . This procedure later deriv ed into
the method of triadic combinations that made acquisition more efficient (T or gerson,
1958 ) b y presenting three stimulus exemplars in all possible triadic combina-
tions. The method of triads in MLDS is an additional simplification of the method
of triadic combinations b y assuming an ordered and monotonically increasing
dimension, which limits the number of combinations to only non-o v erlapping
inter v als (S ection 3 . 1 ). Interestingly , most of the studies using MLDS ha v e used
the method of quadruples (e.g. Maloney and Y ang ( 2003 ); Obein et al. ( 2004 );
Fleming et al. ( 2011 ); Paulun et al. ( 2015 ); Charrier et al. ( 2007 ), see Devinck and
Knoblauch ( 2012 ) for an exception). In this thesis the method of triads w as used
instead of quadruples, because upon introspection the comparison of triads feels
easier than of quadruples.
From a mathematical point of vie w the tw o methods are identical (Knoblauch
& Maloney, 2012 ). In the quadruple case the decision v ariable is specified as Decision variable
in triads
and quadruples ∆ Q = ( Ψ x 4 − Ψ x 3 )−( Ψ x 2 − Ψ x 1 ) and in the triad case as ∆ T = ( Ψ x 3 − Ψ x 2 ) −
( Ψ x 2 − Ψ x 1 ) . It is assumed that for the triad comparison Ψ x 2 is sampled twice
resulting in tw o independent realizations of a random v ariable (for MLDS in its
signal detection theor y for mulation). Accor dingly , the v ariances of ∆ Q and ∆ T
are identical and so is the maximum of the respectiv e unconstrained scales.
Subjectiv ely , the triad comparison feels easier because the compared inter v als
( [ Ψ x 1 , Ψ x 2 ] and [ Ψ x 2 , Ψ x 3 ] ) are adjacent to each other with a common refer ence
stimulus ( Ψ x 2 ) that can be used as anchor . This benefit w ould translate into a
decision model for triad comparisons where Ψ x 1 and Ψ x 3 are compar ed to a
single realization of Ψ x 2 . In this case, ho w e v er , the v ariance of ∆ T will be larger
than of ∆ Q , because the use of a single realization of Ψ x 2 adds co v ariance to the
sum of other wise independent Gaussian random v ariables. The consequence of

13 . 2 anchoring of the scales 1 1 3
using a single realization of Ψ x 2 is hence the introduction of additional v ariability
to the decision v ariable, which is counter-intuitiv e with the fact that the task for
triads feels easier than for quadruples. Whether the equiv alency betw een the
method of triads and quadruples holds true is still an open question that needs
to be addressed experimentally .
13 . 2 anchoring of the scales
MLDS estimates interval scales, and as such they pr o vide representation of interval
differ ences . For inter v al scales any linear transformation of the type x ′ = a · x + b
is allo w ed, which keeps the representation of interv al dif ferences meaningful.
Inter v al scales need – b y definition – an arbitrar y origin or zero v alue (Krantz Scales anchored
at zero
et al., 1971 ). The GLM in MLDS is designed in a w a y that the origin is ef fectiv ely
defined at zero, making per ceptual scales b y default anchored at their minimum
(S ection 3 . 2 ).
The arbitrary nature of the scale’s origin can be problematic when scales need
to be compared with each other , such as in the present case (W iebel et al., 2017 ).
Here scales for dif ferent vie wing contexts w ere measured independently , and
a default origin at zero w as assumed for all of them. In this study an equal
origin translates into assuming that the perception of the lo w est reflectance is
equiv alent among all viewing contexts, i.e. the darkest reflectance is per ceiv ed
equally black. W e discussed in Chapter 8 that, in this case, a problem of anchor -
ing among the scales is unlikely , as the predicted matching data w ere consistent
with the data acquired independently in the matching experiment. Ho w ev er , this
equiv alency ma y not apply to all experimental cases and it should be carefully
considered when designing an experiment.
Other scaling methods, such as pair comparisons in Thurstonian scaling, are
based on the judgment of inter v al differences as w ell, and therefore also pr oduce
inter v al scales with an arbitrar y origin (with the exception of magnitude estima-
tion that produces ratio scales). A potential pr oblem of anchoring is therefore
common to all scaling methods when scales are measured and compar ed among
dif ferent contexts.

1 1 4 general discussion and outlook
It ma y be possible to test the equiv alency of the scales’ origin with other
method when in doubt, e.g. with an matching task (method of adjustment) for
the lo w est stimulus v alue. In W iebel et al. ( 2017 ) w e argued that the good cor -
respondence betw een the estimated scales and the matching data suggests that
there w as no problem with the anchoring of the scales. Ho w ev er , it cannot be
argued that MLDS is a better method than matching in pr obing the inter nal per -
ceptual scale, if w e justify the scales’ anchor using the matching data; this argu-
mentation w ould be circular .
It remains to be tested whether giving the scales other anchor(s) impro v es or
w orsens the predictions deriv ed from them. In principle it is possible to modify
the GLM in MLDS to allo w an origin dif ferent than zero. It w ould require the
definition of an alter nativ e design matrix that drops not the first but any other
column, setting the origin to that corresponding stimulus v alue (S ection 3 . 2 ).
This ma y be adv antageous under theoretical considerations that w ould prefer
another anchor for the scales, e.g. the maximum.
13 . 3 mlds and mul tiplica tive noise
S caling methods that rely on the measurement of JNDs , as Fechnerian scaling,
fail to pro vide meaningful scales if the noise is multiplicativ e (Chapter 2 ). The
fact that MLDS can estimate a perceptual scale in the presence of multiplicativ e
(unequal-v ariance) noise is a clear methodological benefit (Chapter 4 ), giv en that
the type of noise distribution cannot be tested experimentally .
The multiplicativ e noise case tested in these simulations correspond to the Multiplicative noise
most studied case of unequal-v ariance: the noise increases with the stimulus di-
mension as obe ying W eber ’s la w . In discrimination methods, signal detection
theory does pro vides statistical tools to fit multiplicativ e noise models, e.g. in
y es/no or 2 -IFC tasks (DeCarlo, 1998 ; McNicol, 1972 ; Knoblauch & Maloney,
2008 ). MLDS in its current implementation does not allo w the definition and es-
timation of such a statistical model. Ho w ev er , it is feasible to extend MLDS to
include heter oscedastic (unequal-v ariance) noise.
The estimation w ould inv olv e the use of modified link functions and a similar
maximum likelihood estimation procedure. An extension could be de v eloped b y

13 . 4 mlds and obser ver models 1 1 5
Direct report of cues
Image
formation
infer ence
• Pictorial
• Binocular disparity
• Motion pa rallax
• F oreshortening
• Density gradient
• Size gradient
• P erspective
convergence

P er ceptual
?
Report in a
psychophysical
task
Behavior
(Physical) Slant Cues to slant (-depth)
Perceived
slant
?
Distal Proximal Perceptual

Figure 13 . 1 : Distal, proximal and per ceptual dimensions in slant-from-texture.
using existing packages in the R language environment (package glmx , Zeileis,
Koenker , & Doebler, 2015 ), which implement binar y GLMs with heteroscedastic
noise. It remains to be tested whether this extension to MLDS can be used in
the context of perceptual judgments of appearance, and whether it pr o vides
meaningful predictions of sensitivity .
13 . 4 m l d s a n d o b s e r v e r m o d e l s
Ideal obser v er analysis is an useful framew ork for testing potential mechanisms
of perceptual inference in the visual system (Geisler, 2011 ). In the study pr e-
sented in Part II, ideal obser v er models w ere used to simulate tw o possible be-
ha viors in lightness perception: one observ er model that is fully lightness con-
stant, and another that has only access to luminance information. These tw o
models are extremes of possible beha vior giving the ambiguous luminance sig-
nal in the retinal arra y . When these models w ere used to simulate responses
to an ‘mock’ MLDS experiment, the scales predicted b y MLDS w ere sufficiently
dif ferent up to a certain noise lev el limit which w as belo w what is commonly
found experimentally (S ection 6 . 4 ). MLDS could thus distinguish betw een these
models because the mapping betw een the proximal and per ceiv ed dimensions
w as sufficiently dif ferent betw een them (Figure 6 . 1 B).

1 1 6 general discussion and outlook
The case of slant-from-texture in Part III is differ ent. In slant-from-texture,
multiple pictorial cues in the proximal r etinal arra y are a v ailable to the visual Pictorial cues
system (Figure 13 . 1 ). These pictorial cues has been defined as: (i) foreshorten-
ing, the change in the aspect ratio of the texture elements; (ii) size gradient, the
change in the size of the texture elements; (iii) density gradient, the change in
the density of the texture elements; and (iv) perspectiv e conv ergence, the ten-
dency of the texture elements of conv erging to w ards the center due to linear
perspectiv e (Cutting & Millard, 1984 ; Saunders & Backus, 2006 ).
Figure 13 . 2 A sho ws how some of the pictorial cues ar e calculated from a tex-
ture stimulus. Any pictorial cue could be used in isolation to infer slant b y inv ert-
ing the process of image formation. Ho w ev er , it is still an open question which
cues the visual system uses in slant-from-texture (Knill, 1998 ; Saunders & Backus,
2006 ; T odd et al., 2010 ) For many pictorial cues the mapping betw een the proxi-
mal and the distal (slant) dimensions follo w functions with similar cur v ature, as
sho wn in Figure 13 . 2 B.
Figure 13 . 2 C sho ws the results of a pilot experiment using MLDS and slant-
from-texture stimuli for one obser v er . The blue lines and errorbars sho w the
(nor malized) dif ference scale when the observ er w as asked to judge perceiv ed
slant, and the y ello w , red, and orange lines sho w the scales from simulated ob-
ser v er models using the pictorial cues from Figure 13 . 2 A. The scales w ere found
not to correspond to any of the cues in isolation, for fiv e dif ferent obser v ers (Fig-
ure 13 . 2 C). Rather , the results suggest that the scales actually follo ws a mixture
of cues: a foreshortening cue at lo w slant v alues and a scaling contrast cue at
higher slant v alues.
The ‘scaling contrast’ cue is calculated as the ratio differ ence betw een the Scaling contrast
widths of the elements in the upper most and lo w ermost regions of the texture
(Figure 13 . 2 A), and it has been proposed to be a cue used b y obser v ers when
judging slant-from-texture tasks (T odd, Thaler , Dijkstra, Koenderink, & Kappers,
2007 ; T odd et al., 2010 ). It can be seen in Figure 13 . 2 C that the dif ference scale
does not follo w the scaling contrast cue. Ho w ev er , these results does not re-
ject the possibility that scaling contrast is a v ehicle for the judgment of slant
which can be used in the visual system’s computations, and that for obser v ers
it is simply not possible to report pictorial cues directly . T odd et al. ( 2010 ) used

1 1 8 general discussion and outlook
a modified v ersion of a partition scaling task that required the adjustment of
perceptually equal interv als. The use of an adjustment task in their study ma y
account for the dif ferences found in our study .
When the same obser v er in Figure 13 . 2 C w as asked to judge the foreshort-
ening of the upper most elements, keeping stimuli and experimental procedur e
other wise identical, the obtained scale w as not different (Figure 13 . 2 D). It is not
possible to disentangle with a negativ e result whether the visual system is not
using foreshortening or scaling contrast to infer slant, or it is simply not possible
for the obser v er directly report pr oximal cues.
Thus, MLDS can be used with ideal obser v er models but only when the models
are dif ferent enough with respect to the mapping among the distal, pr oximal and
perceptual dimensions, as is the case for lightness. The de v elopment of these
models depend on theoretical considerations and does not ha v e to do per se with
MLDS .
13 . 5 o u t l o o k : tow ards more realistic stimuli
This thesis ev aluated the use of MLDS for the reliable measurement of per ception,
highlighting its adv antages that ha v e been already discussed. As critical as the
psy chophysical method chosen to measure per ceptual phenomena is the type
of stimulus that are used to pr obe the visual system (Figure 1 . 3 ). As outlined
in Chapter 1 , adequate and unambiguous stimuli ma y be more suitable for the
study of vision b y pro viding more realistic stimulation and thus a v oiding un-
w anted strategies and confounds. The issue of realism in the stimulus has been
lately a focus of attention in the field (Fleming, 2016 ; Brainard & Radonjic, 2016 ).
In the follo wing I discuss this issue, outlying some possible directions aiming to
increase the realism of stimuli used in visual per ception research.
13 . 6 simple vs . realistic stimuli
In lightness perception r esearch, stimuli of div erse naturalness and complexity
ha v e been used. On one extreme, simple and flat stimuli ha v e been used, such as

13 . 6 simple vs . realistic stimuli 1 1 9
the simultaneous contrast displa y (Figure 1 . 2 B) that only ha v e luminance infor-
mation without a clear distal structure as its origin. These type of stimulus are
usually presented in flat monitor screens and sho w no cues to depth, conditions
that are far a w a y from naturally-occurring objects and surfaces. On the other
extreme, lightness perception has been also studied using r eal physical stimuli,
made of paper and cardboar d, and real sour ces of illumination (e.g. Gilchrist,
1977 , 1980 ). There is no doubt that these type of stimuli are realistic, ho w ev er
they ar e also impractical. They usually need complicated experimental appara-
tuses and the data collected are usually limited due to practical constraints on
time.
In studies that find lightness constancy to be v ariable or ev en lo w , the finding
is commonly attributed to the procedures or e v en the instructions used (Arend
& Goldstein, 1987 ; Radonji ´ c & Brainard, 2016 ), which w ould not constrain w ell
responses based on lightness. Experimentally it ma y be possible that lightness
constancy is truly lo w or v ariable, but it ma y as w ell be that the stimuli used are
too simple and not realistic enough to stimulate the visual system in a naturally
and appropriate w a y (Koenderink, 1999 ). Realistic stimuli are thus needed in
order to r eliably deter mine in which conditions lightness constancy is lo w , and
dev elop successful models that must predict these deviations from constancy .
It seems to be dif ficult to design stimuli that are at the same time w ell-controlled
and realistic. One solution is the use of computer -generated scenes b y ra y-tracing
techniques (e.g. Heasly , Cottaris, Lichtman, Xiao, & Brainard, 2014 ). Ra y-tracing Ray-tracing
rendering aims to simulate the beha vior of the light realistically , pro viding w ell-
controlled stimuli and ef ficient data collection. A high degree of constancy can
be found in these type of experiments – as in the study in Part II – which tell us
that these stimuli constrain w ell the perceptual response of judgments based on
lightness.
Ho w ev er , ra y-traced scenes are still presented in flat monitor screens, and
some experimental evidence suggests that the use of monitors can be detrimen-
tal for perceiv ed depth (W att, Akeley , Ernst, & Banks, 2005 ; Hoffman, Girshick,
Akeley , & Banks, 2008 ). In nor mal viewing of the natural w orld, our ey es’ v er-
gence is correctly aligned to a specific depth plane at the point of interest. When
the ey es fixate at this depth plane, the lens accommodates its refractiv e po w er to

1 2 0 general discussion and outlook
match the plane, and consequently retinal blur occurs normally for depth planes
that are farther or closer i.e. the e y e’s depth of field. These phenomena of v er- Depth cues conflict
gence, accommodation of the lens, and retinal blur , collectiv ely kno wn as focus
cues, dynamically signal the correct depth while our e y es look at the real w orld.
On computer displa ys, ho w ev er , focus cues signal flatness at a fixed distance to
the screen. Therefore, focus cues ar e in conflict with experimentally added cues
that will signal the desired depth plane (e.g. using pictorial or binocular dispar -
ity cues). Most of depth perception resear ch has treated focus cues as negligible,
ho w ev er it has been sho wn that when cue conflict is a v oided, depth and realism
is impro v ed (Hoffman et al., 2008 ).
In the follo wing I propose tw o w a ys of increasing realism in experiments
studying depth and lightness perception. These ideas w ere de v eloped as re-
search pr ojects for this thesis, and are suggested here as an outlook on ho w
to increase realism in psy chophysical experimentation on depth and lightness.
13 . 6 . 1 Incr easing realism by adding motion parallax
Motion parallax is the relativ e motion of the retinal input due to head mo v e- Motion parallax
ments, and it signals relativ e depth. Computing relativ e depth can be accom-
plished b y comparing the relativ e motion of objects in the retinal input (Rogers
& Graham, 1979 ). Motion parallax is a strong cue to depth, and it is kno wn that
ev en small head mo v ements can result in a detectable signal to be used to re-
triev e depth infor mation (Rogers & Graham, 1979 ; A ytekin & Rucci, 2012 ). How-
ev er , the contribution of motion parallax is omitted in almost all psy chophysical
experiments that w ork with depth. When inevitable small head mo v ements oc-
curs in standard psy chophysical setups, these will signal flatness of the scene,
hindering 3 -D realism. Consequently , adding correct motion parallax to stimuli
presented on screens will most likely increase the r ealism of the scene.
Rogers and Graham ( 1979 ) first studied motion parallax b y dynamically adjust-
ing the visual stimuli according to the observ er ’s head position. The head rested
on a chinrest that could mo v e horizontally in front of the screen, and its posi-
tion w as used to update the stimuli accordingly , simulating correct perspectiv e
transfor mation (Rogers & Graham, 1979 ). This technique, also called head-joked

1 2 2 general discussion and outlook
method, allo w ed the experimenters to use motion parallax as an independent
cue for depth perception. The pr oposed setup dra ws from this method, b y track-
ing digitally the viewpoint of the observ ers and adjusting in real-time the per -
spectiv e projection of the render ed scene according to the observ er ’s viewpoint
(Figure 13 . 3 ).
The first component required in the setup is head-tracking: w e must track and Head tracking
recor d the position of the head and/or ey es of the obser v er . An easy and sim-
ple option for head-tracking is the use of infrared technology (other commercial
technologies based on radio are also a v ailable). The har dw are consist of an in-
frared (IR) camera and infrared emitting diodes mounted on a glasses’ frame
(see Figure 13 . 3 B and C). The IR camera is placed centered belo w the screen
pointing to w ards the observ er , and the obser v er w ears glasses with IR diodes at-
tached to the sides. The distance betw een the diodes is kno wn and fixed, as w ell
as the camera’s field of view . It is thus possible to estimate the head distance to
the camera and consequently to the screen, and deriv e the coordinates in space
using trigonometry and coordinate transfor mations (Figure 13 . 3 C).
The second component for this setup is the rendering of the simulated scene Dynamic rendering
in 3 -D with v ar ying viewpoint that matches the head position (see Figure 13 . 3 B).
This can be accomplish using real-time rendering softw are, such as OpenGL or
ra y-traced pre-rendered scenes. The goal is to generate the illusion of a virtual
scene that spans in depth, which is looked from the observ er ’s point of view
through the screen. The scr een acts as a sort of ’windo w’ to a simulated, virtual
room (see Figure 13 . 3 B). T o accomplish this ef fect, the estimated position of the
middle point betw een the IR diodes (which approximates to the middle point be-
tw een the ey es) is used as the camera position or vie wpoint in the rendering tool.
When the obser v er mo v es, for example to the right as sho wn in Figure 13 . 3 B, the
perspectiv e projection in the rendering changes accor dingly , giving the impres-
sion of a real room in depth with its contents.
The setup for adding motion parallax w as implemented during this thesis
and a basic rendering example w as tested. The example w as a simulated room
in 3 D containing geometric v olumes, with a visible framew ork for the room’s
w alls that pro vided additional linear perspectiv e cues (Figure 13 . 3 B). The added
motion parallax increased the realism of depth of this scene. Ho w ev er , it w as

13 . 6 simple vs . realistic stimuli 1 2 3
Figure 13 . 4 : Adelson’s checker -shado w illusion using a e-ink displa y device. T w o checks, one
in shado w and one in plain view , are equal in luminance but dif ferent in lightness.
The right panel sho ws the image with a superimposed bar betw een the tw o checks;
the left panel sho ws the image untouched.
not possible to increase the realism in the case of slant-fr om-texture, because, as
Figure 9 . 1 sho ws, stimuli in slant-from-texture are seen through flat apertur es
that do not allo w the presentation of the room’s w alls or any other perspectiv e
cues. It seems that the dynamic change of perspectiv e according to the vie wpoint
is necessary for motion parallax to be useful as a depth cue. It remains to be
tested whether the setup increases realism for more complex scenes, such as for
the Adelson checker -shado w stimulus.
13 . 6 . 2 Incr easing realism by using an electr onic ink display
The fact that stimuli are presented in (flat) monitors ma y be critical for the study
of lightness perception, because monitors, unlike r eal surfaces, emit light rather
than r eflect it. This could produce v arious confounds when lightness perception
is probed. Koenderink ( 1999 ) critically expressed the concern that, with the use
of computers, the moder n study of visual perception deals more with computer
graphics pipelines rather than with actual perceptual inference.

1 2 4 general discussion and outlook
An alter nativ e approach is using r eal stimuli, but instead of using actual pa- E-ink displays
per and cardboar d, using electronic ink (e-ink) displa ys. These displa ys aim to
resemble real paper , and the y can be programmed to dynamically change their
content. They do not emit light, rather , they contain microparticles filled with
white or black ink that reflect light as normal matte paper does. These micropar -
ticles contain ink electrically charged, with dif ferent polarity for white and black
ink. By v ar ying the electrical field betw een the device’s surface, the amount of
black and white particles in each pixel can b y controlled and manipulated, and
thus the reflectance of the displa y .
The a v ailability of these devices has gr o wn in recent y ears, being no w commer -
cially a v ailable. During this thesis I explored the possibility of studying lightness
perception using this type of de vice. Figure 13 . 4 sho ws the Adelson’s checker-
shado w illusion constructed with real stimuli. The checkerboar d surface is an
e-ink displa y ( 13 . 3 ’ dev elopment kit, V isionect Ltd ) that allo ws the change of its
content digitally . The light source w as a LED light coming from the left, and a
white plastic cylinder w as 3 -D printed (Ultimaker 2 , Ultimaker B.V . ).
T o construct the illusion, lookup tables w ere measured in order to determine Adelson checkerboard
on e-ink display the mapping betw een luminance and the de vice’s reflectance, for check positions
in shado w and plain vie w . These lookup tables depend critically on the geome-
try of the scene (light source position, cylinder position and other reflecting sur -
faces). Once measured, the checkerboar d image w as rendered with r eflectance
v alues that are thus kno wn to produce equiluminant checks ( 20 cd/m 2 ). These
tw o checks, although equiluminant, are perceiv ed as light (shado w) and dark
(plain view) in lightness. The left panel in Figur e 13 . 4 sho ws a picture of the
setup, and the right panel sho ws the same picture modified with a superim-
posed gra y bar , indicating the equiv alence of pixel gra yv alues.
This setup promises to be useful in studying open questions regar ding mech-
anisms of lightness perception such as assimilation or the ef fect of edges and
contrast in lightness inference under naturalistic conditions One immediate ap-
plication w ould be to measure the ef fect size in the Adelson’s checker -shado w
illusion using this setup, and comparing it to an analogous experiment using a
ra y-traced scene with same geometr y and luminance specification. Subjectiv ely ,
the ef fect in the Adelson checkerboard becomes more pr onounced for the actual

13 . 7 conclusions 1 2 5
real stimulus when compared to a pictur e of the same stimulus (Figure 13 . 4 ),
which is consistent with the finding that the increase of realism makes the ef fect
size more pronounced (Maertens et al., 2015 ). The magnitude of the ef fect change
with this setup remains to be quantified; and more generally , it also remains to
be deter mined whether the use of real stimuli indeed intr oduces more realism,
and in this w a y a more reliable pr obing of the processes inv olv ed in lightness
perception.
13 . 7 conclusions
I propose in this doctoral dissertation the use of MLDS as a r eliable tool for mea-
suring perception. MLDS is a method based in appearance judgments of clearly
visible stimulus dif ferences, and it allo ws the estimation of perceptual scales in
an ef ficient and reliable w a y , b y using the method of triads which pro vides an in-
tuitiv e and easy task for the obser v er . It a v oids the shortcomings of performance-
based (discrimination) methods that use judgments of small, near -threshold stim-
ulus dif ferences in dif ficult and unintuitiv e tasks.
In this thesis numerical simulations w ere first used to establish the accuracy
and precision of MLDS , as w ell as the ef fects of violations on its model assump-
tions. MLDS w as tested experimentally in the domain of lightness perception,
where it w as successful in estimating scales that reflected lightness constancy .
The task required only within-context comparisons, and MLDS seems superior
to asymmetric matching procedures as the latter pr o vide only indirect measure-
ments of the underlying scales. Additionally , matching data predicted b y the per -
ceptual scales follo w ed closely the data acquired in an independent asymmetric
matching procedure, further suggesting that MLDS indeed pr obed the inter nal
representation of lightness.
MLDS w as also framed in a signal detection theor y for mulation, and its perfor-
mance for deriving sensitivity w as established using simulations. MLDS w as more
ef ficient and quantitativ ely analogous than traditional perfor mance methods in
the estimation of sensitivity , but only when MLDS model assumptions w ere sat-
isfied. This equiv alency w as tested experimentally in a slant-from-texture task,
from which sensitivity has been pre viously studied in the literature. W e found

1 2 6 general discussion and outlook
v ar ying degrees of agreement that could be due to truly differ ences in the per-
ceptual representation, or alternativ ely violations of model assumptions.
The use of MLDS seems promising for the reliable measur ement of perceptual
phenomena. T ogether with the use of realistic stimuli, MLDS offers a method
to measure the perceptual dimension, and in that w a y enabling the testing of
theoretical models of perceptual infer ence.

A
A P P E N D I X
This appendix contains supplementary infor mation, figures and tables from the
studies presented in Part II and Part III.
a . 1 appendix to p art ii : using mlds to measure lightness
a . 1 . 1 Goodness of fit of differ ence scales
In general for all our obser v ers the fitted models had acceptable goodness of
fit in all viewing contexts (T able A. 1 ). In only tw o out of 50 cases the goodness
of fit w as significantly different that the expected under the model assumptions
(O 7 and O 8 for plain view context). The ‘de viance accounted for ’ (DAF) v aried
greatly betw een observ ers, in the range betw een approx. 17 % to 63 % and it
w as directly related to the lev el of noise estimated for each observ er . Obser v ers
w ere sorted accor ding to their noise lev el, from lo w er to higher (as sho wn as the
scale’s height in Figure A. 1 ), and from T able A. 1 it is e vident that the DAF on
a v erage decreases as the estimated noise of the obser v er is higher .
127

1 2 8 appendix
Obser v er plain
high
transp.
dark
lo w
transp.
dark
high
transp.
light
lo w
transp.
light
mean
O 1 63 . 0 66 . 6 59 . 6 61 . 4 62 . 0 62 . 5
O 2 57 . 7 58 . 5 55 . 3 57 . 6 52 . 2 56 . 2
O 3 53 . 6 58 . 8 49 . 5 54 . 3 50 . 9 53 . 4
O 4 44 . 6 54 . 8 51 . 5 50 . 4 52 . 8 50 . 8
O 5 48 . 9 50 . 9 45 . 3 43 . 6 46 . 8 47 . 1
O 6 49 . 2 50 . 1 45 . 3 50 . 5 43 . 9 47 . 8
O 7 45 . 8 48 . 6 44 . 1 48 . 7 45 . 3 46 . 5
O 8 38 . 8 44 . 6 42 . 7 47 . 0 51 . 7 45 . 0
O 9 40 . 6 38 . 9 32 . 6 40 . 4 30 . 7 36 . 7
O 10 17 . 6 18 . 9 14 . 4 19 . 0 13 . 8 16 . 7
T able A. 1 : ‘Deviance accounted for ’ (DAF) as a measure of goodness of fit of the dif-
ference scaling model, for each vie wing context and obser v er . High and lo w
indicate high and lo w transmittance, and dark and light indicate a lo w er and
a higher reflectance of the transparent medium.

A. 1 appendix to p art ii : using mlds to measure lightness 1 2 9
Obser v er plain
high
transp.
dark
lo w
transp.
dark
high
transp.
light
lo w
transp.
light
O 1 0 . 17 0 . 52 0 . 32 0 . 31 0 . 19
O 2 0 . 06 0 . 37 0 . 35 0 . 09 0 . 01
O 3 0 . 17 0 . 05 0 . 05 0 . 07 0 . 48
O 4 0 . 02 0 . 40 0 . 11 0 . 25 0 . 59
O 5 0 . 03 0 . 06 0 . 14 0 . 35 0 . 12
O 6 0 . 05 0 . 49 0 . 06 0 . 34 0 . 05
O 7 0 . 003 * 0 . 06 0 . 03 0 . 10 0 . 30
O 8 0 . 002 * 0 . 06 0 . 17 0 . 03 0 . 25
O 9 0 . 14 0 . 03 0 . 02 0 . 62 0 . 13
O 10 0 . 30 0 . 38 0 . 07 0 . 65 0 . 16
T able A. 2 : P-v alue from Monte-Carlo simulations as a measure of goodness of fit of the
dif ference scale model (as explained in S ection 3 . 5 ) for each viewing context
and obser v er . Asterisk indicate p<0 . 01 . High and lo w indicate high and lo w
transmittance, and dark and light indicate a lo w er and a higher reflectance of
the transparent medium.

1 3 0 appendix
Luminance [cd/m²]
Difference scale
high transparent dark
high transparent light
low transparent dark
low transparent light
plain view
Luminance [cd/m²]
Luminance [cd/m²] Luminance [cd/m²]

Figure A. 1 : Individual dif ference scales from all (n= 10 ) observ ers. Difference scales w ere deriv ed using MLDS
for each viewing condition independently (colors). Despite that the vie wing condition change
the presented luminance range, the dif ference scales co v er a similar range as in plain view .
Error bars indicate 95 % confidence interv al obtained using bootstrap techniques. Obser v ers are
sorted b y the scales’ maximum in decreasing or der . The Munsell scale for plain view is also
depicted (dashed black line).

A. 1 appendix to p art ii : using mlds to measure lightness 1 3 1
Difference scale
high transparent dark
high transparent light
low transparent dark
low transparent light
plain view
Reflectance [povray units]
Reflectance [povray units]

Figure A. 2 : Individual dif ference scales from all (n= 10 ) observ ers. Same data from Figure A. 1 but plotted
as a function of reflectance.

1 3 2 appendix
high transparent dark
high transparent light
low transparent dark
low transparent light
plain view
T arget Luminance [cd/m²]
Matched Luminance [cd/m²]
T arget Luminance [cd/m²]
0
10 0
20 0
30 0
40 0
50 0
O1 O2 O3 O4
0 10 0 20 0 30 0 40 0 50 0
0
10 0
20 0
30 0
40 0
50 0
O9
0 10 0 20 0 30 0 40 0 50 0
O8
0
10 0
20 0
30 0
40 0
50 0
O5 O6
0 10 0 20 0 30 0 40 0 50 0
O10
0 10 0 20 0 30 0 40 0 50 0
O7

Figure A. 3 : Individual matching data of all (n= 10 ) obser v ers. Matched luminance is plotted as a function
of target luminance for each vie wing condition (colors). For each viewing condition a linear
regression w as computed and the fit is sho wn as a dash line.

A. 1 appendix to p art ii : using mlds to measure lightness 1 3 3
high transparent dark
high transparent light
low transparent dark
low transparent light
plain view
T arget Luminance [cd/m²]
T arget Luminance [cd/m²]
Predicted Match Luminance [cd/m²]
O1
0
10 0
20 0
30 0
40 0
50 0
O2 O3 O4
O5
0
10 0
20 0
30 0
40 0
50 0
O6 O7
0 10 0 20 0 30 0 40 0 50 0
O8
0 10 0 20 0 30 0 40 0 50 0
O9
0 10 0 20 0 30 0 40 0 50 0
0
10 0
20 0
30 0
40 0
50 0
O10
0 10 0 20 0 30 0 40 0 50 0

Figure A. 4 : Predicted matching data from MLDS of all (n= 10 ) observ ers. Same la y out and legend as in Fig-
ure A. 3 .

1 3 4 appendix
a . 1 . 2 Simulated observer models and noise
W e w ere interested in testing ho w good MLDS could distinguish the lightness
constant obser v er model and the luminance-based obser v er . These tw o models
represent tw o extremes of beha vior for lightness judgments. W e ran multiple
simulations of the tw o obser v er models, and analyzed ho w often they could
be told apart with increasing le v els of noise. W e aimed to establish an upper
limit of noise at which the tw o models can reliably be separated using MLDS .
W e found that this limit of noise w as considerably higher than the v alues found
experimentally .
The simulated sensory functions w ere linear functions Ψ ( x ) = ax + b . For the
lightness constant obser v er , there w ere fiv e dif ferent sensor y functions, one for
each transparency condition, which w ere the inv erse of the atmospheric transfer
functions (A TFs, Figure 5 . 1 C). Thus, the functions ‘undo’ the process of image
for mation b y mapping the compressed range of luminance to the full range be-
tw een 0 and 1 (Figure 6 . 1 B top panels) For the luminance-based observ er , a single
sensory function w as set which linearly mapped the full range of luminance v al-
ues ( x ) to a range betw een 0 and 1 (Figure 6 . 1 B bottom panels). The response to
each of these functions w as corrupted b y Gaussian noise, with a fixed v ariance
( σ 2 ).
The simulation proceeded using the same pr ocedure as in the actual psy-
chophysical experiment. The obser v er models responded to the method of tri-
ads, b y dra wing their response to each stimulus v alue in the triad. T riads w ere
constructed as non-o v erlapping inter v als using the ten reflectance v alues used
in the experiment. In total n = 120 unique triads w ere simulated per block, and
n b = 10 blocks w ere simulated. The generated data w as fed into the MLDS anal-
ysis procedure, and scales and their v ariability w ere estimated. Each simulation
w as repeated N= 100 times per obser v er model, and the noise v alue σ w as v aried
systematically in the range betw een 0 . 01 and 1 . 2 .
W e defined a measure of constancy in order to quality the type of output that
simulations produced. For a giv en simulation w e compared the scale’s maximum
deriv ed for each transparency condition with the scale’s maximum deriv ed for
plain view . When the 95 % confidence inter v als of these tw o scale’s maxima

A. 1 appendix to p art ii : using mlds to measure lightness 1 3 5
o v erlapped, the simulation w as qualified as with a ‘lightness constant’ result.
This means that a lightness constant result is expected when the maxima of the
scales coincide on their range in the y-axis. As the scales resulting from MLDS
are - b y design - anchored at zero on its minimum, only a comparison of the
maximum is needed. In Figure 6 . 1 B the o v erlap of scale’s maxima is evident for
the lightness constant model for all transparency conditions, and for none in the
luminance-based obser v er .
The proportion of simulations that r esulted in a constancy result (y-axis) is
sho wn for v ar ying lev els of noise (x-axis) in Figure A. 5 . W e found that as the
noise lev el increases, the luminance-based model more frequently cannot be told
apart from the lightness constant model. A complete separation betw een the
tw o occurs when the estimated noise lev el is below ˆ σ = 0 . 4 . In this w a y , w e
established that when the scale’s noise estimated with MLDS falls belo w this
v alue, w e can successfully distinguish betw een these tw o obser v er models.
a . 1 . 3 Normalized Contrast model
In this w ork w e applied the nor malized contrast model introduced b y Zeiner and
Maertens ( 2014 ) and W iebel et al. ( 2016 ). In brief, in a first step the Michelson
contrast is computed betw een a target and its eight surr ound checks ( x ). That is,
the target intensity X is normalized relativ e to its a v erage local surround S 1 ,..., 8
b y
x = X − mean ( S 1 ,..., 8 )
X + mean ( S 1 ,..., 8 ) (A. 1 )
In a second step, the target Michelson contrast ( x ) is normalized relativ e to
the contrast range in the region of transparency ( t max − t min ), and this range is
subsequently mapped to the contrast range in plain view ( p max - p min ). The so
nor malized Michelson contrast ( NMC ) relates to the target contrast x accor ding
to the follo wing equation:
NMC = x − t min
t max − t min ∗ ( p max − p min ) + p min (A. 2 )

1 3 6 appendix
Estimated Noise Estimated Noise
% simulations with a
lightness constant result
observer
model
luminance-based
lightness constant
high transparent dark high transparent light
low transparent dark low transparent light

Figure A. 5 : Ability of distinguish betw een the tw o obser v er models. Percentage of sim-
ulations that produced a ‘lightness-constant’ output as a function of the esti-
mated noise b y MLDS and for both observ er models.

A. 1 appendix to p art ii : using mlds to measure lightness 1 3 7
V iewing context Slope
empirical predicted t ( 9 ) p
high transparent dark 2 . 4 ± 0 . 7 3 . 1 ± 0 . 7 - 2 . 71 0 . 02
high transparent light 2 . 4 ± 0 . 8 2 . 2 ± 0 . 6 0 . 94 0 . 34
lo w transparent dark 4 . 4 ± 1 . 5 4 . 5 ± 0 . 1 - 0 . 16 0 . 88
lo w transparent light 4 . 6 ± 1 . 8 3 . 7 ± 1 . 4 1 . 42 0 . 19
V iewing context Intercept
empirical predicted t ( 9 ) p
high transparent dark − 25 . 1 ± 15 . 6 − 46 . 4 ± 17 . 6 3 . 08 0 . 01
high transparent light − 146 . 6 ± 75 . 7 − 142 . 1 ± 44 . 9 - 0 . 17 0 . 87
lo w transparent dark − 64 . 9 ± 23 . 8 − 75 . 0 ± 27 . 2 0 . 90 0 . 39
lo w transparent light − 385 . 0 ± 191 . 5 − 316 . 4 ± 128 . 1 - 1 . 04 0 . 33
T able A. 3 : Linear regression results for data obtained in the matching experiment (em-
pirical) in comparison to matching predictions deriv ed using the estimated
scales from the MLDS experiment (predicted). Mean ± S.D., p : p-v alue of
paired t-tests betw een the empirical and predicted parameters.

1 3 8 appendix
Reflectance Luminance [ cd/m 2 ]
r plain
high
transp.
dark
high
transp.
light
lo w
transp.
dark
lo w
transp.
light
0 . 06 15 18 69 21 88
0 . 11 25 22 73 23 90
0 . 19 40 28 79 25 94
0 . 31 60 36 87 29 98
0 . 46 89 48 99 35 104
0 . 63 120 60 111 41 110
0 . 82 155 74 125 49 116
1 . 05 199 92 144 57 126
1 . 29 242 108 159 65 134
1 . 50 281 125 176 73 142
1 . 67 312 137 188 79 148
1 . 95 365 157 209 90 160
2 . 22 415 177 229 100 169
T able A. 4 : T arget luminance v alues in differ ent viewing conditions. The luminance v al-
ues (in cd/m 2 ) corresponding to the 13 reflectance v alues (in povray units, first
column) are sho wn for each viewing condition. Reflectance v alues from 0 . 11
to 1 . 67 w ere used as targets in the both experiments ( MLDS and matching).

A. 2 appendix to p art iii : using mlds to measure sensitivity 1 3 9
a . 2 a p p e n d i x t o p a r t i i i : using mlds to measure sensitivity
Di ff e re n ce s cal e [d' un its]
Sla nt [de g]
− 2
0
2
4
6
8
10
12
14 O 1
− 1
0
1
2
3
4
5
6
7
8 O2
− 5
0
5
10
15
20 O 3
0
2
4
6
8
10
12
14 O 4
0 10 20 30 40 50 60 70
− 5
0
5
10
15
20
25 O 5
0 10 20 30 40 50 60 70
− 5
0
5
10
15
20
25 O 6
0 10 20 30 40 50 60 70
− 5
0
5
10
15
20
25
30 O 7
0 10 20 30 40 50 60 70
− 5
0
5
10
15
20
25
30 O 8

Figure A. 6 : Dif ference scales estimated b y MLDS for all obser v ers. Markers indicate discrete scale v alues
obtained from MLDS , err or bars indicate confidence inter v als of scale estimates obtained b y
bootstrap. The continuous line indicate the cubic spline fitted to the discrete scale v alues.
a . 2 . 1 Goodness of fit of differ ence scales
The goodness of fit w as acceptable only for three obser v ers (O 2 , O 3 , O 4 ) when
the data w as fitted with the MLDS default parameters (probit link function with
zero asymptotes). Consequently , w e applied a w orkflo w consisting on a series of
refitting attempts until a satisfactory goodness of fit w as achiev ed (as suggested
in Knoblauch & Maloney, 2012 , pp. 219 - 222 ). The goodness of fit statistics at

1 4 0 appendix
0.0 0 .14 0.29 0.4 3 0 .57 0. 71 0.8 6 1 .0
stim u lus va lue
0
20
40
60
80
10 0
%
= 0. 03 5
= 0. 07
= 0. 14

Figure A. 7 : Co v erage analysis: % of simulations on which the ‘true’ v alue w as included
in the confidence inter v al of the estimated scale. At 95 % confidence, 95 %
of the simulations (black dashed line) should include the true v alue. T ested
at three dif ferent noise lev els σ .
each step of the w orkflo w for each obser v er is sho wn in T able A. 5 . First, w e
estimated the error rates fr om the data (analogous to guess and lapse rates in
psy chometric functions) and refitted the model with those err or rates as non-
zero asymptotes of the link function. W e obtained non-significant p-v alues for
the models of obser v ers O 5 and O 7 , and these scales w ere kept. For obser v ers
O 1 , O 6 and O 8 , ho w ev er , p-v alues w ere still significant. For these remaining
obser v ers, w e then proceeded to manually divide the data into tw o halv es, and
w e refitted the model for each half independently ( 420 trials each). When the
goodness of fit of a scale from any of the tw o halv es of data w as found to be
appropriate, it w as kept for further analysis; this w as the case for observ er O 1
and O 6 . For obser v er O 8 , both halv es of the data w ere non-significant; in this
case w e choose the one that had a higher ‘deviance accounted for ’.
The data and goodness of fit statistics that w ere finally considered for the
estimation of near -threshold perfor mance are marked with a gra y background
in T able A. 5 , and the corresponding scales are sho wn in Figure A. 6 . Figure A. 8
sho ws the goodness of fit plots as in Figure 3 . 3 .

A. 2 appendix to p art iii : using mlds to measure sensitivity 1 4 1
− 4 − 3 − 2 − 1 0 1 2 3 4
Dev ianc e res i du al
0.0
0.5
1.0
Cum ulat ive de n sity func tion
50 60 70 80 90 10 0 110 12 0 1 30
Num ber of R uns
0.00
0.05
0.10
Fr eque n cy
(a) O1
− 4 − 3 − 2 − 1 0 1 2
Dev ianc e res i du al
0.0
0.5
1.0
Cum ulat ive de n sity func tion
180 200 220 2 40 260 2 80 30 0
Num ber of R uns
0.00
0.01
0.02
0.03
Fr eque n cy
(b) O2
− 4 − 3 − 2 − 1 0 1 2 3 4
Dev i an ce r esi dua l
0.0
0.5
1.0
Cum ulat ive den s ity f un c tion
80 90 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0
Num be r o f R un s
0.00
0.02
0.04
0.06
Fr equen cy
(c) O3
− 4 − 3 − 2 − 1 0 1 2 3 4
Dev i an ce r esi dua l
0.0
0.5
1.0
Cum ulat ive den s ity f un c tion
10 0 12 0 14 0 16 0 18 0 20 0 22 0 24 0
Num be r o f R un s
0.00
0.02
0.04
Fr equen cy
(d) O4
− 4 − 3 − 2 − 1 0 1 2
Dev ianc e res i du al
0.0
0.5
1.0
Cum ulat ive de n sity func tion
100 12 0 140 1 60 180 200
Num ber of R uns
0.00
0.02
0.04
0.06
Fr eque n cy
(e) O5
− 4 − 3 − 2 − 1 0 1 2 3
Dev ianc e res i du al
0.0
0.5
1.0
Cum ulat ive de n sity func tion
30 40 50 60 70 8 0 90 1 00
Num ber of R uns
0.00
0.02
0.04
0.06
Fr eque n cy
(f ) O6
− 3 − 2 − 1 0 1 2 3
Dev ianc e res i du al
0.0
0.5
1.0
Cum ulat ive de n sity func tion
60 8 0 100 12 0 140 1 60
Num ber of R uns
0.00
0.02
0.04
0.06
Fr eque n cy
(g) O7
− 4 − 3 − 2 − 1 0 1 2 3
Dev ianc e res i du al
0.0
0.5
1.0
Cum ulat ive de n sity func tion
30 40 50 60 70 80
Num ber of R uns
0.00
0.02
0.04
0.06
Fr eque n cy
(h) O8

Figure A. 8 : Goodness of fit of dif ference scales obtained b y Monte-Carlo simulation, as described in S ec-
tion 3 . 5 and in Knoblauch and Maloney ( 2012 ). Each panel sho ws the tw o diagnostic graphs
produced b y the MLDS package for each obser v er .

1 4 2 appendix
Asymptotes GoF measure
lo w er upper AIC DAF p-v alue
O 1
all - - 687 42 < 10 − 3
all 0 . 01 0 . 03 632 47 < 10 − 2
1 / 2 - - 271 56 0 . 01
2 / 2 - - 299 50 0 . 28
O 2 all - - 714 40 0 . 08
O 3 all - - 383 68 0 . 23
O 4 all - - 560 53 0 . 80
O 5 all - - 520 57 0 . 01
all 0 . 33 0 . 01 469 47 0 . 18
O 6
all - - 508 58 < 10 − 3
all 0 . 06 0 . 01 492 57 < 10 − 2
1 / 2 - - 201 68 0 . 10
2 / 2 - - 261 58 < 10 − 3
O 7 all - - 435 64 < 10 − 3
all 0 . 03 0 . 02 400 66 0 . 43
O 8
all - - 390 68 < 10 − 3
all < 10 − 3 0 . 01 389 68 0 . 02
1 / 2 - - 222 64 0 . 05
2 / 2 - - 171 73 0 . 05
T able A. 5 : Goodness of fit measures and v alues of link function asymptotes used for the
fitting of the dif ference scale model, for each obser v er and part of data con-
sidered. S ee text for a description of the w orkflo w emplo y ed to achiev e ap-
propriate goodness of fit. AIC: Akaike information criterion; DAF: ‘deviance
accounted for ’; p-v alue: p-v alue statistic based on Monte-Carlo simulation
that accessed goodness of fit (see S ection 3 . 5 ). Gra y background indicate data
that w ere considered for the analysis and the estimation of thresholds.

A. 2 appendix to p art iii : using mlds to measure sensitivity 1 4 3
-25 0 25 -25 0 25 - 25 0 25 -25 0 25
0.5
1.0
0.5
1.0
0.5
1.0
0.5
1.0
O2
O3
O4
O1
Fr ac ti on c o rr e ct
Sl an t [ deg ]
26 deg 37 deg 53 deg 66 deg

Figure A. 9 : Psy chometric functions obtained in Experiment 2 for all ev aluated standards v alues (columns)
and obser v ers (ro ws). Blue (red) lines indicate functions for comparisons below (abo v e) the
standard. Straight horizontal lines indicate 95 % confidence interv als for the thresholds calcu-
lated at perfor mance of d ′ 0 . 5 , 1 and 2 .

1 4 4 appendix
Fr ac ti on c o rr e ct
Sl an t [ deg ]
26 deg 37 deg 53 deg 66 deg
0.5
1.0
-25 0 25 -25 0 25 - 25 0 25 -25 0 25
0.5
1.0
0.5
1.0
0.5
1.0
O5
O6
O7
O8

A. 2 appendix to p art iii : using mlds to measure sensitivity 1 4 5
C .I. wi dth o f t h res ho l d f ro m 2-IF C [ deg ]
C .I. wi dth o f t h res ho l d f ro m ML DS [ deg ]
0
5
10
15
O1 O 2 O3
0
5
10
15
O4 O 5
0 5 10 15 20
O6
0 5 10 15 20
0
5
10
15
O7
0 5 10 15 20
O8
20
20
20

Figure A. 10 : Comparison of the v ariability in the threshold estimation. The width of the confidence in-
ter v als from Figure 11 . 2 and Figure 11 . 3 are plotted against each other for each observ er
individually . All standard v alues are plotted together .

1 4 6 appendix
− 0.1 0
− 0.0 5
0.0 0
0.0 5
0.1 0
0 100 0 200 0 300 0
nu m be r o f t ria ls
− 0.1 0
− 0.0 5
0.0 0
0.0 5
0.1 0
0
20
40
60
80
10 0
0 100 0 200 0 300 0
0
20
40
60
80
10 0
B A
D
C
= 0 . 03 5
= 0 .07
= 0 .14
E F
0.0 0
0.0 5
0.1 0
0.1 5
0.2 0
0 100 0 200 0 300 0
0.0 0
0.0 1
0.0 2
0.0 3
0.0 4
0.0 5
0.0 6
0.0 7
Bia s (es tim at ed - t r ue )
Percentage
Width CI
Bia s (es tim at ed - t r ue )
Percentage
Width CI

Figure A. 11 : Simulation results for threshold estimation using MLDS with v ariable num-
ber of trials. Bias (A, D), confidence inter v als’ cov erage (B, E) and width (C,
F) for the estimation of thresholds using MLDS as a function of the number
of simulated trials, for three dif ferent noise v alues ( σ ). Panels A-C corre-
spond to results for a standar d stimulus v alue st = 0 . 2 ; panels D-F for
st = 0 . 8 . The v ertical dashed line indicate the number of trials used in the
actual experiments. The horizontal dashed lines in panel B and E indicate
the expected co v erage percentage. Errorbars indicate mean ± S.D. acr oss
n= 100 simulations.

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Why organizations use Identific for document trust, entry 46

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