Maximum likelihood difference scales represent perceptual
magnitudes and predict appearance matches
Christiane B. Wiebel*
Modeling of Cognitive Processes,
Department of Software Engineering and Theoretical
Computer Science, Technische Universit¨
at Berlin,
Berlin, Germany
Guillermo Aguilar*
Modeling of Cognitive Processes,
Department of Software Engineering and Theoretical
Computer Science, Technische Universit¨
at Berlin and
Bernstein Center for Computational Neuroscience,
Berlin, Germany
Marianne Maertens
Modeling of Cognitive Processes,
Department of Software Engineering and Theoretical
Computer Science, Technische Universit¨
at Berlin,
Berlin, Germany
One central problem in perception research is to
understand how internal experiences are linked to
physical variables. Most commonly, this relationship is
measured using the method of adjustment, but this has
two shortcomings: The perceptual scales that relate
physical and perceptual variables are not measured
directly, and the method often requires perceptual
comparisons between viewing conditions. To overcome
these problems, we measured perceptual scales of
surface lightness using maximum likelihood difference
scaling, asking observers only to compare the lightness
of surfaces presented in the same context. Observers
were lightness constant, and the perceptual scales
qualitatively and quantitatively predicted perceptual
matches obtained in a conventional adjustment
experiment. Additionally, we show that a contrast-based
model of lightness perception predicted 98% of the
variance in the scaling and 88% in the matching data. We
suggest that the predictive power was higher for scales
because they are closer to the true variables of interest.
Introduction
One major objective in the scientific study of
perception is to understand how psychological experi-
ences are linked to physical variables in the world
(Fechner, 1860). Devising proper methods to quantify
this relationship has turned out to be challenging
because psychological variables, contrary to physical
ones, cannot be observed directly and must be inferred
from observers’ responses to properly chosen stimuli
(e.g., Gescheider, 1988). In the absence of a well-
established measurement theory (Krantz, Luce, Suppes,
& Tversky, 1971), Fechner’s (1860) simple method of
adjustment (matching) is hard to beat and remains
widely used (Koenderink, 2013).
To illustrate the problem, let’s say we are interested
in the perceived lightness of the target check (Figure
1A, red outline) presented behind a transparent
medium. Introducing a transparent medium between a
surface and the observer (Figure 1B) changes the
mapping between surface reflectance and retinal
luminance in a characteristic way (Figure 1C). The
luminance range of surfaces seen through a transparent
medium is substantially reduced and potentially shifted
relative to the luminance range for surfaces seen in
plain view. To be invariant against such changes, the
visual system has to ‘‘undo’’ these changes by
appropriate computations (e.g., Singh, 2004; Singh &
Anderson, 2002; Wiebel, Singh, & Maertens, 2016).
This approximate invariance of perceived lightness
across varying luminance is known as lightness
constancy. We know from experience and from
Citation: Wiebel, C. B., Aguilar, G., & Maertens, M. (2017). Maximum likelihood difference scales represent perceptual
magnitudes and predict appearance matches. Journal of Vision,17(4):1, 1–14, doi:10.1167/17.4.1.
Journal of Vision (2017) 17(4):1, 1–14 1
doi: 10.1167/17.4.1 ISSN 1534-7362 Copyright 2017 The AuthorsReceived November 21, 2016; published April 3, 2017
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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empirical studies that human observers are indeed
largely invariant against such fluctuations in retinal
luminance. However, we still lack a theoretical model
of how the visual system accomplishes lightness
constancy. To develop such a model, we must be able
to measure the relationship between retinal luminance
and perceived lightness in a reliable and comprehensive
way. To that end, we ideally want to estimate the
functions describing this relationship, which are known
as transducer functions or perceptual scales (e.g.,
Kingdom & Prins, 2010).
The most commonly used method for measuring this
relationship is the method of adjustment even though it
does not provide a direct estimate of the transducer
functions and it presumes a number of operations on
the part of the observer. Figure 2 depicts the processes
involved in adjustment or matching procedures for
perceived lightness. An observer adjusts the intensity of
a test stimulus so that it looks identical to a given
standard. It is assumed that the observer internally
compares magnitudes of perceived lightness for the
target (W[x
T
]) and the match (W[x
M
]). What is being
Figure 1. Experimental stimuli. (A) The basic stimulus is a 10 310 checkerboard composed of checks with 13 possible reflectance
values. In an asymmetric matching task, observers adjust the luminance of an external test field so that it matches the perceived
lightness of a specified target check (here I2). Observers are said to be lightness constant when their matches indicate the inversion of
the various reflectance-to-luminance mappings that are introduced by different transparent media (see panels B and C). (B)
Checkerboards were also presented behind different transparent media that varied in reflectance (dark and light) and in
transmittance (high and low). (C) ATFs relate target reflectance (x-axis) to target luminance (y-axis) (Adelson, 2000). The color scheme
corresponds to the images in panel B. In the transparency conditions, the luminance range is compressed and/or shifted with respect
to plain view. This is reflected in corresponding slope and intercept changes of the respective ATFs.
Figure 2. Perceptual processes underlying matching procedures. (A) At each position, match and target, there is a transducer function
that relates retinal luminance (x
M
,x
T
) to perceived lightness [W(x
M
), W(x
T
), insets on the stimulus]. (B) What is measured in a
matching procedure are the luminances x
M
and x
T
that correspond to equal perceived lightness at both positions [W(x
M
)¼¼ W(x
T
)].
After Maertens and Shapley (2013).
Journal of Vision (2017) 17(4):1, 1–14 Wiebel, Aguilar, & Maertens 2
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measured, however, are not the transducer functions
relating the two but the corresponding luminances of
the target and the match (x
T
and x
M
, Figure 2B).
Another problem with the method arises when, as in
the above case, test and match are presented in
different contexts (asymmetric matching). In most
cases, researchers are interested in such asymmetric
comparisons because they allow one to quantify the
degree of perceptual constancy. Such asymmetric
comparisons become problematic, however, when the
difference in context causes appearance differences that
cannot be compensated along the dimension of the
adjustment (Brainard, Brunt, & Speigle, 1997; Ekroll &
Faul, 2013; Foster, 2003; Logvinenko & Maloney,
2006; Logvinenko, Petrini, & Maloney, 2008). The
consequence would be an inaccurate or even invalid
measurement that does not capture the perceptual
representation of the stimulus.
Recently, there have been attempts to tackle the
problems associated with matching (Logvinenko &
Maloney, 2006; Logvinenko et al., 2008; Radonji´
c&
Brainard, 2016; Radonji´
c, Cottaris, & Brainard, 2015b;
Umbach, 2013). Although it is widely accepted that
observers are relatively lightness constant under natural
viewing conditions, many experiments still find varying
amounts of constancy for different viewing conditions,
stimuli, task types, or even instructions (Foster, 2011;
Gilchrist et al., 1999). Such deviations might either be a
consequence of methodological problems such as the
ones just outlined or a meaningful deviation from
constancy, which then would need to be explained by
any successful lightness model. Progress in revealing
the underlying mechanisms for lightness perception is
therefore tightly coupled with choosing appropriate
and robust experimental methods that allow the
comprehensive testing of theoretical models.
As an effort in this direction, we address the
limitations of matching procedures by adopting the
following approach. We measure the transducer
functions directly using maximum likelihood difference
scaling (MLDS, Maloney & Yang, 2003). MLDS is a
scaling method that allows the efficient estimation of
perceptual scales, i.e., the transducer functions relating
retinal luminance and perceived lightness (Figure 1). It
has been used to study various perceptual dimensions
(e.g., Fleming, J¨
akel, & Maloney, 2011; Obein,
Knoblauch, & Vi´
enot, 2004). Furthermore, it is based
on a signal detection model, which potentially allows
one to relate measurements of appearance with
measurements of discriminability (Aguilar, Wichmann,
& Maertens, 2017; Devinck & Knoblauch, 2012). Here
we used MLDS to measure perceptual scales in
different contexts using only within-context compari-
sons in order to avoid the procedural problems of
asymmetric matching. The estimated scales are con-
structed from the judgment of perceived stimulus
differences and not from the adjustment of a reference
as in other scaling methods, such as magnitude
estimation (Gescheider, 1988) or partition scaling
(Whittle, 1994). MLDS requires a straightforward
perceptual judgment and is thus less susceptible to
strategic influences.
To scrutinize whether MLDS provides reliable
perceptual scales of lightness, we validate the scales
empirically and theoretically. First, we use the esti-
mated scales to predict perceptual matches and
compare them to matches gathered in an independent
asymmetric matching experiment. Second, we compare
the predictive power of a contrast-based lightness
model (Wiebel et al., 2016; Zeiner & Maertens, 2014)
for scaling and matching data. To anticipate, we found
that (a) the empirical perceptual scales for different
contexts were consistent with lightness constancy, (b)
matching data were well predicted by the perceptual
scales, and (c) human lightness perception followed a
difference scale that corresponds to a normalized
contrast metric. The predictive power of the contrast-
based lightness model was higher for the scaling than
for the matching data, suggesting that estimating
perceptual scales has the advantage of probing more
directly the internal dimension under study.
Methods
Observers
Ten naive observers participated in the study; five of
them were female. Observers’ ages ranged from 19 to 32
years. All observers had normal or corrected-to-normal
visual ability and were reimbursed for participation.
Informed written consent was given by all observers
prior to the experiment.
Stimuli and apparatus
Stimuli were presented on a linearized 21-in. Siemens
SMM2106LS monitor (400 3300 mm, 1024 3768 px,
130 Hz). Presentation was controlled by a DataPixx
toolbox (Vpixx Technologies, Inc., Saint-Bruno, QC,
Canada) and custom presentation software (http://
github.com/TUBvision/hrl). Observers were seated 110
cm away from the screen in a dark experimental cabin.
Observers’ responses were registered with a Response-
Pixx button-box (VPixxTechnologies).
The stimuli were images of customized checker-
boards composed of 10 310 checks (Figure 1). The
images were rendered using Povray (Persistence of
Vision Raytracer Pty. Ltd., Williamstown, Victoria,
Australia, 2004). The position of the checkerboard, the
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light source, and the camera were kept constant across
all images. Checks were assigned one out of 13 surface
reflectance values according to the experimental design
(see below). In the transparency conditions, a trans-
parent layer was placed between the checkerboard and
the camera (Figure 1B). It was positioned so as to cover
all targets and their surrounding checks in both the
MLDS and the matching experiment. The transparency
was created using alpha blending (Metelli’s episcotister
model). The image luminances of the background B
and the foreground Fare combined according to some
weighting factor aso as to result in a new image
luminance at the position of transparency T¼a3Bþ
(1 a)3F.Anavalue of 0 corresponds to an opaque
foreground T¼F;aof 1 corresponds to a fully
transparent foreground T¼B. The transparent layer
varied in transmittance and reflectance. The dark
transparency had a value of 0.35 in povray reflectance
units (19 cd/m
2
) and the light transparency of 2 (110 cd/
m
2
). Values of a¼0.4 and 0.2 were used in the high and
low transmittance conditions, respectively. The ren-
dered images were converted to grayscale images. The
background luminance was 141 cd/m
2
. Detailed values
of luminance for each transparent medium can be
found in Supplementary Table S3).
In the matching experiment, an adjustable test field
was presented above the checkerboard to assess
observers’ lightness matches (Figure 1A). The test field
was embedded in a coplanar surround checkerboard
that was composed of 5 35 checks. The size of the test
field was 1.2831.28visual angle and that of the
surround checkerboard was 38338. The luminances of
the checks in the surround checkerboard were fixed
throughout the experiment, and the luminances were
chosen so that two adjacent checks did not have the
same luminance. The mean luminance of the surround
checks was 178 cd/m
2
, which is identical to the mean
luminance of the 13 checks in the main checkerboard in
plain view. The surround checkerboard was presented
in four different spatial arrangements, resulting from
clockwise rotation of the original in steps of 908.A
configuration was assigned randomly to each trial.
Design and procedure
Perceptual scales and asymmetric matching func-
tions were measured for five different viewing condi-
tions, a plain view condition, and four transparency
conditions (Figure 1).
MLDS experiment
We used MLDS with the methods of triads (Figure
4A; Knoblauch & Maloney, 2008, 2012). We used 10
out of the 13 reflectance values to construct the triads.
The lowest and the two highest reflectance values were
omitted to achieve a feasible number of trials. With p¼
10 reflectance values, the total number of unique triads
was n¼p!/((p3)! 33!)¼10!/(7! 33!)¼120. Each triad
contained three values that were selected so as to
enclose nonoverlapping intervals. They were presented
in ascending (x
1
,x
2
,x
3
) or descending (x
1
.x
2
.
x
3
) order (Knoblauch & Maloney, 2008). The reference,
x
2
(check I2 in Figure 4A), was located between the two
comparisons, x
1
and x
3
(checks B2 and I9 in Figure
4A). In each trial, observers judged which comparison
check, x
1
or x
3
, was more different in lightness from the
reference. Observers used a left or right response button
to indicate their choice. No time limit was imposed.
Figure 3. Estimation of scales using MLDS with triads. (A) Hypothetical scale relating perceptual experiences W(x
i
) to stimulus values
x
i
. For an example stimulus triad (x
2
,x
5
,x
8
), observers are asked to compare which pair is more different, (x
2
,x
5
)or(x
5
,x
8
). The
decision model for this example triad is D¼[W(x
8
)W(x
5
)] [W(x
5
)W(x
2
)] or D¼W(x8) 2W(x
5
)þW(x
2
). (B) The weights for
each term in the decision model are the covariates in a design matrix, X, in a binomial GLM. Each row in the design matrix Xindicates
the elements of a triad in one trial. The shaded rows are two repetitions of the same triad, which is the example triad in panel A. The
design matrix contains all triads shown in one experiment. (C) Empirical scale that results from solving the GLM and obtaining the
coefficients b. The coefficients correspond to the scale values at different levels of the physical variable. Error bars are estimates for
the error associated with each coefficient and were obtained using bootstrap.
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To keep the local context comparable for the
elements of a triad, we controlled the luminances of the
eight checks surrounding each triad element. The same
eight luminance values were used for each triad
element, but they differed in spatial arrangement. Their
mean luminance was 178 cd/m
2
, which was identical to
the mean luminance of all checks seen in plain view.
The remaining 73 checks were drawn randomly without
replacement from a set consisting of six repeats of the
13 different reflectance values. This resulted in a slight
variation of the mean luminance of those checks
between trials (up to 6 cd/m
2
). The checks were
positioned so that two neighboring checks did not have
the same reflectance.
Each triad was repeated 10 times, resulting in 1,200
trials per viewing condition and 6,000 trials in total.
Trials were randomized across viewing condition, triad,
and target reflectance. The experiment was divided into
several sessions. A new image was created for each trial.
Matching experiment
Target reflectances and viewing conditions were
identical to those in the MLDS experiment. The target
check was presented at the position of the reference
(check I2 in Figure 1) in the MLDS experiment.
Observers adjusted the luminance of the external test
field to match the perceived lightness of the target
check. The luminance was adjusted by pressing one of
four buttons, two of them for coarse adjustments (610
cd/m
2
) and the other two for fine adjustments (61 cd/
m
2
). The maximum luminance of the monitor was 550
cd/m
2
. Satisfactory matches were confirmed with a fifth
button that initiated the next trial. No time limit was
imposed on the adjustment procedure.
The eight checks surrounding the target were
assigned in the same way as in the MLDS experiment.
The remaining 91 check reflectances were drawn
randomly without replacement from a set consisting of
eight repeats of all 13 reflectance values. Thus, the
mean luminance across trials was comparable to that in
the MLDS experiment. Again, neighboring checks had
to have different reflectances.
Each combination of target reflectance and viewing
condition was repeated 10 times, resulting in a total of
500 trials. A new image was created for each trial, and
trials were randomized across experimental conditions.
Figure 4. Method of triad procedure and observer models. (A) In the triad comparison, observers compared the lightness of three
specified checks (B2,I2, and I9, marked with a red outline). The upper panel shows a triad comparison in plain view, the lower panel a
comparison behind one of the transparent media. (B) Simulation for a lightness-constant (upper panels) and luminance-based
observers (lower panels). For the lightness-constant observer, the perceptual scales (upper left panel) correspond to an inverse
mapping of the ATFs (Figure 1C). For the luminance-based observer (lower left panel) the luminance-to-lightness mappings in
different contexts coincide on a single function. We generated data for each of the models in simulations, and the estimated
perceptual scales are shown on the right panels. See text for details.
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MLDS analysis
In the following, we explain how we used the MLDS
routines (Knoblauch & Maloney, 2008) in Rto analyze
the data. The left panel in Figure 3 depicts a
hypothetical perceptual scale that relates psychological
experience, W(x), to a physical variable, x. The central
panel illustrates how the decision model translates into
the statistical model that is used to estimate scale
parameters, and the right panel depicts the estimated
scale values. Observers perform the triad judgments for
different levels (x
i
) of the physical variable (e.g., x
2
,x
5
,
x
8
in Figure 3). They judge whether the difference D¼
(W[x
8
]W[x
5
]) (W[x
5
]W[x
2
]) is smaller or larger
than zero. The decision model for all possible triads is
summarized in the design matrix X, which contains
separate columns for each x-value (Figure 3B). Each
row of the design matrix contains the weights for the
decision model of that respective triad. The coefficients
(b) are estimated in a (binomial) generalized linear
model (GLM) to account for the observed responses
(Y) using maximum likelihood, and they represent the
scale values for all levels of the physical variable. The
linear predictors X*bare related to the observed
responses by using a link function g(), which maps the
range of the linear predictors to a range of the response
probabilities E[Y]. The decision model in MLDS is
stochastic, and it assumes a single Gaussian-distributed
noise source ethat corrupts the decision variable. By
default, the GLM estimation assumes a variance of the
noise source of one ðr2¼r2
D¼1Þ, and as a conse-
quence, the amount of noise estimated by the model is
inversely related to the scale’s maximum, with a higher
maximum when the estimated noise is low (so-called
‘‘unconstrained’’ scales in Knoblauch & Maloney,
2008). However, alternative parameterizations are also
possible. Within the framework of the GLM, the
scaling can be controlled fairly simply by prescaling the
design matrix. For example, dividing the weights in the
matrix Xin Figure 3 by two—giving 0.5, 1, and 0.5—
yields a scale for which r2
D¼4 that corresponds to
r2
b¼1 for each lightness level. This would parameterize
the scale in terms of d0(as shown in more detail in
Aguilar et al., 2017; Devinck & Knoblauch, 2012).
Simulation of observer models
We used an ideal observer analysis to test whether
MLDS could distinguish between different generative
models. In particular, we tested a lightness constant
against a luminance-based observer, two extremes of
behavioral judgments. The model comparison is done
as follows. We define internal scales for each of the two
models (Figure 4B). For a luminance-based observer,
the luminance-to-lightness mappings in different con-
texts coincide on a single function and differ only in the
range of luminance values (Figure 4B, lower left panel).
Formally, the sensory representation function was
defined as
WlumðxÞ¼axþb;
where xis luminance, and a,bare linear coefficients
calculated to map the range of luminance in plain view
[L
min
,L
max
] to the range [0, 1].
For a lightness-constant observer, the mapping
functions in different contexts should ‘‘undo’’ the
transformations of image formation in which equal
surface reflectances are mapped onto different lumi-
nance ranges (Figure 1C). Thus, we model this observer
by using internal mapping functions that are the inverse
functions of the atmospheric transfer functions (ATFs)
shown in Figure 1C. Formally, the sensory represen-
tation function was defined as
WlightðxÞ¼aixþbii21:::5;
where xis luminance, and a
i
,b
i
are linear coefficients
calculated to map the range of luminance for each
viewing condition to the range [0, 1] (for simplicity, we
used linear functions, but power functions could be
used as well and would not change our ideal observer
results).
Each of the two observer 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
variable was calculated as
D¼½Wðx3ÞWðx2Þ ½Wðx2ÞWðx1Þ þ
ð1Þ
with ;N(0, r
2
), and W
*
is either W
lum
or W
light
.
Simulated responses were generated choosing the triad
(x
2
,x
3
) when D.0 and (x
1
,x
2
) otherwise. Finally, the
simulated data were subjected to the MLDS analysis to
obtain the coefficients bthat constitute the scale values.
Figure 4B shows the model perceptual scales (left) and
the estimated scales (right), and it is evident that for the
chosen noise level (r¼0.15) the method recovers the
underlying scale.
We repeated the ideal observer analysis for a range
of different noise levels (r, minimum ¼0.01 and
maximum ¼1.2, see Supplementary Material). The two
observer models were distinguishable for a broad range
of noise levels up to approximately 0.4. This upper-
bound value was much higher than the noise levels that
have been observed in previous experiments (Devinck
& Knoblauch, 2012; Knoblauch & Maloney, 2008). We
therefore concluded that MLDS could be used to derive
meaningful scales because they would allow us to
distinguish between these two different observer
models.
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Results
Figure 5 shows the perceptual scales measured in
different viewing conditions aggregated across all
observers. The scales are interval scales with the
minimum anchored at zero and the maximum being
inversely proportional to the estimated noise (in MLDS
terminology referred to as ‘‘unconstrained scales’’;
Knoblauch & Maloney, 2012).
The empirical scales are consistent with a lightness-
constant observer and not with a luminance-based
observer. This is evident from a comparison between
the model predictions (Figure 4B) and the observed
result pattern (Figure 5A). Although the estimated
scales are not linear, they share crucial features with the
hypothetical scales. First, there is a difference in
‘‘intercept’’ between perceptual scales in the light and
dark transparency conditions (blue vs. green lines in
Figure 5A). Second, the scales are steeper for trans-
parent media with lower transmittance than with higher
transmittance (light vs. dark colored lines in Figure
5A). Figure 5A also plots the Munsell neutral value
scale (Munsell, Sloan, & Godlove, 1933) that would be
predicted for our choice of luminances (dashed black
line in Figure 5A).
The Munsell scale represents the expected scale that
relates equal steps in perceived lightness to luminance
(Whittle, 1994). It was calculated by setting the highest
luminance in the plain view stimulus as the white
reference, i.e., to the maximum of one (Pauli, 1976). It
is evident from Figure 5A that the Munsell scale is
consistent with the perceptual scale estimated in our
plain view condition. The typical nonlinear shape
indicates higher sensitivity for differences between
checks with low reflectances than for checks with high
reflectances. This has indeed been reported in previous
work (e.g., Chubb, Landy, & Econopouly, 2004). To
aggregate scales across observers, we normalized the
scales of each individual observer relative to the
maximum scale value in the plain view condition. The
ranges of the scales differed between observers because
different observers have different noise levels. The data
for individual observers are provided in Supplementary
Figure S1.
Scales as a function of reflectance
To better illustrate the degree of lightness constancy
across conditions, we replaced luminance by reflectance
at the x-axis of the perceptual scales. In such a
perceived lightness versus reflectance plot, the scales of
a lightness-constant observer should coincide on a
single function. Figure 5B shows that this was indeed
that case.
To assess the agreement between scales in different
conditions quantitatively, we compared the functions
that were fit in each condition against what we call a
global fit in which the data from all conditions are fitted
Figure 5. MLDS difference scales in different viewing conditions. (A) Difference scales as a function of luminance. The functions depict
the aggregated scales across observers (n¼10). For each observer, the scales were normalized 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). Error
bars indicate M6SD. (B) Same as in panel A, but scales are plotted as a function of reflectance. Scale values (markers, M6SD)
were fitted with a power function (lines) individually for each viewing condition.
Journal of Vision (2017) 17(4):1, 1–14 Wiebel, Aguilar, & Maertens 7
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by a single function. If the data in different viewing
conditions can be explained by one internal model, then
the global fit should account for the data as well as the
individual fits for each viewing condition. We fitted the
scale parameters in each condition and the global scale
with a power function W(x)¼ax
e
þbusing a nonlinear
least squares method (Ritz & Streibig, 2008). To evalu-
ate the goodness of fit, we computed R
2
values for linear
fits to the data. The average R
2
was already reasonably
high (0.86). We then performed Ftests on nested models
(power function vs. its linear submodel with e¼1),
which revealed that the power functions fitted the data
significantly better than the linear ones, F
min
(1, 97) ¼
15.6, p,0.001. From this, we concluded that the power
functions captured the data sufficiently well.
We used a GLM to test whether applying single
models to the data in the five different viewing
conditions would result in better fits than applying a
global model to all data. We compared the respective
sum of squares for the global model with three
parameters (a,b,e) and for the separate models with 5 3
3 parameters. There was a benefit for the separate
model fits relative to the global model, F(12, 497) ¼
18.57, p,0.001. To explore the cause for this
difference, we computed one-way repeated-measures
ANOVAs for each of the three parameters of the power
functions. We found a significant difference between
scales for the exponent parameter, e,F(4, 36) ¼16.6, p
,0.001, which determines the curvature of the func-
tion. Post hoc tests on the exponents revealed significant
differences between each of the light transparency
conditions and the plain view and the dark transparency
with high transmittance (Bonferroni corrected p,
0.05). The main difference between the light transpar-
ency conditions and the plain view and the dark trans-
parency (high transmittance) conditions is the difference
in curvature between these functions (Figure 5B).
The light transparency conditions are special insofar
as during image formation the reflectance-to-luminance
mapping undergoes a range reduction and a range shift
(see Figure 1). This means that checks seen through a
light transparent medium undergo the greatest com-
pression in its contrast range. The Michelson contrast
for targets in plain view range from 0.84 to 0.4
whereas in the low transparent light condition they
range from 0.16 to 0.16 (the contrast is computed
relative to the mean luminance in the region of
transparency). Therefore, sensitivity might be lower for
this range of the stimuli.
Perceptual scales and matching functions
We illustrated in Figure 2 how the data recorded in
matching procedures are related to perceptual scales.
Here, we show to what extent the theoretical relation-
ship can be corroborated by experimental data. To
predict matching data from perceptual scales, we
needed to first find the scale value W(x
T
) that
corresponds to a particular target luminance x
T
in one
of the transparency conditions. In the next step, we
needed to find the luminance value x
M
that corresponds
to the scale value at the match position W(x
M
),
assuming that observers match the lightness of the
match region to that of the target region according to
W(x
M
)¼¼ W(x
T
). We did not measure a perceptual
scale at the match position but instead adopt the plain
view scale to represent the scale for the matches. In
order to be able to read out x-values corresponding to
any possible W-value and vice versa, we fitted the scales
with power functions, w(x)¼ax
e
þb, using a nonlinear
least squares method. We derived the predicted
matching data from the ‘‘unconstrained’’ scales indi-
vidually for each observer, and we then aggregated
them in the same way as the empirical data obtained
from the matching experiment.
In Figure 6, empirical and predicted matches are
plotted next to each other (panels A and B, respec-
tively), and it can be seen that they share some
characteristic features. The matching functions, like the
scales (Figure 5A), differ in slope and intercept between
the different transparency conditions. Differences in
transmittance are accompanied by differences in slope,
and differences in reflectance are accompanied by
differences in intercept. Unlike the scales, the matching
functions are linear.
For a quantitative evaluation of the degree of
similarity between empirical and predicted matching
data, we computed linear regressions for each of the
viewing conditions. We used within-subject ttests to
compare slopes and intercepts between predicted and
empirical functions. The average slope and intercept
values are listed in Supplementary Table S4 together
with the relevant test statistics. We found significant
differences between the predicted and the empirical
functions only for the dark transparent medium with a
high transmittance.
Predictive power of a contrast-based model
The estimated perceptual scales are an interesting
test case for lightness models because they represent a
more direct measurement of perceived lightness than
the matching data. In particular we compared how well
our previously suggested normalized-contrast model
(Zeiner & Maertens, 2014) could account for both the
scaling and the matching data.
The normalized contrast model was initially moti-
vated by the observation that the introduction of a
transparent medium leads to a systematic change in
contrast range of the respective image region. It was
Journal of Vision (2017) 17(4):1, 1–14 Wiebel, Aguilar, & Maertens 8
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suggested that this change in contrast range might serve
as a cue to segregate the region from regions seen in
plain view (Anderson, 1999; Singh, 2004; Singh &
Anderson, 2002). It has been subsequently shown that
the accompanying contrast statistics can be used to
accurately predict perceived lightness (Singh, 2004;
Singh & Anderson, 2002; Wiebel et al., 2016; Zeiner &
Maertens, 2014). The normalized contrast model
engages two processing steps: First, the target intensity
is normalized relative to its local surround by
computing the Michelson contrast between target and
surround. Second, this target contrast is normalized
relative 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 Supple-
mentary material). The so-derived normalized contrast
predicts observers’ lightness matches in contrast units.
Figure 7 shows the aggregated data of both
experiments as a function of the model predictions. If
the computed normalized contrast accounts well for
differences 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).
Transforming the x-axis into units of normalized
Michelson contrast did indeed linearize the perceptual
scales. To test how well the normalized contrast model
accounts for the variability between the different
context conditions, we computed a global R
2
value. As
described before, we treat all data as if they were
coming from one underlying model. The normalized
contrast measure accounts for 98%of the variance in
the scaling data and for 88%of the variance in the
matching data. This indicates that the normalized
contrast measure is a better predictor for the scales
than the matching data by explaining more variance.
The residuals of these fits are provided in Supplemen-
tary Figure S6.
Discussion
The goal of this work was to better understand how
psychological experiences are linked to physical vari-
ables. We studied the question in the domain of lightness
perception, but the observed principles equally apply to
other domains of perceptual appearance. To make
progress toward that goal, we measured perceptual
scales that link perceived lightness to image luminance
using MLDS. Our results show that the estimated
perceptual scales (a) are consistent with a lightness-
constant observer model in all viewing contexts, (b)
predict perceptual equality across different viewing
contexts, (c) indicate that human lightness perception
follows a difference scale that corresponds to a
normalized contrast metric. The normalized contrast
model accounted for more of the variance in the scaling
(98%) than in the matching data (88%), suggesting that
estimating perceptual scales has the advantage of
probing more directly the internal lightness scale.
Figure 6. Empirical and predicted matching data. (A) Results of the matching experiment. The luminance adjusted in the matching
field (y-axis) is plotted as a function of target luminance (x-axis) in each viewing condition. Data were aggregated across observers (n
¼10). Error bars indicate M6SD. (B) Same as in panel A but for matches predicted from the estimated MLDS scales.
Journal of Vision (2017) 17(4):1, 1–14 Wiebel, Aguilar, & Maertens 9
Downloaded from jov.arvojournals.org on 11/20/2020
MLDS-based lightness scales
The estimated perceptual scales were in close
correspondence with each other (Figure 5B), i.e.,
perceived lightness followed the actual check reflec-
tances despite substantial variations in check luminance
across viewing conditions. This reflects a high degree of
lightness constancy. This was corroborated by the
simulated observer models because the empirical scales
were consistent with the lightness-constant and not the
luminance-based observer. The shape of the perceptual
scales followed the shape of the classical Munsell scale.
The perceptual scales are an estimation of the
transducer functions, which cannot be uncovered using
matching (see Figure 2).
In addition to the MLDS experiment, we conducted
a conventional asymmetric matching experiment. We
tested to what extent the postulated relationship
between perceptual scales and matching (Figure 2)
would be evident in the data. Predicted and empirical
matching functions were consistent with each other
(Figure 6). The high degree of consistency is notewor-
thy because triad comparisons and matching require
different perceptual judgments. Asymmetric matching
can be likened to measuring a rod of unknown length
with a ruler whereas in triads rods of different lengths
would be compared among each other. The consistency
between both types of measurements indicates that the
stimulus suitably constrains the perceptual response to
judgments based on lightness and not on luminance.
This cannot be taken for granted (Arend & Goldstein,
1987; Radonji´
c& Brainard, 2016), in particular
because observers were not explicitly told what
dimension to judge.
A potential challenge when comparing perceptual
scales measured in different contexts is the necessary
assumption of how scales are anchored. Two percep-
tual scales might have the same shape but cover a
different range, implying a different anchoring.
Classical scaling experiments did not confront this
problem because perceptual scales were measured in
only one context, i.e., plain view. The default of
MLDS is to anchor the perceptual scales at zero. This
is an arbitrary choice, and any linear transformation
of the scale would be a valid outcome of the analysis.
The good correspondence between the estimated
scales and the matching data in the present case
suggests that there was no substantial anchoring
problem.
As described by Knoblauch and Maloney (2012),
MLDS assumes that observers are stochastic in their
judgments with the noise originating at the decision
level (as shown in Equation 1). This assumption implies
that observers are worse at judging interval differences
that are small, i.e., when (W[x
3
]W[x
2
]) ;(W[x
2
]
W[x
1
]). This critical assumption in MLDS is different
than other scaling methods, such as Fechnerian scaling
that uses integration of just-noticeable differences or
other discrimination-based scaling methods (Baird,
1978). These scaling methods assume a noise source at
an early sensory representation level and not at a late
decision level. Here, we compared perceptual lightness
Figure 7. Perceptual scales (Figure 5A) and matching data (Figure 6A) plotted as a function of the normalized Michelson contrast.
Dashed lines indicate a linear fit to the data for all viewing contexts (R
2
¼0.98 for MLDS, R
2
¼0.88 for matching). Error bars indicate
M6SD across observers.
Journal of Vision (2017) 17(4):1, 1–14 Wiebel, Aguilar, & Maertens 10
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scales that were measured in different viewing condi-
tions and hence could have been associated with
different amounts of decision noise. This was not what
we observed. Although individual observers differed in
their overall noise level, all scales measured for one
observer had comparable estimated noise levels.
However, these assumptions must be considered
carefully, and ultimately their validity must be ad-
dressed experimentally (Aguilar et al., 2017).
The estimated noise level is critical for the interpre-
tation of scales as well as with respect to the distinction
between our two observer models (lightness constant
vs. luminance-based). This is possible only up to a limit
at which observers’ noise is too large for the models to
be distinguished. We established in simulation that this
upper bound is at an estimated noise level ˆ
r¼0.4
(Supplementary Material). In our observers the esti-
mated ˆ
rvalues varied from 0.13 to 0.21 for observers
O1 to O8, i.e., values below the upper limit of model
discriminability. For observer O9, ˆ
r¼0.39 was at the
boundary of discriminability, and for observer O10, ˆ
r¼
0.71 was beyond the upper limit. Thus, the noise level
of observer O10 did not allow a definite selection of
either of the two models. The estimated noise level also
must be considered carefully when comparing scales
against ideal observer models.
Alternatives to asymmetric matching
Asymmetric matching has been criticized in the past
for two main reasons: First, observers’ matches reflect
the underlying perceptual magnitudes only indirectly
(Gescheider, 1988; Maertens & Shapley, 2013). Second,
observers’ matches might not reflect perceptual identity
but merely the best possible match (Brainard et al.,
1997; Ekroll & Faul, 2013; Foster, 2003; Logvinenko &
Maloney, 2006). In particular, the question whether
lightness is represented by more than one dimension
across different contexts has been tackled using
different methods (Logvinenko & Maloney, 2006;
Logvinenko et al., 2008; Umbach, 2013). Beyond
methodological shortcomings, asymmetric matching
tasks have also been criticized for their lack of realism
because in real life we rarely adjust the color of an
object but rather select objects based on their color. In
two recent studies, Radonji´
c, Cottaris, and Brainard
(2015a, 2015b) measured color constancy in a color
selection paradigm in which they asked observers to
select which of two competitors was more similar to a
given target.
Their task was analogous to the triad comparison
used in MLDS, but the design was different from the
standard MLDS design. A limited number of targets
was presented as anchor for a respective set of
competitors, but these competitors were not compared
with each other. MLDS would involve triad compar-
isons of all possible combinations of targets and
competitors.
The data were analyzed with a customized version
of MLDS. The crucial difference from our approach is
that in their critical condition target and competitors
were presented in different illuminations. As a
consequence, observers’ judgments were subject to the
same comparison problem as in asymmetric matching.
To estimate a perceptual scale, it was assumed that
target and competitors are represented on a common
underlying dimension. In our way of thinking, this
means to skip the step of estimating the different
transducer functions (scales), which map luminance to
perceived lightness in different contexts (Figure 4B),
andtocomparestimulidirectlyontheinternalaxis.
As we have outlined above, this assumption is valid
only for a lightness constant observer, i.e., for
observers whose perceptual scales in different viewing
situations have comparable scale maxima. The au-
thors reported moderately high color constancy
indices, which were comparable to asymmetric
matches for the same type of stimuli (Radonji´
cet al.,
2015a, 2015b). We suggest including such cross-
context comparisons only to validate predictions from
the MLDS-based scales as we did here with the
asymmetric matches.
Models of lightness perception
We claim that perceptual scales are an important test
case for models of lightness perception because they
offer a direct estimate of the transducer functions that
we are interested in. A successful model should be able
to explain both characteristics of lightness appearance:
perceptual equality across contexts as well as sensitivity
differences manifested in the shape of perceptual scales
(Hillis & Brainard, 2007).
If we assume that the goal of the visual system is to
accurately represent surface reflectance, then reflec-
tance would be the best predictor of perceived surface
lightness. Thus, for a perfect lightness-constant ob-
server, the perceptual scales measured in different
contexts should perfectly overlap when plotted against
reflectance. Our empirical scales are consistent with a
lightness-constant observer; however, they reveal small
deviations, especially for the two lighter transparent
media (Figure 5B). When we plotted the scales as a
function of normalized contrast (Wiebel et al., 2016;
Zeiner & Maertens, 2014) instead of reflectance, the
differences between scales were substantially reduced
(Figure 7A). This means that the normalized contrast
metric does not perfectly capture veridical surface
reflectances but is rather tightly correlated with them.
One might be tempted to conclude that the predictive
Journal of Vision (2017) 17(4):1, 1–14 Wiebel, Aguilar, & Maertens 11
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power of the contrast-based model ‘‘exceeds’’ that of
physical surface reflectances because it accounts for the
deviations from lightness constancy that we observed in
the data.
This finding is consistent with the idea that the
visual system, instead of doing inverse optics (e.g.,
Barrow & Tenenbaum, 1978; D’Zmura & Iverson,
1993), might use a set of readily available but
imperfect cues to infer stable properties of objects
(e.g., Anderson, 2011; Fleming, 2014). The involved
computations might not always lead to a veridical
percept with respect to the physical world, but to an
overall reliable estimate of the appearance of objects
(e.g., Marlow, Kim, & Anderson, 2012). The esti-
mated scales were linearized by the transformation to
contrast units, which implies that the model accounts
for the sensitivity differences between low and high
reflectances (e.g., Lu & Sperling, 2012), a feature that
cannot be quantitatively captured with matching. The
higher agreement between the model and the percep-
tual scales (compared to matching) supports the idea
that the perceptual scales are a more direct and
informative measure of the internal variable of
lightness and subject to fewer sources of variability.
General conclusions
In this paper, we show that a scaling method is more
powerful than matching in elucidating the perceptual
representation of surface lightness. MLDS provides a
direct estimate of the transducer functions that relate
the physical dimension of reflectance to the psycho-
logical dimension of perceived lightness. In addition,
MLDS avoids the practical difficulties associated with
asymmetric matching tasks because all perceptual
comparisons are made within the same viewing context.
Observers confirmed that subjectively the triad com-
parison required by MLDS was a natural and
straightforward task.
So why is it then that asymmetric matching remains
the method of choice despite the obvious benefits of
MLDS. We suspect that experimenters feel slightly
uneasy about explicitly making and committing to the
various assumptions that are required by MLDS in
order to statistically estimate the perceptual scales.
However, as we illustrate in Figure 2, asymmetric
matching procedures also assume the presence of
internal scales, but they are hidden, and their shape
cannot be inferred from observers’ matches. We think
that the present results are encouraging and advocate
the estimation of scales because they provide a more
direct estimate of internal variables against which we
can test our theoretical models of appearance.
Keywords: lightness,perceptual scales,MLDS,
asymmetric matching
Acknowledgments
This work has been supported by an Emmy-Noether
research grant of the German Research Foundation
(DFG MA5127/1-1) and by the Research Training
Grant ‘‘Sensory Computation in Neural Systems’’ of
the German Research Foundation (GRK1589/1-2). We
would like to thank Michael Landy, Kenneth
Knoblauch, Bart Anderson, Richard Murray, Felix
Wichmann, Frank J¨
akel, and David Higgins for their
constructive suggestions that helped in improving this
manuscript. CBW has moved in the meantime to the
Honda Research Institute Europe in Offenbach,
Germany.
*CBW and GA contributed equally to this article.
Commercial relationships: None.
Corresponding author: Guillermo Aguilar.
Email: [email protected].
Address: Modeling of Cognitive Processes, Technische
Universit¨
at Berlin and Bernstein Center for
Computational Neuroscience, Berlin, Germany.
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