Na u al scene s a is ics media e he pe cep ion o image complexi y
This is a p ep in (d a ) e sion o a pape o appea in Visual Cogni ion. Please ead and ci e he published
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Nicolas Gau i , Fe nando Sole -Toscano, Hec o
Zenil — 28 h July, 2014
Abs ac
Humans a e sensi i e o complexi y and egula i y in
pa e ns (Yamada, Kawabe & Miyazaki, 2013; Falk
& Konold, 1997). The subjec i e pe cep ion o
pa e n complexi y is co ela ed o algo i hmic (o
Kolmogo o -Chai in) complexi y as de ined in
compu e science (Li & Vi anyi, 2008), bu also o
he equency o na u ally occu ing pa e ns (Hsu,
G i i hs & Sch eibe , 2010). Howe e , he possible
media ional ole o na u al equencies in he
pe cep ion o algo i hmic complexi y emains
unclea . He e we eanalyze Hsu e al. (2010)
h ough a media ional analysis, and complemen
hei esul s in a new expe imen . We conclude ha
human pe cep ion o complexi y seems pa ly
shaped by na u al scenes s a is ics, he eby
es ablishing a link be ween he pe cep ion o
complexi y and he e ec o na u al scene s a is ics.
KEY WORDS: isual complexi y; isual pe cep ion;
algo i hmic complexi y; andomness
Humans a e ex emely sensi i e o pa e ns and
egula i ies (Yamada, Kawabe & Miyazaki, 2013).
Ou b ains de ec sligh depa u es om
andomness. Someone h owing 3 dice and ge ing
h ee ‘6s’ o he pa e n ‘1, 2, 3’ is likely o be
s unned by he ac ha hese combina ions a e
egula , o be inc edulous in he ace o his meage
e idence ha he dice a e ai (Falk & Konold,
1997). This gene al human ea u e— he
disce nmen o ules go e ning he wo ld—may be
hough o as he cogni i e basis o science, bu also
as an adap i e abili y shaped by na u al e olu ion o
a oid p edic able dange s.
Psychologis s ha e linked ou na u al pe cep ion o
andomness o he ma hema ical heo y o
algo i hmic complexi y (also known as Kolmogo o -
Cha in complexi y): he mo e complex he s imulus,
he mo e andom i will be pe cei ed o be.
Fo mally, he algo i hmic complexi y o a sequence
is he leng h o he sho es p og am ha p oduces
he sequence in ques ion and hal s (Li & Vi ányi,
2008). In his de ini ion, he said p og am doesn’
in ol e a speci ic compu e , bu a he a gene al
Uni e sal Tu ing Machine, an abs ac compu e .
Algo i hmic complexi y is ela ed o he p obabili y
ha such a machine, ed wi h a andom p og am,
will p oduce a pa icula sequence and hal , a link
o mally p o en by he coding heo em (Le in, 1974)
and ini ially concei ed o sol e he p oblem o
induc ion—which i does in a e y gene al and
powe ul way (Solomono , 1964a, 1964b)—so
powe ul ha he measu e is indeed ul ima ely
uncompu able, hough app oxima ions a e possible.
The main in ui ion behind algo i hmic p obabili y is
ha i a sequence is no andom hen i will con ain
some egula i y ha can be encoded in a compu e
p og am o leng h sho e han he sequence ha
can gene a e i by mechanis ic means. And sho e
p og ams a e mo e likely o occu , and he e o e
mo e equen han longe ones i each p og am
ins uc ion is uni o mly andomly chosen, which
es ablishes a powe ul connec ion be ween
complexi y and equency. Wi hin he amewo k o
algo i hmic complexi y heo y, “ andomness” and
“complexi y” a e in e changeable concep s: he
o mal de ini ion o andomness elies on
complexi y, and complexi y is a di ec measu e o
andomness.
The hypo hesis ha he human pe cep ion o
andomness is linked o algo i hmic complexi y
could no be e i ied p io o ecen de elopmen s in
compu e science. Indeed, i me hods ha e long
exis ed ha allow sa is ac o y es ima ions o he
algo i hmic andomness o long sequences, such as
comp ession algo i hms (Zi & Lempel, 1978), un il
ecen ly no such me hods we e a ailable o assess
he algo i hmic complexi y o sho sequences
(Sole -Toscano, Zenil, Delahaye & Gau i , 2013,
2014; Zenil, Sole -Toscano, Delahaye & Gau i ,
2012; ).
In ligh o algo i hmic complexi y, we unde ook an
in es iga ion in o how humans pe cei e
andomness, and how we lea n (i we do) o
pe cei e complexi y. Hsu, G i i hs and Sch eibe
(2010) ad anced an in e es ing hypo hesis: he
equency wi h which a pa e n appea s in eal wo ld
scenes could explain how we pe cei e andomness,
pe mi ing us o in e complexi y om he wo ld we
see. Hsu e al. (2010) scanned a se o pho og aphs
o eal wo ld na u al scenes and ex ac ed e e y
possible 4×4 a ay om hese images. Then hey
compu ed he esul ing p obabili y dis ibu ion, and
de i ed he andomness o each a ay x, de ined as
andom(x) = log(P(x| )/P(x|n)), P(x|n) being he
ela i e equency o he a ay in he na u al scene
da abase, and P(x| ) he p obabili y ha his a ay
appea s by chance i e e y cell in he a ay is
selec ed a andom (ei he whi e o black). They
chose 100 balanced a ays wi h p obabili ies o
occu ence in eal scenes anging om low o high.
They hen had 77 subjec s decide whe he hese
a ays looked andom o no . This led o a measu e
o subjec i e andomness ( he p opo ion o
pa icipan s decla ing he a ay andom) o each
a ay.
They ound ha subjec i e p obabili y and na u al
andomness we e posi i ely co ela ed on hese
pa icula 100 a ays ( = .75, p < .0001). We
compu ed ha Kolmogo o -Chai in complexi y is
also signi ican ly linked o he subjec i e pe cep ion
o andomness. Wi h he a ays published in Hsu e
al. (2010), we ound a co ela ion o = .52 (p <
.0001) be ween wo-dimensional algo i hmic
complexi y as de ined in Zenil e al. (2012)—see
also Gau i , Zenil, Delahaye and Sole -Toscano
(2013)—and subjec i e p obabili y.
This pa e n o co ela ions is no su p ising.
Because he wo ld can be hough o as a gene a o
o pa e ns, like a andom compu e p og am, he
p obabili y ha an a ay will occu in he wo ld is
hen linked o i s algo i hmic complexi y. Indeed, as
compu ed wi h he 100 a ays o Hsu e al. (2010),
we also ound a posi i e co ela ion be ween
algo i hmic complexi y and na u al scene s a is ics (
= .50, p < .0001).
Could na u al scene s a is ics accoun o human
pe cep ion o algo i hmic complexi y?
Na u al scene s a is ics media e he pe cep ion o image complexi y
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This would be in line wi h ecen esul s in
neu oscience epo ed by Be ks, O ban, Lengyel
and F ise (2011). They analyzed co ical ac i i y in
e e s, and compiled e idence in a o o he
hypo hesis ha ou b ain lea ns an op imal in e nal
p obabilis ic model o he en i onmen , based on
na u al wo ld equencies—see also Teglas,Vul,
Gi o o, Gonzales, Tenenbaum and Bona i (2011)
o examples o child en’s apid adap a ion o
na u al equencies.
Bu how much o ou pe cep ion o andomness is
a ibu able o lea ning h ough he na u al wo ld?
To answe his ques ion, we pe o med a media ion
analysis using scaled da a. A eg ession o
subjec i e andomness on bo h algo i hmic
complexi y and na u al scenes s a is ics gi es an
adjus ed R-squa ed equal o .58 (p < .0001). Figu e
1(A) displays he coe icien s linking complexi y o
subjec i e andomness (.19, p = .013) and na u al
scenes s a is ics o subjec i e andomness,
con olling o algo i hmic complexi y (.66, p <
.0001). In his igu e, “Algo i hmic complexi y”
s ands o he Kolmogo o -Chai in complexi y o he
a ays, as app oxima ed by he me hod desc ibed in
Zenil e al. (2012). “Na u al s a is ics” e e s o he
andom unc ion de ined abo e, in which P(x|n)
s ands o he equency o a ay x in he na u al
scenes da ase . Las , “subjec i e andomness” is a
sho hand o log(p(x)), whe e p(x) designa es he
p opo ion o pa icipan s who indica e ha x is
seemingly andom. A Sobel es con i ms he
media ional ole o na u al scene s a is ics (z = 4.78,
p < .0001).
The esul sugges s ha ou pe cep ion o
complexi y is pa ially d i en by he pe cep ion o
na u al scenes. Howe e , i is ai o unde sco e wo
poin s ha may p ejudice he alues ound he e.
Fi s , we canno con ol he link be ween “na u al
scene s a is ics” (i.e. he “ andom” unc ion) and he
choice o he se o pic u es. Second, because he
100 a ays chosen o use he e all wi hin ce ain
pa ame e s ( hey a e all balanced, and ha e been
chosen in such a way ha hey a e e enly
dis ibu ed on he na u al scene s a is ics scale),
a iance o na u al scene s a is ics could be
a i icially high.
In he ollowing expe imen , we o e come hese wo
possible d awbacks in o de o ge a clea iew o
he possible media ional ole o na u al scene
s a is ics in he pe cep ion o complexi y.
Me hod'
We pe o m an expe imen simila o ha p esen ed
abo e, bu eleasing some cons ain s ha could
a ec he esul s. We do no impose ha e e y
pa e n is balanced in e ms o whi e and black cells.
We do no choose s ill na u e sho s only. Ou
hypo hesis is ha e en when hese cons ain s a e
elea ed, na u al scene s a is ics will play a
media ional ole.
Pa icipan s'
A sample o 100 pa icipan s (59 male, 41 emale)
was ec ui ed ia he Amazon Mechanical Tu k.
Hi ed “wo ke s” om he Mechanical Tu k we e
equi ed o ha e a 90% app o al a ing on p e ious
Mechanical Tu k asks (HITs) and a leas 50
p e ious HITs app o ed. Ages in yea s anged
be ween 19 and 55 (mean ± sd = 30.6 ± 8).
Pa icipan s we e paid 0.30 USD o hei
pa icipa ion. The expe imen du a ion anged om
84s o 289s (mean ± sd = 212.2 ± 45). Olde
pa icipan s in his sample showed a sligh endency
o need mo e ime ( = .15).
S imuli'
Hsu e al. (2010) used a se o 62 pic u es
p e iously used by Doi, Inui, Lee, Wach le and
Sejnowski (2003) o compu e he na u al scenes
s a is ics. All pic u es we e s ill na u e sho s,
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Figu e 1 Media ion analysis, pe o med wi h scaled da a as compu ed om he da ase o Hsu e al. (2010)
[subplo A] and wi h ou expe imen al da a [subplo B]. In each subplo , he op g aph displays he
s anda dized co ela ion coe icien . The bo om g aph displays (1) he s anda dized eg ession
coe icien be ween na u al s a is ics and algo i hmic complexi y, (2) he s anda dized eg ession
coe icien be ween complexi y and subjec i e andomness, and (3) he pa ial s anda dized eg ession
coe icien be ween na u al scenes s a is ics and subjec i e andomness, con olling o algo i hmic
complexi y. * p < .05 , *** p < .001.
Na u al scene s a is ics media e he pe cep ion o image complexi y
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including no aces, u ban scenes o a i icial objec s.
The e o e, he andom unc ion may a y i
compu ed wi h o he se s o pic u es. To es his
hypo hesis, we applied he me hod used by Hsu e
al. (2010) o a new se o 100 andom pic u es,
aken om he Wikimedia Commons da abase1. The
sample included na u al scenes bu also animals
and non-na u al objec s such as buildings. Then we
bina ized he pic u es o black and whi e pixels
using he median as he h eshold. We hen di ided
each image in o 4×4 adjacen bina y squa e a ays
and calcula ed ( o he whole se o 100 images) he
p obabili y o each squa e.
The esul ing “ andom” unc ion is s ongly
co ela ed o he da a ob ained by Hsu e al. when
compu ed on hei choice o 100 a ays ( = .91, p <
.0001), which alida es he me hod. The co ela ion
be ween na u al scenes s a is ics ( unc ion andom)
and algo i hmic complexi y was .50 when compu ed
on he 100 a ays chosen by Hsu e al. Howe e ,
because he choice o a ays was no andom, he
co ela ion could well be o e es ima ed. When
compu ed on e e y possible 4×4 a ay ound while
scanning he 100 andom pic u es, he co ela ion
emains highly signi ican , al hough sligh ly lowe (
= .42, p < .0001), con i ming he p e ious esul .
We hen picked a andom a sample o 100 a ays
om among all he a ays ound in ou se o 100
images, using he sample unc ion in R. We did no
con i e o ob ain balanced a ays, a depa u e om
he design o Hsu e al. (2010). Figu e 2 displays he
100 a ays ob ained by andom selec ion.
P ocedu e'
The p ocedu e mi o ed he one used in Hsu e al.
(2010), al hough ou expe imen ook place online.
Pa icipan s illed ou a ques ionnai e simila o ha
used by Hsu e al. (2010). They we e in o med ha
a se ies o a ays would appea on he sc een, and
ha hei ask was o decide whe he he a ays
we e p oduced by a andom p ocess o by a non-
andom p ocess. Fo each a ay, hey we e asked o
p ess a bu on, ei he “ andom” o “no andom”
acco ding o hei pe cep ion.
Resul s'
The da a we e analized wi h he same me hod as
Hsu e al. (2010). Algo i hmic complexi y is
posi i ely co ela ed wi h na u al s a is ics ( = .46, p
< .0001) and subjec i e andomness ( = .36, p <
.0001), as a e subjec i e andomness and na u al
s a is ics ( = .56, p < .0001).
A mul iple eg ession o subjec i e andomness on
algo i hmic complexi y and na u al scene s a is ics
yields an ajus ed R-squa ed o .31 (p < .0001).
Figu e 1(B) displays he coe icien s linking
complexi y o subjec i e andomness (.14, p = .15)
and na u al scenes s a is ics o subjec i e
andomness, con olling o algo i hmic complexi y
(.47, p < .0001). A Sobel es con i ms he
1
h p://commons.wikimedia.o g/wiki/Special:Rand
om/Image
media ional ole o na u al scene s a is ics (z = 3.66,
p < .001).
Discussion'
Pe haps as an upsho o he eschewal o cons ain s
as compa ed wi h he Hsu e al. (2010) s udy
(balanced a ays sca e ed along he na u al
p obabili y ange), he coe icien s a e now smalle .
Howe e , he pa e ns o co ela ions a e
ema kably simila .
These esul s sugges ha na u al scene s a is ics
a e indeed an impo an elemen in he pe cep ion o
complexi y. The co espondence be ween he
eanalysis and he subsequen expe imen also
sugges s ha he e is some objec i e na u al
p obabili y o a ays, linked bo h o ou pe cep ion o
complexi y and o he o mal de ini ion o complexi y
a ising om Kolmogo o -Chai in heo y.
Al hough ou pe cep ion o complexi y may be
la gely explained by na u al scene s a is ics, his
does no p eemp he possibili y o a complemen a y
means o pe cep ion, which could e en ually u n
ou o be inna e. Howe e , u he s udies would be
needed o con i m his assump ion.
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e sion. h p://www. and online.com/ oc/p is20/cu en #.U9S99lZDs7s
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Na u al scene s a is ics media e he pe cep ion o image complexi y
This is a p ep in (d a ) e sion o a pape o appea in Visual Cogni ion. Please ead and ci e he published
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Figu e 2 The 100 a ays used in ou expe imen oge he wi h hei algo i hmic complexi y (abo e each
a ay). A ays a e o de ed acco ding o hei na u al scene equency ( om mo e o less equen a ays).