Natural scene statistics mediate the perception of image complexity

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 e sion. h p://www. and online.com/ oc/p is20/cu en #.U9S99lZDs7s 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 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 e sion. h p://www. and online.com/ oc/p is20/cu en #.U9S99lZDs7s 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,                  !     "   # $ !! "! "% 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 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 e sion. h p://www. and online.com/ oc/p is20/cu en #.U9S99lZDs7s 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. 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Please ead and ci e he published e sion. h p://www. and online.com/ oc/p is20/cu en #.U9S99lZDs7s 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).