Perceiving Geometry: Geometrical Illusions Explained by Natural Scene Statistics

127 Pages·2005·14.115 MB·English

by  Catherine Q. Howe, Dale Purves (auth.)

Perceiving Geometry
Geometrical Illusions Explained by
Natural Scene Statistics
Perceiving Geometry
Geometrical Illusions Explained by
Natural Scene Statistics
by
Catherine Q. Howe and Dale Purves
LibraryofCongressCataloging-in-PublicationData
PerceivingGeometry:GeometricalIllusionsExplainedbyNaturalSceneStatistics/
byCatherineQ.Howe,DalePurves
AC.I.P.Cataloguerecordforthisbookisavailable
fromtheLibraryofCongress.
ISBN-10:0-387-25487-0 (alkpaper) e-ISBN-10:0-387-25488-9
ISBN-13:978-0387-25487-6 e-ISBN-13:978-0387-25488-3
(cid:1)C 2005SpringerScience+BusinessMedia,Inc.
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout
thewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233Spring
Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviews
orscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,
electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknow
orhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,even
iftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhether
ornottheyaresubjecttoproprietaryrights.
PrintedintheUnitedStatesofAmerica.
9 8 7 6 5 4 3 2 1 SPIN11385059
springeronline.com
Contents
Acknowledgment...................................................... vii
1. Introduction ....................................................... 01
2. TheGeometryofNaturalScenes..................................... 15
3. LineLength ....................................................... 25
4. Angles ............................................................ 37
5. Size............................................................... 47
6. Distance .......................................................... 63
7. TheMu¨ller-LyerIllusion............................................ 73
8. ThePoggendorffIllusion............................................ 85
9. Implications ....................................................... 97
References............................................................ 107
Glossary.............................................................. 115
Index................................................................. 125
Acknowledgment
WearegratefultoourcolleaguesShuroNundyandZhiyongYangfortheircollaboration
insomeoftheworkdiscussedinthebook,andformuchusefuladviceanddiscussion.
HelpfulcriticismandadvicewasalsoprovidedbyJimVoyvodic.
Chapter 1
Introduction
Understandingvision—whetherfromaneurobiological,psychologicalorphilosophi-
calperspective—representsadauntingchallengethathasbeenpursuedformillennia.
Duringatleastthelastfewcenturies,naturalphilosophers,andmorerecentlyvisionsci-
entists,haverecognizedthatafundamentalprobleminbiologicalvision—andindeed
afundamentalprobleminperceptiongenerally—isthatthesourcesunderlyingvisual
stimuliareunknowableinanydirectsense.
Thereasonforthisquandaryistheinherentambiguityofthestimulithatimpinge
onsensoryreceptors.Inthecaseofvision,thelightthatreachestheeyefromanyscene
conflatesthecontributionsofreflectance,illuminationandtransmittance,aswellasa
hostofsubsidiaryfactorsthataffecttheseprimaryphysicalparameters(Figure1.1A).
Evenmoreimportantwithrespecttothetopicunderconsiderationhere,spatialprop-
ertiessuchasthesize,distanceandorientationofphysicalobjectsarealsoconflated
in light stimuli (Figure 1.1B). As a result, the provenance of light reaching the eye
at any moment—and therefore the significance of the stimulus for visually guided
behavior—isprofoundlyuncertain.Inmoreformalterms,thisquandaryisreferredto
astheinverseopticsproblem.
Thesebasicfactsabouttherelationshipoftherealworldandtheinformationcon-
veyedbylightreachingtheretinapresentadifficultproblem.Successfulbehaviorina
complexandpotentiallyhostileenvironmentclearlydependsonrespondingappropri-
atelytothephysicalsourcesofvisualstimuliratherthantothephysicalcharacteristics
ofstimuliassuch(whichasindicatedinFigure1.1,areofuncertainsignificance).If
theretinalimagesgeneratedbylightstimulicannotspecifytheunderlyingrealityan
observer must deal with, how then does the visual system produce behavior that is
generallysuccessful?
EXPLORINGVISIONINTERMSOFPERCEIVEDGEOMETRY
In the chapters that follow, we consider the evidence that, with respect to space, the
humanvisualsystemsolvesthisproblembyincorporatingpasthumanexperienceof
2 PerceivingGeometry:GeometricalIllusionsExplainedbyNaturalSceneStatistics
A
SSttiimmuulluuss
Stimulus
Illumination Reflectance Transmittance
B
Retinal projection
Figure1.1 Thefundamentalprobleminbiologicalvisionisthenecessarilyuncertainrelationshipbetween
theinformationintheimagesthatfallontheretinaandtheirreal-worldsources.A)Conflationofthefactors
thatdeterminetheamountandspectralqualityoflightfallingontheretina.Illuminationdependsonthe
propertiesofasourcelikethesun;thereflectanceofobjectsdependsontheirphysicalcomposition;and
transmittancedependsontheamountandqualityoftheatmosphereinterveningbetweenanobjectandthe
observer(aswellasbetweenthesourceofilluminationandtheilluminatedobjects).Thesebasicfactorsthat
togetherdeterminetheluminanceandspectraldistributionofanystimulusattheeyecannotbedisentangled
byanalysisoftheretinalimage.B)Theproblemismuchthesameintheperceptionofgeometry,sincethe
spatialpropertiesofthree-dimensionalobjectsarealsoconflatedwhenlightarisingfromthemisprojected
ontoaplane.Thediagramshowsthatthesameretinalprojectioncanbegeneratedbyobjectsofdifferent
sizesatdifferentdistancesfromtheobserver,andindifferentorientations.Again,thereisnologicalwayto
disentanglethesefactorsbyanalysisoftheretinalimage.(AfterPurvesandLotto,2003)
Introduction 3
with what projected patterns of light on the retina have typically corresponded to
in the real world. Indeed, we consider this not simply an adjunct to vision, but the
fundamental scheme that determines spatial vision. This empirical strategy explains
manyotherwisepuzzlingaspectsofwhatwesee,andtheseexplanations—whetherin
qualitative or quantitative terms—provide the best indication to date of how human
perceptionsofthegeometricalaspectsoftheworldareactuallygenerated.
Thecruxoftheargumentisthatthelinkbetweenstimuliandperceptswithrespect
tovisualspace—i.e.,thewayweexperiencesize,distanceandorientation—canonly
be understood in a statistical framework in which the perceptions generated by light
patternsprojectedontotheretinaaredeterminedbytheprobabilitydistributionsofthe
possiblesourcesofthoseprojections.Thisframeworkcanrationalizemanyotherwise
puzzlingdiscrepanciesbetweenvisualperceptsandthephysicalparametersofvisual
stimuli. These discrepancies—often presented in the form of “geometrical illusions”
(seeFigure1.2)—havelongpresentedachallengetoanyoneinterestedinthenatureof
vision,andattemptstounderstandthemcanbetracedbackseveralcenturiesormore.
Giventheinherentambiguityofretinalimages,thebiologicalrationaleforseeing
the geometrical aspects of retinal stimuli in terms of the probability distributions of
their possible sources is not difficult to understand. Much to the advantage of the
observer,thisvisualstrategycontendswiththeproblemofstimulusambiguitybytaking
A B C
D E
Figure1.2 Examplesofsomemuch-studiedgeometricalillusions.A)TheT-illusion.B)TheHering
illusion.C)ThePonzoillusion.D)TheMu¨ller-Lyerillusion.E)ThePoggendorffillusion.
4 PerceivingGeometry:GeometricalIllusionsExplainedbyNaturalSceneStatistics
advantageofeonsoftrialanderrorinhumanexperience.Inthiswaywe,andpresumably
allotheranimalswithsophisticatedvision,ensurethatvisuallyguidedresponseswill
usuallydealsuccessfullywithobjectsandconditionsthatareunknowablebyanydirect
means. A consequence of generating percepts on a statistical basis, however, is that
whatobserversactuallyseedoesnotalwayscorrespondtothephysicalcharacteristics
ofthestimulusortheparticularphysicalconditionsthatgeneratedthestimulus;rather,
whatisseencorrespondstotheempiricalsignificanceofthestimulus,i.e.,whatithas
typicallymeantforvisuallyguidedbehavior.
Hereweexaminethevalidityofthisgeneralidea,usingasexamplesaseriesof
classicalgeometricalstimuliandthe“illusions”theygenerate(thequotationmarksare
tosuggestthatthisisnotaparticularlyaptwordsince,inthepresentframework,all
visualperceptsareequallyconstructedfromthestatisticalinformationacquiredthrough
experience). The approach in each instance is to examine the statistical relationship
between the relevant retinal images and their real-world sources, asking whether the
perceptsreportedbyhumansubjectsaccordwiththepredictionsmadeonthebasisof
the statistics derived from a database of natural scenes. The database includes mea-
surements of the distance and direction of the physical sources of each point in the
images,andiseffectivelyaproxyfortheaccumulatedvisualexperienceofourspecies.
Assuch,thedatabase(seeChapter2)canbeusedtorevealthestatisticalregularities
betweenretinalimagesandreal-worldsourcesthatmusthavedeterminedtheevolution
ofhumanvision.
Of course, a variety of other visual perceptual qualities can be (and have been)
exploredinthisframework,includingbrightness/lightness,colorandsomeaspectsof
motion(reviewedinPurvesandLotto,2003).Wehavechosentofocushereonperceived
spatial relationships because understanding the way we see geometry is intrinsically
interestingandmuchdebated;afurtherreasonisthat,fromatechnicalperspective,the
probabilitydistributionsofthepossiblesourcesofgeometricalstimulicanbederived
directly by an analysis of range images of natural scenes. The ability to determine
the statistical relationship between geometrical projections and their sources is an
enormousadvantage,andmeetingthisgoalismuchmoredifficult(althoughcertainly
notimpossible)withrespecttootherstimuluscharacteristics.
Of course, the fact that geometrical stimuli and the perceptual anomalies they
elicit have challenged the imaginations of so many thinkers in a variety disciplines
oversomanyyearsisstrongmotivationaswell.Intheend,however,thepointofthe
worksummarizedhereissimplytounderstandhowandwhyweperceivevisualspace
andthegeometryoftheobjectsthereinthewaywedo.
EXAMPLESOFWELL-KNOWNGEOMETRICALILLUSIONS
Numerousobservershavepointedoutthatmeasurementsmadewithrulersorprotrac-
tors of a variety of simple visual stimuli are often at odds with the perceptions they
elicit,frequentlyinamostengagingway.Constructingstimulithatproduceapartic-
ularlyintriguinggeometricalillusionwassomethingofacottageindustryinthe19th
and early 20th centuries; although some of these demonstrations were described by
thepreeminentvisionscientistsofthetime,agoodgeometricalillusionhasprovided