Handbook of Psychology, Volume 4: Experimental Psychology

114Foundations of Visual Perception

withthereferenceelements;thatis,themostlikelyedgeele-
mentscanbeconnectedthroughthecontoursofminimal
changeofcurvature.Geisleretal.(2001,p.713)concluded
thattheabsoluteECstatistic“reflectstherelativelysmooth
shapesofnaturalcontours,and...providesdirectevidence
thattheGestaltprincipleofgoodcontinuationhasageneral
physicalbasisinthestatisticsofthenaturalworld.”
The authors reported that the same “basic pattern” as in
Figures 4.19B and 4.19C occurred in the statistics obtained
from all the images, as well as in the analysis of edges under
different spatial scales. As a control, the authors ascertained
that in the images containing random patterns (white noise),
the absolute statistic of EC was random.

Bayesian Edge Co-occurrence

Beforeweexplainthisstatistic,letusbrieflyrecalltherele-
vantideasofBayesianinference,whichwehavealready
encounteredinthesectiononsignaldetectiontheory.Inthe
contextofadetectionexperiment,wesawthatwhenobservers
generatetwohypothesesaboutthestateofaffairsintheworld
(“noisetrial”vs.“signalplusnoisetrial”)therelevantevi-
dencecanbemeasuredbytakingtheratioofthelikelihoodsof
eventsassociatedwiththetwohypotheses(Figure4.8E).The
resultingquantity(thelikelihoodratio)canbecomparedwith
anotherquantity(thecriterion)toadjudicatebetweenthe
hypotheses.
Similartotheconditionsofadetectionexperiment,inmea-
suringtheECstatisticsonecanpittwohypothesesagainst
eachotherwithrespecttoeverypairofedgeelements:C,“the
elementsbelongtothesamecontour”and~C,“theelements
donotbelongtothesamecontour.”Therelevantevidencecan
beexpressedintheformofalikelihoodratio:

(d,,),(3)

wherep(d,,
C) and p(d,,
~C) are the conditional
probabilities of a particular relationship {d,,} between
edge elements to occur, when the elements belong or do not
belong to the same contour, respectively. (We explain how to
obtain the criterion in a moment.)
Geisler et al. (2001) measured the likelihood ratio for
every available relationship {d,,} as follows. In every
image, observers were presented with a set of highlighted
pixels (colored red in the example image in Figure 4.19A)
corresponding to the centers of edge elements detected in the
image. Using a computer mouse, observers assigned sets of
highlighted pixels to the perceived contours in the image.
Thus observers reported about the belongingness of edge
elements to contours in every image. With this information

p(d,,
C)

p(d,,
C)

Geisler et al. conditionalized the absolute probabilities of EC

by whether the edge elements within every pair belonged to

the same contour or not, that is, to obtain the likelihoods

p(d,,
C) and p(d,,
~C).

TheresultingdistributionofL(d,,)isshowninFig-

ure4.19D,againusingacolorscale,averagedacrossall

thesampleimagesandtwoobservers.(Thetwoobservers

largelyagreedabouttheassignmentofedgestocontours,

withthecorrelationcoefficientbetweenthetwolikelihood

distributionsequalto.98.)Incontrasttotheplotsofabsolute

statisticsinFigures4.19Band4.19C,theplotofconditional

ECinFigure4.19Dshowsall36orientationsateveryloca-

tioninthesystemofcoordinates(d,).Thedistributionof

L(d,,)showsthatedgeelementsaremorelikelyto

belongtothesamecontourthannot(whenL[d,,]>1.0,

labeled from green to red in Figure 4.19D), within two sym-

metrical wedge-shaped regions on the sides of the reference

edge element.

Why measure the Bayesian statistic of EC in addition to the

absolute statistics? The Bayesian statistic allows one to con-

struct a normative model (i.e., a prescriptive ideal observer

model; Figure 4.4) of perceptual grouping of edge elements.

Besides informing us on how the properties of element rela-

tions covary in natural images (which is already accomplished

in absolute statistics), the Bayesian statistic tells us how the

covariance of edge elements connected by contours differs

from the covariance of edge elements that are not connected.

As a result, the Bayesian statistic allows one to tell whether

human performance in an arbitrary task of perceptual group-

ing by continuation is optimal or not. Human performance in

such a task is classified as optimal if human observers assign

edge elements to contours with the same likelihood as is pre-

scribed by the Bayesian statistic. In the next section we see

how Geisler et al. (2001) constructed the ideal observer model

of grouping by continuation and how they compared its per-

formance with the performance of human observers.

Predicting Human Performance from the Statistics

of Natural Images

Psychophysical Evidence of Grouping

by Good Continuation

To find out whether human performance in grouping by good

continuation agrees with the statistics of EC in natural im-

ages, Geisler et al. (2001) conducted a psychophysical exper-

iment. They used a stimulus pattern for which they could

derive the predictions of grouping from their statistical data

and pit the predictions against the performance of human

observers. An example of the stimulus pattern is shown in

Figure 4.21A.