The“Structure” of the Visual Environment and Perception115

Thestimulusconsistsofasetofalignedlinesegments
(arrangedinthisexampleinanearlyverticalpathtotherightof
theverticalmidline),embeddedinthebackgroundofrandomly
orientedlinesegments.Observersviewedsuccessivepresenta-
tionsoftwopatterns,onecontainingboththetargetpathand
thebackgroundnoiseandonecontainingonlythebackground
noise.Theirorderwasrandomized.Thetaskwastotellwhich
ofthetwopresentationscontainedthetargetpath.
Geisler et al. (2001) varied the length of the path, the
amplitude of the path deviations from a straight line, and path
noisiness (due to the range of random orientations of the line
segments comprising the path) to generate up to 216 classes
of random contour shape. The data from the psychophysical
experiments provided the authors with a “detailed parametric
measurement of human ability to detect naturalistic contours
in noisy backgrounds” (p. 717).
To generate the predictions of contour grouping from the
EC statistics, Geisler et al. (2001) needed a function that
determines which pairs of edge elements group together.
Theauthors derived two such local grouping functions(Fig-
ures4.19E and 4.19F)—one based on the absolute statistic
and one based on the Bayesian statistic—which we explore in
detail in a moment. Because Geisler et al. measured EC for
pairs of edge elements, they used a transitivity rule to con-
struct contours consisting of more than two elements: “if
edge element abinds to b, and bbinds to c, then abecomes
bound to c.” Using this rule, Geisler et al. could predict which
target paths are seen in their stimuli, using the local grouping
functions derived from the statistics of natural images: An
example of grouping by continuation from image statistics
isshown in Figure 4.21B. We consider the Bayesian local
grouping function first, because it requires fewer parameters
than does the absolute local grouping function.

Bayesian Local Grouping Function

Aswesawearlier,thelikelihoodratioateverylocationinthe
(d,)spaceinFigure4.19Dtells,for36orientationdiffer-
ences,howlikelyitisthattheedgeelementsbelongtothe

samecontourasthereferenceelement.Todecidewhethertwo

edgeelementsbelongtothesamecontour,foranyparticular

relationshipd,,betweentheelements,thecorresponding

likelihoodratiocanbecomparedwithacriterion,which

Geisleretal.(2001)calledabindingcriterion,.Asthesig-

naldetectiontheoryprescribes,theidealbindingcriterionis

equaltotheratioofpriorprobabilities(calledpriorodds,as

discussedearlier):

(4)

wherep(C) and p(~C) are the probabilities of two edge ele-

ments to belong or not to belong, respectively, to the same

contour.

TheprioroddswereavailabletoGeisleretal.(2001)di-

rectlyfromtheBayesianECstatistic.Inatrueidealobserver

modelofgroupingbygoodcontinuation,thelocalgrouping

functionwouldhavetocompletelydeterminewhichedgeele-

mentsshouldgroupsolelyfromthestatisticsofnaturalim-

ages(i.e.,withnofreeparametersinthemodel).However,it

turnedoutthatGeisleretal.couldnotusethisoptimalstrategy

becausetheyfoundthatthemagnitudeofvariedasthey

variedtheareaofanalysisintheimage.Inotherwords,the

authorscouldnotfindaunique—anideal—magnitudeof.

Instead,Geisleretal.decidedtoleaveasa(single)freepa-

rameterintheirmodel,justastheobservercriterionisafree

parameterinmodelinghumandataobtainedinadetectionex-

periment.Byfittingthesinglefree-parametermodeltohuman

data,Geisleretal.foundthatthebestresultsareachievedwith

=0.38;theBayesianlocalgroupingfunctionshowninFig-

ure 4.19F was constructed using that best-fitting magnitude of

. Thus, the local grouping function was not truly ideal.

Absolute Local Grouping Function

Because absolute EC statistics do not convey information

about belongingness of edge elements to contours, Geisler

et al. (2001) had to introduce a second parameter, in addition

to binding criterion , in order to derive a local grouping

function from the absolute EC statistics. This new parameter,

calledtolerance,determined how sharply the probabilities of

element grouping fell off around the most likely parameters

of EC shown in Figure 4.19C. For example, low tolerance

implies that grouping occurs only when the parameters

are close to the most common values evident in the absolute

EC statistics. Different values of tolerance result in different

absolute local grouping functions; one is shown in Fig-

ure 4.19E. When fitting the predictions of the two-parameter

absolute local grouping functions to human data, Geisler et

al. were able to obtain almost as good a correlation between

the predicted and the observed accuracies (r=.87) as they

^1 p(C)

p(C)

p(C)

p(C)

Figure 4.21 (A) An example of the path stimulus. (B) The prediction of
grouping in A, from the EC statistics shown in Figure 4.19. Source:Copy-
right 2001 by Elsevier Science Ltd. Reprinted with permission.

[Image not available in this electronic edition.]