Psychonomic Bulletin & Review 1995,2 (3), 279-321

Illusory contours: Toward a neurally based perceptual theory

GREGORY W. LESHER Enkidu Research, Rochester, New York

Although illusory contours were first described nearly a century ago, researchers have only recently begun to approach a consensus on the processes underlying their formation. Neurophysiological and psychophysical evidence indicate that neural mechanisms of the early visual cortex subserve illusory contour generation, although cognitive factors play important roles in determining the fmal percept. I summarize experiments concerning the determinants of illusory contour strength and form, concen­ trating on findings particularly relevant to modeling. After establishing arguments for the early gener­ ation of illusory contours, I provide an overview of formation theories, culminating with descriptions of neural models. The constraints that experimental data place on models are outlined, and neural mod­ els are evaluated with respect to these constraints. Throughout the review,I indicate where further ex­ perimental and modeling research are critical.

Schumann introduced the first illusory contour, depicted ing particular theories of contour formation (although in Figure 1a, in 1900, noting that a central "white rectan­ Parks's "(Mostly) Atheoretical Review" is something ofan gle with sharply defined contours appears, which objec­ exception to this rule). Within the last half decade, how­ tively are not there." Schumann had discovered and noted ever, as neural models ofillusory contour formation have two salient features ofillusory figures: sharp edges in re­ become predominant, most ofthese earlier theories have gions of homogeneous luminance, and a brightening fallen out offavor-although no single neural model has within the figure. Perhaps because this stimulus failed to received universal acceptance. Strong neurophysiological yield particularly convincing illusory contours, it was not evidence, combined with a body of convincing psy­ until more than 50 years later that illusory contours be­ chophysical data, has caused strictly retinal and strictly came a topic for activeresearch. AlthoughEhrenstein (1941) cognitive factors to be less important to the attempt to ex­ demonstrated stimuli with salient illusory contours, it is plain contour formation. Although cognitive factors can remarkable that he failed to note their existence in the text playa profound role in determining the salience of illusory ofhis article explicitly, commenting instead only on illu­ figures, it appears that low- to intermediate-level mecha­ sory brightening effects. Kanizsa's (1955) stunning figures, nisms are both necessary and sufficient to establish the examples ofwhich are depicted in Figures Iband 1c, pro­ perceptual foundation from which the bulk of both psy­ duced extremely sharp, salient contours along with obvi­ chophysical and neurophysiological data can be ade­ ous brightening, initiating a wave ofresearch. quately explained. Although there have been a number ofexcellent illusory Most neural models of contour formation are still in contour review papers (Halpern & Salzman, 1983; Meyer their infancy, lacking the sophistication needed to model & Petry, 1987; Parks, 1984), research advances in the psy­ any but the simplest of illusory phenomena. Even more chophysical, neurophysiological, and modeling domains mature models, such as the boundary contour system have been more than sufficient to mandate a new summary (Grossberg & Mingolla, 1985a, 1985b, 1987a, 1987b),have ofthe literature. Inthis review, I will not only present more yet to demonstrate, through simulations, all the effects that recent data, but also employ an approach to the summaries they are potentially capable ofemulating. However, it is to that is markedly different from those previously taken. At be expected that the additional focus on neural models the time ofthe earlier reviews, determination ofthe locus will greatly accelerate the pace of their development. offormation ofillusory contours was the central issue in Therefore, a comprehensive review ofthe psychophysical illusory contour research, a fact reflected in the structure and neurophysiological data relevant to such modeling ofthe review papers. Experimental results were presented would be ofsignificant value. The purpose ofthis review primarily in the historical context of supporting or refut- is twofold: first, to establish a checklist and summary ofthe determinants ofillusory contour strength, and second, to review and critically examine the various neural models of illusory contour formation. Rather than review the entire I would like to thank Ennio Mmgolla ofBoston University for gener­ body ofillusory contour literature-in their bibliographi­ ously providing many helpful comments and suggestions during the de­ cal review, Purghe and Coren (1992b) have cited some velopment ofthis review. Correspondence should be addressed to G. W Lesher, Enkidu Research, 2250 N. Triphammer Road, #N3A, Ithaca, NY 445 references-I concentrate on summarizing and ana­ 14850 (e-mail: [email protected]). lyzing the experimental findings that one can reasonably

279 Copyright 1995 Psychonomic Society, Inc. 280 LESHER

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Figure 1. (a) Schumann's (1900) original illusory figure; (b) Kanizsa's (1955) triangle; and (c) example ofcurved illusory contours. Figures Ib-lc are from "Margini Quasi-Percettivi in Campi con Stimulazione Omogenea," by G. Kanisza, 1995, Rivista di Psicologia, 49, pp, 7-30 (Figures 10-11). Copyright 1955 by Giunti Gruppo Editoriale. Reprinted with permission. expect to be modeled within the next decade. In areas uating Neural Models, containing an analysis ofthe con­ where our current knowledge is incomplete, the research straints that the psychophysical data place on neural mod­ and experiments necessary to fill the gaps are indicated. els, as well as an evaluation ofthe degree to which the cur­ The general complexity ofneural models ofcontour for­ rent models satisfy these constraints. Finally, in the mation precludes complete computational specification Summary, I present an overview ofthe current state ofre­ within this review.I therefore provide only briefsummaries search and modeling. ofthe models, with a concentration on the motivation and unique characteristics ofeach model. Although the review TYPES OF ILLUSORY CONTOURS is structured with the the neural or computational modeler in mind, I have endeavored also to make it informative for An illusory contour is defined as the percept ofa clear empirical and theoretical researchers. boundary in regions where there is no corresponding lu­ The remainder of the review is partitioned into seven minance gradient, as in the stimuli ofFigures 1 and 2. Illu­ sections, as outlined below. In the section on Types ofIl­ sory contours are modal in nature (Kanizsa, 1955, 1976}­ lusory Contours, I present a definition ofillusory contours they are perceptually salient, appear to belong to the fig­ and analyze the different stimuli that generate these con­ ure rather than the ground, and have a real phenomeno­ tours, with emphasis on monocular stimuli defined solely logical presence. This is an important, if somewhat am­ in the luminance domain. The three major perceptual char­ biguous, characteristic. Immediately after presentation of acteristics of illusory contours and figures-strength, Figure l a, Schumann (1900, 1904) goes on to observe a brightness, and depth-are examined in Characteristics of "subjective contour" connecting the line segments ofFig­ Illusory Figures. In the next section, Determinants ofIl­ ure 3a. Kanizsa refers to this percept and to the lines or lusory Contour Strength, I introduce and summarize the arcs connecting the dots ofFigure 3b as "virtual lines." He various factors that affect the strength ofillusory contours notes that while virtual lines have a definite perceptual and the degree of illusory brightening. Neurophysiologi­ presence, such lines are not nearly as salient, as real, as il­ cal and psychophysical evidence that illusory contours are lusory contours-they are amodal in character. While the instantiated early in the visual processing stream are pre­ distinction is perhaps one of degrees, vision researchers sented in Illusory Contours as Early Vision Phenomena. today would generally not refer to the groupings of Fig­ In Phenomenological Models ofIllusory Contour Forma­ ure 3 as illusory contours. tion, a history ofvarious theories ofcontour generation is The term "boundary" was employed in the definition of tendered, during the course of which additional psycho­ illusory contours to avoid the loaded connotation oflumi­ physical data are presented and a basis for subsequent nance difference associated with the term "edge." In all of neural models is established. These computational models the stimuli ofFigures 1and 2, the illusory contours are boun­ are reviewed in Neural Models of Illusory Contour For­ daries between two perceptual regions. Specifically, in all mation, with the subsequent section, Developing and Eval- but Figure 2c, the illusory contour serves as a boundary b c

Figure 2. (a) Edge-induced illusory contour; (b) line-end induced illusory contour; (c) line-end induced illusory contour without illusory figure. PERCEPTUAL THEORY OF ILLUSORY CONTOURS 281

a b pecially apparent under scotopic conditions, where figure­ ground contrasts are often insignificant. As Rock and Anson (1979) have stated, illusory figures are the solu­ tions to complex perceptual problems: What are the most probable organizations that account for the stimuli? The ecological relevance ofillusory contours renders it highly unlikely that they simply represent epiphenomena ofother visual processes. Rock and Anson defined their ecologi­ Figure3. (a) The other"subjective contour" of Schumann (1900); cal problem in the context ofa cognitive approach to illu­ (b) a series of dots, like Schumann's lines, give rise to a "virtual sory contour formation, but there is no inherent link be­ line" (circle) that does not have the perceptual salience of an illu­ sory contour. tween the perceptual challenge and anyone method of arriving at the answer. We can, for the time being, atheo­ retically pursue the significance ofillusory forms. between a partially specified figure and the inducing ele­ ments that it occludes. In these cases the illusory figure is Edge-Induced Illusory Contours modally observed, with a clear brightness difference be­ The class of illusory contours defined solely by lumi­ tween the figure and the identically contrasted background. nance variations can be further divided into two subclasses. In the dark-on-light examples provided here, there is an il­ Whether or not these two classes indicate distinct under­ lusory brightening, but for light inducers on a dark back­ lying mechanisms for contour completion will be dis­ ground an illusory darkening is observed (Ehrenstein, cussed later in the review. The first ofthese types, typified 1941; Kanizsa, 1955). In Figure 2c, there is no illusory fig­ by Figures 1a and 1b and Figure 2a, consist of solid in­ ure and no consistent brightening or darkening-although ducing elements containing edges or gaps locally consis­ there may be some local brightness effects near the ends tent with a noncoincidental occluding figure ofthe same ofthe individual lines-but the illusory contour still marks luminance as that of the background. That is, without a perceptual boundary between the two regions. needing to postulate a coincidental layout ofthe occluding The definition of illusory contours and figures refers figure(s), one can easily imagine that each gap arises from only to their phenomenological appearance, with no refer­ a simple occlusion. The illusory contour is collinear to those ence to the underlying stimuli responsible for generating the edges of the inducers that are consistent with the afore­ contours. Therefore illusory contours are not bound to any mentioned occlusion, and it persists in regions ofhomo­ one class ofinducing stimuli, but may in fact be generated geneous luminance between edges of adjacent inducers. by a range ofdifferent stimuli configurations in the lumi­ Since completion is in directions roughly collinear to the nance, stereo, and kinetic domains. I will be primarily con­ inducer edges, these types of inducers are referred to as cerned with stimuli defined solely in the luminance do­ edge inducers. The inducer edges that specify the illusory main, such as those in Figures 1 and 2, providing only brief figure by real luminance gradients are referred to by overviews of illusory contours generated in the other do­ Lesher and Mingolla (1993) as "supporting edges," while mains. There are relatively few studies ofintermodally gen­ the portions ofthe illusory figure adjacent to supporting erated illusory contours, but current research appears to in­ edges are referred to as "supported." In Figure 2a, for ex­ dicate a gratifying symmetry between domains-inducers ample, the illusory figure is supported within the missing and the resulting illusory contours appear to interact in sim­ sector ofeach circular inducer. The illusory contour itself ilar ways regardless ofthe inducer modalities (Kellman & is by definition unsupported. Loukides, 1987; Prazdny, 1986). These intermodal similar­ The constraint that edge-inducing stimuli be locally ities, the relative immaturity ofillusory contour research in consistent with occlusion dovetails nicely with the eco­ nonluminance domains, the breadth of research in the lu­ logical motivation for illusory forms, but it should not be minance domain, and the need to limit this paper to a man­ taken as implying that any such occlusion need be per­ ageable scope warrant the concentration on luminance-de­ ceived for illusory contour formation to occur. The con­ fined contours. Nevertheless, a basic understanding ofhow sistency constraint is simply a convenient and intuitive way illusory contours defined in nonluminance domains are re­ to relate a characteristic shared by all edge inducers. Ship­ lated to the more familiar luminance-defined contours will ley and Kellman (1990) may have succeeded in defining be essential to future modeling efforts. local consistency in more concrete, geometric terms by The ecological significance ofillusory figures is clear. showing that discontinuities (comers and line ends) in the Creatures with the ability to perceive illusory forms, to inducer structure are both strong indications ofocclusion disambiguate figure from ground under conditions ofnear and strong determinants ofillusory contour formation. As equiluminance and equicolor, are provided with a strong Figure 4a indicates, supporting edges may be on the inside advantage over creatures that lack this ability. Illusory con­ ofthe illusory form, indicating local self-occlusion ofmark­ tours represent a powerful method for defeating camou­ ings on the figure surface by the edges ofthe figure. Local flage of both predator and prey (Ramachandran, 1986a, consistency with occlusion implies that not all supporting 1987). They doubtless also play important roles in navi­ edges need be consistent with a single coherent occluding gation, providing a more accurate model of the external form, as is evidenced by the examples ofFigure 4b, which world. The benefits ofillusory contour perception are es- have illusory contours supported on both the inside and 282 LESHER

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Figure 4. (a) Inducers on the inside of the illusory figure can also drive illu­ sory contour formation (Kennedy & Lee, 1976). (b) Combinations of inducers on the inside and outside ofthe figure indicate that there need be no global con­ sistency with occlusion. Figures 4a and 4b (left) are from "A Figure-Density Hypothesis and Illusory Contour Brightness," by J. M. Kennedy and H. Lee, 1976, Perception, 5, pp. 387-392 (Figures 4-5). Copyright 1995 by Pion Lim­ ited, London. Adapted with permission. Figure 4b (right) is from "On the Filtered-Components Approach to Illusory Visual Contours," by T. E. Parks and L. Pendergrass, 1982, Perception & Psychophysics, 32, pp, 491-493 (Fig­ ure 2). Copyright 1982 by Psychonomic Society, Inc. Adapted with permission.

the outside ofthe resulting illusory figure (Kennedy, 1979; edge-induced illusory contours also play roles in deter­ Kennedy & Lee, 1976; Parks, 1986; Parks & Pendergrass, mining the strength ofcontours induced by line ends. 1982). In all the stimuli ofFigure 4, the supporting regions Illusory contours may persist if colored line segments are are also locally consistent with a configuration in which added which span the linear or curvilinear gaps between the illusory contour represents a hole in the foreground, corresponding line ends (e.g., a colored cross inserted in through which partially occluded inducers are visible, the center ofan Ehrenstein figure) (van Tuijl, 1975; van even though this may not necessarily be the percept. Tuijl & de Weert, 1979; Varin, 1971). In such stimuli, the While local consistency with occlusion marks a general line color may spread to fill the illusory figure-a phe­ constraint on the existence of illusory contours, there are nomenon known as neon color spreading. There is a cor­ more specific characteristics that profoundly influence il­ relation between the degree ofneon color spreading and the lusory contour formation. Most importantly, the shape of strength of the bounding illusory contour (Watanabe & the illusory figure typically must be well specified by the Takeichi, 1990), although the degree ofneon color spread­ inducers-points ofhigh curvature must fall within the part ing is also dependent on stimulus characteristics unrelated ofthe figure supported by inducer edges. This implies that to illusory contour formation, such as the degree ofalign­ multiple supporting edges are necessary, and that adjacent ment between inducing lines and colored segments supporting edges must be at least partially aligned with one (Redies, 1989; Redies & Spillman, 1981; Redies, Spill­ another. Additional factors that affect the formation ofillu­ man, & Kunz, 1984; Watanabe & Sato, 1989). sory contours and their resulting strengths include the de­ There is no clear classification boundary between edge gree ofalignment ofsupporting edges, number ofinducers, and line-end induced contours. A line always has some degree of support, inducer contrast, and inducer shape. In width and therefore provides edge information, albeit at a addition, such cognitive determinants as perceptual set and very small scale. Conversely, an edge inducer has two cor­ memory may influence contour formation. These, and ners, the nonsupporting edges ofwhich are often roughly other, factors will be discussed in detail later in the review. perpendicular to the supporting edges, and thus provides something akin to line-end information. Gradually increas­ Line-End Induced Illusory Contours ing the width ofa line results in a smooth progression from As the region ofsupport becomes smaller in relation to line end to edge inducer, with clear perception ofan illu­ the illusory contour length, the inducing elements progress sory contour throughout the entire transition (Lesher & from edge inducers to line-end inducers, examples of Mingolla, 1993; Petry, Harbeck, Conway, & Levey, 1983). which are depicted in Figures 2b and 2c. Line-end induc­ The role ofedges and line ends in illusory contour forma­ ers must fulfill the same basic criteria as edge inducers re­ tion poses one ofthe fundamental questions in modeling garding sufficient specification ofthe illusory figure shape, these phenomena: Do the two types ofinducers imply the but the resulting illusory contours are not collinear with existence of two completely separate mechanisms or ex­ the regions ofsupport, but rather in directions roughly per­ tremes of the same mechanism? Alternatively, we might pendicular to the supporting lines. Whereas with edge in­ ask: Do both parallel and perpendicular completion occur ducers the illusory figure is tightly constrained in both po­ during illusory contour formation? sition and orientation within supported regions, in line-end illusory contours the inducers primarily constrain posi­ Other Types ofIllusory Contours tion-orientation tends to be approximately perpendicular The definition ofillusory contours provided above-a to the inducing lines, but it may vary significantly from this clear boundary in a region ofhomogeneous luminance­ angle (Gillam, 1987; Gillam & Goodenough, 1994; Parks, captures several phenomena not traditionally considered 1980). Most ofthe factors that influence the perception of illusory contours. In particular, illusory contours induced PERCEPTUAL THEORY OF ILLUSORY CONTOURS 283 in the stereo and kinetic domains have not typically been tion exists when the inducers are static-they have as­ recognized as such. Ithas long been known that the strength pects ofedge or line-end stimuli-but not until the induc­ of illusory contours induced by both edge and line-end ers move does the illusory contour become salient. Note stimuli can be enhanced by providing additional stereo or that an experienced observer may report the illusory con­ motion cues, such as to increase the salience of the per­ tours even without motion. A number of researchers ceptual boundary between illusory figure and inducing (Kanizsa, 1979; Petter, 1956) have noted a related class of elements (Gregory & Harris, 1974;Lawson, Cowan, Gibbs, luminance-defined stimuli, referred to as "spontaneously & Whitmore, 1974; Prazdny, 1985; Ramachandran, 1986a, splitting figures" by Kellman and Shipley (1991), that 1987; Ramachandran & Cavanagh, 1985; Whitmore, Law­ have multiple potentials for completion that are only real­ son, & Kozora, 1976). In these examples, contours were ized under certain perceptual (as opposed to stereo or ki­ already established in the monocular, luminance domain­ netic) conditions-the perceived depth ordering of the stereo and kinetic cues only mediated their strengths. How­ figures determines the contour visibilities. ever, illusory contours can be induced without any sup­ An understanding ofillusory contours defined in non­ porting luminance discontinuities whatsoever. luminance domains is essential to the development of a In the stereo domain, Julesz's (1960) random dot stere­ global model of illusory contour formation and of early ograms give rise to strong illusory contours at the edges of vision in general. Using a tilt aftereffect paradigm, Para­ the disparate regions, although they were not recognized diso, Shimojo, and Nakayama (1989) showed that there is as such until much later (Julesz & Frisby, 1975). In these some cross-adaptation between real and illusory contours. cases, illusory contours are observed only in regions where Berkley, Debruyn, and Orban (1994) have significantly there are local depth disparities (i.e., areas proximal to el­ extended this line ofstudy to establish additional interac­ ements with definite disparity), but Prazdny (1986) .and tions with motion-defined contours. Only by examining Mustillo and Fox (1986) have reported illusory contour interactions between different modalities can we hope to formation in random dot stimuli in which only the edge in­ understand how the brain organizes complex scenes into ducers ofa Kanizsa figure are defined by local disparity. unified percepts. The design of a model for luminance­ Contour completion occurs between these stereoscopically defined illusory contour formation requires that one be defined inducers across areas with no local luminance or aware ofthe larger visual scheme, allowing for future de­ depth discontinuities-a remarkable fact when one con­ velopments and extensions to integrate contours defined siders that in the completed regions there is substantial ev­ in various domains. idence, in the form of dot pairs specifying depth behind the inducing elements, against the existence ofan occlud­ CHARACTERISTICS OF ing illusory figure. Similarly, in the kinetic domain, illu­ ILLUSORY FIGURES sory contours can be specified by local motion cues (An­ dersen & Cortese, 1989; Bruno & Bertamini, 1990; Bruno At least three measurable perceptual characteristics are & Gerbino, 1991; Kaplan, 1969; Kellman & Loukides, associated with illusory figures: contour strength, figure 1987; Shipley & Kellman, 1994; Stappers, 1989) or can be brightness, and figure depth. Strength is defined as the induced with no local cues, but with motion-defined in­ salience or clarity ofthe illusory boundary itself. Illusory ducers (Kellman & Loukides, 1987; Prazdny, 1986). Cases brightness (or darkening) often accompanies illusory fig­ ofillusory contour generation in the absence oflocal stereo ures and is typically perceived as a homogeneous light­ or kinetic cues are examples ofsecond-order contour com­ ness increase within the illusory figure. Illusory figures pletion. Local cues giverise to inducers, themselves illusory, are often perceived nearer to the observer than their in­ which then behave analogously to luminance-defined in­ ducers, thus the depth ofthe figure marks another consis­ ducers, resulting in illusory contourcompletion across re­ tently measurable characteristic. gions with no local stereo or kinetic cues. There have been Of the three measures of illusory contours, illusory no experiments to determine whether such second-order brightening is the easiest to characterize and was the ear­ inducers behave equivalently to luminance-defined in­ liest to be quantified. In the first study ofillusory bright­ ducers under parametric variations ofinducer characteris­ ening, Ehrenstein (1941) noted qualitatively a number of tics (e.g., inducer size and distance between inducers). stimulus variations that affected brightness. A wide range In addition to contours defined solely in the stereo or of quantitative brightness reports followed, including kinetic domains, several classes ofillusory contours must measurements by rating scale and magnitude estimation be defined in both the luminance and the motion domains. (Dumais & Bradley, 1976; Halpern, 1981; Jory & Day, These include moving phantoms (Maguire & Brown, 1979; Petry et al., 1983), matching (Brussel, Stober, & 1987; Mulvanny, Macarthur, & Sekuler, 1982; Tynan & Bodinger, 1977; Spillman, Fuld, & Neumeyer, 1984), Sekuler, 1978; Weisstein, Maguire, & Berbaum, 1977), nulling (Halpern, 1987; Spillman et al., 1984), and thresh­ stereokinetic contours (Bressan & Vallortigara, 1986, 1987, old increment levels(Coren & Theodor, 1975;Dresp, 1993; 1991; Vallortigara, 1987; Vallortigara, Bressan, & Zanfor­ Dresp & Bonnet, 1991; Jory, 1987). Brightness measure­ lin, 1986; Zanforlin & Vallortigara, 1990), and rotation­ ments appear to be consistent across different types of induced contours (Klymenko & Weisstein, 1981, 1983, measurement procedures (Dresp, 1992; Spillman et al., 1984,1987; Klymenko, Weisstein, & Ralston, 1987). All 1984). The most noticeable ofillusory contour character­ these stimuli are similar in that the potential for comple- istics in such common figures as Kanizsa's and Ehren- 284 LESHER

stein's, illusory brightness received a great deal ofearly at­ be invaluable for ensuring the reliability offuture psycho­ tention and was often invoked as the causal factor in illusory physical experiments. Even with carefully defined mea­ contour formation, as opposed to a consequence of illu­ sures, ratings ofillusory contour strength are notorious for sory figure formation. Indeed, researchers have demon­ extravagant intersubject variations, particularly among strated a high degree ofcorrelation between simultaneous naive subjects. Illusory contours represent a unique and brightness contrast and illusory brightening in edge­ unfamiliar form ofvisual stimulation, so it is not surpris­ induced illusory figures (Brigner & Gallagher, 1974; Dresp, ing that many inexperienced subjects have difficulty es­ 1992;Dresp, Lorenceau, & Bonnet, 1990),although bright­ tablishing coherent determinations of contour strength. ness contrast alone is insufficient to explain the phenom­ What exactly are subjects rating in strength/clarity ex­ enon (Coren & Theodor, 1975; Frisby & Clatworthy, 1975). periments? Are all the definitions above synonymous? The depth ofillusory figures was the next feature to re­ "Strength" and "salience" are measures ofperceptual rec­ ceive widespread attention, as Coren (1972) claimed for it ognizability and pronouncedness, while "sharpness" is a an important explanatory role in illusory contour forma­ measure of spatial spread. The other common terms, tion. Subsequent studies using rating and magnitude scales "clarity" and "distinctness," fall somewhere in between (Bradley & Dumais, 1984; Halpem, 1981; Salzman & Hal­ these two extremes. Surely pronouncedness and spatial pern, 1982; Whitmore et al., 1976), matching (Coren & spread are at least somewhat related, perhaps explaining Porac, 1983), and nulling (Gregory & Harris, 1974; Whit­ the consistency between subjects given different defini­ more et aI., 1976) verified that traditional edge-induced il­ tions, but they do not represent the same illusory contour lusory figures (such as Kanizsa's figure) appear to hover measure. Diffuse illusory contours, such as those ofFig­ in front ofthe inducers for most observers. There is some ure 5, have very fuzzy (unsharp) boundaries (Kennedy, evidence that line-end illusory figures also appear strati­ 1976). A weak illusory contour does not appear to have fied in depth, although to a lesser extent than do their edge­ this type offuzziness, as would be required by a "sharp­ induced counterparts (Coren & Porac, 1983). Great store ness" definition, but rather has less perceptual wallop (or was originally placed in the perceptual salience of the salience). Because ofthe apparent consistency across sub­ depth effect, because it was apparently strong enough to jects given different definitions of contour strength, the invoke size constancy for small elements inside the illu­ difference between these two definitions is not of para­ sory figure (Coren, 1972; Parks, 1985; Porac, 1978), but mount importance for psychophysics, but it does have se­ in subsequent studies it was pointed out that the Ebbing­ rious implications for the representation of illusory con­ haus effect-in which the perceived size ofa central form tours in neural models-amplitude (or energy) ofneural is mediated by the size ofsurrounding elements-was at activity and degree ofspatial spread are distinctly differ­ least partially responsible for the original findings (Parks, ent quantities. To date, all neural models have implicitly 1987; Predebon, 1985). Nevertheless, depth effects are gen­ employed the former representation, not even considering erally robust and are associated with most illusory figures. the alternatives. For the purpose of this paper, I will use Most researchers in the modeling community are con­ the terms "strength" and "clarity" interchangeably in dis­ centrating on modeling the strength or clarity ofillusory cussing this illusory contour measure. It is reasonable to contours. Unfortunately, strength is by far the most diffi­ assume that contour sharpness represents a fourth charac­ cult ofthe three measures to define. Whereas brightness teristic of illusory figures. However, because of the ap­ and depth, terms applicable to generic visual scenes, have parent high degree of correlation between strength and standard and well-specified definitions, it seems that the sharpness, and the lack ofempirical studies ofsharpness only consistent definition given to strength is "that char­ per se, there will be no further explicit discussion ofillu­ acteristic which is not brightness or depth." Observers have sory contour sharpness in this paper. been instructed to rate strength (Day & Kasperczyk, 1983), salience (Dumais & Bradley, 1976), clarity (Halpern, 1981; Shipley & Kellman, 1992a), distinctness/sharpness (Petry et aI., 1983), perceived contrast (Banton & Levi, 1992), and several other variants ofthese terms, while matching procedures have used benchmark stimuli ofquestionable equivalence. Subjects in different experiments not only are provided with a variety ofclaritydefinitions, but often are given only vague indications ofwhat exactly they are rat­ ing. In spite of this lack of consistency and specificity, there are few examples of subjects reporting confusion about what to look for or rate. In addition, data appear to be relatively consistent across subjects and experiments Figure5. Diffuse illusory contours have brightness, but no clar­ done with different methods and definitions, although ity or depth. From "Subjective Contours, Contrast, and Assimila­ there has been no systematic study ofthe correlation be­ tion," by J. M. Kennedy, 1979, in C. F. Nodine and D. F. Fisher (Eds.), Perception andPictorialRepresentation (pp. 167-195, Fig­ tween the various definitions and measurement tech­ ure 9.12), New York: Praeger. Copyright 1979 by Praeger Pub­ niques. Establishment ofa standard definition ofcontour lishers. Reprinted with permission of Greenwood Publishing strength and an associated measurement technique would Group, Ine., Westport, CT. PERCEPTUAL THEORY OF ILLUSORY CONTOURS 285

While brightness, depth, and clarity all mark measurable vidual roles in contour formation. An overview ofthe cur­ traits of illusory figures, there has been little agreement rent psychophysical data on low-level determinants ofil­ about their independence or their role in illusory contour lusory contours is provided in the following section. formation. Coren (1972) claimed that perceived differ­ In contrast to low-level determinants, high-level deter­ ences in depth drove illusory contour formation and thus minants consist ofconfigurational factors generally asso­ determined brightness and clarity. Others claimed that ciated with cognitive phenomena. These factors cannot be brightness drove the formation and thus determined depth defined in simple geometric terms, but require reference and clarity (Brigner & Gallagher, 1974;Brussel et aI., 1977; to abstractions and perceptual definitions. The degree of Frisby & Clatworthy, 1975). Cognitivists and Gestaltists completeness ofthe inducing elements is an oft-cited ex­ made the claim that boundaries were formed first, fol­ ample ofa high-level determinant. Also in this class are lowed by depth and brightness (Gregory, 1972; Kanizsa, the perceived depth of the figure and the effects of per­ 1955; Rock & Anson, 1979). We can now safely say that ceptual set, attention, memory, and recognizability ofthe all these narrow views are incorrect. There are illusory illusory figure. Many ofthese complex factors cannot eas­ contours without depth or brightness (Ware & Kennedy, ily be varied parametrically, but still assist in revealing the 1977, 1978a), illusory figures without clarity or depth (as internal representation ofillusory contours. An overview in Figure 5)(Kennedy, 1976, 1978b, 1979, 1981; Meyer & ofhigh-level determinants is provided after the summary Dougherty, 1987; Parks, 1981, 1984; Richardson, 1979), oflow-level factors, followed by a discussion ofthe eco­ illusory contours with clarity and depth but no brightness logical significance ofboth types ofdeterminants. The ex­ (Coren, 1972; Prazdny, 1983; Shapley & Gordon, 1987), isting neural models of illusory contour formation have and illusory contours with clarity and brightness but no been concerned primarily with modeling ofspecific low­ depth (Ware & Kennedy, 1978a). In short, every combi­ level determinants. However, development of the next nation but those involving depth without clarity has been generation of models (or model extensions) will require observed. There is a great deal ofindependence between an understanding ofhigh-level determinants, as well as of clarity, brightness, and depth. Watanabe and Oyama how the models should be integrated into a comprehensive (1988) employed the causal inference method to deter­ visual processing architecture. mine to what degree, ifany, these measures depended on one another. They found depth to be dependent on both Low-Level Determinants clarity and brightness, and brightness to be somewhat de­ The low-level determinants discussed below are, for the pendent on clarity. However, of the three illusory figure most part, well-quantified findings whose specifications characteristics, illusory brightness is most directly related engender a general acceptance in the illusory contour re­ to a reasonably well-understood, low-level phenomena­ search community. In particular, in the domain of edge­ for edge inducers only, the degree ofbrightening is highly induced illusory contours, the effects of inducer spatial correlated with the degree of simultaneous brightness extent and position, inducer luminance, and alignment of contrast near the inducers (Dresp, 1992; Dresp et aI., 1990). inducers are understood to a degree commensurate for quantitative modeling. These parameters have not all been DETERMINANTS OF ILLUSORY well quantified for line-end inducing stimuli, however, CONTOUR STRENGTH limiting the degree to which accurate specification ofthe operational characteristics ofmodels ofcontour formation The definition ofedge inducers included the constraint is possible. In addition, a more thorough understanding of that the layout ofillusory contour inducers must be locally such subtle effects as interscale interactions and the rela­ consistent with occlusion of the inducers by the illusory tion between edges and line ends is required so that the figures (or with self-occlusion). This characteristic, obvi­ structure and mechanisms of models ofillusory contour ously related to the ecological significance of illusory formation can be constrained further. forms, is a general requisite for the formation ofillusory Spatialextentand proximityofinducers. The role of contours, but there are also many specific determinants of the spatial extent (or retinal size) ofthe inducing elements an illusory contour's strength and an illusory figure's has always been a topic ofintense research, but the recent brightness. The structural determinants ofcontour strength publications ofShipley and Kellman (1992b) and Banton can be divided into two classes. The first ofthese encom­ and Levi (1992) indicate a renewed interest in this area. In passes simple, configurational factors ofthe inducing el­ discussing inducer spatial extent, one must recognize that, ements, and thus I refer to these factors as low-level. Low­ for a given stimulus, several spatial factors may be varied. level stimuli determinants are describable in concrete, For example, in a Kanizsa figure, the inducers may be geometric terms without recourse to such perceptual ab­ moved apart with no change in size, the inducers may be stractions as completeness or depth. In general, low-level enlarged without movement, or the entire stimulus may be factors can be varied parametrically. Luminance, spatial enlarged. These variations are depicted in Figure 6. Note alignment, spatial extent, separation, and quantity of in­ that these three types ofspatial variation are not indepen­ ducers are all examples oflow-level determinants. Clarity dent-anyone type can be achieved by manipulations of and brightness typically vary smoothly with incremental the other two varieties. For example, a decrease in the reti­ variations in these parameters, making low-level determi­ nal size ofthe figure can be achieved through decreases of nants suitable for parametric investigation of their indi- appropriate proportions in both inducer spacing and ra- 286 LESHER

a c "Figure 6. Spatial•variations ofedge inducers: (a) inducer spacing, (b) inducer radius, (c) retinal size. dius. Early studies tended to be concentrated on a single set (1976) they observed no effect ofretinal size. Shipley and of spatial variations, but more recently, researchers have Kellman (1992b) verified this controversial finding by attemptedto integrateall three sets intoa single determinant. performing forced choice experiments in which pairs of Dumais and Bradley (1976) first studied the effect of stimuli were presented that differed only in retinal size. retinal size on contour strength, finding that strength - Note that the stimuli employed resulted in illusory figures decreased monotonically for Kanizsa triangles with in­ that ranged in retinal size only from 0.7° to 2.8°, so that creasing retinal size (from 1.2° to 18.9° ofvisual angle). there was little overlap with the larger retinal sizes em­ Employing Ehrenstein figures, Siegeland Petry (1991) pro­ ployed by Dumais and Bradley. By plotting clarity ratings duced similar results for large visual angles-a decrease as a function of the support ratio, which in the case of a in both clarity and brightness-but found that clarity ac­ Kanizsa square is twice the inducer radius divided by the tually increased as retinal size increased from 0.25° to 1.5° center-to-center inducer separation, Shipley and Kellman when the Ehrenstein figure had eight or more inducing (1992b) discovered that clarity was a linearly increasing lines. Watanabe and Oyama (1988) found that clarity and function ofthe support ratio, at least in the case of solid, brightness decreased monotonically as they increased the edge inducers. separation between the sectored circle inducers in a Ka­ Banton and Levi (1992) performed similar experiments nizsa figure without varying any other spatial parameters. by varying Kanizsa figure inducer separation and inducer By varying the width of inducing lines in an Ehrenstein size. They present experimental results which indicate that figure, Petry et al. (1983) and Sambin (1985) found that local support ratio-the support ratio along each side ofa clarity was a monotonically increasing function of the Kanizsa rectangle-is the determining factor of contour width of the inducers, while brightness exhibited an strength, as opposed to a global support ratio which would inverted-U shape. Dresp (1992) investigated the effect of encompass the perimeter of the entire illusory figure. In Kanizsa (edge) inducer separation and size on illusory these experiments, Banton and Levi established that con­ brightness, finding that brightness was an inverse function tour clarity, as measured by a matching procedure, was an of inducer proximity and a positive function of inducer increasing function ofthis local support ratio. This rela­ size. In general, these studies have indicated that both tion was monotonic but not linear, in contrast to Shipley illusory contour strength and illusory figure brightness and Kellman's (1992b) findings. Clarity was roughly a increase with increasing degrees of inducer support and linear function ofsupport ratio when plotted on a log-log decreasing distance between inducers. scale, implying that clarity and support ratio are related by Shipley and Kellman (1992b) have attempted to unify a power function: inducer separation, inducer size, and retinal size ofthe stim­ clarity oc [I - support ratio] -1.48. (1) uli into a single determinant. They posit that the only rel­ evant spatial factor is the ratio of the length of supported This curve matches the linear function ofShipley and Kell­ contour to the length ofthe entire illusory contour, a pa­ man reasonably well for low and intermediate support ra­ rameter referred to by Lesher and Mingolla (1993) as the tios, but it diverges significantly for high support ratios. A support ratio. Shipley and Kellman (1992b) created stim­ possible source ofthis variance is the difference in clarity uli similar to those ofFigure 6 by varying retinal size, in­ measurement techniques-Banton and Levi used an un­ ducer spacing, and inducer size independently, and had conventional technique wherein the luminance of a real subjects rate the resulting illusory contour clarities. Like line was matched to the illusory contour; Shipley and Kell­ Watanabe and Oyama (1988), they found that clarity de­ man used rating scales. Also, Banton and Levi's paramet­ creased with increasing inducer spacing and decreasing ric fit seems unduly biased by several entries at very low inducer size. However, in constrast to Dumais and Bradley support ratios. Although the data ofthe two research groups PERCEPTUAL THEORY OF ILLUSORY CONTOURS 287 do not coincide perfectly, both indicate the same general sory brightness increased monotonically with increasing trend for edge inducers: a monotonic increase in contour line width, for a constant number oflines. However, nei­ clarity as support ratio increases. ther strength nor brightness varied monotonically with The effect ofsupport ratio on line-end induced contour support ratio--support ratio was not a reliable predictor of strength has been insufficiently studied, but the data of either quantity, exhibiting varied trends for different num­ Petry et al. (1983) indicate that it may also play an impor­ bers oflines and inducer radii. Therefore, Shipley and Kell­ tant role in determining contour strength in these cases as man's (1992b) hypothesis that support ratio determines well. Employing stimuli similar to those ofFigure 7, and contour strength is invalid for line-end inducers. The du­ excepting very thin lines, they found clarity to increase to ality ofedges and lines (edges as thick lines, lines as thin a saturated plateau with increasing support ratio, defined edges) leaves some doubt about the validity ofsupport ratio for their Ehrenstein figures as the product ofthe number theories in all but the most constrained circumstances. oflines and line width, divided by the illusory figure cir­ Number of inducers. The number of inducing ele­ cumference. This effect has been replicated by Sambin ments has close relation to the spatial extent ofthe induc­ (1985), using an Ehrenstein figure, but it has been contra­ ers-both factors affect the support ratio, especially for dicted by recent studies done with other forms ofillusory edge inducers. Using solid semicircular inducers ofvary­ stimuli (Lesher & Mingolla, 1993; Soriano, Spillman, & ing number and size,Mast and Fox (1994) demonstrated that Bach, 1994). Employing a phase-shifted line grating as in contour clarity increases monotonically with increasing Figure 2c, Soriano et al. (1994) found that contour saliency support ratio, independently ofthe number ofedge induc­ tended to decrease monotonically with increasing line ers. Although this has been the only explicit study ofthe width. Primary differences between Ehrenstein figures and effect of the number of edge inducers, there have been grating figures include the curvedness of the contour in several investigations ofthe number ofline-end inducers the former and the lack ofillusory brightness in the latter, on clarity and brightness, using both thin and thick lines. but it is not clear whether either of these differences ac­ As is indicated in Figure 7, Petry et al. (1983) varied the counts for the discrepancy in the data. number, as well as the widths, ofinducing lines in Ehren­ Lesher and Mingolla (1993) employed variants offig­ stein figures. They found that increasing the number of ures first presented by Varin (1971), consisting ofa Ka­ lines had very little effect on contour clarity but resulted nizsa square with concentric circles substituted for the in a monotonic increase in brightness. This finding con­ solid circle inducers, as depicted in Figure 8. By indepen­ flicts with those of a number of other researchers, all of dently varying the width (8a ~ 8b) and number oflines whom report a monotonic increase in both clarity and (8b ~ 8c), one can study the effects ofinducer width, sup­ brightness with increases in the number oflines in Ehren­ port ratio, and number of inducers. With many thin in­ stein figures (Ehrenstein, 1941; Frisby & Clatworthy, 1975; ducing lines, the Varin figure consists solely of line-end Sambin, 1985; Siegel & Petry, 1991). The difference inre­ inducers, whereas with a single thick inducing line for each sults probably stems from the particular line widths used. sectored circle, the figure consists solely ofedge inducers. Petry et al. employed a series ofdifferent line widths, most Lesher and Mingolla found that contour strength and illu- ofwhich were thicker than those used by the other exper- a --I-- I I b I I Figure 7. Increasing support ratio by (a) increasing the line widths for a con­ stant number of lines and (b) increasing the number of lines for a constant line width. From "Stimulus Determinants of Brightness and Distinctness of Sub­ jective Contours," by S. Petry, A. Harbeck, J. Conway, and J. Levey, 1983, Per­ ception & Psychophysics, 34, pp.169-174 (Figure I). Copyright 1983 by Psy­ chonomic Society, Inc. Reprinted with permission. 288 LESHER

abc ~~~~Cf'~ ~~~~C0~

Figure 8. Variants of figures presented by Varin (1971) provide a means by which to examine the roles that line ends and edges play in illusory contour formation.

imenters. In the restricted case of very thin lines, Petry the absolute clarity decreased with illumination levels, et al. did observe that the number oflines affected clarity, Warm et al. also observed that the clarity ofillusory con­ although not as pronouncedly as was observed in the other tours increased relative to that ofreal contours. experiments. Soriano et al. (1994) confirm a monotonic in­ Several studies have shown that there need be no global crease in contour saliency with increases in the number of agreement between inducer contrasts to instantiate illu­ lines in phase-shifted grating figures. sory contours (Brigner, 1982; Grossberg & Mingolla, With the stimuli ofFigure 8, however, Lesher and Min­ 1985a; Kellman & Loukides, 1987; Prazdny, 1983;Shap­ golla (1993) found that for constant line widths, both con­ ley & Gordon, 1987). That is, not all the inducers need be tour strength and illusory brightness were inverted-U of the same direction of contrast; alternating dark and functions of the number of inducing lines. Soriano et al. light inducers on an intermediate (gray) background are (1994) have observed a similar trend while holding the also efficacious, as is indicated by the stimuli ofFigure 9. number oflines constant and varying the spacing between Prazdny (1983) originally reported that in the stimuli of adjacent lines. In Lesher and Mingolla's experiment, line Figure 9a, illusory brightness was markedly diminished, if spacing necessarily decreased when the number of lines not completely eliminated. Hershberger and Stallard increased, owing to the constraint that the largest ofthe (1984) reported that, paradoxically, Prazdny's figure actu­ concentric circles always retained the same radius. In the ally appeared brighter than it would if composed of all experiment ofSoriano et al., the extent ofthe contournec­ dark inducers. They defined a parameter, called "contrast essarily increased with increasing line spacing, owing to variability," which takes into account not only the contrast the constraint that the number oflines remained constant. between the inducers and the background, but also that be­ In both experiments, possible confounds with the other tween the individual inducers. Hershberger and Stallard interdependent factors render it impossible to determine reported that illusory brightness increased with increasing which factor or factors are responsible for the inverted U. contrast variability, but they failed to control for the effect Additional experiments with better controls and interde­ ofspatial inducer variations: To change the contrast vari­ pendence analyses are required in order to tease out the ef­ ability they moved the boundaries between the dark and fects of line spacing, number of lines, and extent of the light portions ofthe inducers, thereby altering the spatial contour on illusory contour strength and illusory bright­ configuration as well as the contrast. More recently, Kell­ ening. In any event, the inverted U ranks as an important man and Loukides (1987) have employed stimuli similar and surprising new finding that must be reconciled with to that ofFigure 9b to demonstrate that illusory brightness neural models ofcontour formation. may be completely eliminated by allowing subjects to bal­ Inducer luminance and contrast. The effect of in­ ance the luminance of the positive and negative contrast ducer luminance and contrast on illusory contour clarity inducers. The exact nature and extent ofbrightness effects and brightness has been a topic oflively debate. Dumais in such mixed-contrast stimuli remain an open issue. and Bradley (1976) initially found that contour clarity in­ Although there need be no contrast agreement between creased with decreasing ambient illumination of a Ka­ inducers, local contrast between inducer and background is nizsa figure, a finding supported by Brussel et al. (1977). extremely important. Most researchers have reported the However, these results have been cast into doubt by Warm, absence of illusory contours when inducers are isolumi­ Dember, Padich, Beckner, and Jones (1987), Spillman et al. nant with the background (de Weert, 1983; Gregory, 1977; (1984), and others (Parks & Marks, 1983; Watanabe & Livingstone & Hubel, 1988; Watanabe & Sato, 1989), al­ Oyama, 1988). Warm et al. found that the clarity ofa Ka­ though Ejima and Takahashi (1988) noted that illusory con­ nizsa figure decreased with decreasing illumination, while tours generated by abutting line gratings (as in Figure 2c) Spillman et al. (1984) discovered the same relation be­ can be perceived under certain isoluminant conditions. tween illusory brightness and illumination in an Ehren­ De Weert (1983) presented a detailed study ofthe effect of stein configuration. Warm et al. hypothesize that subjects color and color differences on illusory contour formation. in the experiment ofDumais and Bradley (1976) were rat­ Inducer alignment. Kanizsa (1955) noted that one of ing illusory contour clarity with respect to real contour the conditions for the formation ofillusory contours was clarity/sharpness at the same illumination level. Although proper alignment between the supporting edges ofthe in- PERCEPTUAL THEORY OF ILLUSORY CONTOURS 289

Figure 9. Inducing elements of varying contrast polarity can generate strong illusory contours with limited amounts of illusory brightening. (a) Prazdny's (1983) influential figure. (b) Stimulus similar to those employed by Hersh­ berger and Stallard (1984) to demonstrate that there can be illusory brighten­ ing associated with such figures. Figure 9a is from "Illusory Contours Are Not Caused by Simultaneous Brightness Contrast," by K. Prazdny, 1983, Perception & Psychophysics, 34, pp. 403-404 (Figure 1B). Copyright 1983 by Psychonomic Society, Inc. Figure 9b is adapted from "Contrast Variability Lightens Subjec­ tive Figures," by W. Hershberger and S. Stallard, 1984, Perception & Psy­ chophysics, 36, pp, 92-94. Copyright 1984 by Psychonomic Society, Inc. Fig­ ure 9a is reprinted, Figure 9b adapted, with permission. ducers-that is, that illusory contours are observed only ley & Kellman, 1992a). Misalignment ofstraight inducing when the continuation ofeach supporting edge meets an­ edges, however, weakens the resulting illusory contour. other supporting edge. This continuation reflects the cur­ Bross and Michelangeli (1988) attempted to quantify vature ofthe supporting edges, as is indicated by the curved the effect ofedge misalignment by studying the time that illusory figure ofFigure Ic. Kanizsa presented only stim­ it took subjects to perceive an illusory form when pre­ uli in which the continuations from the different support­ sented with Kanizsa stimuli having inducers rotated so ing edges coincided perfectly. For example, perfectly that the edges ofthe missing sectors did not align. Reac­ aligned straight-edged inducers led to straight illusory tion delays lengthened as alignment decreased, allegedly contours, and perfectly aligned curved-edged inducers led indicating weakened contours. In a more comprehensive to curved illusory contours. By presenting Figure lOa, in and convincing paper, Shipley and Kellman (1992a) pre­ which straight inducers lead to curved illusory contours sented experiments in which the inducers of a Kanizsa rather than straight contours meeting at a vertex, Pastore square were shifted laterally, with concomitant variation (1971) and Gregory (1972) demonstrated that illusory oforientation ofthe missing inducer sectors, as depicted contour formation involved more than simple continua­ in Figure lOb. They found that contour strength decreased tion of supporting edges, a finding confirmed by others with misalignments and that it was highly correlated with (Bross & Michelangeli, 1988; Rock & Anson, 1979; Ship- what they refer to as inducer "relatability," Supporting

a •• b...... f" ~ f" 'If" 'I ".t,,~,,~ ••Figure 10. Curved illusory contours can result when two straight support­ ing edges are misaligned. (a) Gregory's (1972) original stimulus. From "Cog­ nitive Contours," by R. L. Gregory, 1972, Nature, 258, pp. 51-52 (Figure 3). Copyright 1972 by Macmillan Magazines Limited. Reprinted with permis­ sion. (b) Shipley and Kellman (1992a) varied the degree and type of edge mis­ alignment to determine how well subjects' ratings of contour strength corre­ sponded with "relatability" ofthe edges. From "Perception of Partly Occluded Objects and Illusory Figures: Evidence for an Identity Hypothesis," by T. F. Shipley and P.J. Kellman, 1992, Journal ofExperimentalPsychology: Human Perception & Performance, 18, pp. 106-120 (Figure 6). Copyright 1992 by American Psychological Association. Reprinted with permission. 290 LESHER edges are relatable if their extensions intersect with an The relative positions of line ends appear to playa oblique angle, as Figure II depicts (Kellman & Shipley, greater role in determining whether illusory contours will 1991). Contour strengths are determined by the degree to appear than does the orientation of the lines with respect which the relatability criteria are fulfilled, as well as by the to the illusory contours. If an excessive amount ofcurva­ acuteness of the intersection angle-the more acute the ture is required in order to connect the line ends, there will angle, the weaker the contour. Note that Kellman and Ship­ most likely be no illusory contour (Gillam, 1987; Row­ ley do not claim that the illusory contours take the simple bury, 1982; Shipley & Kellman, 1992a). In both edge and piecewise-linear paths used to determine relatability. The line-end induced contours, completion in the form of a role of relatability in Kellman and Shipley's model of il­ smooth curve is apparently preferred to completion with lusory contour formation is discussed further in the pre­ curvature discontinuities (Gregory, 1972; Shipley & Kell­ sent paper in the section on Neural Models of Illusory man, I992a). However,in rare cases, illusory contours with Contour Formation. unsupported corners or discontinuities may be observed For line-end inducers, the resulting illusory contour is (Coren, Porac, & Theodor, 1986). For example, some ob­ roughly perpendicular to the inducing lines. Inducer align­ servers report a square illusory figure in the Ehrenstein ment for this type ofstimulus refers to alignment between stimulus ofFigure 2b, rather than a circle. perpendiculars to the line ends. Kennedy (1978a) varied the SmaU-scale facilitation and interference. Informa­ slant ofthe inducing lines in an Ehrenstein figure from per­ tion on a scale much smaller than that ofthe inducing el­ pendicular to the edge ofthe illusory circle to tangent to the ements can have a profound effect on the resulting illusory circle, noting that illusory brightness and clarity decrease contour. Propitious placement of small dots or short line monotonically with this alteration. However,there are many segments, called anchors, can drastically affect the shape examples ofefficacious line-end inducers that are not per­ ofan illusory figure and may increase contour salience, as pendicular to the illusory contour (Frisby & Clatworthy, is indicated in Figure 12 (Brady & Grimson, 1981; Day & 1975; Gillam, 1987; Gregory, 1987; Halpern, Salzman, lory, 1980; Gerbino & Kanizsa, 1987; Minguzzi, 1987; Harrison, & Widarnan, 1983;Parks, 1980;Rowbury, 1982). Rowbury, 1982), even though by themselves these anchors In fact, recent evidence shows that under certain conditions, are incapable ofgenerating illusory contours (Day & lory, perpendicular elements are inferior to oblique and acute in­ 1980; Kanizsa, 1955; Zucker & Davis, 1988). Gerbino and tersections (Gillam & Goodenough, 1994). Not limiting Kanizsa have noted that unlike normal inducing elements, line-end contour completion strictly to the perpendicular which abut the illusory contour, anchors appear to lie makes ecological sense; an occluded line conveys a de­ within or along the contour. This observation, iftaken in creasing amount ofinformation about the orientation ofthe the context ofthe illusory figure occluding the inducers, occluder as the line thickness approaches zero. However, if may imply that anchors are perceived as belonging to the one makes the reasonable assumption that foreground and illusory figure rather than to the background. As yet, there background position and orientation are independent, the have been no studies to determine whether this is the case, probability that an edge or line in the background intersects although Gregory and Harris (1974) demonstrated that a foreground edge at an angle between ()and ()+ d()is pro­ the stereoscopic depth ofanchor-like elements affects il­ portional to sin ()d()(Heitger & von der Heydt, 1993). This lusory contour strength in a manner consistent with this implies that a perpendicular intersection is ecologically hypothesis. Anchors can relieve the constraint that illu­ most likely,with probabilities decreasing as the angle ofin­ sory contours have only a limited degree ofcurvature be­ tersection becomes more acute or oblique. tween inducing elements by providing information about

b _ -.------~ Figure 11. Two inducers are relatable if linear extensions oftheir supporting edges (or perpendicular extension of line ends) intersect with an oblique angle. The inducers in (a) meet this criterion and are thus relatable, but in (b) the ex­ tensions do not intersect and in (c) the intersection angle is acute, implying that the inducers are unrelatable in both these cases. From "A Theory of Visual In­ terpolation in Object Perception;' by P. J. Kellman and T. F. Shipley, 1991, Cog­ nitive Psychology, 23, pp. 141-221 (Figure 32). Copyright 1991 by Academic Press. Adapted with permission. PERCEPTUAL THEORY OF ILLUSORY CONTOURS 291 n n n u u u Figure 12. Day and Jory (1980) demonstrated that small point or dot "an­ chors" facilitate contour formation and change the shape ofthe illusory figure. From "A Note on a Second Stage in the Formation of Illusory Contours," by R. H. Day and M. K. Jory, 1980, Perception & Psychophysics, 27, pp. 89-91 (Figure 2). Copyright 1980 by Psychonomic Society, Inc. Reprinted with per­ mission. where comers (curve discontinuities) should be located, rious effect ofmultiple lines converging on the illusory bor­ as in the rightmost stimuli ofFigure 12. der at the same location (such as to form a vertex) is also Conversely, small-scale information can also weaken il­ well established, even with very short line segments (Al­ lusory contours, or even prevent their formation altogether. bert & Hoffman, 1992; Kennedy, 1978b; Purghe, 1993). With a series of striking images, similar to that of Fig­ Spatial discontinuities. Kellman and colleagues (Kell­ ure 13a, Kennedy (1988) has demonstrated how altering man & Loukides, 1987; Kellman & Shipley, 1991; Ship­ the shape ofa line end so that it appears beveled, pointed, ley & Kellman, 1990, 1994) have made the strong claim or rounded can eliminate illusory contours (or at least ren­ that discontinuities in the first derivative of the inducers der them diffuse, as in Figure 5). Interference from a small are essential to establishing illusory contours. These dis­ scale-the bevel of the line ends-

a b ~I// ,,\1.0------_. ._- ~I\" /il\'

Figure 13. Small scale interference: (a) Beveled line ends destroy illusory con­ tours. From "Line Endings and Subjective Contours," by J. M. Kennedy, 1988, Spatial Vision, 3, pp.151-158 (Figure 1). Copyright 1988 by VSP International Science Publishers. Adapted with permission. (b) SmaU interfering dots de­ crease the clarity ofthe illusory contour. From "A Dynamic Model for Anom­ alous Figures: The Shape of Line-Induced Brightness Modifications," by B. Pinna and M. Sambin, Perception, 20, pp. 219-232 (Figure 7). Copyright 1991 by Pion Limited, London. Adapted with permission. 292 LESHER

ual the luminance gradient ofthe edges. A similar argument figure (Gregory, 1987; Soriano et aI., 1994). Soriano et al. might be made for sharp-edged inducers: at sufficiently (1994) have shown that relatively small overlaps «0.1°) large scales regions of high curvature are indistinguish­ of shifted gratings significantly decrease the probability able from real discontinuities, so every sharp-edged stim­ of illusory contour detection. Note that if the interfering uli would have discontinuities. Regardless ofwhether we lines ofFigure 14c are perceived as a "wire" square which accept the troublesome scale argument for blurry-edged in­ is weaved around the illusory square, the illusory con­ ducers, or limit the domain to sharp-edged inducers, spa­ tours may remain salient-an indication that the percep­ tial discontinuities evidently represent a salient low-level tual depth ordering ofvarious parts ofthe stimuli plays an determinant ofillusory contour formation. important role in contour formation, as will be discussed Inducer structure and figural interference. Kanizsa further in the next section. Similarly, many researchers (1976) showed that ifthe solid inducers ofhis eponymous have noted that background texture can interfere with il­ figure were replaced by outlined circles (with appropriate lusory contour formation ifit is perceived as being behind missing sectors), illusory contour formation was blocked. the inducing elements (Ramachandran, Ruskin, Cobb, Figure 14 depicts this configuration, along with two other Rogers-Ramachandran, & Tyler, 1994; Reynolds, 1981; configurations in which illusory contours and illusory Rock & Anson, 1979). brightening are either eliminated or weakened considerably (Frisby & Clatworthy, 1975; Kanizsa, 1974). There are no High-Level Determinants examples of illusory contours induction collinear to thin High-level determinants are typically more difficult to lines, as would be required for contour formation in the first quantify than their low-level counterparts.The effects of two examples ofFigure 14, although there have been ex­ stereoscopic depth modification of the illusory figures, amples ofcollinear lines in noninductive roles, such as the suitable for parametric investigation and thus perhaps short line anchors employed by Minguzzi (1987). Ifone more suitably a low-level determinant, are included within considers collinear lines as extremely thin edges, it appears the more abstract categorization of "depth modifica­ that the thickness ofthe inducer (in the direction perpen­ tions"-aspects of which are decidedly high level and dicular to the support region) plays an important role in il­ poorly understood. None of the high-level determinants lusory contour formation, as has been supported by sev­ has yet been addressed in a fully functional model ofillu­ eral parametric investigations (Brigner & Gallagher, 1974; sory contour formation, although certain aspects ofthe ef­ Purghe, 1991; Sambin, 1981). Purghe (1991) found contour fect of depth and stratification of inducers and illusory strength to be an increasing function ofthis inducer thick­ figures on the existence of illusory contours have been ness. For line-end stimuli, the lengths ofthe inducing lines discussed (Grossberg, 1987a, 1994; Kellman & Shipley, are strong determinants of contour strength-longer in­ 1991; Sajda & Finkel, 1995). Although the apparently ducing lines tend to result in stronger illusory contours "cognitive" nature ofsome ofthe high-level determinants (Ehrenstein, 1941;Redies & Spillman, 1981; Sambin, 1985; preclude them from detailed simulation-at least until the Soriano et aI., 1994). cognitive processes themselves are understood better­ Kanizsa (1974) noted that illusory contour strength was many ofthe effects discussed below are amenable to mod­ lessened by the addition oflines that penetrated the con­ eling. Even in cases where the underlying cognitive pro­ tour, as is depicted in Figure 14c. Similar findings have cesses are currently unmodelable, one is not prevented been reported in the context ofline-end contours in which from modeling the effects ofthe results ofthese processes one or more ofthe inducing lines penetrates the illusory on illusory contour formation.

Figure 14. Figural interference effects: (a) Outlining a Kanizsa figure elim­ inates illusory contours. (b) Contour induction collinear to thin inducing lines does not occur. (c) Lines intersecting illusory contours can decrease contour strength. Figures 14a and 14c are from "Contours Without Gradients or Cog­ nitive Contours?" by G. Kanisza, 1974, Italian Journal ofPsychology, 1, pp. 107-123 (Figures 29 and 35). Copyright 1974 by Societa Editrice iI Mulino. Adapted by permission. Figure 14b is from "Illusory Contours: Curious Cases of Simultaneous Brightness Contrast," by J. P. Frisby and J. L. Clatworthy, 1975, Perception, 4, pp. 349-357 (Figure 8b). Copyright 1975 by Pion Limited, London. Adapted by permission. PERCEPTUAL THEORY OF ILLUSORY CONTOURS 293

Figure 1s. Stereoscopic manipulation of illusory figure depth affects the il­ lusory contour strength.

Depth modifications. Observers usually perceive illu­ inducer structure unchanged have been employed to un­ sory figures as being raised from the background, floating equivocally demonstrate the effect ofperceived depth on above (and occluding) the inducing elements (Coren, contour salience. Rock and Anson (1979) constructed a 1972; Kanizsa, 1955). Thus, one might wonder what ef­ display, similar to that ofFigure 16, in which a coarse tex­ fect the perceived depth ofthe figure has on the strength ture was printed over the entire background ofa Kanizsa of associated illusory contours. Figure 15 depicts a Ka­ figure. Under monocular conditions, no illusory contour nizsa figure which, when stereoscopically fused, has dis­ was perceived, apparently because offigural interference parity information indicating that it lies in front or behind from the texture. The same was true ifthe texturewas stereo­ the inducers, depending on which pair ofstimuli are fused. scopically located in back ofthe inducers (uncrossed fu­ In the former case, the illusory contour looks much stronger sion of the middle and right images of Figure 16). If the than the monocular contour, whereas in the latter, it looks texture was stereoscopically placed in front ofthe induc­ much weaker. These types of stimuli were employed by ers (uncrossed fusion ofthe left and middle images), how­ Simmonds (1974), who had subjects rate contour strength ever, the illusory contour became clear. This last case is as disparity was varied. Simmonds found no effect on consistent with a semitransparent texture lying in front of strength when the figure was made to appear in front of both the inducers and the illusory figure, allowing strati­ the inducers, but did find that strength was decreased fication of illusory figure and inducers to occur without when the figure was made to appear in back-when the paradox. When this texture is behind the stimuli, how­ cues were not consistent with an occluding illusory figure. ever, the observer has the perception of looking through However, subsequent research has shown a reliable effect the central region ofthe stimuli, and thus the system is not ofboth crossed and uncrossed disparities (Lawson et al., consistent with an occluding figure (since the illusory fig­ 1974; Nakayama, Shimojo, & Ramachandran, 1990; Whit­ ure would need be transparent to the texture but opaque to more et al., 1976). When one uses a binocular stimulus with the inducer). This example, and others in the same vein no disparity as a baseline, contour strength increases mono­ (Nakayama et al., 1990; Reynolds, 1981), show that the tonically to a saturated maximum as figure depth decreases perceived consistency with global occlusion can have a (nearer to the observer), with the inverse holding true as profound effect on contour strength. Using stimuli similar figure depth increases. Gregory and Harris (1974) report to those ofRock and Anson (l979}-periodic textures over­ similar results with line-end inducer configurations. lying conventional illusory contour inducing stimuli-Ra­ In changing the disparity information of the illusory machandran and Cavanagh (1985; Ramachandran, 1986a, figure, the inducers themselves are slightly altered, lead­ 1986b, 1987) have demonstrated "capture" ofthe texture ing to the possibility (albeit a small one) ofconfound be­ by the illusory figure when the texture has ambiguous tween the effects ofdisparity and those ofinducer struc­ stereoscopic depth. That is, within the illusory figure the ture. However, indirect depth modifications that leave the texture appears to be at the same depth as the figure is, while

Figure 16. Rock and Anson (1979) showed, using the physical equivalent of this stereogram, that the perceived depth of the diagonal texture determines whether or not an illusory contour will be formed. The texture appears in front of the inducers if the left and middle images are fused (uncrossed), and illusory contours appear. When the middle and right images are fused, the texture ap­ pears in back ofthe inducers, and illusory contour formation is blocked. 294 LESHER

elsewhere it appears to be at the same depth as is the back­ 1979; Kellman & Shipley, 1991; Petter, 1956). Illusory ground. Ramachandran (I986a) and Ramachandran et al. contours are perceived along the boundaries of the shape (1994) point out that even in certain monocular stimuli, reg­ that is perceived to be nearer to the observer, with only ular background textures can be captured by the illusory amodal completion along the borders of the further figure to appear at the same depth as that of the figure. shape-a perception that spontaneously reverses with Despite the findings ofRock and Anson (1979), consis­ time. These effects, along with related phenomena in the tency with occlusion ofthe inducers by the illusory figure areas ofperceptual set and memory, indicate that perceived is not a prerequisite for perception of a salient contour. stratification can have a profound influence on the ap­ Strong illusory contours have been demonstrated in the pearance ofillusory contours. absence of perceived depth by employing self-occluding Inducer completeness. The incompleteness ofthe in­ stimuli (Kellman & Shipley, 1991; Kennedy, 1979; Ken­ ducing elements in edge-induced figures has long been nedy & Lee, 1976; Parks, 1986) and three-dimensional held to be a contributing, ifnot the governing, factor in il­ stimulus constructs (Ware & Kennedy, 1977, 1978a). The lusory contour formation. An incomplete inducer is a cue phase-shifted grating of Figure 2c, one of the most fre­ that there maybe an occluding figure. Kanizsa (1955) pre­ quently employed illusory stimuli, is itselfnot entirely con­ sented a display consisting of four inducers shaped like sistent with global occlusion-one side ofthe grating must crosses (Figure 18a) to demonstrate the ineffectiveness of be perceived in front ofthe other, so that side's line ends are complete (in a gestalt sense) stimuli in generating illusory not occluded at all. Purghe (1993) has constructed a clever contours. However, employing the same cross inducers, series ofpictorially 3-D stimuli decidedly inconsistent with along with other inducers whose symmetry would be bro­ illusory figure occlusion, which still give rise to strong il­ ken ifamodally completed behind the illusory figure (as lusory contours and illusory brightening. Purghe's exam­ in Figure 18b), Day and Kasperczyk (1983) showed that ples were designed to refute the claims of Parks and Rock illusory contours were generated by these perceptually (1990), who argued that 3-D stimuli inconsistent with oc­ complete inducers, albeit with less strength than for tradi­ clusion did not give rise to illusory contours. tional incomplete inducers. This finding has been con­ The examples in the literature which seem to indicate firmed by Purghe and his colleagues, in a series of thor­ that contours are destroyed by inconsistency with an oc­ ough examinations of the effects of completeness and cluding illusory figure can be discounted on the grounds symmetry on contour strength (Purghe, 1988, 1991; ofother monocular effects-small-scaleor figural interfer­ Purghe & Coren, 1992a; Purghe & Katsaras, 1991). In a ence, as observed in the stimuli ofParks and Rock (1990) similar vein, Rock and Anson (1979) showed that the per­ and Rock and Anson (1979), respectively. It does appear, ceptual set of the observers affected whether they per­ however, that providing monocular cues that indicate that ceived illusory contours with complete, but asymmetric, the stimuli are indeed consistent with occlusion can facil­ inducers (as in Figure 18c). itate instantiation ofillusory contours which would other­ The concept ofcompleteness cannot easily be defined wise be blocked by these interference effects. Although in simple geometric terms. It certainly has relations to sym­ this asymmetry between contour instantiation and de­ metry, smoothness, and regularity, but one must also take struction may be resolved by demonstration ofcontour de­ into account familiarity ofthe figure, perceptual set, and struction solely on the grounds of(monocular) occlusion a number of other abstract factors. It is indeed a gestalt inconsistency, there have been no such convincing dis­ property. Thus, the apparent effect of inducer complete- plays or experiments yet. As mentioned above, however, there is no such asymmetry in the case ofstereoscopically altered stimuli: illusory contours can be weakened or de­ stroyed by disparity information inconsistent with occlu­ sion. Certain arrangements ofinducers may support several different interpretations ofoccluding illusory figures, lead­ ing to ambiguous illusory figures (Bradley, 1987; Bradley & Dumais, 1975; Scrivener, 1983; Shank & Walker, 1989). Bradley and Dumais present a stimulus similar to that of Figure 17a, which their subjects have reported perceiving as two different illusory figures: a six-pointed star and two triangles rotated 1800 with respect to one another, one Figure 17. Somestimuli can induce"ambiguous" illusory con­ lying atop the other. In addition, either ofthe two triangles tours which vary depending upon the perceived depth ordering of the inducers and illusory figures. (a) Bradley and Dumais's in the latter percept can appear to be in front. At a partic­ (1975) stimulus can supportthree percepts: a six-pointed starand ular time, the locations ofthe illusory contours depend on two layering schemes ofa pair of triangles oriented 180· with re­ which figure and which depth ordering the observer is spect to one another. From "Ambiguous Cognitive Contours," by perceiving. The figure may spontaneously reverse, lead­ D. R. Bradley and S. T. Dumais, 1975, Nature, 257, pp. 582-584 (Figure 3). Copyright 1975 by Macmillan Magazines Limited. ing to a different set of illusory contours. A similar phe­ Adapted with permission. (b) The positions of illusory contours nomenon is observed with spontaneously splitting figures, in a "spontaneously splitting figure" (Kellman & Shipley, 1991) an example ofwhich is depicted in Figure 17b (Kanizsa, depend on which shape is perceived to be nearer to the observer. PERCEPTUAL THEORY OF ILLUSORY CONTOURS 295

a b c

Figure 18. Perceptually complete Inducers can still generate illusory con­ tours, albeit with lesser strength than those that result from perceptually In­ complete Inducers. Figure 18a Is from "Marglnl Quasl-Percettivlln Campi con Stlmulazlone Omogenea," by G. Kanlzsa, 1955, Rivista di Psicotogia, 49, pp, 7-30 (Figure 21). Copyright 1955 by Glunti Gruppo Edltorlale. Adapted by permission. Figure 18b Is from "Amodal Completion as a Basis for Illusory Contours," by R. H. Day and R. T. Kasperczyk, 1983, Perception & Psy­ chophysics,33, pp. 355-364 (Figure 3). Copyright 1983 by Psychonomic Soci­ ety, Inc. Adapted by permission. Figure 18c is from "Subjective Contours and Apparent Depth," by S. Coren, 1972, Psychological Review, 79, pp, 359-367 (Figure 8c). Copyright 1972 by the American Psychological Association. Adapted by permission.

ness on illusory contour clarity is difficult to explain in cepts typically associated with determination ofthe effec­ low-level terms. Van Tuijl and Leeuwenberg (1982) have tiveness of illusory contour inducers-they were able to presented a language coding method that mathematically markedly increase the percentage of observers who per­ captures some elements ofinducer completeness, but that ceived an illusory figure in stimuli like those of Fig­ is incapable ofemulating many ofthe subtler aspects. Al­ ure 18c. Similarly, Coren, Porac, and Theodor(1986, 1987) bert (1993) has pointed out that many stimuli employed in have shown that perceptual set may also have an effect on demonstrating the lack ofillusory contours with complete the shape ofthe perceived illusory figure. By cuing sub­ inducers share a single geometric property-namely, a jects as to what type of illusory shape to expect in an nearby edge parallel to the supporting edge, as in Figure 18a. Ehrenstein figure (circle or square) they could bias the Further, Albert has shown that in many ofthese cases, tilt­ distribution of reported percepts. Indeed, even in the vi­ ing the offending edge away from parallel restores illusory sion research community, many common stimuli that contour strength while maintaining the perceptual com­ yield illusory contours, such as Ehrenstein and Koffka pleteness of the inducing elements. Sajda and Finkel figures, were not recognized as such until many years after (1995) have made similar claims about the ability oflocal their initial presentations (Gregory, 1987; Kanizsa, 1955; mechanisms to emulate gestalt properties. Although these Landauer, 1978). observations certainly do not explain the effects ofinducer Altering subjects' perceptual set may cause them to per­ completeness, they represent examples ofhow simple de­ ceive illusory contours where they previously observed tenninants can give rise to seemingly complex phenomena. none, implying that learning may be involved in illusory The gestalt tendency ofcontinuation has been related to contour perception. Gellatly (1982) provided subjects completeness by several researchers (Kanizsa, 1974; Min­ with stereoscopic cues that the triangle in a Kanizsa con­ guzzi, 1987). Minguzzi points out that inducing elements figuration was behind the inducers, giving rise to the tend to continue (amodally) behind illusory figures. The "porthole effect" wherein subjects have the impression of nonsupporting edges of edge inducers continue in their seeing a triangle through three circular apertures. After curvilinear paths (e.g., circles missing sectors become viewing several ofthese stimuli, subjects reported having amodally completed circles), while the lines of line-end the same percept in the monocular domain with a standard inducers amodally extend through the illusory figure. Al­ Kanizsa triangle. Gellatly and Bishop (1987) demonstrated though completion requires knowledge of the overall in­ a similar example in which motion was used to cue an il­ ducer structure, continuation is a local tendency and thus lusory contour which was then perceived in a static envi­ is more readily definable in low-level terms, However, Davi, ronment, and verbal explanations by Bradley and Petry Pinna, and Sambin (1992) have found that there is no cor­ (1977) were enough to cause subjects to perceive alternate relation between the perceived continuation of lines and illusory contours in an illusory Necker cube. In all these either illusory contour strength or illusory figure brightness. cases, subjects were provided with additional information Perceptual set, learning, and memory. Rock and about the relative depths ofthe illusory figure and the in­ Anson (1979) demonstrated that the mental state ofsub­ ducing elements. Previously unperceived contours be­ jects had a significant effect on whether or not they per­ came visible once the subjects were aware ofthe stratifi­ ceived illusory contours. By pictorially cuing subjects to cation ofthe inducers and the illusory figure. Wallach and such abstractions as incompleteness and alignment---eon- Slaughter (1988), studying the role ofmemory in illusory 296 LESHER

contour formation, found that subjects more readily per­ Complete inducers should be less likely to induce illusory ceived illusory shapes familiar from previous experimen­ forms than should incomplete inducers. Familiar shapes tal presentation than they did novel shapes. In all ofthese should be preferred to novel shapes. Every high-level de­ cases, the stimuli maintain a potential for illusory contour terminant has a clear ecological purpose, easily under­ formation-all low-level determinants are consistent with standable from the problem-solving perspective. contour generation-which is only realized when subjects Some low-level determinants would doubtless find their are cued to the existence of the contours. In stimuli that way to our hypothetical list as well, but prescience would be support multiple illusory figures, the degree to which the necessary to include them all, since some determinants ap­ inducing elements subserving each figure are compatible pear unrelated to, or even incompatible with, the task ofre­ with low-level determinants may determine which illu­ liably creating illusory forms. The reasons for the effect of sory figure will be perceived. For example, in a Kanizsa inducer spatial extent and proximity are obvious enough­ configuration, a triangle is observed more often than three the more the support, the more probable the existence ofan portholes, perhaps because the triangle inducers (the edges occluding object. Ifobjects generally have smooth, contin­ along the missing sector of the circle) are well aligned, uous edges, then the preference for aligned inducer edges whereas the circular porthole inducers (the curved, outside becomes apparent. Anchors reflect the need to grasp at any edges ofthe circle) require curved completion for subjects information that can disambiguate contour paths left am­ to perceive an illusory contour. biguous by insufficient inducer support. Figural interfer­ Attentional effects. Closely related to perceptual set ences, such as the outlining ofan edge inducer, indicate in­ are the effects that attention can have on the formation of compatibility with occluding objects. Small-scale structure illusory contours. In a visual search paradigm, Grabo­ ofan inducer, particularly a line-end inducer, may hint at a wecky and Treisman (1989) have shown that detection of natural termination rather than an occlusion. But why an edge-induced illusory triangle is influenced by the num­ should dots near an inducer weaken an illusory contour? ber ofbackground distractors, inferring from this that illu­ Why should increasing line density eventually result in a sory contour perception is not a preattentive process. In the decrease in contour strength? Why shouldn't support ratio context of a texture segmentation task, Meyer and Fish be an absolute predictor ofcontour strength? It is unclear (1987) have reported similar findings. Comparing the whether these determinants are epiphenomena ofother eco­ times required to discriminate between different shapes of logicalprocesses-unavoidable anomalies caused by the in­ edge-induced illusory forms and between equivalent out­ tegration ofthe illusory contour problem with all the other lined forms, Pritchard and Warm (1983) found that reac­ problems ofvision within a constrained neurophysiological tion times increased more for illusory form discrimination framework--or represent some time-tested kernels ofeco­ than for outlined form discrimination as attentional loads logical truth. In either case, it is possible that perceptual were increased, implying a greater attentional demand for groupings of proximal or otherwise spatially related ele­ illusory contour perception. The high attentional demand ments, ofboth the inducing and the interfering variety,play indicated by these experiments has been used to argue for a significant role in determining illusory contour strength. a cognitive theory ofillusory contour formation. However, Gurnsey, Humphrey, and Kapitan (1992) have pointed out aLUSORY CONTOURS AS that edge type illusory figures were employed in all of EARLY VISION PHENOMENA these studies, and they reason that the attentional findings may be related to the low contrast between illusory figure A great deal ofevidence suggests that illusory contours and background, as opposed to the high contrast between are initially formed early in the visual processing stream, the inducers and background. Eliminating contrast and although higher level processes help to determine the final other interfering factors by using fields oflines with line­ positions and strengths ofthese preliminary contours. The end-induced illusory contours, Gurnsey et at. (1992) have primary evidence for early formation comes from a recent demonstrated parallel search and quick texture segmenta­ series of neurophysiological studies in which cells re­ tion. Nevertheless, the earlier experiments highlighted dif­ sponding to illusory contours were found, but there is also ferences in the saliency ofreal and illusory contours. a convincing body of supporting psychophysical evi­ dence. By demonstrating an equivalence between real and Ecology ofDeterminants illusory contours in sustaining visual illusions such as tilt Given that illusory contour perception developed as a and motion aftereffects, psychophysical studies have sug­ method of disambiguating low-contrast figures-to dis­ gested that illusory contours obtain a neurophysiological cover the most probable organization that would account reality at early levels ofvisual processing. Overviews ofthe for the visual stimuli-high-level determinants are partic­ relevant neurophysiological and psychophysical findings ularly easy to relate to visual ecology. If, given our eco­ are provided below. logical problem, we were told to write down the features that should be important for determining the presence of Neurophysiological Evidence of illusory contours, it is likely that we would come up with Illusory Contours all the high-level determinants discussed in this review. von derHeydt, Peterhans, andBaumgartner(1984). Depth information in conflict with illusory form stratifi­ In an influential paper, von der Heydt et at. (1984) pre­ cation should weaken or destroy the proposed contour. sented results from single cell recordings indicating they PERCEPTUAL THEORY OF ILLUSORY CONTOURS 297 had discovered neural correlates ofillusory contours in V2 humans are extremely sensitive to the alignment of the (but not V I) of macaque monkeys. Subsequent work in line ends: illusory contours are destroyed with very little the same laboratory extended these original findings, not grating shift in either direction. Soriano et al. (1994) have only establishing the existence of"illusory contour cells," reported that an overlap or gap ofas little as a tenth ofa but also providing data on the effect ofa variety ofpara­ degree markedly reduces the probability that subjects will metric stimulus variations on cellular activity (Peterhans report seeing an illusory contour. & von der Heydt, 1989, 1991; Peterhans, von der Heydt, & As with line gratings, a significant fraction ofV2 cells Baumgartner, 1986; von der Heydt & Peterhans, 1989a, (-30%) responded well to notch type stimuli that yield il­ 1989b). In all these studies, stimuli consisting ofabutting lusory contours in humans (Figure 19b), although response line gratings (Figure 19a) and stimuli consisting of two levels were typically lower than for the line gratings. Cell­ parallel bars with rectangular notches cut out (Figure 19b) ular activity fell offwith increasing notch separation and were tested extensively. These two cases will be examined was greatly decreased when only a single notch was pre­ independently below. sented. Placing a very thin bar over the mouth of one of Almost half of the cells examined in V2 by von der the notches significantly attenuated response levels, a Heydt et al. (1984) exhibited sizable responses to drifting finding perhaps related to figural interference effects ob­ bars or edges and the illusory contour induced by drifting served psychophysically. line gratings. For line gratings, they observed cellular re­ The neurophysiological responses to notch stimuli also sponse when the lines of the grating were oriented per­ parallel human psychophysical data. Cells that responded pendicularly to preferred orientation of the cell; the cell to the notch stimuli also tended to respond to the phase­ was not merely responding to the inducing lines. The ori­ shifted grating stimuli, as well as to real bars and edges. entational tuning curve ofa given cell, as determined with This finding led von der Heydt et al. (1984) to conclude that the illusory contour stimulus, closely resembled that cell's they had found a neural correlate of illusory contours. tuning curve as determined with a luminance-defined bar. Further, since these cells responded to both real and illu­ Response was optimal when the lines in the grating were sory stimuli, they believed that they had discovered a uni­ perpendicular to the illusory contours, monotonically de­ fied boundary representation. Their findings led them to creasing as the lines were slanted away from perpendicu­ propose a model ofillusory contour formation, discussed lar. Cells were not simply responding to individual line in the neural modeling section, which serves as the foun­ ends, however; the typical cell would not respond to a grat­ dation for several more advanced architectures. ing with only two or three bars, but it responded more and Redies, Crook, and Creutzfeld (1986). Employing more strongly as additional bars were added, until a satu­ phase-shifted gratings similar to those used by von der rated level ofactivity was reached. Heydt et al. (1984) (Figure 19a), Redies et al. (1986) stud­ For the most part, the cellular recordings of von der ied the response ofcells in visual areas 17 and 18 ofcats. Heydt et al. (1984) revealed cells whose responses to vari­ The researchers found that no simple cells in areas 17 or ations in inducer configurations closely resembled human 18 responded to the illusory contours of the stimuli, but psychophysical responses to those same variations-for that some complex cells in both areas responded to the il­ example, increasing to a saturated response as the number lusory contours. Redies et al. posit that the difference in ofinducing lines increased. However,when the researchers areas ofillusory contour cell location from those ofvon der shifted each halfofthe grating along the inducing lines so Heydt et al. may well be due to a variation across species. that the lines meshed or left a gap, many cells still re­ Careful controls were employed to eliminate the possibil­ sponded strongly,especially in the case ofa gap. This marks ity that cells were merely responding to the ends ofindi­ a divergence from human psychophysical data, because vidual lines, ensuring that some cells were indeed respond­ ing to a nonluminance-defined border-something at least similar to an illusory contour. Mean complex cell re­ sponses dropped off with increased line spacing, as was a b observed by von der Heydt et al. The experiment ofRedies et al. (1986) was ahead ofits time; whether cats could actually perceive illusory con­ tours was not known until 2 years later, when Bravo, Blake, and Morrison (1988) employed a discrimination task to determine that cats could indeed perceive Kanizsa fig­ ures. Soon after, de Weerd, Vandenbusche, de Bruyn, and Orban (1990) performed similar discrimination experi­ ments with phase-shifted gratings. Employing Kanizsa figures rather than gratings, Gregory (1987) failed to find Figure 19. The stimuli employed by von der Heydt et al, (1984) cellular responses in cat visual cortex; but his studies were in their neurophysiological experiments. From "IllusoryContours limited to area 17, and he made no effort to locate com­ and Cortical Neuron Responses:' by R. von der Heydt, E. Peter­ hans, andG. Baumgartner, 1984,Science,224 (June 15), pp.1260­ plex cells. Thus, whether edge-induced illusory contours 1262 (Figure 2). Copyright 1984 by the American Association for can induce activity for cells within area 18 ofthe cat (or the Advancement ofScience. Adapted with permission. area 17 complex cells) remains unknown. 298 LESHER

Grusof, Shapley, and Hawken (1993). Grosof et al. rarefaction (Bressan, 1987), facilitation ofamodal comple­ ( 1993) have recently produced data indicating the existence tion behind a figure (Bruno & Gerbino, 1987), and figure­ of neurons in V I of macaques that apparently respond to ground organization (Peterson & Gibson, 1994) are nec­ line-end stimuli similar to those employed by Redies et al. essary, serving to support the intuitive sense that illusory (1986). Consistent responses were observed in response to figures have the same cognitive significance as real fig­ phase-offset sinusoidal gratings, with controls to ensure ures, particularly with respect to stratification in depth, that cells were not responding to local edges. Cells exhibit­ but such experiments do not provide much assistance in ing responses to the contour tended to be complex, but pinpointing the locus (or loci) ofillusory contour forma­ some simple cells with this characteristic were also dis­ tion. The large number of comparative studies precludes covered. Phase-offset sinusoid gratings give rise to illu­ all but the briefest mention of most: Only comparisons sory contours with support ratios approaching 1.0; there is with profound theoretical implications will be discussed rea/luminance contrast at almost every location along the here in detail. contour, although the direction and strength of contrast Orientation and motion estimations. Some of the varies periodically along the contour. Grosof et al. did earliest and most influential experiments demonstrating find one V I complex cell that also responded to more con­ perceptual equivalence between real and illusory contours ventionalline-end gratings with low support ratio, but the involved the examination ofbiases in motion and orienta­ discovery ofa single cell does not constitute convincing tion estimations. By revealing the similarities between real evidence ofa mechanism for illusory contour formation. and illusory contours in the realms oftilt aftereffects (Smith Redies et al. (1986) and Grosofet al. (1993) found only & Over, 1975, 1976) and motion aftereffects (Smith & cells that responded to very simple illusory contours with Over, 1979; Weisstein et aI., 1977), early researchers made perfectly aligned line ends. As yet, there is no evidence that a compelling case for a similar representation ofthe two these neurons would respond to edge-induced illusory con­ contour types. These aftereffects are often attributed to tours or curved illusory contours. These neurons may not be short-term habituation in early visual cortex (Barlow & signaling illusory contours per se but may represent only Hill, 1963; Movshon, Chambers, & Blakemore, 1972), the early stages of processing-a simple summation of rendering the observed similarities between real and illu­ aligned end-stopped cells. To establish the role that these sory contours theoretically important. In further implication neurons play in illusory contourperception, additional para­ ofa shared internal representation, all ofthese aftereffects metric studies ofinducing stimuli structure must be under­ appear to cross over between real and illusory contours, a taken--ofthe effect ofgrating overlap and gap, response finding supported by more sophisticated recent studies on to edge inducers, and response to curved illusory contours. tilt aftereffects (Berkley et aI., 1994; Paradiso et aI., 1989). Smith and Over (1977) have also demonstrated similari­ Psychophysical Similarities Between Real ties in orientation masking, with full crossover between and Illusory Contours contour types. In the debate between bottom-up and top-down propo­ Vogels and Orban (1987) discovered equivalent biases nents of illusory contour formation, psychophysical simi­ in estimation ofreal and illusory contour orientation, but larities and dissimilarities between illusory contours and real with an asymmetric learning effect in which practice on il­ (luminance discontinuity) contours have played an impor­ lusory contour orientation estimation carried over to real tant, though somewhat tiresome, role. Although the results contour estimation, but not vice versa. A number of re­ ofthese similarity studies are conclusive in determining how searchers have employed illusory contour versions ofclas­ real and illusory contours are related on given tasks, the ex­ sical illusions involving impaired estimations of orienta­ periments have had scant success in altering the views of tions. Itappears that illusory contours facilitate the Bourdon proponents for either ideology. Many ofthe illusions or phe­ illusion (Walker & Shank, 1987, 1988a, 1988b; Wen­ nomena employed in these experiments are not well enough deroth, Criss, & van der Zwan, 1990) but slightly decrease understood in themselves for the phenomena to be pinned the effect ofthe Poggendorffillusion (Day, Dickinson, & down to specific levels ofvisual processing; thus, the estab­ lory, 1977; Goldstein & Weintraub, 1972; Meyer& Garges, lishment ofa correlation between the response to two types 1979). General equivalence in orientation estimation is a of contours specifies only that illusory contours and real strong, though not unequivocal, argument for early repre­ contours have the same perceptual significance at some un­ sentation ofillusory contours, specified level. Slightly more informative is the case ofdis­ Time course studies. Whether the perception of illu­ similarity, in which the implication (by cognitivists at least) sory contours parallels the perception ofreal contours in is that illusory contours gain perceptual significance after (at terms ofthe saliency ofthe contour as a function oftime a higher level than) real contours. is an important theoretical issue. Cognitive theories gen­ For the purposes of determining the point of illusory erally predict a significant lag between detection of illu­ contour generation within the visual processing stream, sory contours and detection ofreal luminance gradients; the most salient ofthe comparisons are those which deal inducer information must make its way to higher levels of with phenomena that have known neural correlates. Ex­ the brain, cognitive contour formation must occur, and periments indicating similarities between real and illusory (perhaps) illusory contour information must make its way contours in higher level characteristics such as reversibil­ back down to earlier levels ofthe visual cortex. However, ity ofambiguous figures (Shank & Walker, 1989), texture this may also be true for neural models that incorporate feed- PERCEPTUAL THEORY OF ILLUSORY CONTOURS 299 back loops. Employing a Kanizsa figure, Reynolds (1981) Such delays would also be expected in neural systems that showed that perception ofillusory contours (under mask­ rely on feedback to establish illusory contours. To explain ing conditions) was not reliably reported unless the induc­ Reynolds's (1981) intriguingresults at a neural levelrequires ing stimuli were presented for longerthan 75 msec, although more sophisticated modeling techniques, but his data are the inducing elements were clearly perceived at 50 msec. not incompatible with recent designs which incorporate In a second experiment, a Kanizsa figure with overlaying stratification and depth effects (Grossberg, 1994; Sajda & bricklike texture did not yield a salient contour with ex­ Finkel, 1995). tended viewing, presumably because ofthe figural inter­ Binocular rivalry. Bradley (1982) has examined the ference ofthe intersecting lines. When flashed, however, effect of binocular rivalry on Kanizsa triangles. When a a contour was perceived with durations between 75 and normal illusory triangle was presented to one eye and an 250 msec; but it was not perceived for durations below or inverted illusory triangle to the other, a stimulus closely above this range, perhaps indicating an initial top-down related to Figure 17a, subjects tended to perceive a fusion hypothesis rejected upon further consideration. A third ofthe two triangles into a six-pointed star which would oc­ experiment, in which the texture was seen as transparent casionally split into two overlying triangles in the manner when observed in steady state (much as in Rock & Anson, observed for other ambiguous illusory contours. Con­ 1979), showed a similar effect, with contours becoming versely, ifthe illusory triangles were replaced by real tri­ evident again after 400 msec-again compatible with a angles ofslightly higher luminance than that ofthe back­ hypothesis process (for further details, see the discussion ground, binocular rivalry occurred, and the six-pointed ofcognitive theories and the related figure in the section star, if observable at all, was highly unstable. Fable and Phenomenological Models ofIllusory Contour Formation). Palm (1991), however, observed binocular rivalry between Reynolds's (1981) results differ markedly from those of illusory contours with dichoptic stimuli presentation, al­ Gellatly (1980), who employed a similar method to study beit less complete than observed between real contours. time course saliency. Gellatly found that the illusory tri­ Other comparisons. In studying flicker haloes­ angles became apparent after 10 msec, well before the in­ streaky bands surrounding stimulus edges when the stim­ ducers themselves were perceived. The variation in results ulus was presented intermittently at frequencies below the undoubtedly arises from the difference in masks used. fusion frequency-Remole et al. (1985) noted strikingly Whereas Reynolds used noisy, solid circles covering the similar halo extents for real and illusory contours as both inducers, Gellatly employed rings around the inducers, the flicker frequency and stimuli luminances were changed. thus employing the metacontrast paradigm along the outer This experiment supports a neural equivalence ofcontour inducer edges but no masking along the illusory contour types, since flicker haloes are thought to have an early supporting edges (edges of the missing sector). The sig­ locus (Remole, 1970). Arguing against early instantiation, nificant effect ofthe mask is confirmed by the use ofan Halpern and Warm (1980,1984) studied image fragmen­ inducer outline as the mask. Gellatly demonstrated that such tation-the disappearance and reappearance of parts of a mask increased the duration necessary for one to per­ the stimuli at extended viewing durations-and found that ceive the contour to several hundred milliseconds. illusory figures fragment far less frequently than their In a study ofvisual persistence, Meyer and Ming (1988) luminance-defined counterparts, although the inducing note that illusory contour perception appears to lag behind elements fragment more frequently. Since fragmentation the perception ofreal contours, while also observing that is commonly linked to neural adaptation and fatigue illusory contours persist much longer than their inducing (MacKinnon, Forde, & Piggins, 1969), Halpern and Warm's elements, once the latter have been replaced by blank data preclude illusory contours from appearing at the same fields. Petry and Gannon (1987) had subjects rate bright­ neural level at which this adaptation is occurring. Since ness as a function oftime as masked figures were flashed, adaptation may be occurring very early in the visual pro­ finding that illusory figure brightness lags behind real cessing stream, however, these fragmentation results are figure brightness but exhibits the same general time course incompatible only with the simplest ofbottom-up models. plot. Besides these studies, a number ofanecdotal confir­ mations of the delay in illusory contour formation have Developmental and Clinical Findings been provided by researchers working with apparent mo­ Several studies ofthe development ofillusory contour tion (Mather, 1988; Petersik, 1987; Ramachandran, 1986a; perception in children have been performed in the last Vallortigara, 1987; von Griinau, 1979). In these studies, decade. Bertenthal, Campos, and Haith (1980), who stud­ stimulus presentation times had to be significantly longer ied the response ofinfants to Kanizsa-type displays, found for illusory figures than for real figures in order to achieve a marked change in detection accuracy between the ages the same degree ofapparent motion. Of course, the sup­ of5 and 7 months. Although the authors argued that the port ofapparent motion itselfis an important indicator of infant was indeed seeing the illusory contour, they admit­ equivalence between the contour types. ted that the infants might have been responding to some Time course studies clearly indicate that the substrate simple figural effect such as local brightness contrast. of illusory contour formation cannot consist solely of Soubitez (1982) noted that children did not tend to per­ bottom-up neural processing. However, the delays in con­ ceive the central portion ofan Ehrenstein figure brighter tour formation are not so lengthy as to be compatible only than that ofan Ehrenstein figure with a real circle along with cognitive approaches to illusory contour perception. the inner line ends, an addition known to reduce bright- 300 LESHER

ness for adults (Ehrenstein, 1941), until 3 years of age. The two paradigms differ markedly not only with re­ Abravanel (1982) found that likelihood ofperception and spect to which aspects of the illusory figure they empha­ recognition of Kanizsa-like illusory figures tended to in­ size, but also with respect to the level ofabstraction from crease with age, with a significant change from ages 3 to 5 which they must be approached. Those who are in the years. All the 3- to 4-year-olds tested in an informal study brightness camp assume that illusory brightening drives il­ by Kennedy (1987) reported seeing what he classified as lusory contour formation, and thus they must attempt to illusory contours. In all these cases, communication prob­ explain in neurophysiological terms whence the brighten­ lems inherent in dealing with young children may have ing derives. Those who are in the form camp argue that the had a significant effect, so it is difficult to determine the re­ task is to explain what particular geometrical aspect ofthe liability ofthese studies. However, one cannot rule out the stimuli causes a viewer to segment the scene so as to give possibility that perception of illusory contours does not rise to an illusory figure. In the earliest days ofillusory con­ develop until several years after birth, to some degree rep­ tour research, the two approaches could safely be classified resenting a learned skill. If this proves to be the case, learn­ as low-level and high-level respectively, but this distinction ing at the ages mentioned above would correspond to is no longer valid. Although the initial concepts of neural changes in cognitive perceptual strategies, as opposed to modeling ofcontour formation did indeed arise within the changes in the mechanisms ofthe lower levels ofthe visual brightness camp, more recent models have recognized the cortex. This would, ofcourse, be a strong argument for an importance offactors previously stressed only by members increased role ofhigher level factors in contour formation. of the form camp-specifically, the role ofcues to depth in Two important clinical studies have been performed in illusory contour formation. Below, I sketch a history ofthe order to establish correlations between certain visual two ideologies: the evolution of the form camp theories deficits and illusory contours. Hamsher (I978) studied the from the phenomenological to the concrete, and the grad­ link between scores on benchmark stereo performance ual relegation ofbrightness to the role ofa secondary pro­ tests and Kanizsa-like illusory contour perception for cess. The neural models ofcontour formation, which give brain-damaged patients with a range ofperceptual disor­ all indications ofrepresenting the future ofillusory contour ders. A strong correlation was found between the inability modeling, are reviewed in a subsequent section. to perceive illusory contours and poor stereo performance, a similarity that Hamsher suggested was attributable to Brightness Theories deficits in the discrimination of subtle difference in the Although the progenitor ofbrightness theories ofillusory spatial location of features. If this is indeed the common contour formation, Ehrenstein's (1941) work does not factor, Hamsher's experiment underscores the deleterious technically fall into the category of illusory contour re­ impact ofboth inducer misalignment and small-scale in­ search. He was primarily interested in explaining the strik­ terference on contour formation. Stevens (I983) studied a ing brightness effects that he noted in the Ehrenstein fig­ single patient who had trouble interpreting interpositions, ure, making no attempt to link illusory brightening to finding that although this patient could not see illusory illusory contours. Indeed, it is not obvious that Ehrenstein contours monocularly, stereoscopically presented illusory even noticed the illusory contours within his stimuli. In contours were seen vividly. This study suggests that, at his brightness research, however, he established many of least for some stimuli, salient illusory contours require that the key configural parameters that would later be used to subjects perceive illusory figures as being separated in study contour clarity, as well as brightness, including depth from the inducing elements. number ofinducing lines, width of lines, size of stimuli, and small-scale shape of the line ends. Ehrenstein's ex­ PHENOMENOLOGICAL MODELS OF planation for illusory brightening was that the inducing el­ ILLUSORY CONTOUR FORMATION ements were laterally blurred into their surround because ofeye movements, resulting in a real contrast difference The first two phenomenological models ofillusory con­ between the central region and the surround. This theory tour formation set the distinctly different paths that re­ was discredited by Spillman, Fuld, and Gerrits (1976), search for the following decades was to take. Ehrenstein who observed illusory brightening when an Ehrenstein (I941), in a highly influential study of variations of his figure was flashed for a duration briefenough to preclude eponymous figures, was primarily interested in the nature eye movements, but Ehrenstein's work stands as a concise and origin ofillusory brightness. Future researchers along examination ofthe parameters that affect illusory bright­ this line, whom I will refer to as members ofthe brightness ening and illusory contour strength. camp, expanded upon Ehrenstein's theory and expounded that illusory brightness actually drove illusory contour Gestalt Theories formation. On the other hand, Kanizsa (I955) stressed the Kanizsa (I955) presented an impressive array ofstim­ importance of understanding illusory forms as a whole, uli containing illusory contours, ofboth the edge and the downplaying the significance ofsuch specific character­ line-end type. Prior to Kanizsa's paper, most examples of istics as brightness and clarity. These features, he believed, illusory contours had been rather weak, consisting pri­ followed naturally from the instantiation of an illusory marily ofstraight contours, like those in Schumann (1900, form. The researchers concentrating on the form of illu­ 1904). Kanizsa, a member ofthe Gestalt school, saw illu­ sory figures I will refer to as members ofthe form camp. sory figures as a consequence ofthe "poorform" ofthe in- PERCEPTUAL THEORY OF ILLUSORY CONTOURS 301 ducers. That is, he saw the inducers as incomplete in the retracted his original explanation ofillusory contours for­ gestalt sense. He posited that in a configuration such as mation (Kanizsa, 1976, 1987), but there can be little doubt that ofFigure 2a, the incomplete circular inducers are re­ that regularity ofthe inducers can have a significant effect organized by the perceptual system so that they become on the strength ofthe resulting contour (see discussion of more complete or regular circles, because there is a ten­ high-level determinants). In a related area, Wasserstein, dency in visual perception toward simplicity and stability: Zappulla, Rosen, and Gerstman (1987) have shown a link­ prdgnanz, Reconciliation ofthis percept with the missing age between poor closure ability and lack ofillusory con­ inducer sectors requires the perception of an illusory tour perception in brain-damaged subjects. The gestalt square occluding parts of each circle. Note that Kanizsa characteristic of continuation, which has obvious bonds considered neither the completion nor the reconciliation with both completion and closure, has also been impli­ to be a conscious decision, but rather an autonomous, self­ cated as a driving force behind contour formation (Kanizsa, regulating process. The concept ofcompletion ofpartially 1974; Minguzzi, 1987). As is the case with completion, occluded objects was not new, but the idea that such however, psychophysical experiments have not supported amodal completion (of the inducers) could occur in the a causal link between inducer continuation and contour absence ofreal (luminance induced) occlusion and thus be strength (Davi et al., 1992). intimately related to illusory contour formation, certainly was. Kanizsa noted that the brightness theory was a possi­ Depth Theories ble alternative, but he discounted it primarily on the Chronologically, the next major explanatory initiative re­ grounds that illusory figures appear brighter than they garding illusory contours came again from the form camp, should if conventional brightness contrast alone was re­ as Coren (1972) proposed his "depth cue" theory. Noting sponsible for the brightening. This argument has been val­ that all illusory figures (known at that time) were perceived idated experimentally in studies showing brightness to be as occluding the inducers, he posited that interposition acts greater in illusory figures than in controls having the same as a cue for illusory contour formation. That is, once depth local configurations (Bradley & Mates, 1985; Coren & cues have been perceived, the visual system supplies ap­ Theodor, 1975; Dresp, 1992). propriate surfaces to account for the apparent depth strati­ Aspects ofthe gestalt approach to illusory contour for­ fication ofthe stimuli. Stimuli that yield weak illusory con­ mation have not held up well to scrutiny. The basic tenet tours do so because depth cues are weak. Whereas Kanizsa ofthis approach is that the (amodal) completion ofirreg­ (1955) argued that completion ofirregular inducers drove u1arinducing elements is primarily responsible for driving contour formation, Coren argued that the implicit depth the formation ofillusory figures. Therefore the existence cues signified by some aspects ofthese irregular inducers ofinducers that are already regular (in the gestalt sense), drove contour formation, which in turn drove amoda1com­ or do not become more regular (or even become more ir­ pletion ofthe inducers behind the illusory figure. regular) upon completion, would refute such a theory. A Unfortunately, Coren failed to specify any concrete number of researchers have demonstrated stimuli with depth cues other than interposition, leaving himselfopen such inducers (Davi et al., 1992; Day & Kasperczyk, 1983; to the criticism of circularity of definition by Rock and Kellman & Shipley, 1991; Purghe, 1988, 1991; Purghe & Anson (1979): there can be no interposition before two Coren, 1992a; Purghe & Katsaras, 1991), several exam­ surfaces are perceived, and in Coren's hypothesis, there will ples ofwhich are depicted in Figure 20. Kanizsa himself not be two surfaces until interposition is noted. Further-

Figure 20. In contradiction with Gestaltist predictions, complete inducing stimuli can generate salient illusory contours, as indicated by these examples of (a) Day and Kasperczyk (1983a), (b) Purghe (1991), and (c) Kellman and Shipley (1991). Figure 20a is from "Amodal Completion as a Basis for Illusory Contours," by R. H. Day and R. T. Kasperczyk, 1983, Perception & Psy­ chophysics, 33, pp. 355-364 (Figure 3). Copyright 1983 by Psychonomic Soci­ ety, Inc. Reprinted by permission. Figure 20b is from "Is Amodal Completion Necessary for the Formation of IUusory Figures?" by F. Purghe, 1991, Percep­ tion, 20, pp, 623-626 (Figure 2). Copyright 1991 by Pion Limited, London. Adapted with permission. Figure 20c is from "A Theory of Visual Interpola­ tion in Object Perception," by P. J. Kellman and T. F. Shipley, 1991, Cognitive Psychology, 23, pp. 141-221 (Figure 41). Copyright 1991 by Academic Press. Reprinted with permission. 302 LESHER

more, not all illusory figures involve interposition, as is ing and memory on contour perception (Coren et aI., 1986, the case with diffuse illusory contours (Kennedy, 1976, 1987; Gellatly, 1982; Gellatly & Bishop, 1987; Landauer, 1978b, 1979), physically three-dimensional constructs 1978;Pritchard & Warm, 1983;Wallach& Slaughter, 1988), (Ware & Kennedy, 1977, 1978a, 1978b), stimuli with in­ and other time course studies (Meyer & Fish, 1987;Pomer­ ducers on the inside of the illusory figure (Kennedy & antz,Goldberg, Golder,& Tetewski, 1981;Takahashi, 1993). Lee, 1976; Parks, 1986; Purghe & Coren, 1992a), and, in­ In spite of a considerable degree of empirical support deed, even the phase-shifted grating first presented by for certain aspects of the problem-solving approaches, Coren himself (Coren, 1972). Coren's formulation ofthe these theories are no longer in favor as the primary expla­ depth-cue theory is untenable, but, as I will point out later, nation ofillusory contour formation. To disprove cognitive his conjecture contains the kernel ofwhat is now a widely theories absolutely would be extremely difficult, because accepted general theory of illusory contour formation. of their relative lack of specificity in the computational sense of affording testable implementations or simula­ Cognitive Theories tions. Given that problem-solving approaches are merely Gregory (1972) authored the next important paper from restatements of the ecological purpose of illusory con­ the form camp, arguing that illusory contour formation tours, we would expect that psychophysical data would was largely a cognitive or cognitive-like process. Kanizsa offer few conflicts with cognitive theories. However, the and Coren likewise believed illusory contour formation to recent neurophysiological data of von der Heydt et al. be a rather high-level operation, but both stressed the pas­ (1984, 1989b) and the similarities observed between real sive and automatic nature of the process. On the other and illusory contours in terms oflow-level perceptual ef­ hand, Gregory placed the emphasis on a postulation method fects indicate a relatively low-level instantiation of illu­ wherein cognitive explanations ofthe stimuli configura­ sory contours. Although not completely incompatible with tion were proffered, presumably subconsciously. In a more cognitive theories because ofthe extensive degree of top­ comprehensive paper, Rock and Anson (1979) referred to down feedback to visual cortex (Zeki & Shipp, 1988), illusory figures as solutions to a perceptual problem, the­ these data strongly suggest that bottom-up mechanisms orizing that the solutions are arrived at through some cog­ can themselves sustain the basic elements ofillusory con­ nitive process. In the case ofa Kanizsa figure, for exam­ tour formation. Neural models indicate a range ofpossi­ ple, the percept ofan occluding triangle ofthe same color ble incarnations that such mechanisms may take. as the background's is more likely (in some sense) than the Although cognitive theories ofillusory contour forma­ veridical configuration of three incomplete circles with tion appear unparsimonious now that local, low-level fea­ perfectly aligned sector edges. VanTuijl and Leeuwenberg tures and processes sufficient to produce basic illusory con­ (1979, 1982) have attempted to quantize the likelihood of tours have been identified, it would be a critical error to a given percept via an energy/information minimization disregard the role that cognitivefactors play in establish­ approach. Using a language code for pattern representa­ ing the perception of these contours. Low-levelneural mod­ tion, they have demonstrated a high degree ofcorrelation els serve as strong foundations for illusory contour for­ between subjects' perceptions and the efficiency of a mation, but they cannot represent the entire process. given perceptual organization. Bottom-up models can never explain the profound effects In support oftheir arguments, Rock and Anson (1979) that perceptual set, memory, and attention have on the show the effect of perceptual set on contour perception course ofillusory contour formation. In addition, illusory (see earlier discussion). Ifa cognitive postulation process contours are not perceptually identical to real contours: we is responsible for instantiation of illusory contours, then can often distinguish between the two types (Ware & altering cognitive expectations should (and does) alter con­ Kennedy, 1978a, 1978b), and they are not always equiva­ tour perception. Another convincing experiment involves lent in perceptual comparisons. Ultimately, the actual per­ the superposition ofa grating pattern over a Kanizsa fig­ ception ofillusory contours is not only a cognitive process, ure, as described in the high-level determinants section but doubtless a process that also manifests itself in lower and depicted in Figure 16 (Rock & Anson, 1979). The per­ level neural representations. ceived depth of the grating has a profound effect on whether subjects perceive an illusory figure or not. In Brightness Theories Revisited studying the time course of illusory contour perception, Brigner and Gallagher (1974) first examined the link Reynolds (1981) employed a similar (monocular) stimu­ between conventional brightness contrast and illusory lus in which a regular pattern was superimposed on a Ka­ contour brightness and strength. Recent studies have nizsa figure, as described in the discussion of psycho­ shown that the degree ofillusory darkening in a Kanizsa physical similarities between contour types. The salience square with light-on-dark inducing elements is propor­ of the illusory contour varied as a function of time, ap­ tional to the degree ofconventional brightness contrast in parently indicating time-varying cognitive hypotheses re­ similar control stimuli (which do not result in illusory fig­ garding the relative depths ofthe inducers, overlayed pat­ ures) under variations of spatial and luminance character­ tern, and illusory figure. Additional support for this form istics ofthe inducing elements (Dresp, 1992; Dresp et aI., of cognitive theory comes from the perception of re­ 1990). Using flickering stimuli, Coren (1991) determined versible and ambiguous illusory figures (Bradley, 1982, that figure brightness and contour strength were depen­ 1987;Bradley,Dumais, & Petry, 1976),the effects ofleam- dent on the level oflateral inhibition in the retina-inter- PERCEPTUAL THEORY OF ILLUSORY CONTOURS 303 actions responsible for brightness contrast effects (Ratliff, stimuli inducing illusory contours actually contained the 1965). Dresp (1992) has demonstrated an increased sen­ illusory form information in their spatial frequency spec­ sitivity in detection ofsmall targets near the inducers ofa trum-that the illusory forms were not, in fact, illusory if Kanizsa square but not in the center ofthe illusory figure, one considered only the low-frequency information ofthe implying that local brightness mechanisms may be respon­ inducer configuration. If the early stages of visual pro­ sible for generating illusory brightness but cannot be re­ cessing had various frequency bands, these low-frequency sponsible for the filling in ofthe illusory figure that results illusory forms could easily be perceived directly. This the­ in the perception ofa form with homogeneous brightness. ory was discredited by Tyler (1977), who attacked Gins­ Although simultaneous contrast may explain illusory burg's methodology; Parks and Pendergrass (1982), who brightness effects (at least for edge inducers), proponents showed the existence of illusory contours without low­ ofthis explanation for illusory contours failed to address frequency correlates; and Parks (1983), who showed that a key issue: Namely, how are the actual illusory contours existence of a low-frequency form did not necessarily generated? It seems odd, in retrospect, how cavalier early yield an illusory form. Nevertheless, it is important that re­ researchers were about relegating contour clarity to sec­ searchers, especially in neurophysiological studies and re­ ondary status. The typical explanation ofthe contours them­ search involving animals and infants where it is difficult selves was that once illusory brightness was established, to ascertain exactly what the subjects are perceiving and the information was "then used together with physically to what they are responding, ensure that low spatial fre­ present brightness gradients to generate perceptions" quency content ofdisplays does not playa significant role (Frisby & Clatworthy, 1975)-arather opaque exposition. in what they believe to be illusory contour perception (Para­ Another major problem with the conventional bright­ diso et aI., 1989; Vogels & Orban, 1987). Lack of such ness contrast approach is that it fails to explain the strik­ controls casts doubt on the reliability ofmany ofthe ani­ ing brightening in figures containing very little contrast, mal and infant studies ofillusory contour perception. such as the Ehrenstein figure. By slanting the inducing lines of an Ehrenstein figure, Kennedy (1978a) showed Multistage Theories that more conventional contrast (due to inducers lying tan­ Day and lory (1980) proposed that illusory contour for­ gentially to the illusory circle) did not result in stronger il­ mation was a two-stage process, the first stage being the lusory brightening and in fact weakened both clarity and instantiation ofillusory brightness via brightness contrast brightness considerably. Frisby and Clatworthy (1975) ad­ and dissimilation, and the second being the formation of dressed the issue of brightness in line-end induced fig­ the sharp illusory contour itself. Here, then, was explicit ures, noting the existence of increased brightness at the recognition from the brightness camp that contour forma­ ends oflines-a phenomenon later referred to as "bright­ tion itselfwas not a trivial, obvious process. However, no ness buttons" by Kennedy (1979). In revised brightness explicit method was proposed to explain contour forma­ theories, these buttons play the same role as conventional tion. Halpern (1981) was one ofthe first to note that the brightness contrast, spreading out in some undefined way solution to the problem of illusory contour formation to form the illusory figure. Similar approaches were put would not come from one camp or the other, but rather from forth by Kennedy (1978a, 1979, 1988) and lory and Day an amalgamation ofdifferent theories. (1979; Day & lory, 1980), the latter ofwhom introduced Soon after, theories considering illusory brightening as the widely used term dissimilation to capture the bright­ the primary factor in illusory contour formation received ness button effect. Frisby and Clatworthy went so far as to a fatal blow from the short paper by Prazdny (1983), who propose crude receptive fields capable ofgenerating dis­ displayed figures similar to that in Figure 9a, consisting of similation, a noteworthy predecessor to the neural models equal numbers ofdark and light inducers on a gray back­ that would follow. lory (1987) reported an increase in lu­ ground which still yielded a salient boundary. Many sim­ minance sensitivity near the ends oflines, a finding bol­ ilar qualitative displays have been demonstrated (Gross­ stered by a recent comprehensive study by Dresp (1993). berg & Mingolla, 1985a; Hershberger & Stallard, 1984; Dresp found that detection thresholds for light dots were Shapley & Gordon, 1985, 1987), while Kellman and Lou­ lowered when they were in the proximity ofthe ends oflight kides (1987) have confirmed with quantitative psycho­ lines, but not for dots near "T" or 'T' junctions, an indi­ physics that such inducing stimuli can indeed induce clear cation that only in the absence ofperpendicular elements illusory contours in the absence ofany illusory brightness. do line ends generate brightness buttons. The dissimilation Advancing interdisciplinary approach one step further, theories, which invariably included simultaneous bright­ Day (1986, 1987) attempted to reconcile bottom-up and ness contrast effects as well, could explain most brightness top-down theories. De-emphasizing the importance ofdis­ effects in both edge and line-end illusory figures, but they similation per se, Day conjectured that it was because line suffered from the same lack ofattention to actual contour ends were an example ofCoren 's depth cues that they were formation as did conventional brightness theories. important in illusory contour formation. A link between neurophysiological data-previouslythe stronghold ofthe Spatial Frequency Theories brightness camp-and the form camp was thereby estab­ During the debate between the brightness and form lished. This theoretical vein has been extended by Kell­ camps, a third party briefly entered the fray. Ginsburg man and Loukides (1987), who posited that spatiotempo­ (1975, 1987) and Becker and Knopp (1978) proposed that ral discontinuities-abrupt changes in object borders, 304 LESHER

positions, luminances, binocular depth, and so forth-are enough to truly explain the formation ofany contours­ the most general ofdepth cues and therefore drive illusory straight or curved-across extended regions ofhomoge­ contour formation. Shipley and Kellman (1990) have neous luminance. shown that contour formation is dependent on the pres­ Ullman (1976) presented the first true computational ence of spatial discontinuities in the inducing stimuli, at model ofillusory contour formation. His model is neural least for inducers containing sharp luminance edges. Heit­ in the sense that it consists ofseveral layers oflocally in­ ger, Rosenthaler, von der Heydt, Peterhans, and Kiibler terconnected cells. Based on completion collinear to edges (1992) have since proposed a specific neural model for the and lilies, Ullman's model consists ofthree layers ofcells determination of spatial discontinuities in the luminance connected in such a way as to achieve a preference for domain. Examples ofother spatiotemporal discontinuities straight-line completion, but with an allowance for curved that may produce illusory contours include jumps in binoc­ completion when straight-line completion is not possible. ular depth and accretion/deletion ofbackground elements. These completion constraints are accomplished solely by A basic, although generally unspoken, tenet of many local connections: cells excite neighbors that are roughly cognitivists had been that low-level mechanisms were in­ collinear and have similar orientations. Within the first two capable of solving the ecological problem posed by illu­ system layers, these local interactions establish a number sory contour inducing stimuli. The recognition that depth of fuzzy curves. Competition in the third layer forces a cues can be assigned geometric definitions, the develop­ choice ofspecific local curves. In the case ofinducers that ment ofneural mechanisms to integrate these cues, and the are not aligned collinearly, the illusory contours created by strong neurophysiological and psychophysical evidence Ullman's model have minimal curvature, consisting ofcir­ for early instantiation ofillusory contours have combined cular arcs tangent to the inducer edges. Although Ullman to legitimize neural models of contour formation in the made no attempt to locate a particular neural substrate of eyes ofmany researchers. Within the last 3 years, the num­ his model, he showed that locally connected "neurons" ber ofsuch models has more than doubled, and significant could produce curved illusory contours. enhancements have been made to the earlier systems. As Ullman's (1976) model pioneered neural models ofil­ I will show in the next section, however, the apparent res­ lusory contour formation but received little attention from olution ofthe brightness-form conflict has not led to a less the research community, probably because (1) it only ex­ contentious field of research; there are fundamental dif­ plained completion parallel to edges, ignoring the per­ ferences in the approaches to neural modeling as well. In pendicular completion of line-end inducers, (2) it was a addition, there are still many unresolved issues regarding computational vision model in a time oflimited computer the integration ofhigher level determinants and cognitive power, defying widespread simulation, and (3) it appeared factors into neural models. to have little basis in real neural structures. Itwas not until the influential papers ofGrossberg and Mingolla (1985a, NEURAL MODELS OF 1985b) and Peterhans et a\. (1986) that neural systems re­ ILLUSORY CONTOUR FORMATION ceived widespread attention. These models had solid bases in physiology and could generate edge- and line­ Although neural systems represent the newest genera­ end-induced illusory contours, explaining the rudiments tion of models of illusory contour formation, they cer­ ofboth parallel and perpendicular completion. tainly do not lack in sophistication or complexity. These The approach ofPeterhans et a\. (1986), which has in­ architectures consist of networks of cells with weighted spired a variety of more sophisticated models (Finkel & connections between units, and, to varying degrees, they Edelman, 1989; Kellman & Shipley, 1991), bases all illu­ are meant to describe the structure as well as the function sory contour induction on line ends. Line-end (or discon­ ofthe networks in the visual cortex that are responsible for tinuity) models rest on the supposition that illusory con­ illusory contour formation. Unlike most ofthe phenome­ tours are generated solely by ends oflines and the corners nological models discussed in the previous section, neural ofedge inducers, with completion primarily in directions models contain analogues ofbiological substrates as well perpendicular to the inducing lines or corner elements. as a theoretical basis ofcontour formation. Neural mod­ Although their classification might seem to limit their ex­ els were rejected out ofhand by early researchers, because planatory role to line-end inducers, these models still main­ they believed such models were incapable of emulating tain the capability ofgenerating illusory contours with edge even such relatively simple phenomena as curved illusory inducers. Unlike the parallel completion approach ofUIl­ contours (Gregory, 1972). Early evaluations were made on man (1976) or the perpendicular completion approach of the basis of ideas such as those of Jung and Spillman Peterhans et a\., hybrid models encompass both forms of (1970), who proposed that line-end illusory contours might completion. More importantly, however, illusory contours result from brightness induced by cells with elliptical re­ may be generated by either line ends or edges in hybrid ceptive fields at the ends oflines (also Frisby & Clatwor­ systems. Although they employ markedly different para­ thy, 1975), and Stadler and Dieker (1972), who proposed digms, both the boundary contour system of Grossberg that illusory contours were the result ofpartial activation and Mingolla (1985a, 1985b)and the newly proposed model of the same feature detectors that were responsible for ofHeitger and von der Heydt (1993) include mechanisms perception ofluminance-defined contours. Indeed, many for induction by both line ends and edges, along with par­ of these early model outlines were not specified well allel and perpendicular illusory contour completion. PERCEPTUAL THEORY OF ILLUSORY CONTOURS 305

The Line-End Models lusory contours (and for strengthening ofdegraded occlu­ Peterhans et al. (1986). On the basis of their neuro­ sions). This dichotomy, although shared by most other physiological studies, Peterhans et aI. (1986) proposed a models of illusory contour formation, is not a requisite model ofillusory contour formation in which real and il­ characteristic for such modeling, as has been shown by lusory contour representations are unified in early visual Grossberg and Mingolla (1985a, 1985b, 1987a, 1987b). cortex (V2). Illusory contours are generated by end­ Finkel and Edelman (1989). The system of Finkel stopped cells whose receptive fields lie along a line roughly and Edelman (1989) is founded on the preliminary work perpendicular to their preferred orientation, as depicted in ofPeterhans et aI. (1986), but it represents a more sophis­ Figure 21a. Pairs ofthese cells multiplicatively gate one ticated model, with significantly greater explanatory another, ensuring that a single inducer cannot trigger for­ power. Whereas Peterhans et aI. only attempted to explain mation of an illusory contour. The output ofeach gating the simplest of illusory contour phenomena, Finkel and is summed with all other gated outputs, as well as with the Edelman have explored subtle issues such as the integra­ activity of complex cells whose positions and preferred tion ofcontour signals from different modalities and fig­ orientations match the line along which the end-stopped ural interference between real and illusory contours. How­ cells lie. The gating ofend-stopped cells signals the pres­ ever, as in the Peterhans et aI. model, end-stopped cell ence of illusory contours, while the complex cells signal activity drives all illusory contour formation. A diagram the presence ofreal contours. A unified contour represen­ of the occlusion modules of Finkel and Edelman's com­ tation is achieved by summing these two representations. plex model is presented in Figure 22. The full model in­ The model of Peterhans et aI. (1986) is based on per­ cludes a motion-processing subsystem, which is fully in­ pendicular completion between line ends. How can edge­ tegrated with the occlusion system. Although the motion induced illusory contours be explained in this context? aspects oftile system will not be discussed in detail, it is Peterhans et aI. posit that the cornersof edge inducers trig­ important to note that such a multimodal model represents ger end-stopped cell activity, as depicted in Figure 21b. a first step toward simulating the interactions between il­ Since the same end-stopped cells that responded to line lusory contours generated in different domains. Ofpartic­ ends can be triggered by the comers ofedge inducers, con­ ular interest in this model, motion-induced contour infor­ tour completion can occur with both inducer types in di­ mation feeds back into the luminance-defined channels, rections perpendicular to the preferred orientations ofthe allowing for emulation of second-order contour induc­ individualend-stopped cells.This system introduces a para­ tion, in which motion-defined inducers generate illusory digm common to many models ofcontour formation: the contours (Kellman & Loukides, 1987; Prazdny, 1986). existenceof separatepools of cells for real and illusory con­ After generating oriented and end-stopped cell re­ tour detection. Complex cells are employed for detection sponses via simple receptive field models, Finkel and of real luminance gradients, whereas cells that respond to Edelman (1989) pooled end-stopped cell activity over pairs ofend-stopped cells are employed for detection of il- similar orientations at the same location into a "wide- a b

Figure 21. (a) In the model of Peterhans, von der Heydt, and Baumgartner (1986), signals for real and illusory contours additively converge at the same neuron. (b) End-stopped cells respond to the corners in a Kanizsa figure lead­ ing to illusory contour formation. From "Mechanisms of Contour Perception in Monkey Visual Cortex," by E. Peterhans and R. von der Heydt, 1989, Jour­ nal ofNeuroscience, 9, pp. 1749-1763 (Figures 15-16). Copyright 1989 by the Society for Neuroscience. Adapted with permission. 306 LESHER

Finkel and Sajda (1992). Finkel and Sajda (1992, Occlusion - 1994; Sajda & Finkel, 1992a, 1992b, 1995) have pre­ Module I sented an intermediate level visual model for segmentation Termination Occlusion ofscenic elements, including illusory figures, into differ­ - Discontinuity Conflict CeUs Module ent depth planes. The primary mechanism for illusory con­ tour formation is based on perpendicular completion be­ I tween line ends and comers, much as the earlier model of Common Wide• Angle Termination Termination Finkel and Edelman (1989). As such, one would expect Module - CeUs that the model of Finkel and Sajda should behave in a ~ manner qualitatively similar to that ofFinkel and Edelman • Reentrant when confronted with variations in low-level inducer de­ Termination /~ Conflict terminants. However, Finkel and Sajda's model is geared CeUs Module toward representation of sophisticated surface percep­ t tions, rather than the effect oflow-level factors, and there­ Oriented __1 fore may be able to emulate certain high-level determi­ CeUs nants, especially those related to the effects ofperceived depth on illusory contour formation. t Nakayama et al. (1990) have pointed out that local edge Image information can be used to determine whether boundaries Figure 22. The luminance-based processing modules of the are "intrinsic" to a particular object or whether they are model of Finkel and Edelman (1989). From "Integration of Dis­ "extrinsic" to the object-signifying occlusion by another tributed Cortical Systems by Reentry: A Computer Simulation object. In Finkel and Sajda's (1992) model, local and of Interactive Functionally Segregated Visual Areas," by L. H. Finkel and G. M. Edelman, 1989, Journal ofNeuroscience, 9, global cues to boundary "ownership" are linked through a pp. 3188-3208 (Figure 1). Copyright 1989 by the Society for series ofcompletion mechanisms, resulting not only in a Neuroscience. Adapted with permission. determination ofwhich contours belong to the same sur­ face, but also in full representations ofall surface contours, both modal and amodal, stratified into distinct depth angle" representation. They then specified that termina­ planes. Cues to contour ownership employed by Finkel tion discontinuities-ecological cues to occlusion---eould and Sajda include closure, similarity and proximity, local arise only ifat least three such wide-angle cell pools were concavities, and direction of line terminations. By com­ active along a line perpendicular to the center (mean) ori­ bining these cues through a series oflocal interactions the entation ofthese wide-angle pools. Finally, perpendicular model arrives at a stratified representation of surfaces, contour completion between termination discontinuities which may be refined via feedback of real and hypothe­ occurred at the occlusion module. Finkel and Edelman sized (either illusory or amodal) contours to earlier stages. speculated that the contour representation at the occlusion Kellman and Shipley (1991). Kellman and Shipley module fed back to early levels via a "reentrant" system. have abstracted the line-end theory ofcontour induction a Reentrant signals would allow the system to mediate con­ step further, claiming that first-order spatial and spatio­ flicts between real and illusory contours, sharpening or temporal discontinuities are the generating features ofil­ eliminating contours as appropriate. Conflict resolution lusory contours. These discontinuities can lead to illusory would be required, for example, when a real line crossed contour formation ifthey are relatable-that is, ifthey sat­ an illusory contour. Besides addressing this form of fig­ isfy certain constraints concerning the intersection ofthe ural interference,Finkel and Edelman's model also included extensions of lines perpendicular to the inducing line or a mechanism that may account for some small-scale inter­ parallel to the edge associated with inducer comers, as is ference effects. At the termination discontinuity level, ac­ indicated by Figure II. Unlike in the earlier line-end mod­ tivity was inhibited ifthe common termination module de­ els, Kellman and Shipley propose that contour completion termined that some ofthe end-stopped activity was actually can proceed in directions parallel to the supporting edges arising from a natural comer or vertex-features known to of the inducers, provided that these edges are associated inhibit contour formation-rather than a line termination. with discontinuities. Because contours are initially in­ The importance ofline terminations as occlusion indi­ duced only by discontinuities, this model is not technically cators for scene segmentation is the basic tenet ofFinkel considered a hybrid system, although its use ofboth paral­ and Edelman's (1989) model. The structure ofthe result­ lel completion and mediation ofcontour strength by edge ing model mandates that illusory contour formation is me­ information positions the model at the border between diated only by these terminations. Thus illusory contours line-end and hybrid systems. formed by edge inducers must result from end-stopped ac­ The degree to which discontinuities are relatable deter­ tivity arising from termination cell activity at the comers mines the contour strength, a hypothesis supported by sev­ ofinducing elements, as Figure 21b indicates for the sim­ eral recent experiments (Shipley & Kellman, 1992b). As pler model ofPeterhans et al. (1986). There is neither par­ in the Finkel and Edelman model, edge-induced illusory allel completion nor mediation ofcontour strength by par­ contours are induced by the discontinuities at the comers allel (edge) information within this model. ofthe inducing elements (Figure 2Ib). Shipley and Kell- PERCEPTUAL THEORY OF ILLUSORY CONTOURS 307 man (1992a) presented data indicating that contour clar­ tend to respond most strongly to the ends oflines or cor­ ity is a linear function ofthe support ratio over a range of ners. In the second competitive stage, cells at the same lo­ 0.3 to 0.8, where the length of the supported region and cation, but with different orientational preferences, com­ the length of the unsupported region were varied inde­ pete. The first step in end-cut generation occurs in the first pendently. Banton and Levi (1992) have reported a mono­ competitive stage as strong cell activations along the edges tonic, but nonlinear, relation. By scaling the size oftheir ofa line inhibit the weaker activations near the line end. stimuli, with inducers and gaps increasing proportion­ In the second competitive stage, these weakened activa­ ately, Shipley and Kellman showed that clarity is a fixed tions disinhibit perpendicular cell activations near the end function ofsupport ratio over a range ofabsolute stimulus ofthe.line, generating the perpendicular end cut. sizes. Their theory holds that illusory contour strength is The actual illusory contours in the BCS ate generated directly proportional to the support ratio, regardless ofthe via a process oforiented long-range completion. In con­ absolute degree ofsupport, the proximity ofthe inducers, trast with the previously discussed researchers, Grossberg or the size of the inducers. Kellman and Shipley (1991; and Mingolla believe that completion always occurs in di­ Shipley & Kellman, 1992a) noted that the support ratio rections approximately parallel (collinear) to the orienta­ must be incorporated into the mechanics ofthe illusory con­ tion preferred by the inducing units of the second com­ tour interpolation process, but they provided no further petitive stage. Edges directly activate these inducing units, computational details. Note that this model is not strictly whereas line ends rely on the first and second competitive neural-Shipley and Kellman presented no details ofhow stages to indirectly activate inducing units through end their system should be implemented-but could be linked cutting. To accomplish contour completion, bipole filters to a neural architecture with modest efforts. with bowtie-shaped receptive fields combine inputs from spatially disparate cells whose receptive field centers fall The Hybrid Models roughly along a common line or curve, and whose orien­ Grossberg and Mingolla (1985a, 1985b). Grossberg tational preferences are consistent with that line or curve. and Mingolla (1985a, 1985b, 1987a, 1987b) proposed the In discussing their recent psychophysical experiments, existence of two parallel systems in the visual stream, a Field, Hayes, and Hess (1993) have recently referred to boundary contour system (BCS) for establishing the boun­ such a mechanism as an "association field." The relative daries ofobjects, and a feature contour system (FCS) for contribution ofa cell to a bipole filter depends on that cell's establishing the color and brightness ofthese objects. Il­ position, alignment, and orientation with respect to the bi­ lusory contour formation is performed by the BCS through pole. A bipole cell fires only ifboth sides ofits bipartite interactions between spatial arrays of cells which repre­ receptive field are sufficiently excited. These cells feed sent orientations ofboundaries at every position. Itshould back to earlier stages, providing orientational information be noted that in Grossberg and Mingolla's approach, the to locations where there are no bottom-up signals sup­ visibility ofillusory contours depends on the FCS's estab­ ported by image contrast. By tuning the extent to which lishing local brightness differences between the regions on oriented cells in various positions, alignments, and orien­ either side ofthe contour. As Figure 23 indicates, the first tations (with respect to the bipole) contribute to bipole ac­ stage ofthe BCS consists ofan array oforiented contrast tivity,the discrete (on or off) long-range completion scheme detectors at each spatial location. The inputs to these de­ employed by Peterhans et al. (1986) and Finkel and Edel­ tectors consist not ofthe raw image data, but rather ofthe man (1989) can be accomplished with a bipole filter, as output of a competitive on-center, off-surround prepro­ can a measure similar to the relatability of Kellman and cessing stage designed to emulate the partial brightness Shipley (1991). normalization performed within the retina. In the second Once boundaries have been determined by the BCS, the BCS stage, first-stage cells with the same orientational filling-in process ofthe FCS establishes brightness levels preference but opposite contrast preference are pooled to throughout the image. Within the FCS, retinal brightness yield a contrast-independent boundary representation. signals spread out diffusively until blocked by BCS bound­ Because units sensitive to oriented image contrasts may ary signals, restoring brightness information removed dur­ be activated by stimuli falling anywhere within their ex­ ing BCS processing. Illusory contours are perceived only tended receptive fields, a degree ofpositional uncertainty when the FCS indicates that BCS contours represent boun­ is inevitably present in the distribution of responses of daries between regions of two locally different bright­ fields of such units. This is particularly true at line ends nesses. Note that in cases ofstimuli with opposite-contrast and comers, where the lack ofdefinite orientation infor­ inducing elements, local brightness effects will not nec­ mation leads to severe ambiguity. Grossberg and Mingolla essarily result in a global brightness percept. posited the existence of a two-stage competitive mecha­ Grossberg (1994). Drawing upon a large body ofearlier nism designed to overcome these uncertainties oflocal mea­ work (Grossberg, 1987a, 1987b; Grossberg & Marshall, surement. These processes establish "end cuts" at the end 1989; Grossberg & Mingolla, 1985a, 1985b, 1987a, 1987b; ofeach line-activityat cells with orientation preferences Grossberg & Rudd, 1989, 1992), Grossberg (1994) has re­ perpendicularto the line itself. In the first competitive stage, cently synthesized an advanced theory of human vision cells with the same orientational preference in neighbor­ model called FACADE (from form and color and depth). ing locations compete, functionally generating end stop­ As the name indicates, this BCSIFCS-based theory incor­ ping in the sense that survivors of the competition will porates not only form, the one aspect that has been dis- 308 LESHER

O Second COl!lPl\titive Yv--'-J+ Sfage 00 First COl!lPl\titive Sfage

Figure 23. The boundary contour system of Grossberg and Mingolla (1985a, 1985b, 1987a, 1987b). From "Neural Dynamics ofPerceptual Grouping: Textures, Boundaries, and Emergent Seg­ mentations," by S. Grossberg and E. Mingolla, 1985, Perception & Psychophysics, 38, pp. 141-171 (Figure 30). Copyright 1985 by the Psychonomic Society, Inc. Adapted with permission.

cussed above, but also color and stereo depth (and also, Heitger and von der Heydt (1993). Heitger and von though not indicated by the name, motion). Ofparticular der Heydt (1993) have recently presented a model, loosely interest to this review are the advances that Grossberg has based on the earlier line-end model of Peterhans et a1. made in the area ofhow depth interacts with contour for­ (1986), which employs the sophisticated edge and dis­ mation. The basic premise behind depth perception in the continuity detectors described in Heitger et a1. (1992). FACADE system is that local depth information from Rather than the simple linear grouping ofthe line-end mod­ binocular and monocular cues forces individual bound­ els, however, signals from these detectors are combined aries onto independent BCS stratification planes. Inhibi­ via a two-lobed grouping field which closely resembles tion from large to small scales effects resolution of am­ Grossberg and Mingolla's bipole filter. As for the bipole biguous depths and eliminates boundaries from appearing filter, the weights ofthe grouping field can be assigned so in incorrect stratification planes. Once boundaries have that they mirror psychophysical data regarding several been stratified, modal (illusory) and amodal contour com­ low-level determinants. A multiplicative gating of lobe pletion are subserved by the same mechanisms acting on pairs ensures that completion is always in an inward di­ different planes. After all boundaries have been established, rection. By allowing "bent" combinations ofgrouping field boundary signals from large scales are reintegrated with lobes, the system can produce curved illusory contours. smaller scales for the purpose ofbrightness and color de­ After grouping, and a competitive stage that eliminates termination. This process ensures that subsequent FCS contradictory orientations at the same position, fuzzy filling in leads to veridical perception, preventing the as­ bands of activity indicate the presence and strength of signment ofindividual brightnesses and colors to multiple both illusory and real contours. In a collection ofimpres­ objects at the same position, but on different depth planes­ sive simulations, the maximal ridges along these bands a mishap that would make all occluding objects appear closely approximate human illusory contour perception. transparent. Reciprocal interactions between processing Unlike the line-end models discussed above, the model streams ensure that BCS boundaries and FCS surfaces are of Heitger and von der Heydt (1993) allows grouping in mutually consistent. Both modal and amodal contours are both parallel and perpendicular directions. Inputs to the established within the FACADE system, but only modal grouping field are not simple line-end activities, but a com­ contours are visible. plex combination ofortho(line-end) andpara (edge or line) PERCEPTUAL THEORY OF ILLUSORY CONTOURS 309

activity. The degree to which ortho and para activity con­ For each low- and high-level determinant, I will outline tribute to grouping is dynamically determined by a weight­ and justify the constraints that the psychophysical data ing factor K:, which indicates the "cornerness" ofthe stim­ place on neural models ofillusory contour formation. An uli at each point in space. The larger IC, the higher the relative analysis ofthe degree to which current models fulfill these contribution ofthe para signals. This model makes explicit constraints will also be provided. Because no modelers distinction between parallel and perpendicular comple­ have attempted to reconcile their neural models with the en­ tion, with different degrees ofcompletion being employed tire body ofpsychophysical data, some models support ex­ as determined by the computation of K:. This approach is perimental data overlooked by their originators. More often, fundamentally different from that taken in the BCS, in which however, the models fall short ofexplaining previously un­ parallel and perpendicular completion result from the considered phenomena, and, in some cases, are the subjects same mechanism operating in different spatial environ­ ofunsupported explanatory claims. Before dealing with the ments-an implicit distinction between the two forms of modeling constraints ofspecific determinants, I will discuss completion. Noting that line ends do not specify the exact some general issues regarding the inherent differences be­ orientation of the occluding object, but merely suggest tween line-end and edge inducers and how these differences the most probable orientation, Heitger and von der Heydt influence modeling ofillusory contour formation. have included an orientational tolerance for ortho comple­ tion but not for para completion, since the latter is tightly Edge and Line-End Inducers constrained by inducer edge orientations. The supposition ofline-end and discontinuity models is A potentially powerful aspect of Heitger and von der that all contour completion is in a direction roughly per­ Heydt's (1993) model is the manner in which figure­ pendicular to the inducing element or to that specific part ground assignments are made. By maintaining tags indi­ ofthe stimulus doing the inducing (e.g., a comer). This as­ cating the direction of termination of edges and corners sumption can be discounted on two grounds. First, numer­ and allowing completions only between certain combina­ ous researchers have shown that the extent ofedge support tions of termination directions, the output of the system (or more recently, the support ratio) has a significant ef­ indicates the likely depth relations between adjacent ob­ fect on illusory contour strength and illusory brightness. jects. Not only does this mechanism facilitate the inter­ The models of Peterhans et al. (1986) and Finkel and pretation ofillusory figures, it makes the system a poten­ Edelman (1989) determine contour strength solely by re­ tially powerful image-understanding tool for real-world lation of perpendicular elements. Conversely, contour imagery, as is evidenced by several examples presented by strength in the model ofKellman and Shipley (1991) is de­ the authors. Note, however, that precluding completion termined by inducer relatability, a concept that applies to across certain termination directions may destroy the abil­ both line-end and edge inducers, and support ratio, which ity ofthe network to model illusory contours with induc­ applies only to edge inducers. Thus Kellman and Shipley's ers on both the inside and the outside ofthe illusory fig­ model, but none ofthe stricter line-end models, survives ure (as in Figure 4b). this first finding. The documented effects ofthe degree of edge support precludes any models that do not at least in­ DEVELOPING AND EVALUATING clude mediation ofcontour strength by something akin to NEURAL MODELS support ratio. The second argument for discounting the basic premise Neural systems have evolved to a sophistication com­ of line-end models is that a number of researchers have mensurate for modeling advanced illusory contour phe­ demonstrated illusory-contour-inducing stimuli contain­ nomena only within the past few years. Widespread im­ ing no line ends, comers, or other such discontinuities plementation ofthe models has lagged behind their initial (Grossberg & Mingolla, 1985a; Lesher & Mingolla, 1993; theoretical formulation, but impressive simulations have Shapley & Gordon, 1987). The existence ofcontours in the recently surfaced. In particular, recent BCS/FCS simula­ absence of discontinuities precludes models that rely on tions and interactions have been employed to reproduce such features for contour induction, such as that ofKell­ results on the effect of line spacing and support ratio on man and Shipley (1991), although the use ofmultiple spa­ contour strength (Lesher, 1993), illusory brightness ef­ tial scales might allow these systems to sidestep this ap­ fects in line-end figures (Gove, Grossberg, & Mingolla, parent discrepancy, as described earlier. Without significant 1994a, 1994b), and visual persistence ofillusory contours modifications, none of the strict line-end models can be (Francis, Grossberg, & Mingolla, 1994). Using their hy­ reconciled with basic psychophysical data. The model of brid model, Heitger and von der Heydt (1993) have Heitger and von der Heydt (1993; which for the sake of demonstrated simulations ofcurved illusory contours and brevity will henceforth be referred to as the CMNCP­ contours generated by inducers on the inside of the illu­ computational model of neural contour processing-as sory figure. Sajda & Finkel (1992a, 1992b, 1995; Finkel taken from the title oftheir paper) suggests the form that & Sajda, 1994) have demonstrated depth ordering ofillu­ the necessary modifications might take. The completion sory figures and inducers. Yetdespite these advances, cur­ mechanisms ofKellman and Shipley's model might be con­ rent models only explicitly support a limited set of the sistent with rudimentary illusory contour data, but devel­ low-level determinants discussed in this paper, with al­ opment ofneural mechanisms and ofmethods ofscale inter­ most no support for the high-level determinants. actions are required before further evaluation is possible. 310 LESHER

Both ofthe hybrid models discussed in this article, the Low-Level Determinants BCS/FCS and the CMNCp, contain mechanisms that pro­ Before discussion ofindividual low-level determinants, vide them with the potential to model illusory contour for­ it will be useful to introduce a construct shared by the mation with both edge and line-end inducers. However, more sophisticated models ofillusory contour formation. the CMNCP perhaps has an advantage over the standard The bipole filter of the BCS, the grouping field of the BCS/FCS in modeling line-end contour formation. Lesher CMNCp, and the association field ofField et aI. (1992) all and Mingolla (1993) have shown that line-end inducers in share a common purpose-determining when contour a Varin configuration lead to significantly higher illusory completion should occur and establishing the resulting il­ contour strengths than do solid edge inducers. Reconcil­ lusory contour strength. These filters, which I will refer to ing the fact that a few highly localized inducers can yield by the generic name ofcompletionfilters, are similar from stronger contours than can an extended edge is difficult in a mechanistic standpoint as well: they integrate edge and a model such as the BCS, which contains a single pathway line-end information over two separate receptive fields for completion and a filter that spatially integrates over re­ (lobes) and fire only when both fields receive sufficient gions of completion. One must suppose that the end-cut excitation. The psychophysical constraint that illusory signals generated by competitive mechanisms are signifi­ contours appear only between two inducing regions and cantly stronger than the signals corresponding to extended do not extend outward from a single inducer mandates such edges. In the CMNCp, where the relative contributions of a dual-lobe arrangement. The known effects of inducer perpendicular and parallel completion can easily be bi­ separation, support ratio, and degree ofalignment mandate ased, there is no corresponding problem. Recent advances that the relative contribution ofeach cell to a completion in the BCS/FCS system, however, may rectify this appar­ filter depends on the relative position and orientation be­ ent shortcoming. Gove and colleagues (Gove, 1993; Gove tween the cell and the completion filter. Specific con­ et aI., 1994a, 1994b) have proposed a mechanism to ac­ straints placed on the completion filter's receptive field complish dissimilation-brightening at the ends oflines­ profile will be discussed below in the relevant sections. which also has the effect of strengthening the functional The relation between any pair ofcells, or between a cell end stopping performed by the competitive stages of the and a completion filter, can be uniquely characterized by BCS. When combined with the surround inhibition ofthe three parameters, as is indicated by the schematic ofFig­ first competitive stage, which weakens edge responses ure 24 (Grossberg & Mingolla, 1985a, 1985b). These pa­ more than end-cut responses, BCS end-cut activities can rameters are the Euclidean distance r between the center easily surpass edge activities in magnitude. A more radi­ ofthe cells' receptive fields, the angle 0 ofone cell with cal approach has been taken by Manjunath and Chellappa respect to the other cell's orientational preference, and the (1993), who have added a separate pool of end-stopped angle f/> between the orientational preferences of the two cells to a RCS-like system for texture discrimination. cells. Any receptive field can be characterized as some functionf(r,O,f/» ofthese three parameters. Simple recep­ Brightness Effects tive fields, such as those of like-oriented cells competing The mention ofmechanisms for modeling dissimilation across space (as in the first competitive stage ofthe BCS), raises the most significant shortcoming in the field ofthe can be characterized by a separable function: modeling of illusory forms-namely, the failure of any f(r,O,f/» = f,.(r)fe( O)f~( f/», (2) model except the BCS/FCS to address the issue ofillusory brightness. Even with edge inducers-letalone line ends­ with some degenerate or trivial.fx-for this case, some­ Dresp (1992) has shown that simultaneous brightness con­ thing along the lines of trast is insufficient to explain the homogeneous appear­ f,.(r) = difference ofGaussians, (3) ance ofillusory figures. There must be additional mecha­ nisms to fill in these forms, as well as a mechanism that can fe(O) = 1, (4) produce dissimilation effects. The feature contour system !¢(f/» = o(f/», (5) ofthe BCS/FCS allows local brightness contrast and dis­ similation mechanisms to determine the global appearance where Ohas no effect on the receptive field. Starting from ofillusory figures via a diffusive process with boundaries the constraints ofthe simplest oflow-Ievel determinants, to the diffusion determined by the BCS. No quantitative I will construct the receptive field profile ofthe comple­ studies involving the effects oflow-Ievel stimuli variations tion filter and show that it must depend nontrivially on all on FCS brightness levels have yet been undertaken, so it three relational parameters. is unclear how well this system will perform. However, Spatial extent and proximity of edge inducers. For given its ability to reflect a number ofsimultaneous bright­ edge-inducing elements, illusory contour strength appears ness contrast effects (Grossberg & Todorovic, 1988), one to be closely related to the support ratio. With all other stim­ can predict that it will make accurate brightness predictions ulus aspects kept constant, this implies that the strength of under parametric stimulus variations for which there is a completion must increase as inducing elements move high correlation between brightness contrast and illusory closer together and as their region ofsupport grows larger. brightness, such as inducer proximity and size (Dresp, 1992; These constraints are satisfied by completion mechanisms Dresp et aI., 1990). that employ weighted spatial integration along the com- PERCEPTUAL THEORY OF ILLUSORY CONTOURS 311

Figure 24. The relation between any two cells in a neural model can be uniquely characterized by three parameters. A completion filter with weights f(r, e, I/J) depending upon these parameters can be defined so that it closely models the effects ofinducer alignment, proximity, and support on iUusory con­ tour strength. pletion path, with weights decreasing monotonically from Proximity and number of line-end inducers. Vary­ the completion centroid outward. That is, a completion fil­ ing the spacing, number, and support ratio ofline-end in­ ter with a decreasingf(r) can reproduce these effects. De­ ducers affects illusory contour strength quite differently creasing the distance between inducers will cause the in­ than the equivalent manipulations ofedge inducers. When ducers to fall in more highly weighted regions, thereby a constant spacing is maintained between line-end induc­ increasing completion strength. Increasing inducer support ers, illusory contour strength increases monotonically as will cause the additional integration ofpreviously uninte­ the number ofline ends increases. The weighted integration grated regions, again increasing completion strength. The ofthe completion filter remains consistent with this psycho­ grouping field ofthe CMNCP and the bipole filter ofthe physical data: additional lines can only increase the com­ BCS both use weighted integrations ofthis form, whereas pletion filter response. However, as the number oflines in­ other models ofillusory contour formation do not include creases within a fixed region (necessarily leading to a such mechanisms. Note that both ofthese models contain decreased spacing between lines), an initial increase in con­ competitive, nonlinear processes prior to completion, tour strength is psychophysically observed, followed by a which may significantly alter the strengths and spatial dis­ strength decrease. This finding cannot be reconciled with tributions of the signals that act as inputs to the comple­ just the basic completion filter; an increased number of tion mechanisms, with the BCS containing a feedback lines should always result in increased contour strength, as loop that can further transform the signals as described should decreased distance between neighboring lines. below. Thus, even if the grouping weights fall off as de­ The inverted-U observed with line proximity can be ex­ scribed above, mathematical analysis (or, if intractable, plained by introducing a distance-dependent interference extensive simulation) of the entire system is required in (competition) between parallel lines or line ends, or, more order to establish the appropriate support and proximity accurately, between detectors signaling these features trends. Lesher and colleagues (Lesher, 1993;Lesher, Gross­ (Lesher & Mingolla, 1993). Note that the CMNCp, as well berg, & Mingolla, 1994) have recently verified these trends as all strict line-end models, would require such competi­ in the BCS through a series ofone-dimensional simulations. tive interactions only between end-stopped cells, since the Reconciling completion strength with the support ratio edge detectors do not participate in perpendicular com­ findings ofShipley and Kellman (1992b) and Banton and pletion. Conversely, the BCS would require such interac­ Levi (1992) requires more than just modeling the effects tions only between edge-sensitive cells since it contains no ofinducer proximity and amount ofabsolute support. The distinct pool ofend-stopped cells, but functionally gener­ fall-off in grouping weights must reflect the psychophys­ ates end stopping via competition between these edge­ ically observed trends for the ratio ofsupported length to sensitive cells. This dichotomy indicates a possible ad­ the length ofthe entire contour (the sum ofdistance between vantage for the former model, since it is the illusory contour inducers and supported length). With an infinite number rather than the lines themselves that are degraded when ofdistinct distance/support pairs leading to the same sup­ the inducing lines are close together. port ratio, and thus the same completion strength, it is a Fortunately, one need not posit novel mechanisms to ex­ relatively simple process to show that no single set (scale) plain interference effects between proximal lines or line ofdiscrete completion filter weights can fulfill the support ends, but rather may exploit existing mechanisms that sub­ ratio constraint, although there may be ways to combine serve other purposes. Many models ofend-stopped cells completion filters of multiple scales in order to remain employ some form ofimplicit or explicit lateral inhibition consistent with this constraint. On the basis of the con­ to achieve their end-stopping capabilities (Dobbins, Zucker, flicting data between edge and line-end inducers, however, & Cynader, 1989; Heitger et al., 1992). Similarly, the first it is not yet clear that support ratio represents an invariant competitive stage ofthe BCS consists ofspatial competi­ determinant ofillusory contour strength, even for edge in­ tions between like-oriented cells. Note, however, that the ducers. Additional research into the relationship between deleterious effect ofneighboring parallel lines on illusory support ratio and inducer type is necessary before explicit contour strength has been observed at spacings many times modeling problems can be fully addressed. the width ofthe lines themselves-beyond the distance at 312 LESHER

which conventional end-stopping models would include but ofdifferent orientations, as in Figure 25c, both I/> and 8 lateral inhibitions and beyond the spatial extents employed become nonzero. The effect ofvarying I/> can be isolated by in most previous BCS implementations (but see Gross­ altering the inducer configuration as in Figure 25d, in berg, Mingolla, & Williamson, 1994; Pessoa, Mingolla, & which 8= O. There is no contour apparent at this nontangen­ Neumann, 1994). With line interactions in place, employ­ tial orientation, so the completion filter response must drop ing the completion filter to model the inverted-U effect of offwith increasing magnitude of 1/>, as it did above for 8. line proximity (or number of lines if their extent is kept The simple constraints outlined above for independent constant) becomes a matter ofbalancing the tendency of variations in I/> and 8 have been defined with reference to completion strength to increase as lines move into regions the optimal completion filter, with 1/>= 8= O. However, 1/>= ofhigher filter weighting with the tendency ofthe strength 8 = 0 only describes the optimal filter for the special case of each line's signal to decrease as neighboring lines in­ in which the inducer edges lie along a common line. Fig­ hibit one another. The inverted-U has recently been qual­ ures 25e and 25f depict optimal completion filters for itatively modeled within a one-dimensional BCS imple­ more complex inducing stimuli, under straight-line and mentation (Lesher, 1993; Lesher et aI., 1994). parabolic completion paradigms, respectively. In Fig­ Inducer luminance and contrast. It appears that illu­ ure 25e, the optimal filter is defined as having I/> = 8, such sory contour strength decreases with decreasing inducer that the linear extensions ofthe inducing edges intersect contrast. Assuming a correlation between inducer contrast at the centroid ofthe completion filter. In Figure 25f, the and strength ofedge and line-end signals, both the BCS and optimal filter is tangent to the parabola defined by the two CMNCP should be able to reproduce these results. Since inducing edges. In the parabolic case, I/> must be greater various inducers need not be ofthe same contrast, contour than 8 in the optimal completion filter. Of course, one completion should be polarity independent, as it is in both could just as easily define optimality in terms ofcircular models. The lack ofillusory contours under conditions of contours (as Ullman did), or any other curvilinear con­ inducerlbackground isolurninance indicates that contour struct. In any event, the response ofthe completion filter formation may occur solely in luminancechannels, although must fall offas 8 and I/> vary from their optimal combina­ there is some evidence that inducerlbackground color choice tion, as indicated by Figure 25g for parabolic completion. can affect luminance difference detection thresholds for il­ As yet, we have only considered how the response ofa lusorycontours (de Weert, 1983).Grossberg (1987a, 1987b) completion filter changes relative to the optimal relative has extended the description ofthe original BCS/FCS to parameter set (8, 1/» for a given inducing stimulus. How do employ separate sets of double-opponent (red-green, we relate our psychophysical knowledge about the effects blue-yellow, white-black) channels. By allowing comple­ ofdegree ofalignment on absolute contour strength as the tion only between boundary signals generated by luminance inducing stimulus is altered? Contour strength decreases differences (as opposed to color differences), the BCS ap­ with decreasing alignment, or alternatively, with increas­ proximates the data on isoluminant contour formation. ing degrees ofcurvature, as is indicated by Figures 25fand Induceralignment. Psychophysical data on the align­ 25h. This effect is emulated by the completion filter ifits ment ofedge inducers mandate that completion between response is greatest for 8 = 0 (straight line completion) adjacent inducers be curvilinear, with decreasing strength and falls offmonotonically as the magnitude of8 increases. as the degree ofcurvature increases, although under cer­ Given all ofthe completion filter constraints discussed tain conditions the contour may contain a discontinuity or above, the strength of the filter response must decrease comer. In addition, contour strength decreases as increas­ with increasing r, increasing 181, and increasing distance ing degrees of curve inflection are required for smooth from the optimal (I/>, 8) pair as defined by the particular completion (the completion of Figure l lb would require curve scheme being employed (e.g., in a linear scheme, an inflection). In the completion filter paradigm, the com­ I/> = 8 would be optimal and contour strength would de­ pletion strength or relatability between two edges or line crease with increasing 11/>-(1). The sophisticated comple­ ends can be veridically modeled by forcing the filter's re­ tion filter of the BCS uses a separable response profile: ceptive field to depend nontrivially on all three relational f(r,8,1/» = f,(r)fe(8)f'l,(8-1/», (6) parameters-r, 8, and 1/>. By definition, a completion filter should fire if it lies as does the simpler filter ofthe CMNCP. Although a sep­ tangent to an illusory (or real) contour. The strongest illu­ arable filter is easier to define and implement, it has the sory contour strength should result when the inducing serious drawback of eliminating all curvature schemes edges lie along a common line, as depicted in Figure 25a. other than the piecewise linear completion ofFigure 25e. For completion filters tangent to this line, 8 = I/> = O. For More sophisticated methods require joint knowledge of I/> completion filters ofthe same orientation, but in different and 8 (and, for some schemes, even r). positions, as in Figure 25b, 8becomes nonzero while I/> re­ The bipole filter of Grossberg and Mingolla (1985a, mains zero. Psychophysically, we observe that there is no 1985b) encompasses all the constraints provided above, illusory contour apparent at the completion filter's posi­ with the maximal nonaligned contributions occurring at tion: The response ofthis filter must be less than that of I/> = 8. As indicated by the recent data ofField et aI. (1992) the tangential filter ofFigure 25a. Thus, for this value of and the propensity for curved contours rather than those I/> = 0, the receptive field profile must fall off with in­ with discontinuities, this configuration is probably sub­ creasing 181. For completion filters at the same position, optimal because it tends to lead to piecewise linear con- PERCEPTUAL THEORY OF ILLUSORY CONTOURS 313

a~~~e

b~~(~f

c C;-~~-r::::::-~ (~g d~~h

Figure 25. The effect of various alignments on completion filter response: (a) optimal tangential completion filter with e= ¢=0, (b) suboptimal fllter with e;t0, ¢= 0, (c) suboptimal nIter with ¢;to, 8=0, (d) suboptimal filter with ¢;to, B> 0, (e) optimal filter for piecewise linear completion, ¢= e, (f) optimal filter for parabolic completion, ¢ > e, (g) suboptimal mter for parabolic completion, ¢ = e, (h) optimal nIterfor parabolic completion, butweakerresponse than for (f). tours intersecting at bipole centroids-although the joint system like the CMNCP can accomplish only through (cur­ action ofmany bipole filters and the competitive interac­ rently) artificial postprocessing methods. The sharpening tions within the feedback loop may restore true curvature. and selection ofillusory contours in the CMNCP involves As described above, a more propitious choice for the oc­ the identification and extraction of "ridges" ofhigh activ­ currence ofmaximal contribution would lead to contours ity-a process yet to be neurally defined by Heitger and in the form ofparabolas or circles, but at the cost oflos­ von der Heydt (1993). Besides sharpening contours, the ing filter separability. The grouping field employed by feedback loop ofthe BCS also serves to eliminate spuri­ Heitger and von der Heydt (1993) is simpler than the bi­ ous and overlapping contours that would be maintained by pole filter, allowing cells to contribute only ifthey have the the CMNCP and other models. While CMNCP simulation same orientational preference as the grouping filter; that results indicate that multiple illusory percepts are visible is, weights are set to zero when l/J * O. By itself, allowing at the same time (e.g., both the Kanizsa square and the cir­ only two nontrivial degrees of filter freedom (r and e) cular inducers are completed), the competitive feedback would have the effect ofprecluding curved illusory con­ of the BCS renders such multiple percepts unstable and, tours, but in a subsequent step filter lobes from a specific in agreement with psychophysical data, ensures that at set of orientations are combined in a weighted fashion. any given time, only a single percept will be visible. The This process brings the third degree offreedom (l/J) back primary drawback ofthe feedback loop (and ofthe BCS/ into the completion mechanism, but only to a limited ex­ FCS in general), from the perspective ofactually simulat­ tent: the orientation combinations are coarse, with most l/J ingillusory contour formation, is that it is computationally information lost during the grouping filter process. Al­ expensive. This fact has allowed the CMNCP to hold the though computationally more efficient than the bipole fil­ advantage ofactually demonstrating a number oftangible ter, this system lacks the flexibility and modeling power of illusory contour simulations that have yet to be emulated the bipole filter, gained through the finer dependence on with the more computationally complex BCS. As the ex­ eand l/J in determining completion strength. Like the bi­ penses ofcomputing resources continue to decline, how­ pole filter, piecewise linear groupings are preferred to true ever, it seems likely that the BCSIFCS will be more widely curved completion, although the conglomerate action of implemented and will start to realize the more advanced many grouping fields does lead to curved completion. simulation results ofwhich it is theoretically capable. Both the BCS and the CMNCP would benefit from com­ Small-scale facilitation and interference. The small­ pletion filters explicitly tuned to produce curved contours. scale facilitation effected by small dots and lines in localiz­ In the BCS, implementation ofsuch filters would merely ing contours suggests that these features may provide involve replacement ofthe standard filter weights, whereas unoriented contributions to the oriented completion mech­ in the CMNCP, the system would require modifications to anisms. That is, it suggests that features which contain little handle completion filters with 3 nontrivial degrees offree­ or no orientational information may contribute to com­ dom (r, e, and l/J) rather than the 2 degrees offreedom cur­ pletion filter activity irrespective ofstandard, orientation­ rently supported. Gove and colleagues (Gove, 1993; Gove dependent weighting factors. Since the edge and line-end et al., I994a) have recently demonstrated BCS bipole fil­ detectors used to detect the inducing elements may be ofa ters with circular completion preferences. scale too large to respond to small anchors, these contribu­ One ofthe most powerful features ofthe BCS is its feed­ tions may arise from detectors tuned to smaller scales. In back loop, which theoretically allows for the capture ofdis­ the FACADE theory ofperception (Grossberg, 1994), each tinct, curved illusory contours which a solely feed-forward bipole filter receives contributions from multiple scales. 314 LESHER

Orientational activity at large scales serves to establish the tional-only inhibition from small scales to larger scales-> generalboundaryposition,whilesmall-scaleactivityserves to or whether large scale to smaller scale interactions should refine the position, especially in regions ofhigh curvature. also be implemented. However, the current psychophysi­ This mechanism may also explain small-scale facilitation ef­ cal data seem to indicate an asymmetric relation. fects ofshort linesand dots. Unoriented stimuli,such as dots, Inducerstructure and figural interference. The fail­ can be expected to excite cells of all orientations at some ure of thin lines to induce collinear illusory contours re­ small scales, thus providing an "unoriented" contribution for quires that parallel completion only be initiated when edge inclusion in completion filters ofall orientations. It is im­ inducers have sufficient thickness. A simple method to ac­ portant that such small-scale components not beable to trig­ complish this is to disallow completion when such a com­ ger completion filters by themselves-a series ofdots (as in pletion would lead to conflicting orientations or orienta­ Figure 3b) does not induce illusory contours-but merely tional cues at the same position. Thus a line could not be bias the position ofcompletion initiated by larger scale in­ extended collinearly through a line end-an orientational ducers. In systems containing feedback loops, such as the cue for an occluding object perpendicular to the line itself. BCSIFCS and the model ofFinkel and Sajda (1992), a small The concept of "spatial impenetrability" in the BCS ac­ bias is sufficient to induce robust illusory contours as reen­ complishes this function by allowing perpendicular ori­ trant activity reinforces the initial bias. entations to inhibit bipole filter activity, canceling out par­ The converse effect offacilitation-the interference ef­ allel completion. Although the CMNCP does have a fect ofsmall details near inducing elements-may result mechanism to suppress the presence of multiple orienta­ from a combination offactors. The fundamental difference tions at the same position after grouping, it apparently between facilitating and interfering elements is the spatial maintains no mechanism that would disallow parallel proximity ofthe element to the inducers, and to the illu­ completion with thin line inducers: the comemess para­ sory contour itself. A facilitating element does not lie in meter K prohibits line ends from participating in para the immediate area ofany inducer but does lie on or near completion but does not prevent central line regions from the illusory contour. Conversely, an interfering element lies contributing. To rectify this shortcoming, end-stopped in the immediate area ofthe inducer but does not lie on the cells would need to inhibit para completion in a fashion illusory contour. In defining mechanisms by which these analogous to that employed by the spatial impenetrability opposite effects can be captured, one must concentrate on mechanisms ofthe BCS. these basic dissimilarities. Figural interference, in which lines crossing an illusory In modeling interference effects, one possibility is that contour destroy that contour, can be handled with a mech­ the elements are simply frustrating the filters being used anism similar to that used to manage the effect ofinducer for edge and line-end detection, altering the spatiallumi­ structure, forcing near-perpendicular orientation informa­ nance distribution and thus leading to degraded or spuri­ tion to inhibit completion. Spatial impenetrability allows ously tilted responses that could weaken completion the BCS to model figural interference; the CMNCP con­ strengths. Given the small scale ofthe interfering elements, tains no suitable mechanism. The combination ofspatial however, this proposition is dubious. A more plausible hy­ impenetrability and the feedback loop of the BCS also pothesis is that these elements are providing small-scale serves to eliminate illusory contours with different orien­ orientational information that conflicts with orientational tations but with the same position. These mechanisms force information at larger scales. For example, in Kennedy's the system to choose between conflicting illusory percepts (1988) demonstrations of the effect of line-end shape, and thus is essential to reproducing the psychophysical large-scale information about the orientation ofthe whole findings regarding the perception of ambiguous illusory line is contradicted (at the line ends) by smaller scale in­ figures. formation about the shape ofthe line end. Cross-orientation, Sophisticated mechanisms for segregating boundaries cross-scale competition might explain the interference ef­ and surfaces into multiple depth planes, as in Finkel and fect, as small-scale cell responses inhibit responses at Sajda's (1992) model and Grossberg's (1994) FACADE sys­ larger scales at the same or nearby positions. As described tem, may also play significant roles in explaining the above in the context offacilitation, unoriented interfering effects of both inducer structure and figural interference, elements would provide tonic inhibition at all orientations, since these determinants are dependent on the perceived while oriented elements would provide more selective in­ depth ofeach scenic element. For example, although mo­ hibition. Competitive interactions between scales within dal collinear completion of thin lines is precluded when the FACADE system cause the optimal scale at each po­ the lines appear in the foreground, amodal collinear com­ sition to inhibit weaker scales, allowing fine details and pletion should be allowed when the lines are perceived in regions ofhigh curvature to be perceived veridically. The the background. Both the intermediate level vision mod­ combination of this mechanism, absent from all other els mentioned above support this kind ofdepth-dependent models ofcontour formation, and interorientation compe­ completion. tition yields a system consistent with small-scale interfer­ ence effects. Since competition is spatially localized, this High-Level Determinants mechanism does not preclude facilitation effects of ele­ Until very recently, neural models have all but ignored ments not in the immediate area ofthe inducers. It is unclear high-level determinants. Even now, most ofthe high-level whether interscale cellular interactions should be unidirec- determinants discussed earlier are well beyond the scope PERCEPTUAL THEORY OF ILLUSORY CONTOURS 315 ofcurrent systems, and, at the levelofdetailedneural model­ lusory figure in front of, or behind, the inducers?-and by ing, promise to remain so for the near future. We simply absolute stereoscopic information-how far in front of have insufficient knowledge ofthe dynamics ofcognition (or behind) the inducers is the illusory figure? Research to model such cognitive determinants as the interactions into the modeling and psychophysics of the effects of between memory, perceptual set, and illusory figure per­ depth ordering on illusory contour formation represents the ception. An incomplete understanding of the underlying next major step in advancing neural systems. cognitive processes does not, however, preclude us from Cognitive factors. Cognitive factors are more difficult modeling how the results ofthese processes influence il­ to integrate into neural models than depth effects, primar­ lusory contour formation. ily because we have only a vague understanding ofthe un­ Depth modifications. The one high-level determinant derlying processes themselves, let alone their neural sub­ that has been the topic ofsubstantial modeling research is strates. Presumably, as our knowledge base becomes more the effect of depth modifications. As mentioned earlier, complete, determinants with progressively higher loci of determination ofthe surface to which each contour belongs origin will be incorporated into neural systems. For now, is the crucial step in determining how these surfaces however, neural models ofillusory contour formation are occlude one another (Nakayama et aI., 1990). This form left with fundamental limitations on the modeling ofmany of depth information, necessarily based solely on real, high-level determinants. This is not to say that the effects luminance-defined contours, can be propagated to create ofcognitive factors have been-or should be-completely a depth ordering ofboth real and illusory contours (Finkel ignored by neural modelers, although coverage to date has & Sajda, 1992, 1994; Grossberg, 1994; Heitger & von der been undeniably sparse. Until neural systems directly ad­ Heydt, 1993; Sajda & Finkel, 1992a, 1992b, 1995). Kell­ dress the daunting problems ofincorporating the various man and Shipley (1991) have posited that both modal (il­ cognitive determinants, they cannot truly be considered lusory contour) and amodal completion are subtended by general models ofthe contour formation process. the same mechanism, with depth ordering determining As mentioned during the discussion of inducer com­ which contours are perceived as modal. This hypothesis pleteness, some high-level determinants may not be as in­ has been supported by recent data indicating that certain herently cognitive as initially thought. Certain effects of low-level determinants have equivalent effects on illusory the seemingly cognitive factor ofinducer completeness may contour and amodal contour perception (Shipley & Kell­ be closely related to local mechanisms (Albert, 1993; man, 1992a). Like Kellman and Shipley's model, Gross­ Sajda & Finkel, 1995). Further psychophysical quantifi­ berg's (1994) FACADE system employs a depth stratifi­ cation ofhigh-level determinants, where possible, may re­ cation scheme with a single completion mechanism for veal additional low-level determinants. However, it is both modal and amodal completion, but it also includes a doubtful that the complex roles ofmemory and perceptual definite mechanism for determination of which contour set in illusory contour generation will be demoted to sim­ should be visible-the filling in ofthe FCS. The FACADE pler explanations. system bases initial depth ordering primarily on binocular In presenting their models of contour formation, re­ cues, but many depth-related illusory contour effects in­ searchers almost invariably mention top-down mediation volve purely monocular stimuli. This system must be ex­ ofcontour strength by "higher level" processes. Grossberg tended to include depth-ordering schemes based on and Mingolla (1985a, 1985b, 1987a, 1987b; Grossberg, monocular cues, such as the local cues exploited by Heit­ 1987a, 1987b, 1994) have been particularly meticulous in ger and von der Heydt (1993) and Sajda and Finkel (l992a, stressing the importance ofthese top-down effects, propos­ 1992b, 1995; Finkel & Sajda, 1994): line ends and corners ing that the BCSIFCS interacts with an object recognition as occlusion indicators, local concavities, similarity and system (ORS) responsible for contour reinforcement, at­ proximity offigural elements, and figure closure. tentional effects, and priming, as well as for primarily feed­ An important aspect of depth-ordering formalisms is forward tasks such as object recognition and image un­ that they explain how changes in the perception ofdepth derstanding. The BCSIFCS .f-7 ORS interactions have not ordering can facilitate illusory contour formation in the yet been simulated; but they appear to be theoretically presence of (what would normally be) interfering ele­ sound, and they are sufficiently well specified for attempts ments, such as the overlayed textural elements ofRock and at qualitative simulation ofsome ofthe simpler high-level Anson (1979) (Figure 16). Such systems can also explain determinants. how flips in the perceptual depth ordering ofambiguous The most promising means ofincorporating cognitive illusory contours can lead to instantiation ofdifferent sets factors is through the use oftop-down priming. A look at of illusory contours. Determination of illusory contour any ofthe figures included in this papermakes it clear that strength depends not only on the relative stratification of illusory contours never achieve the perceptual salience of illusory figure and inducing elements, but is also affected real luminance edges. Some observers may not perceive by the absolute depths; as illusory figures are stereoscop­ every contour, and for most observers, foveation on a con­ ically forced closer to the observer, contour strength in­ tour will weaken or eliminate the contour (Coren, 1972). creases. Itis unknown whether increasing depth via monoc­ Perhaps, then, illusory contours tend to lie near the thresh­ ular cues leads to the same effect. Reconciling depth effects old ofperception, as is indicated by the attention studies with neural models will require mechanisms to mediate ofGurnsey et al. (1992). Slight indications from cognitive contour strength by both relative stratifications-is the il- processes may suppress or strengthen contours, or alter 316 LESHER

their paths completely. For a perceptual set process, a top­ interactions, the relation between edges and line ends, the down prime might be something like "pay particular at­ effects of line spacing, and the effects of depth ordering tention to edge alignments" (as in Rock & Anson, 1979) are just some of the areas in which additional study is or "there might be corners at these positions" (as in Coren required. et aI., 1986). The alternative scheme-that of cognitive The evaluation of current neural models provided in the processes sorting through a web of tentative contours, sup­ previous section indicates the vast degree to which im­ plied by neural mechanisms, to arrive at a single percept provements are possible. Not even the simplest of low-level is also possible, but it lacks the economy and elegance of determinants have been quantitatively modeled, and only a system that reuses a single mechanism to both generate recently have simulations of qualitative effects been pub­ and refine illusory contours. Neural models that already lished. Further evaluations ofthe BCS, CMNCp, and other include feedback, such as the BCS/FCS, FACADE, and neural models will require that they progress from theory Finkel and Sajda's (1992) system, are particularly amen­ to demonstration. At the same time, these models must be able to priming; the slightest top-down hint about which reconciled with low-level determinants that they cannot features might be important to contour formation or where presently emulate, by extension or modification. In par­ to look for an illusory contour could be propagated by the ticular, interscale interactions are conspicuously absent feedback mechanisms to completely reconfigure the per­ from current models, preventing the emulation of small­ cept. Indeed, Sajda and Finkel (1995) have proposed a scale facilitation and interference effects. As quantitative clever paradigm that essentially hypothesizes contours simulations appear and neural models are committed to a based on secondary cues, which are implicitly verified single parameter set, more psychophysically testable pre­ only when contours generated by the feedback process dictions will be possible. All models, with the notable ex­ trigger primary cues. ception of the BCS/FCS, must be extended to rectify an The exact form that top-down primes might take will even more glaring omission: the lack of mechanisms to depend on the particular neural model, but likely candi­ simulatebrightness effects.Although the amount of psycho­ dates include attentional mechanisms that raise the mod­ physical illusory brightness data exceeds that concerning el's sensitivity within specific regions, direct excitation (or illusory contour strength and depth effects, there have inhibition) of oriented or end-stopped cells, and global bi­ been no qualitative BCS/FCS simulations ofthe effect of asing ofparticular model parameters. An effective system low-level inducer variations on brightness-an area in will probably need to accommodate multiple types of which this model appears particularly well qualified. primes. Although it would be more satisfying to model the Of the many high-level determinants, the pronounced cognitive processes as well, we can assign generation of effects of depth ordering on contour strength are most the prime or attentional determinant to a cognitive dae­ amenable to detailed neural modeling. Although simple mon without compromising the neural models, because depth-ordering mechanisms have found their way into these higher level processes are at least partially indepen­ some current models (such as the CMNCP), much more dent ofthe modeling ofillusory contourformation. With sophisticated methods will be required in order to repro­ this technique, the explanatory power ofnext-generation duce psychophysical results. Such methods have recently models should extend well into the domain ofhigh-level been formulated (Grossberg, 1994; Kellman & Shipley, determinants. 1991; Sajda & Finkel, 1995), but further development is still required, as are more extensive simulations. Purely SUMMARY cognitive factors, such as the effects ofmemory and per­ ceptual set, are difficult to model explicitly, but integration Twentyyears ofdebate over the nature and dynamics of ofthese determinants into neural models may be possible illusory contour formation have gradually resolved to a through the use oftop-down primes to represent the out­ determined focus on neural models. Qualitative studies come ofcognitive processes. The overdue incorporation of designed to yield insight into the general form of the these cognitive primes represents an intriguing and promis­ underlying process--eognitive or retinal, gestalt or geo­ ing area for future research. Finally, we must be aware that metric-have for the most part given way to quantitative a model of monocular, luminance-defined illusory con­ studies formulated in order to tabulate the specific deter­ tour formation must fit within the larger framework ofil­ minants ofillusory contour strength, illusory brightening, lusory contour formation in other domains, which in turn and perceived depth of the illusory figure. Recognition must represent a subsystem ofcomprehensive human vi­ that neural systems represent, at least to a rough approxi­ sion. To this end, the incorporation and integration ofcon­ mation, the substrate of illusory contour formation has tour formation systems into models of other visual do­ led to recent experiments designed to refine our under­ mains must represent the ultimate goal of our research. standing ofneural models and constrain the possible types ofinteractions and mechanisms. 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