A measure of closure James Elder ? Steven Zucker y 93-2 March 2,1994 McGill Research Centre for Intelligent Machines McGill University Montr?eal , Qu?ebec , Canada ? e-mail : elder@mcrcim.mcgill.edu ye-mail : zucker@mcrcim.mcgill.edu Postal Address : 3480 University Street , Montr?eal , Qu?ebec , Canada H3A 2A7 Telephone : -LRB-514-RRB- 398-6018 FAX : -LRB-514-RRB- 398-7348 Network Address : mcrcim@mcrcim.mcgill.edu A measure of closure James Elder Steven Zucker Abstract In images of real visual scenes , contours are often fragmented by occlusion , shadows and low reflectance contrast . In order to infer shape from contour , the human visual system must selectively integrate fragments projecting from a common object while keeping fragments from different objects separate . In a previous paper we showed that contour closure has a strong influence on the speed of this process . In this paper we employ a visual search method to evaluate potential measures for the perceptual closure of fragmented shapes . We show that while certain intuitive measures are psychophysically inconsistent , a measure based on the L2 norm of the lengths of contour gaps is appropriate for both polygonal and smooth shapes in that response time can be expressed as a function of this measure . The L2 norm is shown to emphasize large gaps relative to small , and to embody a principle of perceptual regularity . Our results also have implications for the shape of perceptually natural contour interpolants . Acknowledgements This research was supported by grants from NSERC and the AFOSR . We thank Michael Langer and Laurie Wilcox for useful comments on the manuscript . Sommaire Lorsque l'on consid?ere des sc?enes visuelles r?eelles , l'image des contours des objects sont souvent fragment?es par des occlusions , des ombres , et de faibles contrastes . Pour inf?erer la forme des objects ? a partir de leur contour , le syst?eme visuel doit int?egrer les segments de contour qui sont projet?es d'un m ^ eme objet alors qu'il tient ? a l' ? ecart les segments d'autres objets . Nous montrons qu'une mesure de la fermeture qui est en accord avec nos r?esultats peut ^ etre d?eriv?ee de la norme L2 de l'ensemble des intervalles entre les segments . 1 . Introduction The human visual system infers surface shape from many image properties : motion gradients , binocular disparities , shading , texture and blur gradients . Generally these properties are computed from measurements made over twodimensional subsets of the image domain : surface projections of objects in a scene . Multi-dimensional shape properties are also inferred from the one-dimensional image subsets projecting from space curves in a scene -LRB- Fig. 1 -RRB- . These space curves and their projections are of three types : occlusion curves demarcate the visible portions of surfaces , reflectance curves outline surface regions which are distinct in some material properties , and shadow curves bound surface patches which are eclipsed by other surfaces in the scene . All three types of contour embody information about surfaces in the scene . In a previous paper -LRB- Elder & Zucker -LSB- 1993 -RSB- -RRB- , we proposed that contour closure plays a key role in this inference of shape from contour . This hypothesis was based on the fact that the boundary of an unoccluded object with a simply connected surface projects as a simple , closed contour in a retinal image . A classical result in topology known as the Jordan curve theorem states that such a contour partitions the plane into two sets : an inside -LRB- figure -RRB- and an outside -LRB- ground -RRB- . This partition allows the definition of two-dimensional shape prop - -LRB- a -RRB- -LRB- b -RRB- Figure 1 a : Nude , Henri Matisse -LRB- Clark -LSB- 1956 -RSB- -RRB- . b : Contour drawing of figure from Studies for the Libyan Sibyl , Michelangelo -LRB- Nicola ?? des -LSB- 1941 -RSB- -RRB- . erties such as curvature sign , concavities , convexities , narrowings and bulges . These properties play a critical role in computational theories of planar shape perception -LRB- Blum -LSB- 1973 -RSB- ; Hoffman & Richards -LSB- 1985 -RSB- ; Kimia , Tannenbaum & Zucker -LSB- 1990 -RSB- ; Leyton -LSB- 1989 -RSB- -RRB- and moreover impose strong constraints on the shape of the three-dimensional surfaces in the scene -LRB- Biederman -LSB- 1988 -RSB- ; Koenderink -LSB- 1984 -RSB- ; Lowe -LSB- 1985 -RSB- ; Marr -LSB- 1982 -RSB- -RRB- . In topology there is no concept of partial contour closure -LRB- a curve is either closed or it is not -RRB- . Such a view has also been taken within the computer vision community -LRB- Ullman -LSB- 1984 -RSB- -RRB- . In real visual scenes , however , image contours are often fragmented by occlusion , shadows and low reflectance contrast , yet shape can still be perceived . The visual system is somehow able to selectively integrate fragments projecting from a common object while keeping fragments from different objects separate . If the property of contour closure is used in this process , this perceptual closure must be quite different from our topological notion of closure . In our previous work , we investigated the exploitation of closure in the inference of shape through a series of visual search experiments in which subjects discriminated outline figures by means of their two-dimensional shape -LRB- Elder & Zucker -LSB- 1993 -RSB- -RRB- . The basic stimuli used in these experiments were composed of two unconnected but nearby contours -LRB- Fig. 2 -RRB- . The contour segments were the same for both the target and distractor , which thus differed only in how the segments were placed relative to each other . In the target stimulus , they were arranged to bend inward , forming a ` spindle ' shape , while in the distractor stimulus they bent outward , forming a ` barrel ' shape . The stimuli were chosen so that the discrimination was based on measurements which were two-dimensional and extrinsic to the contour segments . The task was thus a discrimination of two-dimensional shape . Target Distractor Figure 2 Basic stimuli used in closure experiments reported in Elder & Zucker -LSB- 1993 -RSB- . By modulating the degree of closure of the target and distractors in tan - dem , we showed that two-dimensional shape processing is rapid for closed stimuli but slow for open stimuli . We further showed that search speed can be characterized as a monotonic function of the degree of closure -LRB- Fig. 3 -RRB- . Unlike topological closure , perceptual closure appears to be a continuous function of certain geometric properties of image curves . -20 -16 -8 -4 4 8 12 16 20 24-12 Closing pixels 4000 3500 3000 2500 2000 1500 1000 500 Target : Response time -LRB- msec -RRB- Figure 3 Response time as a function of closure for a two-dimensional shape discrimination task . From Elder & Zucker -LSB- 1993 -RSB- . We have also studied some elementary photometric properties of perceptual closure -LRB- Elder & Zucker -LSB- 1993 -RSB- -RRB- . Briefly , we found that perceptual closure is highly sensitive to contrast polarity : on the time scale of our experiments subjects were unable to take advantage of reversed contrast contour segments which closed a figure . On the other hand , subjects were very little affected by the precise contrast of the closing segments as long as contrast did not reverse . With the exception of this sensitivity to contrast reversal , we are encouraged by our previous results to think of perceptual closure as an essentially geometric issue . Our principal goal in this paper is to find a measure ? for perceptual closure based solely on the geometric properties of a fragmented figure . Figures may be fragmented in many ways . Gaps in the bounding contour may be few or many , large or small , uniform or widely varying in size and spacing , etc. . . To be more precise , suppose that we represent a simple , differentiable , planar closed curve by the regular parameterization : -LSB- 0 ; T -RRB- ! < 2 . If we represent a particular fragmentation by the subset of -LSB- 0 ; T -RRB- at which there is actually contour present , then the set of all possible fragmentations is the infinite dimensional power-set of the interval -LSB- 0 ; T -RRB- . Our goal is to collapse this infinite dimensional set into a one-dimensional measure : a real-valued function C representing the closure of an infinite number of different but \ closureequivalent '' fragmentations . By thus establishing these equivalence classes , we hope in turn to understand the interplay of factors -LRB- e.g. number of gaps , gap size , gap regularity -RRB- which determine our ability to integrate fragmented contours in complex scenes . To evaluate potential measures for closure , we again employ visual search experiments in which the subject must rapidly discriminate a target shape from distractor shapes which differ from the target in their two-dimensional ? We use measure here in the broad sense -LRB- syn . amount , degree -RRB- , not in the measure theory sense . properties . Within an experiment , the target and distractor shapes remain the same , however the type and degree of fragmentation applied to each shape is systematically varied . We stress that the target and distractor are varied in tandem : in any particular stimulus condition , the target and distractor have the same fragmentation pattern . Our previous work -LRB- Elder & Zucker -LSB- 1993 -RSB- -RRB- provides strong evidence that response time in such a visual search task is directly related to the closure of the figures involved . We can thus evaluate a potential measure of closure by comparing the response time data for different experiments using the same shapes , but different types of fragmentation . If two different fragmentations have the same closure measure C but result in different response times , then the closure measure is judged to be psychophysically inconsistent . In other words , we demand that response time be a function of the closure measure for any given experiment . To summarize , let X be the set of possible fragmentations of a figure . In our experiments we vary the fragmentation x of the stimuli and measure response time r for discrimination as a function of the fragmentation : r -LRB- x -RRB- ; x 2 X . We posit the existence of functions C : X ! -LSB- 0 ; 1 -RSB- and R : -LSB- 0 ; 1 -RSB- ! < + satisfying R -LRB- C -LRB- x -RRB- -RRB- = r -LRB- x -RRB- 8x 2 X. Based on our previous results -LRB- Fig . 3 -RRB- and our intuition about closure , we expect R to be monotonically decreasing on -LSB- 0 ; 1 -RSB- . Thus our hypothesis has two parts : 1 . A consistent closure measure exists . 2 . Response time is decreasing on the range of this closure measure . The closure measure can be considered an extremely compressed representation of a fragmented figure which predicts the effect of the fragmentation on the speed of shape perception . If closure had little to do with how the brain computes the shape of objects in complex scenes , it would be unlikely that such a compressed predictor would exist . Support for the existence of such a measure is thus an indication that the computation of closure is an important step in this perceptual process . We begin in section 2 by employing polygonal figures based on those in Fig. 2 . By experimenting with different fragmentations of these figures , we will rule out certain intuitive measures of closure as psychophysically inconsistent , and will propose a more appropriate alternative -LRB- an L2 measure -RRB- . In section 3 we report the results of experiments with a set of smooth stimuli , quite different from the polygonal figures , and test our proposed L2 measure against these results . In section 4 we discuss the L2 measure in terms of the equivalence classes of fragmented figures it creates and its relation to principles of perceptual organization . In section 5 we report additional psychophysical results pertaining to the shape of perceptual contour interpolants . 2 . Closing Polygonal Figures 2.1 Methods Visual search displays were created on a 60 Hz , non-interlaced color Amtron monitor , driven by a Symbolics 3640 computer . Subjects sat in a dimly lit room , approximately 1 m from the screen . A 7ffi ? 7ffi square display window of luminance 11 cd = m2 was positioned in the center of the screen against an unilluminated background . Stimuli were drawn in the display window at 72 cd = m2 . All stimuli were approximately :5 ffi ? :5 ffi in size . Placement of the stimuli within the display window was based on a regular 5 ? 5 grid with nodes separated by 1:4 ffi in the horizontal and vertical directions . Nodes were selected pseudo-randomly , and the precise position of each stimulus was pseudorandomly selected from the set of positions within a :3 ffi horizontal and vertical distance from the selected node . Stimuli were thus separated by at least :8 ffi from centre to centre , and by at least :3 ffi from tip to tip . Each stimulus was randomized in orientation . The procedure is illustrated in Fig. 4 . First , an example of the target for which the subject will be searching is shown -LRB- Fig. 4 -LRB- a -RRB- -RRB- . The subject then presses a mouse button to trigger a block of visual search trials . Within a block , the shape and fragmentation of both the target and distractor remains the same . In each trial , a display is presented which always contains exactly one target and 15 distractors -LRB- Fig. 4 -LRB- b -RRB- -RRB- . When the target is detected , the subject clicks a mouse button and the response time for detection is recorded . At the same time , the visual search display is replaced by a validation display in which the stimulus positions are represented by small reference dots -LRB- Fig. 4 -LRB- c -RRB- -RRB- . The subject must correctly identify the target location -LRB- by clicking on the appropriate dot -RRB- for the trial to be considered valid . If an error is made , the trial is rejected . This sequence is continued until the subject has correctly identified the target in 10 trials . Target -LRB- a -RRB- -LRB- b -RRB- -LRB- c -RRB- Figure 4 Visual search sequence Our procedure differs from the traditional visual search method -LRB- Treisman & Gelade -LSB- 1980 -RSB- -RRB- , which we believe may be subject to a systematic bias arising from a correlation between subject errors and response time . A comparison of the two procedures for simple closure experiments and a discussion of the advantages of our method can be found in Elder & Zucker -LSB- 1993 -RSB- . For these experiments using polygonal figures , the target was always the narrow \ spindle '' shape , and the distractor was always the fatter \ barrel '' shape , as in Fig. 4 . For each experiment , a set of 7 stimulus pairs was employed over which the degree of fragmentation increased according to a specified fragmentation pattern . A single block of trials was performed for each stimulus pair , in random order . Before each experiment , subjects completed a practice sequence identical to the recorded session , but including only 2 trials for each block . In addition , for every block in the recorded session , the first trial for each block was used as practice , and the response times were not used in computing mean results , which are thus averaged over 9 trials for each stimulus condition . 2.2 Subjects A total of 14 subjects were used for these experiments involving polygonal stimuli . Either 10 or 11 subjects were used for each experiment : the exact number is stated with the results . All subjects had normal or corrected vision . Results were averaged over all participating subjects : error bars indicate standard error of the mean . Data points differing from the mean by more than 6 standard deviations were pruned . 2.3 Data Analysis All models used in this paper are either linear -LRB- y = a + bx -RRB- or exponential -LRB- y = a + becx -RRB- . The goodness of fit of these models is represented by the probability Q that , given the model and an estimate of the variance of the data , the O2 of a sampled data set would exceed the computed value . Linear models are used unless a higher Q value can be obtained with an exponential model . Q ranges from 0.36 to 0.81 for the data reported in this paper . 2.4 Two Extremes of Fragmentation A subset of the figures used in Elder & Zucker -LSB- 1993 -RSB- are shown in Fig. 5 . Each figure consists of only two fragments , separated by two gaps . To begin , we select a fragmentation pattern which is as different as possible from this first pattern -LRB- Fig. 6 -RRB- . In these figures , fragments are simply dots , regularly spaced over the gaps in the figure boundary . We refer to the first set as \ two-gap figures '' and the second set as \ regular-gap figures '' . Two-gap fragmentations could be caused by a single occluding object or shadow . Regular fragmentations could be caused by a periodic structure -LRB- e.g. a picket fence -RRB- interposed between the figure and the eye , or between the figure and a light source , thus dappling the image . 460 388 3016 2224 1432 640 46 Number of dots : Total Gap : Figure 5 Two-gap polygonal stimuli used in Elder & Zucker -LSB- 1993 -RSB- . Fig. 7 shows the results of a visual search experiment using these regulargap stimuli , plotted with the results for the two-gap stimuli from Elder & 42 4 40 6 36 10 32 14 460 46 44 2Number of dots : Total Gap : Figure 6 Regular-gap polygonal stimuli Zucker -LSB- 1993 -RSB- , as a function of the total boundary gap . In both cases , response time increases as the total gap increases . However , the two closure functions are very different . It seems that adding evenly spaced contour fragments along a boundary gap has a far more dramatic effect on closure than adding contour in a connected fashion to existing fragments . Even adding just one isolated pixel in the centre of each of the two gaps yields a substantial improvement in search performance . 2.5 The Lp Norm Figure 7 shows that the total boundary gap is a poor indication of the closure of a fragmented contour , since two different fragmentations with the same total gap can yield very different psychophysical results . For the two-gap stimuli , reaction time increases gradually as a function of the total gap . For the regulargap stimuli , very little psychophysical effect is observed until the total gap surpasses 40 pixels , at which point the task rapidly becomes difficult . This behaviour suggests a nonlinear dependence of perceptual closure on the size of individual gaps in a fragmented contour : large gaps are strongly overweighted 5 10 15 20 25 30 35 40 45 1000 1200 1400 1600 1800 Response time -LRB- msec -RRB- | | Boundary Gap | | = Total Gap -LRB- pixels -RRB- 1 600 800 regular-gap two-gap Figure 7 New results for regular-gap polygonal stimuli are shown dashed -LRB- 10 subjects -RRB- . Previous results for the two-gap experiment of Elder & Zucker -LSB- 1993 -RSB- are shown solid . relative to small gaps . Given a fixed total boundary gap , shape discrimination is far slower if this total gap is consolidated in a few large gaps than if it is distributed over many small gaps . A natural way to formalize this behaviour is to collect the lengths of the individual gaps into a single vector ~ g , and then represent the closure of the figure as a function of a length or norm of that vector . We choose from the standard class of norms called Lp norms : jj ~ gjjp = -LRB- nX i = 1 gip -RRB- 1p Three values of p are particularly important , as they represent qualitatively different measures : p = 1 : The L1 or linear norm is the linear sum of the vector components . It is insensitive to how any given set of fragments is distributed around a specified closed path . p = 2 : The L2 or Euclidean norm is the square root of the sum of the squares of the vector components . It has the property of emphasizing large gaps relative to small . It is also related to the regularity of the fragmentation . We discuss this regularity property further in section 4 . p = 1 : The L1 norm selects the largest component of the vector , ignoring the rest . An L1 measure of closure is insensitive to all but the largest gap in a fragmented figure . The total gap measure used to compare the results of the two-gap and regular-gap experiments -LRB- Fig. 7 -RRB- is an L1 norm . As we noted , response time is not a function of this L1 measure of the total gap : it can not lead to a psychophysically consistent measure of closure . Fig. 8 compares the data for these two experiments under the L1 norm . The qualitative difference between the curves for the two-gap and regular-gap experiments shown in Fig. 8 indicates that the L1 norm also can not lead to a psychophysically consistent measure for perceptual closure . While the L1 norm fails to capture the overweighting of large gaps , the L1 norm fails by going too far in ignoring all but the largest gap . Fig. 9 compares the data for the two-gap and regular-gap experiments under the L2 measure . The psychometric curves for these two independent 5 10 15 20 25 regular-gap two-gap 600 800 1000 1200 1400 1600 | | Boundary Gap | | = Maximum Gap -LRB- pixels -RRB- Response time -LRB- msec -RRB- 1 Figure 8 Two-gap and regular-gap results for polygonal figures , plotted as a function of the L1 norm -LRB- maximum -RRB- of the boundary gap . experiments are in rough alignment , suggesting that the L2 norm approximately captures the relative weighting of small and large gaps in determining the perceptual closure of a fragmented figure . 2.6 Testing the L2 hypothesis for polygonal figures The two-gap and regular-gap polygonal stimuli , while quite different , are both highly regular and symmetric . To test the generality of our L2 model for closure , we created the irregularly fragmented stimuli shown in Fig. 10 . These stimuli were created by adding dots one at a time at pseudo-random locations along the line segments connecting the endpoints of the open polygonal figures , until the figures were closed . After repeating this process 5 times , we selected the 6 fragmented figures which most evenly sampled the gap length 600 800 1000 1200 1400 1600 1800 5 10 15 20 25 30 35 Response Time -LRB- msec -RRB- | | Boundary Gap | | -LRB- pixels -RRB- 2 regular-gap two-gap Figure 9 Two-gap and regular-gap results for polygonal figures , plotted as a function of the L2 norm of the boundary gap . Under this measure of closure , the data for the two independent experiments are in rough alignment . 2 | | Boundary Gap | | -LRB- pixels -RRB- 0.04.79.314.018.523.327.932.5 Figure 10 Polygonal random-gap stimuli between open and closed figures according to the L2 measure under consideration . This method thus reduced our involvement in the selection of the test stimuli , and created highly irregular and asymmetric figures which provide information about the hypothesis evenly over the range of interest . Fig. 11 shows that the data for this random-gap experiment agrees well with those for the two-gap and regular-gap experiments under the L2 measure of gap length . 2 | | Boundary Gap | | -LRB- pixels -RRB- 600 800 1000 1200 1400 1600 1800 5 10 15 20 25 30 35 regular-gap Response Time -LRB- msec -RRB- 35 random-gap two-gap Figure 11 Results for two-gap , regular-gap and random-gap experiments , as a function of the L2 measure of gap . Data are also included from two experiments reported in Elder & Zucker -LSB- 1993 -RSB- employing the figures shown . 11 subjects were used for the random-gap experiment . Fig. 11 also shows data points from two other experiments reported in Elder & Zucker -LSB- 1993 -RSB- which agree well with the two-gap , regular-gap and random-gap data . Note that the connectedness of the contour does not seem to play a strong role in speeding the shape discrimination task . 2.7 From a Measure of Gap to a Measure of Closure We can now define a closure measure C based on the L2 measure of gap : C = 1 ? 1l qPn1 g2i , where gi is the length of the gap in the figure and l is the total arclength of the boundary . Note that C 2 -LSB- 0 ; 1 -RSB- . Fig. 12 shows that response time is a decreasing function of closure , as expected . 600 800 1000 1200 1400 1600 1800 0.7 0.75 0.8 0.85 0.9 0.95 1 Response Time -LRB- msec -RRB- two-gap regular-gap random-gap L2 Closure = 1 ? 1l -LRB- Pn1 g2i -RRB- 12 Figure 12 Results for two-gap , regular-gap and random-gap experiments , as a function of the L2 measure of closure . 3 . Closing Smooth Figures 3.1 Stimuli To test whether our L2 closure model can generalize over figure shape , we devised the stimuli shown in Fig. 13 . We designed these stimuli to be as smooth as possible for several reasons . First , this makes them very different from the polygonal figures on which the model is based . Second , we sought figures with strong two-dimensional features without excessive local curvature , so that the figures would be perceived as continuing naturally even when fragmented . Finally , starting with the goal of optimally smooth figures reduced the number of arbitrary -LRB- and perhaps biased -RRB- decisions made in designing the stimuli . DistractorTarget Shape Spline Closing Spline Figure 13 The smooth stimuli are composed of a shape spline and a closing spline . Each spline is constrained to pass through the points indicated by vertical bars , and is then optimized for maximum smoothness . The stimuli are constructed to differ only in their two-dimensional properties . The mathematics of optimally smooth planar curves has received recent attention in the computer vision community -LRB- Horn -LSB- 1983 -RSB- ; Mumford -LSB- 1990 -RSB- ; Ullman -LSB- 1976 -RSB- -RRB- . This problem is classical and has been posed elegantly -LRB- Euler -LSB- 1744 -RSB- ; Mumford -LSB- 1990 -RSB- -RRB- as the minimization of R -LRB- < = 2 + -RRB- ds , where < = and s are the curvature and arclength of the curve , respectively , and is a constant which determines a compromise between smoothness and total arclength . As boundary conditions one typically fixes the endpoints and unit tangent vectors of at the endpoints . The curves which solve this minimization problem are called elastica . While this particular formalization of the problem may seem arbitrary , it does have a physical basis . Since the total elastic energy stored in a thin beam is proportional to R < = 2ds , the path of such a beam fixed at its endpoints is that which minimizes this integral -LRB- Courant & Hilbert -LSB- 1953 -RSB- ; Horn -LSB- 1983 -RSB- -RRB- . This integral has also been used as a measure of the internal energy of interpolating \ snakes '' used to model image curves -LRB- Kass , Witkin & Terzopoulos -LSB- 1987 -RSB- -RRB- . An approximation to this integral , in which a curve is assumed to be a slowly varying graph in the plane , is the classical approximation of the potential energy of a bent rod -LRB- Courant & Hilbert -LSB- 1953 -RSB- -RRB- . Many of the regularization techniques used to interpolate noisy data in computer vision are based on this or related integrals -LRB- Poggio & Koch -LSB- 1985 -RSB- ; Terzopoulos -LSB- 1988 -RSB- -RRB- . Ullman -LSB- 1976 -RSB- has modelled the path of a subjective contour by a pair of arcs of constant curvature which meet the inducers smoothly . He reports psychophysical evidence of a qualitative correspondence between the perceived path of the subjective contour and the path formed by the two arcs which minimize R < = 2ds . We simplified the computationally intensive task of computing elastica by limiting our domain of curves to the set of cubic splines with fixed endpoints and endpoint tangent directions . This approach has been adopted by others in the computation of smooth amodal interpolants which complete contours occluded by foreground surfaces -LRB- Nitzberg -LSB- 1991 -RSB- -RRB- . Each smooth stimulus is composed of a shape spline and a closing spline -LRB- Fig. 13 -RRB- . The target and distractor for each experiment are identical except that for the target , the shape spline forms a large protrusion flanked by smaller hollows , while in the distractor the shape spline is a large hollow flanked by smaller protrusions . The distractor is obtained from the target by flipping the shape spline about the line connecting its endpoints . To avoid introducing tangent discontinuities , the endpoint tangent vectors of the shape spline and the closing spline are designed to be colinear . Fixing the shape and closing splines to pass through the points indicated in Fig. 13 determines the two splines up to a single parameter each . These parameters were then selected to minimize R < = 2ds for the two curves . Note that the closing spline has been made slightly asymmetric . By design , there is a point-by-point equivalence in the unsigned curvatures of the smooth target and distractor . Thus , as for the polygonal figures , discrimination requires an identification of curvature sign or some other twodimensional property which we conjecture depends upon a computation of closure . Fig. 14 shows 2-gap , regular-gap and random-gap fragmentations of these smooth figures , created as for the polygonal figures in section 2 . Note that in determining the regularity of the fragmentation and the length of each gap , gaps are measured along the trace of the original -LRB- optimally smooth -RRB- contour . We discuss this choice further in section 5 . L Closure 2 0.79 0.84 0.87 0.91 0.95 Figure 14 Smooth stimuli with two-gap -LRB- top -RRB- , regular-gap -LRB- middle -RRB- and random-gap -LRB- bottom -RRB- fragmentations . By the L2 measure , the closures of the stimuli within each column agree to within 2 decimal places . 3.2 Methods Visual search displays were created on the same Symbolics system described in section 2 . Subjects sat in a dimly lit room , approximately 1 m from the screen . Stimuli were drawn on an unilluminated screen at luminance 72 cd = m2 . Each display contained 8 stimuli : 7 distractors and 1 target . The stimuli were evenly spaced in a circular geometry , 2:8 deg from a central fixation dot -LRB- Fig. 15 -RRB- . The stimuli were thus separated by about 2:1 deg from centre to centre or 1:3 deg from tip to tip . Each stimulus subtended about 500 of visual angle . The angular position of the target about the central dot , and the rotation of each stimulus about its centre were selected pseudo-randomly for each trial . Figure 15 Display geometry for visual search experiments using smooth stimuli . The same visual search procedure used in section 2 was employed . One block of 14 trials was used for each stimulus condition . 3.3 Subjects 9 subjects were used for each of these experiments . All subjects had normal or corrected vision . Results are averaged over all participating subjects , with error bars indicating standard error of the mean . 3.4 Results Figures 16 , 17 and 18 show the results for these experiments as a function of closure using the L1 , L1 , and L2 metrics , respectively . As for the polygonal figures , the L1 measure is not psychophysically consistent . Interestingly , under the L1 measure , results for two-gap and random-gap stimuli fall into correspondence . This suggests that the many smaller gaps in the random-gap stimuli , which are ignored by the L1 measure , roughly balance one of the large gaps in the two-gap stimuli . However , the poor correspondence with the regular-gap results illustrates how badly the L1 measure underestimates closure for fragmentations with several gaps of comparable arclength . While response time is not a consistent function of closure by either the L1 or the L1 measures , under the L2 measure the curves are in rough allignment . Within the uncertainty of our data , an L2 measure of closure appears to be consistent in that response time can be predicted as a function of this measure . 25 two-gap regular-gap random-gap 600 800 1000 1200 1400 1600 1800 2000 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950.55 L Closure = 1 - Response Time -LRB- msec -RRB- 1 gi / l S Figure 16 Results for two-gap , regular-gap and random-gap experiments using smooth figures , as a function of the L1 measure of closure . two-gap regular-gap random-gap 600800 1000 1200 1400 1600 1800 2000 2200 0.86 0.88 0.9 0.92 0.94 0.96 0.98 10.84 Response Time -LRB- msec -RRB- L1Closure = 1 ? maxi -LRB- gi -RRB- = l Figure 17 Results for two-gap , regular-gap and random-gap experiments using smooth figures , as a function of the L1 measure of closure . 600 800 1000 1200 1400 1600 two-gap regular-gap random-gap Response Time -LRB- msec -RRB- 0.8 0.84 0.88 0.92 0.96 L2 Closure = 1 ? 1l -LRB- Pn1 g2i -RRB- 12 Figure 18 Results for two-gap , regular-gap and random-gap experiments using smooth figures , as a function of the L2 measure of closure . Under this measure of closure , the data for the three independent experiments are in rough alignment . 4 . Implications of an L2 Measure of Closure The results of sections 2 and 3 support our hypothesis that a consistent closure measure exists : response time can be expressed as a function of this L2 measure . Further , our prediction that response time would be monotonically decreasing on the range of this measure is confirmed . However , we have not attempted to find an \ optimal '' norm for perceptual closure which minimizes the spread of our data . There are several factors which limit the value of such an attempt , but above all we must keep in mind that although we have designed our experiments to test interesting and representative cases , these represent only a fraction of the possible situations . It is thus quite possible that the class of Lp norms is appropriate only as a first approximation . In spite of these limitations , there are several qualitative observations that follow from the L2 approximation . In the Introduction , we wondered which attributes of a fragmentation would matter most : the number of gaps , gap size , or gap regularity . The L2 measure of closure suggests answers to these questions . While adding new gaps to a fragmented figure clearly reduces the closure , the L2 norm embodies our finding that the redistribution of existing gaps into a larger number of smaller gaps increases the closure . Thus the connectedness of the contour , which has been proposed by others as an important grouping factor -LRB- Rock & Palmer -LSB- 1990 -RSB- -RRB- , does not play a positive role here , and the number of gaps in a fragmented contour is not in itself a useful indicator of the closure of a figure . Clearly , as existing gaps are increased in size , closure decreases . The L2 norm embodies our finding that , while all gaps in the contour matter , the largest gaps matter most . Thus the size of each gap is very important . There is psychophysical evidence that the mechanisms responsible for long-range visual tasks are qualitatively different from those for short-range tasks -LRB- Burbeck & Yap -LSB- 1990 -RSB- ; Levi & Klein -LSB- 1990 -RSB- ; Zucker & Davis -LSB- 1988 -RSB- -RRB- . The overweighting of large gaps embodied by the L2 measure of closure suggests that mechanisms for grouping distant contour fragments are more complex than the local operations which bridge small gaps . The overweighting of large gaps relative to small also leads naturally to a principle of regularity : given any set of fragments and specified boundary , the maximum closure arrangement is the most regular arrangement -LRB- that for which each gap has the same arclength -RRB- . While for the L2 measure , this regularity property follows directly from Cauchy 's inequality , note that the L1 measure does not have this property , and the L1 measure is sensitive only to the maximum gap . Regularity can be viewed as the most basic principle of perceptual organization from which others are derived -LRB- Witkin & Tenenbaum -LSB- 1983 -RSB- -RRB- : it is interesting that it can be embodied in something as simple as a norm . The L2 measure of closure can be thought of as a highly compressed representation of a fragmented figure , which predicts how the speed of shape perception will be impeded by the fragmentation . Since the space of possible fragmentations is infinite-dimensional , and the closure measure is onedimensional , without prior assumptions a consistent closure measure would be unlikely to exist . Thus we interpret the consistency of the L2 measure in predicting response time as evidence that the brain computes closure as an intermediate step in the formation of shape representations from contour fragments in an image . 5 . Paths of Completion In section 2 , we constrained the closing fragments to lie on the straight lines connecting the endpoints of the open figure . In section 3 we constrained the fragments to lie on a smooth curve : an approximation to a curve of the family of elastica which minimize R -LRB- < = 2 + -RRB- ds . Note that we could have chosen to model the closure of the polygonal shapes by elastica as well , creating figures with billowed ends . Then again , although elastica have some physical basis and have been used as interpolants in computer vision , there is little evidence that they are the most natural for human perception . To evaluate our choice of interpolants we devised two new sets of visual search experiments : one employing the polygonal figures of section 2 and the other employing the smooth figures of section 3 . For both experiments we used the same display geometry , the same experimental procedure and the same subjects described in section 3 . The polygonal stimuli employed are shown together with the experimental results in Fig. 19 . Only the target spindle stimuli are shown : the distractor barrel stimuli have identical fragmentations . Each stimulus consisted of the flanking segments and 4 short -LRB- 2-pixel -RRB- evenly spaced closing fragments , either along the straight line connecting the endpoints of the side fragments , or displaced orthogonally to this straight line . The results show a strong sensitivity to this orthogonal displacement , with a clear minimum in response time when the fragments lie on the straight line connecting the endpoints ? . These results clearly support our choice of interpolant for the polygonal figures . The smooth stimuli employed are shown together with the results in Fig. 20 . Again , only the target stimuli are shown . As for the polygonal stimuli , these also possess 4 short -LRB- 2-pixel -RRB- evenly spaced closing fragments . At the displacement labelled 2 , these fragments lie on the smooth interpolant employed for our experiments in section 3 . At displacement 0 , these fragments have been displaced along the normal to the smooth curve until they lie on the straight line interpolant between the top and bottom segments . The three other displacements also lie at even intervals along this normal direction . Although response time is again highly sensitive to the orthogonal displacement of these fragments , the results are more difficult to interpret . Response time is not at a minimum when the fragments lie on the smooth interpolant , rather , the minimum appears to occur when the fragments are closer to the body of the figure . However , the minimum also does not occur when the fragments lie on the straight line interpolant : response time decreases again when the fragments are placed well within the figure -LRB- displacement -1 -RRB- y . This may ? We compare data using a one-tailed t-test with a significance level of 0.05 . By this measure , response time is a minimum at displacement -LRB- p < :025 -RRB- . yResponse times at displacements of -1 and 1 do not differ significantly -LRB- p > :1 -RRB- . Response times at displacements of 0 , 2 and 3 are significantly longer -LRB- p < :005 -RRB- . 500 550 600 650 700 750 800 -8 -4 4 8 Dot Displacement -LRB- pixels -RRB- Response time -LRB- msec -RRB- Figure 19 Response time as a function of the displacement of the closing fragments orthogonally to the straight path connecting the endpoints of the side fragments . Response time is a minimum when the dots lie on the straight connecting path . -1 1 2 3 1100 1200 1300 1400 1500 1600 1700 Dot Displacement Response time -LRB- msec -RRB- Figure 20 Response time as a function of displacement of dots to either side of the straight path connecting the top and bottom fragments . Response time increases as dots are displaced further from the figure . A minimum occurs when the dots lie on a smooth path connecting the fragments . As the dots are displaced further inward , the figure is truncated -LRB- see text -RRB- . be due to a perceptual truncation of the end segments of the shape spline , so that the closing fragments connect the bottom segment directly with the downward sloping sides of the upper protrusion . Were it not for this complication , the minimum response time would likely be attained at an intermediate displacement between the smooth elastica approximation and the straight line path . This uncertainty in the natural path of completion for the smooth figures prompted us to reanalyze our results by measuring gap length by the Euclidean distance between fragment endpoints , rather than length along a smooth path . With this measure , the total arclength l of the figure becomes a function of the fragmentation . The results of our smooth-figure experiments using the L2 closure metric , but with a Euclidean measure of gap length , are shown in Fig. 21 . Note that the results are little affected by this change . Ullman -LSB- 1976 -RSB- has proposed a \ locality hypothesis '' for contour interpolation : that \ the operation by which a boundary edge is extended to generate a subjective contour is assumed to be local in nature . That is , it depends only on the end part of the given edge to be extended and not on the shape of the entire edge . '' In contrast , the results shown in Fig. 19 and 20 suggest an interesting dependence of the ideal completing path on the global nature of the existing curve . While smooth interpolants are most powerful for speeding the discrimination of smooth figures , straight-line interpolants are more effective for polygonal figures . These straight-line interpolants introduce sharp tangent discontinu - two-gap regular-gap random-gap 0.8 0.84 0.88 0.92 0.96 Response Time -LRB- msec -RRB- 600 1600 1400 1200 1000 800 L2 Closure = 1 ? 1l -LRB- Pn1 g2i -RRB- 12 Figure 21 Results of experiment with smooth shapes plotted as a function of L2 closure , where gap length is determined by the Euclidean distance between fragment endpoints . ities in the figure : the optimal smoothness criteria suggested by Mumford -LSB- 1990 -RSB- and Ullman -LSB- 1976 -RSB- are thus inappropriate for these polygonal shapes . Moreover , the dependence of the natural interpolant on global properties of the image contours provides psychophysical evidence against Ullman 's locality hypothesis for this form of contour interpolation . 6 . Conclusion While data and models for the sensitivity of the human visual system to basic visual signals -LRB- luminance , colour , motion and stereopsis are examples -RRB- have become increasingly quantitative , data and models for how humans perceive supra-threshold geometric relationships are typically qualitative in nature : the Gestalt principles of perceptual organization are still a starting point for many interesting questions in perception research . Our technique of evaluating closure in the context of shape perception provides a quantitative alternative to subjective -LRB- phenomenological -RRB- evaluations of contour completion . Our goal in this paper was to evaluate potential measures for the closure of a set of fragmented contours in an image . Employing both polygonal and smooth contours and three classes of fragmentation , we showed that while L1 and L1 measures for closure are psychophysically inconsistent , an L2 measure is appropriate in that response time can be expressed as a decreasing function of this measure . This L2 measure of closure deemphasizes the connectedness of the contour and , in overweighting the importance of large gaps relative to small , leads naturally to a principle of perceptual regularity . The existence of a consistent and monotonic closure measure suggests that the computation of closure is an intermediate step in the inference of shape from contour . Our need to measure the length of gaps between fragments prompted an investigation of the shape of contour interpolants . This has led to the interesting finding that the shape of the most effective interpolant depends upon global aspects of figure shape . 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O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O 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O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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