{"title": "Orientation, Scale, and Discontinuity as Emergent Properties of Illusory Contour Shape", "book": "Advances in Neural Information Processing Systems", "page_first": 831, "page_last": 837, "abstract": null, "full_text": "Orientation, Scale, and Discontinuity as \nEmergent Properties of Illusory Contour \n\nShape \n\nKarvel K. Thornber \nNEC Research Institute \n\n4 Independence Way \nPrinceton, NJ 08540 \n\nLance R. Williams \n\nDept. of Computer Science \nUniversity of New Mexico \nAlbuquerque, NM 87131 \n\nAbstract \n\nA recent neural model of illusory contour formation is based on \na distribution of natural shapes traced by particles moving with \nconstant speed in directions given by Brownian motions. The input \nto that model consists of pairs of position and direction constraints \nand the output consists of the distribution of contours joining all \nsuch pairs. In general, these contours will not be closed and their \ndistribution will not be scale-invariant. In this paper, we show \nhow to compute a scale-invariant distribution of closed contours \ngiven position constraints alone and use this result to explain a \nwell known illusory contour effect. \n\n1 \n\nINTRODUCTION \n\nIt has been proposed by Mumford[3] that the distribution of illusory contour shapes \ncan be modeled by particles travelling with constant speed in directions given by \nBrownian motions. More recently, Williams and Jacobs[7, 8] introduced the notion \nof a stochastic completion field, the distribution of particle trajectories joining pairs \nof position and direction constraints, and showed how it could be computed in a \nlocal parallel network. They argued that the mode, magnitude and variance of \nthe completion field are related to the observed shape, salience, and sharpness of \nillusory contours. \n\nUnfortunately, the Williams and Jacobs model, as described, has some shortcom(cid:173)\nings. Recent psychophysics suggests that contour salience is greatly enhanced by \nclosure[2]. Yet, in general, the distribution computed by the Williams and Jacobs \nmodel does not consist of closed contours. Nor i:; it scale-invariant-doubling the \ndistances between the constraints does not produce a comparable completion field of \n\n\f832 \n\nK. K. Thornber and L. R. Williams \n\ndouble the size without a corresponding doubling of the particle's speeds. However, \nthe Williams and Jacobs model contains no intrinsic mechanism for speed selec(cid:173)\ntion. The speeds (like the directions) must be specified a priori. In this paper, we \nshow how to compute a scale-invariant distribution of closed contours given position \nconstraints alone. \n\n2 TECHNICAL DETAILS \n\n2.1 SHAPE DISTRIBUTION \n\nConsistent with our earlier work[5, 6], in this paper we do not use the same dis(cid:173)\ntribution described by Mumford[3] but instead assume a distribution of completion \nshapes consisting of straight-line base-trajectories modified by random impulses \ndrawn from a mixture of two limiting distributions. The first distribution consists \nof weak but frequently acting impulses (we call this the Gaussian-limit). The dis(cid:173)\ntribution of these weak impulses has zero mean and variance equal to (7~. The weak \nimpulses act at Poisson times with rate R g . The second distribution consists of \nstrong but infrequently acting impulses (we call this the Poisson-limit). Here, the \nmagnitude of the random impulses is Gaussian distributed with zero mean. How(cid:173)\never, the variance is equal to (72 (where (7~ \u00bb \n(7;). The strong impulses act at \nPoisson times with rate Rp < < kg. Particles decay with half-life equal to a param(cid:173)\neter T. The effect is that particles tend to travel in smooth, short paths punctuated \nby occasional orientation discontinuities. See [5, 6]. \n\n2.2 EIGENSOURCES \n\nLet i and j be position and velocity constraints, (xi,id and (xj,Xj). Then P(jl i) \nis the conditional probability that a particle beginning at i will reach j. Note that \nthese transition probabilities are not symmetric, i.e., P(j 1 i) 1= P(i 1 j). However, by \ntime-reversal symmetry, P(j 1 i) = P(I 1 J) where I = (Xi, -Xi) and J = (Xj, -Xj). \nGiven only the matrix of transition probabilities, P, we would like to compute the \nrelative number of closed contours satisfying a given position and velocity constraint. \nWe begin by noting that, due to their randomness, only increasingly smaller and \nsmaller fractions of contours are likely to satisfy increasing numbers of constraints. \nSuppose we let s~l) contours start at Xi with Xi. Then \n\n(2) _ ~ P( 'I .) (1) \n- ui J '/, Si \nSj \n\nis the relative number of contours through Xj with Xj, i.e., which satisfy two con(cid:173)\nstraints. In general, \n\n- ui \nNow suppose we compute the eigenvector, \n\nSj \n\n(n+1) _ ~ P( 'I .) (n) \n\nJ \n\n'/, Si \n\nwith largest, real positive eigenvalue, and take s~1) = Si. Then clearly si n+1) = AnSi. \nThis implies that as the number of constraints satisfied increases by one, the number \nof contours remaining in the sample of interest decreases by A. However, the ratios \nof the Si remain invariant. Letting n pass to infinity, we see that the Si are just \nthe relative number of contours through i. To summarize, having started with all \npossible contours, we are now left with only those bridging pairs of constraints at \nall past-times. By solving AS = Ps for s we know their relative numbers. We refer \nto the components of s as the eigensources of the stochastic completion field. \n\n\fEmergent Properties of Illusory Contour Shape \n\n833 \n\n2.3 STOCHASTIC COMPLETION FIELDS \n\nNote that the eigensources alone do not represent a distribution of closed contours. \nIn fact, the majority of contours contributing to s will not satisfy a single additional \nconstraint. However, the following recurrence equation gives the number of contours \nwhich begin at constraint i and end at constraint j and satisfy n - 1 intermediate \nconstraints \n\np(n+1) (j I i) = Lk P(j I k)p(n) (k I i) \n\nwhere p( 1) (j I i) = P(j I i). Given the above recurrence equation, we can define an \nexpression for the relative number of contours of any length which begin and end \nat constraint i: \n\nCi = limn -+ oo p(n)(i I i)/ Lj p(n)(j I)) \n\nUsing a result from the theory of positive matrices[l}, it is possible to show that \nthe above expression is simply \n\nCi = Si 8d Lj Sj8j \n\nwhere sand s are the right and left eigenvectors of P with largest positive real \neigenvalue, i.e., AS = Ps and AS = pTs. Because of the time-reversal symmetry \nof P, the right and left eigenvectors are related by a permutation which exchanges \nopposite directions, i.e. , 8i = St. \nFinally, given sand s, it is possible to compute the relative number of closed \ncontours through an arbitrary position and velocity in the plane, i.e., to compute \nthe stochastic completion field. If\", = (x, x) is an arbitrary position and velocity \nin the plane, then \n\nC(\",) = >.s~s Li P(\", I i)Si . Lj P(j I \",)8j \n\ngives the relative probability that a closed contour will pass through \",. Note, that \nthis is a natural generalization of the Williams and Jacobs[7] factorization of the \ncompletion field into the product of source and sink fields. \n\n2.4 SCALE-INVARIANCE \n\nUnder the restriction that particles have constant speed, the transition probability \nmatrix, P, becomes block-diagonal. Each block corresponds to a different possible \nspeed, 'Y- Since the components of any given eigenvector will be confined to a single \nblock, we can consider P to be a function of, and solve: \n\nA(r) s(r) = P(r)s(r) \n\nLet Amax (r) be the largest positive real eigenvalue of P(r) and let ,max be the speed \nwhere Amax (r) is maximized. Then Smax (rma x), i.e., the eigenvector of P (rmax) \nassociated with Amax (rma x), is the limiting distribution over all spatial scales. \n\n3 EXPERIMENTS \n\n3.1 EIGHT POINT CIRCLE \n\nGiven eight points spaced uniformly around the perimeter of a circle of diameter, \nd = 16, we would like to find the distribution of directions through each point and \nthe corresponding completion field (Figure 1 (left)). Neither the order of traversal , \ndirections, i.e., xdlxil, or speed, i.e. , , = IXil. are specified a priori. \nIn all of \nour experiments, we sample direction at 5\u00b0 intervals. Consequently, there are 72 \ndiscrete directions and 576 position-direction pairs, i.e., P(r) is of size 576 x 576. 1 \nlThe parameters defining the distribution of completion shapes are T = Rga~ = 0.0005 \n\nand 'T = 9.5. For simplicity, we assume the pure Gaussian-limit case described in [6] . \n\n\f834 \n\n\u2022 \n\u2022 \n\u2022 \n\n\u2022 \n\n\u2022 \n\na \n\n\u2022 \n\u2022 \n\u2022 \n\n/ \nI \n\n\" I \n\" / \n\nc \n\nK. K. Thornber and L. R. Williams \n\nEight Point Circle (two sizes) \n\nI \ni \n\n,..,,, \no \nw~ \n,0 \nI \n. \no \n\n.-\nw \n\n\u2022 a \n\nIl. \n\n'0 \nxo \nog~~~~~~~~~~~~~~~ \n:l~ \no \n\n30 \n\n,0 \n\n0 \n\n\" \n\n'\" \n\n20 \n\nb \n\nd \n\nFigure 1: Left: (a) The eight position constraints. Neither the order of traversal, direc(cid:173)\ntions, or speed are specified a priori. (b) The eigenvector, Smax (,max) represents the lim(cid:173)\niting distribution over all spatial scales. (c) The product of smaxC!max) and smaxC!max). \nOrientations tangent to the circle dominate the distribution of closed contours. (d) The \nstochastic completion field, C, due to smaxC!max). Right: Plot of magnitude of maximum \npositive real eigenvalue, >'max, vs. logl.l (1/,) for eight point circle with d = 16.0 (solid) \nand d = 32.0 (dashed). \n\n== \n\n~ \n~ \n\n~ ==::J \n\n11 \nU \n\nFigure 2: Observers report that as the width of the arms increases, the shape of the \nillusory contour changes from a circle to a square[4]. \n\nFirst, we evaluated Amax b) over the velocity interval [1.1- 1 , 1.1-3oJ using standard \nnumerical routines and plotted the magnitude of the largest, real positive eigenvalue, \nAmax vs. logl.l(l/,). The function reaches its maximum value at '\"'(max:::::: 1.1- 2\u00b0. \nConsequently, the eigenvector, Smax (1.1 - 2\u00b0) represents the limiting distribution over \nall spatial scales (Figure 1 (right)). \n\nNext, we scaled the test Figure by a factor of two, i.e., d' = 32.0 and plotted \nA~axb) over the same interval (Figure 1 (right)). We observe that A~ax(1.1-x+7) \n:::::: Amax (1.1- X ), i.e., when plotted using a logarithmic x-axis, the functions are \nidentical except for a translation. It follows that '\"'(~ax :::::: logl.1 7 x '\"'(max:::::: 2.0 x '\"'(max' \nThis confirms the scale-invariance of the system-doubling the size of the Figure \nresults in a doubling of the selected speed. \n\n3.2 KOFFKA CROSS \n\nThe Koffka Cross stimulus (Figure 2) has two basic degrees of freedom which we call \ndiameter (i.e. , d) and arm width (i.e., w) (Figure 3 (a)). We are interested in how \n\n\fEmergent Properties of Illusory Contour Shape \n\n(a) \n\n(e) \n\n835 \n\n(d) \n\nr---~ .--......, \n\no \nU \n\nd \n\n(b) \n\n(-0 5w . O.5d) \n\n( O.5w ,O.Sd ) \n\n(--O.Sd , 05w) \n\n(--OSd, --05w) \n\n(--05w. -O.5d) \n\n(O.5w. --OSd ) \n\n(OSd , 05w) \n\nn \n( 0 Sd ,-O.5w ) u \n\nr - - - - -\n\n- - - - - - , \n\nFigure 3: (a) Koffka Cross showing diameter, d, and width , w. (b) Orientation and \nposition constraints in terms of d and w. The normal orientation at each endpoint \nis indicated by the solid lines while the dashed lines represent plus or minus one \nstandard deviation (i.e. , 12.8\u00b0) of the Gaussian weighting function. (c) Typically \nperceived as square. (d) Typically perceived as circle. The positions of the line \nendpoints is the same. \n\nthe stochastic completion field changes as these parameters are varied. Observers \nreport that as the width of the arms increases, the shape of the illusory contour \nchanges from a circle to a square[4]. The endpoints of the lines comprising the \nKoftka Cross can be used to define a set of position and orientation constraints \n(Figure 3 (b)). The position constraints are specified in terms of the parameters, d \nand w. The orientation constraints take the form of a Gaussian weighting function \nwhich assigns higher probabilities to contours passing through the endpoints with \norientations normal to the lines. 2 The prior probabilities assigned to each position(cid:173)\ndirection pair by the Gaussian weighting function form a diagonal matrix, D: \n\nwhere P(r) is the transition probability matrix for the random process at scale \n\" A(r) is an eigenvalue of Q(,), and s(r) is the corresponding eigenvector. Let \nAmax(r) be the largest positive real eigenvalue of Q(r) and let ,max be the scale \nwhere Amax(r) is maximized. Then smax(rmax), i.e., the eigenvector of Q(rmax) \nassociated with Amax (rma x), is the limiting distribution over all spatial scales. \n\nFirst, we used a Koffka Cross where d = 2.0 and w = 0.5 and evaluated Amax (r) over \nthe velocity interval [8.0 x 1.1- 1 , 8.0 x 1.1-8\u00b0] using standard numerical routines. 3 \nThe function reaches its maximum value at ,max::::; 8.0 X 1.1-62 (Figure 4 (left)). \nObserve that the completion field due to the eigenvector, smax(8.0 x 1.1-62 ), is \ndominated by contours of a predominantly circular shape (Figure 4 (right)). We \nthen uniformly scaled the Koffka Cross Figure by a factor of two, i.e., d' = 4.0 and \n\n20bserve that Figure 3 (c) is perceived as a square while Figure 3 (d) is perceived as a \ncircle. Yet the positions of the line endpoints is the same. It follows that the orientations \nof the lines affect the percept. We have chosen to model this dependence through the use \nof a Gaussian weighting function which favors contours passing through the endpoints of \nthe lines in the normal direction. It is possible to motivate this based on the statistics of \nnatural scenes. The distribution of relative orientations at contour crossings is maximum \nat 90\u00b0 and drops to nearly zero at 0 0 and 180 0 \u2022 \n\n3The parameters defining the distribution of completion shapes were: T = RgO'~ = \n0.0005, T = 9.5, \u20acp = O'~/T = 100.0 and Rp = 1.0 X 10- 8 . As an anti-aliasing measure, the \ntransition probabilities, P(j I i) , were averaged over initial conditions modeled as Gaussians \nof variance 0'; = 0'; = 0.00024 and O'J = 0.0019. See [6]. \n\n\f836 \n\nK. K. Thornber and L. R. Williams \n\nKoffke Crosses (TWO sizes) \n\n_ -_=0>== \n\n... \n'\"'~ \nI a , \nLIl~ \no \n\nWo \n\n-o \n> c \nCI~ - . \n00 \n-o , \nIra ... a \na \nD.. \n\n~o \n\ni \n/ \n\n; \n\n/ \n\n~/--\n\nX 0 \nog-HTnTrnTnTrnTnTrnTnTnTrnTn~~ \n~ci \n.. 0 \n\n60 \n\na \n\n20 \n\n'0 \nx \n\nFigure 4: Left: Plot of magnitude of maximum positive real eigenvalue, >'max, vs. \nlogl.l (1h) for Koffka Crosses with d = 2.0 and w = 0.5 (solid) and d = 4.0 and w = 1.0 \n(dashed). Right: The completion field due to the eigenvector, smax (8 .0 x 1.1-62 ) . \n\nw' = 1.0 and plotted Anax (,) over the same interval (Figur~ 4 (left)) . Observe \nthat A~ax (8.0 X 1.1-x+ ) :::::: Amax(8.0 x 1.1- X). As before, thls confirms the scale(cid:173)\ninvariance of the system. \n\nNext, we studied how the relative magnitudes of the local maxima of Amax (,) \nchange as the parameter w is varied. We begin with a Koffka Cross where d = 2.0 \nand w = 0.5 and observe that Amax(r) has two local maxima (Figure 5 (left)). \nWe refer to the larger of these maxima as ,circle . As previously noted, this max(cid:173)\nimum is located at approximately 8.0 x 1.1-62 . The second maximum is located \nat approximately 8.0 x 1.1 -32. When the completion field due to the eigenvector, \nsmax(8.0 x 1.1-32 ), is rendered, we observe that the distribution is dominated by \ncontours of predominantly square shape (Figure 5(a)). For this reason , we refer \nto this local maximum as ,square. Now consider a Koffka Cross where the widths \nof the arms are doubled but the diameter remains the same, i.e., d' = 2.0 and \nw' = 1.0. We observe that A~ax (r) still has two local maxima, one at approxi(cid:173)\nmately 8.0 x 1.1-63 and a second at approximately 8.0 x l.1-29 (Figure 5 (left)). \nWhen we render the completion fields due to the eigenvectors, s~ax(8.0x 1.1-63 ) and \ns~ax(8.0 x 1.1- 29 ), we find that the completion fields have the same general char(cid:173)\nacter as before-the contours associated with the smaller spatial scale (i.e., lower \nspeed) are approximately circular and those associated with the larger spatial scale \n(Le., higher speed) are approximately square (Figure 5 (d) and (c)). Accordingly, \nwe refer to the locations of the respective local maxima as '~ircle and ,~quare ' How(cid:173)\never, what is most interesting is that the relative magnitudes of the local maxima \nhave reversed. Whereas we previously observed that Amax(,circle) > Amax(rsquare), \nwe now observe that A~ax(r~quare) > A~ax(r~ircle)' Therefore, the completion field \ndue to the eigenvector, s~ax(r~quare ) [not s~ax(r~ircle)!l represents the limiting \ndistribution over all spatial scales. This is consistent with the transition from circle \nto square reported by human observers when the widths of the arms of the Koffka \nCross are increased. \n\n\fEmergent Properties of Illusory Contour Shape \n\n837 \n\nKoffke Crosses (two \n\nwidths) \n\n-\n\n--, ~:...-\n\nb \n\na \n\nc \n\nd \n\no \n> \n( \nGl o \nOI~ \n\n.- . wo \n-o \u2022 (l:o \u2022 o \n\n~o \n\no \nQ. \n\n20 \n\n40 \nX \n\n60 \n\n80 \n\na \n\nc \n\nb \n\nd \n\nFigure 5: Plot of magnitude of maximum positive real eigenvalue, Ama x, vs. log 1.1 (1/\"'() \nfor Koffka Crosses with d = 2.0 and w = 0.5 (solid) and d = 2.0 and w = 1.0 (dashed) . \nStochastic completion fields for Koffka Cross due to (a) Smax (\"'(.quar e ) is a local optimum \nfor w = 0.5 (b) Sma x (\"'(ci rcl e ) is the global optimum for w = 0.5 (c) s~ax(\"'(~quar e ) is the \nglobal optimum for w = 1.0 (d) s~a x (\"'(~quar e ) is a local optimum for w = 1.0. These \nresults are consistent with the circle-to-square transition perceived by human subjects \nwhen the width of the arms of the Koffka Cross are increased. \n\n4 CONCLUSION \n\nWe have improved upon a previous model of illusory contour formation by show(cid:173)\ning how to compute a scale-invariant distribution of closed contours given position \nconstraints alone. We also used our model to explain a previously unexplained \nperceptual effect. \n\nReferences \n\n[1] Horn, R.A ., and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press , p. 500, \n\n1985. \n\n[2] Kovacs, I. and B. Julesz, A Closed Curve is Much More than an Incomplete One: \nEffect of Closure in Figure-Ground Segmentation, Pmc. Natl. Acad. Sci. USA, 90, \npp. 7495-7497, 1993. \n\n[3] Mumford, D., Elastica and Computer Vision, Algebraic Geometry and Its Applica(cid:173)\n\ntions, Chandrajit Bajaj (ed.) , Springer-Verlag, New York, 1994. \n\n[4) Sambin, M., Angular Margins without Gradients, Italian Journal of Psychology 1, \n\npp. 355-361, 1974. \n\n[5] Thornber, KK and L.R. Williams, Analytic Solution of Stochastic Completion \n\nFields, Biological Cybernetics 75, pp. 141-151, 1996. \n\n[6] Thornber , KK and L.R. Williams, Characterizing the Distribution of Completion \nShapes with Corners Using a Mixture of Random Processes, Intl. Workshop on \nEnergy Minimization Methods in Computer Vision, Venice, Italy, 1997. \n\n[7] Williams, L.R. and D.W . Jacobs, Stochastic Completion Fields: A Neural Model of \nIllusory Contour Shape and Salience, Neural Computation 9(4) , pp. 837-858, 1997. \n[8) Williams, L.R. and D.W. Jacobs, Local Parallel Computation of Stochastic Com(cid:173)\n\npletion Fields, Neural Computation 9(4), pp. 859-881 , 1997. \n\n\f", "award": [], "sourceid": 1528, "authors": [{"given_name": "Karvel", "family_name": "Thornber", "institution": null}, {"given_name": "Lance", "family_name": "Williams", "institution": null}]}