Orientation Scale and Discontinuity as 
Emergent Properties of Illusory Contour 
Shape 
Karvel K. Thornbet 
NEC Research Institute 
4 Independence Way 
Princeton, NJ 08540 
Lance R. Williams 
Dept. of Computer Science 
University of New Mexico 
Albuquerque, NM 87131 
Abstract 
A recent neural model of illusory contour formation is based on 
a distribution of natural shapes traced by particles moving with 
constant speed in directions given by Brownian motions. The input 
to that model consists of pairs of position and direction constraints 
and the output consists of the distribution of contours joining all 
such pairs. In general, these contours will not be closed and their 
distribution will not be scale-invariant. In this paper, we show 
how to compute a scale-invariant distribution of closed contours 
given position constraints alone and use this result to explain a 
well known illusory contour effect. 
I INTRODUCTION 
It has been proposed by Mumford[3] that the distribution of illusory contour shapes 
can be modeled by particles travelling with constant speed in directions given by 
Brownian motions. More recently, Williams and Jacobs[7, 8] introduced the notion 
of a stochastic completion field, the distribution of particle trajectories joining pairs 
of position and direction constraints, and showed how it could be computed in a 
local parallel network. They argued that the mode, magnitude and variance of 
the completion field are related to the observed shape, salience, and sharpness of 
illusory contours. 
Unfortunately, the Williams and Jacobs model, as described, has some shortcom- 
ings. Recent psychophysics suggests that contour salience is greatly enhanced by 
closure[2]. Yet, in general, the distribution computed by the Williams and Jacobs 
model does not consist of closed contours. Nor is it scale-invariant--doubling the 
distances between the constraints does not produce a comparable completion field of 
832 K. K. Thornber and L. R. k'lliams 
double the size without a corresponding doubling of the particle's speeds. However, 
the Williams and Jacobs model contains no intrinsic mechanism for speed selec- 
tion. The speeds (like the directions) must be specified a priori. In this paper, we 
show how to compute a scale-invariant distribution of closed contours given position 
constraints alone. 
2 TECHNICAL DETAILS 
2.1 SHAPE DISTRIBUTION 
Consistent with our earlier work[5, 6], in this paper we do not use the same dis- 
tribution described by Mumford[3] but instead assume a distribution of completion 
shapes consisting of straight-line base-trajectories modified by random impulses 
drawn from a mixture of two limiting distributions. The first distribution consists 
of weak but frequently acting impulses (we call this the Gaussian-limit). The dis- 
tribution of these weak impulses has zero mean and variance equal to cr'. The weak 
impulses act at Poisson times with rate R 9. The second distribution consists of 
strong but infrequently acting impulses (we call this the Poisson-limit). Here, the 
magnitude of the random impulses is Gaussian distributed with zero mean. How- 
ever, the variance is equal to cr 2 (where Crp 2 ;>;> cr2). The strong impulses act at 
Poisson times with rate Rp  9' Particles decay with half-life equal to a param- 
eter r. The effect is that particles tend to travel in smooth, short paths punctuated 
by occasional orientation discontinuities. See [5, 6]. 
2.2 EIGENSOURCES 
Let i and j be position and velocity constraints, (xi, ci) and (xj, kj). Then P(j] i) 
is the conditional probability that a particle beginning at i will reach j. Note that 
these transition probabilities are not symmetric, i.e., P(jli) - P(i ]_j). However, by 
time-reversal symmetry, P(jli) = P(i ]j) where; = (xi,-:i) and j = (xj,-cj). 
Gi.ven only the matrix of transition probabilities, P, we would like to compute the 
relative number of closed contours satisfying a given position and velocity constraint. 
We begin by noting that, due to their randomness, only increasingly smaller and 
smaller fractions of contours are likely to satisfy increasing numbers of constraints. 
(1) contours start at xi with i. Then 
Suppose we let i 
s(2) 
j = -iP(Jl 
is the relative number of contours through xj with j, i.e., which satisfy two con- 
straints. In general, 
8 n+l) z i P(J [ i)s? ) 
Now suppose we compute the eigenvector, 
,$j z -i P(j [ i)si 
(1) (n+l) 
with largest, real positive eigenvalue, and take s i = si. Then clearly s 
This implies that as the number of constraints satisfied increases by one, the number 
of contours remaining in the sample of interest decreases by ,. However, the ratios 
of the si remain invariant. Letting n pass to infinity, we see that the si are just 
the relative number of contours through i. To summarize, having started with all 
possible contours, we are now left with only those bridging pairs of constraints at 
all past-times. By solving ,s = Ps for s we know their relative numbers. We refer 
to the components of s as the eigensources of the stochastic completion field. 
Emergent Properties of Illusory Contour Shape 833 
2.3 STOCHASTIC COMPLETION FIELDS 
Note that the eigensources alone do not represent a distribution of closed contours. 
In fact, the majority of contours contributing to s will not satisfy a single additional 
constraint. However, the following recurrence equation gives the number of contours 
which begin at constraint i and end at constraint j and satisfy n - I intermediate 
constraints 
P(n+l)(J [ i) = -],k P(J l k)P(n)(k [ i) 
where P(')(jli)- P(j l i). Given the above recurrence equation, we can define an 
expression for the relative number of contours of any length which begin and end 
at constraint i' 
ci = limn-P()(ili)/j P)(j]j) 
Using a result from the theory of positive matrices[I], it is possible to show that 
the above expression is simply 
i --- 8ii/ j 8jj 
where s and  are the right and left eigenvectors of P with largest positive real 
eigenvalue, i.e., As - Ps and A - pT. Because of the time-reversal symmetry 
of P, the right and left eigenvectors are related by a permutation which exchanges 
opposite directions, i.e., i = s;. 
Finally, given s and , it is possible to compute the relative number of closed 
contours through an arbitrary position and velocity in the plane, i.e., to compute 
the stochastic completion field. If ] - (x, ) is an arbitrary position and velocity 
in the plane, then 
C(7]) -- 1 
' i P(rl l i)si ' j P(j l rl)J 
gives the relative probability that a closed contour will pass through /. Note, that 
this is a natural generalization of the Williams and Jacobs[7] factorization of the 
completion field into the product of source and sink fields. 
2.4 SCALE-INVARIANCE 
Under the restriction that particles have constant speed, the transition probability 
matrix, P, becomes block-diagonal. Each block corresponds to a different possible 
speed, % Since the components of any given eigenvector will be confined to a single 
block, we can consider P to be a function of "/and solve: 
/k("/) s(7) = 
Let/kmax ("/) be the largest positive real eigenvalue of P('y) and let ')'max be the speed 
where /kmax("/) is maximized. Then Smax("/max), i.e., the eigenvector of P('Ymax) 
associated with /kmax("/max), is the limiting distribution over all spatial scales. 
3 EXPERIMENTS 
3.1 EIGHT POINT CIRCLE 
Given eight points spaced uniformly around the perimeter of a circle of diameter, 
d - 16, we would like to find the distribution of directions through each point and 
the corresponding completion field (Figure I (left)). Neither the order of traversal, 
directions, i.e., ci/]:[, or speed, i.e., 7 = [k[, are specified a priori. In all of 
our experiments, we sample direction at 5  intervals. Consequently, there are 72 
discrete directions and 576 position-direction pairs, i.e., P(7) is of size 576 x 576.1 
The parameters defining the distribution of completion shapes are T = Rga - 0.0005 
and - ---- 9.5. For simplicity, we assume the pure Gaussian-limit case described in [6]. 
834 K. K. Thornber and L. R. V'lliams 
v 
b 
 Eight Point Ci rcle (two sizes) 
0o 
3o 
x 
Figure 1: Left: (a) The eight position constraints. Neither the order of traversal, direc- 
tions, or speed are specified a priori. (b) The eigenvector, S,a(',,a) represents the lim- 
iting distribution over all spatial scales. (c) The product of Sma (',) and m (',)- 
Orientations tangent to the circle dominate the distribution of closed contours. (d) The 
stochastic completion field, C, due to S('ma). Right: Plot of magnitude of maximum 
positive real eigenvalue, A,,, vs. log.(1/,) for eight point circle with d = 16.0 (solid) 
and d = 32.0 (dashed). 
J  J [ l I 
] 
Figure 2: Observers report that as the width of the arms increases, the shape of the 
illusory contour changes from a circle to a square[4]. 
First, we evaluated A,ax ("/) over the velocity interval [1.1 -, 1.1-3] using standard 
numerical routines and plotted the magnitude of the largest, real positive eigenvalue, 
Amax vs. lOgl.l(1/"/). The function reaches its maximum value at "/max  1.1 -e 
Consequently, the eigenvector, s,ax (1.1 -e) represents the limiting distribution over 
all spatial scales (Figure I (right)). 
Next, we scaled the test Figure by a factor of two, i.e., d  - 32.0 and plotted 
A',a("/) over the same interval (Figure I (right)). We observe that A',a(1.1 -+7) 
 A,a(1.1-x), i.e., when plotted using a logarithmic x-axis, the functions are 
identical except for a translation. It follows that  
"/max  1091.1 7 X "/rnax  2.0X %nax. 
This confirms the scale-invariance of the system--doubling the size of the Figure 
results in a doubling of the selected speed. 
3.2 KOFFKA CROSS 
The Kofftm Cross stimulus (Figure 2) has two basic degrees of freedom which we call 
diameter (i.e., d) and arm width (i.e., w) (Figure 3 (a)). We are interested in how 
Emergent Properties of Illusory Contour Shape 835 
(a) 
I 
(b) (-o 5w, o. sd) ( o. 5w, o. sd) 
 (.5d, 05w) t 
(-05w, -0.5d) (0.5w, -05d) 
(c) (d) 
(05d,OSw)  
I 
( 05d ,-0.5w) / 
Figure 3: (a) Koffka Cross showing diameter, d, and width, w. (b) Orientation and 
position constraints in terms of d and w. The normal orientation at each endpoint 
is indicated by the solid lines while the dashed lines represent plus or minus one 
standard deviation (i.e., 12.8 ) of the Gaussian weighting function. (c) Typically 
perceived as square. (d) Typically perceived as circle. The positions of the line 
endpoints is the same. 
the stochastic completion field changes as these parameters are varied. Observers 
report that as the width of the arms increases, the shape of the illusory contour 
changes from a circle to a square[4]. The endpoints of the lines comprising the 
Koffka Cross can be used to define a set of position and orientation constraints 
(Figure 3 (b)). The position constraints are specified in terms of the parameters, d 
and w. The orientation constraints take the form of a Gaussian weighting function 
which assigns higher probabilities to contours passing through the endpoints with 
orientations normal to the lines? The prior probabilities assigned to each position- 
direction pair by the Gaussian weighting function form a diagonal matrix, D: 
,k(7)s("r) = DP(7)Ds(7) = q(7)s(?) 
where P(,) is the transition probability matrix for the random process at scale 
?, ,k('),) is an eigenvalue of Q(,), and s(,) is the corresponding eigenvector. Let 
,kin,x(?) be the largest positive real eigenvalue of Q(,) and let "/,ax be the scale 
where ,kmax("/) is maximized. Then s,ax(',ax), i.e., the eigenvector of 
associated with ,k,ax (',x), is the limiting distribution over all spatial scales. 
First, we used a Koffka Cross where d = 2.0 and w = 0.5 and evaluated ,k,ax (') over 
the velocity interval [8.0 x 1.1-1, 8.0 x 1.1-8] using standard numerical routines. 3 
The function reaches its maximum value at ',ax  8.0 x 1.1-62 (Figure 4 (left)). 
Observe that the completion field due to the eigenvector, Smax(8.0 x 1.1-62), is 
dominated by contours of a predominantly circular shape (Figure 4 (right)). We 
then uniformly scaled the Koffka Cross Figure by a factor of two, i.e., d' - 4.0 and 
2Observe that Figure 3 (c) is perceived as a square while Figure 3 (d) is perceived as a 
circle. Yet the positions of the line endpoints is the same. It follows that the orientations 
of the lines affect the percept. We have chosen to model this dependence through the use 
of a Gaussian weighting function which favors contours passing through the endpoints of 
the lines in the normal direction. It is possible to motivate this based on the statistics of 
natural scenes. The distribution of relative orientations at contour crossings is maximum 
at 90  and drops to nearly zero at 0  and 180 . 
3The parameters defining the distribution of completion shapes were: T = Raa  - 
0.0005, r -- 9.5, p = O'p/T = 100.0 and Rp -- 1.0 x 10 -8. As an anti-aliasing measure, the 
transition probabilities, P(jli), were averaged over initial conditions modeled as Gaussians 
2 2 = 0.00024 and a -- 0.0019. See [6]. 
of variance rr: = o' v 
836 K. K. Thornber and L. R. Williams 
Kof fka Crosses ('two sizes) 
Figure 4: Left: Plot of magnitude of maximum positive real eigenvalue, ,)imax, S. 
log.(1/-y) for Koffka Crosses with d = 2.0 and w = 0.5 (solid) and d = 4.0 and w = 1.0 
(dashed). Right: The completion field due to the eigenvector, Smax(8.0 X 1.1-62). 
W' = 1.0 and plotted Xx('/) over the same interval (Figure 4 (left)). Observe 
that X'ax(8.0 x 1.1 -+7)  X,a(8.0 x 1.1-). As before, this confirms the scale- 
invariance of the system. 
Next, we studied how the relative magnitudes of the local maxima of 
change as the parameter w is varied. We begin with a Koffka Cross where d = 2.0 
and w = 0.5 and observe that l,x('/) has two local maxima (Figure 5 (left)). 
We refer to the larger of these maxima as '/circle. As previously noted, this max- 
imum is located at approximately 8.0 x 1.1-62. The second maximum is located 
at approximately 8.0 x 1.1-32. When the completion field due to the eigenvector, 
s,a(8.0 x 1.1-32), is rendered, we observe that the distribution is dominated by 
contours of predominantly square shape (Figure 5(a)). For this reason, we refer 
to this local maximum as '/square- Now consider a Koffira Cross where the widths 
of the arms are doubled but the diameter remains the same, i.e., d' - 2.0 and 
w' - 1.0. We observe that A'a('/) still has two local maxima, one at approxi- 
mately 8.0 x 1.1-63 and a second at approximately 8.0 x 1.1-29 (Figure 5 (left)). 
When we render the completion fields due to the eigenvectors, s'a (8.0 x 1.1-63) and 
sna(8.0 x 1.1-29), we find that the completion fields have the same general char- 
acter as before--the contours associated with the smaller spatial scale (i.e., lower 
speed) are approximately circular and those associated with the larger spatial scale 
(i.e., higher speed) are approximately square (Figure 5 (d) and (c)). Accordingly, 
we refer to the locations of the respective local maxima as ?' and 
circle '/square' How- 
ever, what is most interesting is that the relative magnitudes of the local maxima 
have reversed. Whereas we previously observed that X,ax ('/circle) > A,ax (7square), 
we now observe that A' (%quare) > A'a (circle)' Therefore, the completion field 
due to the eigenvector, s'(7's uare) [not S',a(f'circle) ] represents the limiting 
q  
distribution over all spatial scales. This is consistent with the transition from circle 
to square reported by human observers when the widths of the arms of the Koffka 
Cross are increased. 
Emergent Properties of Illusory Contour Shape 83 7 
Io 
0 
:30 
0 
0 
o 
Kof fro Crosses (two widths) 
b 
x 
& 1:> 
Figure 5: Plot of magnitude of maximum positive real eigenvalue, Amax, VS. 1ogl.x (1/if) 
for Koffka Crosses with d = 2.0 and w = 0.5 (solid) and d = 2.0 and w = 1.0 (dashed). 
Stochastic completion fields for Koffka Cross due to (a) Smax ("square) is a local optimum 
for w 0.5 (b) sa("/cict) is the global optimum for w 0.5 (c) t t 
= = Sm(%qu) is the 
global optimum for w 1.0 (d)   
: Sraax(square ) is a local optimum for w = 1.0. These 
results are consistent with the circle-to-square transition perceived by human subjects 
when the width of the arms of the Koffka Cross are increased. 
4 CONCLUSION 
We have improved upon a previous model of illusory contour formation by show- 
ing how to compute a scale-invariant distribution of closed contours given position 
constraints alone. We also used our model to explain a previously unexplained 
perceptual effect. 
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