Lance R. Williams

Stochastic completion fields: a neural model of illusory contour shape and salience

We describe an algorithm and representation-level theory of illusory contour shape and salience. Unlike previous theories, our model is derived from a single assumption: that the prior probability distribution of boundary completion shape can be modeled by a random walk in a lattice whose points are positions and orientations in the image plane (i.e., the space that one can reasonably assume is represented by neurons of the mammalian visual cortex). Our model does not employ numerical relaxation or other explicit minimization, but instead relies on the fact that the probability that a particle following a random walk will pass through a given position and orientation on a path joining two boundary fragments can be computed directly as the product of two vector-field convolutions. We show that for the random walk we define, the maximum likelihood paths are curves of least energy, that is, on average, random walks follow paths commonly assumed to model the shape of illusory contours. A computer model is demonstrated on numerous illusory contour stimuli from the literature.

Segmentation of multiple salient closed contours from real images

Using a saliency measure based on the global property of contour closure, we have developed a segmentation method which identifies smooth closed contours bounding objects of unknown shape in real images. The saliency measure incorporates the Gestalt principles of proximity and good continuity that previous methods have also exploited. Unlike previous methods, we incorporate contour closure by finding the eigenvector with the largest positive real eigenvalue of a transition matrix for a Markov process where edges from the image serve as states. Element (i, j) of the transition matrix is the conditional probability that a contour which contains edge j will also contain edge i. We show how the saliency measure, defined for individual edges, can be used to derive a saliency relation, defined for pairs of edges, and further show that strongly-connected components of the graph representing the saliency relation correspond to smooth closed contours in the image. Finally, we report for the first time, results on large real images for which segmentation takes an average of about 10 seconds per object on a general-purpose workstation.

A Comparison of Measures for Detecting Natural Shapes in Cluttered Backgrounds

We propose a new measure of perceptual saliency and quantitatively compare its ability to detect natural shapes in cluttered backgrounds to five previously proposed measures. As defined in the new measure, the saliency of an edge is the fraction of closed random walks which contain that edge. The transition-probability matrix defining the random walk between edges is based on a distribution of natural shapes modeled by a stochastic motion. Each of the saliency measures in our comparison is a function of a set of affinity values assigned to pairs of edges. Although the authors of each measure define the affinity between a pair of edges somewhat differently, all incorporate the Gestalt principles of good-continuation and proximity in some form. In order to make the comparison meaningful, we use a single definition of affinity and focus instead on the performance of the different functions for combining affinity values. The primary performance criterion is accuracy. We compute false-positive rates in classifying edges as signal or noise for a large set of test figures. In almost every case, the new measure significantly outperforms previous measures.

Perceptual Completion of Occluded Surfaces

Researchers in computer vision have primarily studied the problem of visual reconstruction of environmental structure that is plainly visible. In this paper, the conventional goals of visual reconstruction are generalized to include both visible and occluded forward facing surfaces. This larger fraction of the environment is termed theanterior surfaces. Because multiple anterior surface neighborhoods project onto a single image neighborhood wherever surfaces overlap, surface neighborhoods and image neighborhoods are not guaranteed to be in one-to-one correspondence, as conventional “shape-from” methods assume. The result is that the topology of three-dimensional scene structure can no longer be taken for granted, but must be inferred from evidence provided by image contours. In this paper, we show that the boundaries of the anterior surfaces can be represented in viewer-centered coordinates as alabeled knot-diagram. Where boundaries are not occluded and where surface reflectance is distinct from that of the background, boundaries will be marked by image contours. However, where boundaries are occluded, or where surface reflectance matches background reflectance, there will be no detectable luminance change in the image. Deducing the complete image trace of the boundaries of the anterior surfaces under these circumstances is called thefigural completionproblem. The second half of this paper describes a computational theory of figural completion. In more concrete terms, the problem of computing a labeled knot-diagram representing an anterior scene from a set of contour fragments representing image luminance boundaries is investigated. A working model is demonstrated on a variety of illusory contour displays. The experimental system employs a two-stage process of completion hypothesis and combinatorial optimization. The labeling scheme is enforced by a system of integer linear inequalities so that the final organization is the optimal feasible solution of an integer linear program.

Local parallel computation of stochastic completion fields

We describe a local parallel method for computing the stochastic completion field introduced in the previous article (Williams and Jacobs, 1997). The stochastic completion field represents the likelihood that a completion joining two contour fragments passes through any given position and orientation in the image plane. It is based on the assumption that the prior probability distribution of completion shape can be modeled as a random walk in a lattice of discrete positions and orientations. The local parallel method can be interpreted as a stable finite difference scheme for solving the underlying Fokker-Planck equation identified by Mumford (1994). The resulting algorithm is significantly faster than the previously employed method, which relied on convolution with large-kernel filters computed by Monte Carlo simulation. The complexity of the new method is O (n3m), while that of the previous algorithm was O(n4m2 (for an n n image with m discrete orientations). Perhaps most significant, the use of a local method allows us to model the probability distribution of completion shape using stochastic processes that are neither homogeneous nor isotropic. For example, it is possible to modulate particle decay rate by a directional function of local image brightnesses (i.e., anisotropic decay). The effect is that illusory contours can be made to respect the local image brightness structure. Finally, we note that the new method is more plausible as a neural model since (1) unlike the previous method, it can be computed in a sparse, locally connected network, and (2) the network dynamics are consistent with psychophysical measurements of the time course of illusory contour formation.

Multiple target tracking with lazy background subtraction and connected components analysis

Background subtraction, binary morphology, and connected components analysis are the first processing steps in many vision-based tracking applications. Although background subtraction has been the subject of much research, it is typically treated as a stand-alone process, dissociated from the subsequent phases of object recognition and tracking. This paper presents a method for decreasing computational cost in visual tracking systems by using track state estimates to direct and constrain image segmentation via background subtraction and connected components analysis. We also present a multiple target tracking application that uses the technique to achieve a large reduction in computation costs.

Segmentation of salient closed contours from real images

Using a saliency measure based on the global property of contour closure, we have developed a method that reliably segments out salient contours bounding unknown objects from real edge images. The measure also incorporates the Gestalt principles of proximity and smooth continuity that previous methods have exploited. Unlike previous measures, we incorporate contour closure by finding the eigen-solution associated with a stochastic process that models the distribution of contours passing through edges in the scene. The segmentation algorithm utilizes the saliency measure to identify multiple closed contours by finding strongly-connected components on an induced graph. The determination of strongly-connected components is a direct consequence of the property of closure. We report for the first time, results on large real images for which segmentation takes an average of about 10 secs per object on a general-purpose workstation. The segmentation is made efficient for such large images by exploiting the inherent symmetry in the task.

Perceptual organization of occluding contours

The mechanics of occlusion of one surface by another are described by a set of integer linear constraints. These constraints insure that the output of a contour grouping process is physically valid and consistent with the image evidence. Among the many feasible solutions, the most compelling is the solution which best explains the presence and form of image structure. The problem of computing a complete and consistent surface boundary representation is reduced to solving an integer linear program.<<ETX>>

Topological Reconstruction of a Smooth Manifold-Solid from Its OccludingContour

This paper describes a simple construction for building a combinatorial model of a smooth manifold-solid from a labeled-figure representing its occluding contour. The motivation is twofold. First, deriving the combinatorial model is an essential intermediate step in the visual reconstruction of solid-shape from image contours. A description of solid-shape consists of a metric and a topological component. Both are necessary: the metric component specifies how the topological component is embedded in three-dimensional space. The paneling construction described in this paper is a procedure for generating the topological component from a labeled-figure representing the occluding contour. Second, the existence of this construction establishes the sufficiency of a labeling scheme for line-drawings of smooth solid-objects originally proposed by Huffman (1971). By sufficiency, it is meant that every set of closed plane-curves satisfying this labeling scheme is shown to correspond to a generic view of a manifold-solid. Together with the Whitney theorem (Whitney, 1955), this confirms that Huffmans labeling scheme correctly distinguishes possible from impossible smooth solid-objects.

Orientation, Scale, and Discontinuity as Emergent Properties of Illusory Contour Shape

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 scaleinvariant. In this article, 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.