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.

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 ( n 3 m ), while that of the previous algorithm was O ( n 4 m 2 (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.