LOCAL PARALLEL COMPUTATION OF STOCHASTIC COMPLETION FIELDS

We describe a local parallel method for computing the stochastic compl etion field introduced in the previous article (Williams and Jacobs, 1 997). The stochastic completion field represents the likelihood that a completion joining two contour fragments passes through any given pos ition and orientation in the image plane. It is based on the assumptio n that the prior probability distribution of completion shape can be m odeled as a random walk in a lattice of discrete positions and orienta tions. The local parallel method can be interpreted as a stable finite difference scheme for solving the underlying Fokker-Planck equation i dentified by Mumford (1994). The resulting algorithm is significantly faster than the previously employed method, which relied on convolutio n with large-kernel filters computed by;Monte Carlo simulation. The co mplexity of the new method is O(n(3)m), while that of the previous alg orithm was O(n(4)m(2)) (for an n x n image with m discrete orientation s). Perhaps most significant, the use of a local method allows us to m odel 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 brightne ss structure. Finally, we note that the new method is more plausible a s a neural model since (1) unlike the previous method, it can be compu ted in a sparse, locally connected network, and (2) the network dynami cs are consistent with psychophysical measurements of the time course of illusory contour formation.