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.