Local parallel computation of stochastic completion fields

We describe a local parallel method for computing the stochastic completion field introduced in an earlier paper Williams and Jacobs (1995). 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 upon 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 Of(n³m) while that of the previous algorithm was 0(n⁴m²) (for an n x n image with m discrete orientations). Perhaps most significantly, the use of a local method allows us to model the probability distribution of completion shape using stochastic processes which are neither homogenous nor isotropic