Computing stochastic completion fields in linear-time using a resolution pyramid

Abstract

We describe a linear-time algorithm for computing the likelihood that a completion joining two contour fragments passes through any given position and orientation in the image plane. Our algorithm is a resolution pyramid based method for solving a partial differential equation characterizing a distribution of short, smooth completion shapes. The PDE consists of a set of independent advection equations in (x, y) coupled in the 0 dimension by the diffusion equation. A previously described algorithm used a first-order, explicit finite difference scheme implemented on a rectangular grid. This algorithm required O(n3m) time for a grid of size n × n with m discrete orientations. Unfortunately, systematic error in solving the advection equations produced unwanted anisotropic smoothing in the (x, y) dimension. This resulted in visible artifacts in the completion fields. The amount of error and its dependence on 0 has been previously characterized. We observe that by careful addition of extra spatial smoothing, the error can be made totally isotropic. The combined effect of this error and of intrinsic smoothness due to diffusion in the 0 dimension is that the solution becomes smoother with increasing time, i.e., the high spatial frequencies drop out. By increasing Ax and At on a regular schedule, and using a second-order, implicit scheme for the diffusion term, it is possible to solve the modified PDE in O(n2m) time, i.e., time linear in the problem size. Using current hardware and for problems of typical size, this means that a solution which previously took one hour to compute can now be computed in about two minutes.