A Lattice-Preserving Multigrid Method for Solving the Inhomogeneous Poisson Equations Used in Image Analysis

The inhomogeneous Poisson (Laplace) equation with internal Dirichlet boundary conditions has recently appeared in several applications ranging from image segmentation [1–3] to image colorization [4], digital photo matting [5, 6] and image filtering [7, 8]. In addition, the problem we address may also be considered as the generalized eigenvector problem associated with Normalized Cuts [9], the linearized anisotropic diffusion problem [10, 11, 8] solved with a backward Euler method, visual surface reconstruction with discontinuities [12, 13] or optical flow [14]. Although these approaches have demonstrated quality results, the computational burden of finding a solution requires an efficient solver. Design of an efficient multigrid solver is difficult for these problems due to unpredictable inhomogeneity in the equation coefficients and internal Dirichlet boundary conditions with unpredictable location and value. Previous approaches to multigrid solvers have typically employed either a data-driven operator (with fast convergence) or the maintenance of a lattice structure at coarse levels (with low memory overhead). In addition to memory efficiency, a lattice structure at coarse levels is also essential to taking advantage of the power of a GPU implementation [15, 16, 5, 3]. In this work, we present a multigrid method that maintains the low memory overhead (and GPU suitability) associated with a regular lattice while benefiting from the fast convergence of a data-driven coarse operator.