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

Abstract

Abstract. 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.

Combinatorial Preconditioners and Multilevel Solvers for Problems in Computer Vision and Image Processing

“…It is interesting that the Maximally Connected Neighbor (MCN) algorithm proposed in [19] comes very close to the clustering algorithm proposed in [29]. Of course, it is imaginable that there are instances where MCN may not induce bad clusterings.…”

“…The question then is whether the proposed clustering provides the guarantees that by Theorem 2.3 are necessary to construct a good Steiner preconditioner. In the following Figure, we replicate Figure 2 of [19], with a choice of weights that force the depicted clustering. Figure 2 contains exactly one black/coarse node.…”

“…In recent work [19], Grady proposed a multigrid method where the construction of the "coarse" grid follows exactly the construction of the quotient graph in the previous section. Specifically, Grady proposes a clustering such that every cluster contains exactly one of certain pre-specified "coarse" nodes.…”

“…As the resolution of imaging hardware has pushed at the limits of computational feasibility, researchers inevitably arrived at the study of iterative and hybrid solvers. Recently, vision and graphics researchers have developed specialized solvers [44,16,4,31], and heuristic solvers with impressive empirical performance [39,31,21,19]. In either case, the methods place strict requirements on the system, such as unit weight edges or 4-connectivity.…”

Hierarchical Diagonal Blocking and Precision Reduction Applied to Combinatorial Multigrid

“…CMG constructs the restriction operator R i by grouping the variables/nodes of A i into dim(A i+1 ) disjoint clusters and letting R(i, j) = 1 if node i is in cluster j, and R(i, j) = 0 otherwise. This simple approach is known as aggregate-based coarsening, and it has recently attracted significant interest due to its simplicity and advantages for parallel implementations [22,36]. Classic AMG constructs more complicated restriction operators that can be viewed as (partially) overlapping clusters.…”