A heat flow based relaxation scheme for n dimensional discrete hyper surfaces

We consider the problem of relaxing a discrete (n-1) dimensional hyper surface defining the boundary between two adjacent n dimensional regions in a discrete segmentation. This problem often occurs in computer graphics and vision, where objects are represented by discrete entities such as pixel/voxel grids or polygonal/polyhedral meshes, and the resulting boundaries often expose a typical jagged behavior. We propose a relaxation scheme that replaces the original boundary with a smoother version of it, defined as the level set of a continuous function. The problem has already been considered in recent years, but current methods are specifically designed to relax curves on triangulated discrete 2-manifolds embedded in R^3 and do not clearly scale to multiple discrete representations or to higher dimensions. Our biggest contribution is a smoothing operator entirely based on three canonical differential operators: namely the Laplacian, gradient and divergence. These operators are ubiquitous in applied mathematics, are available for a variety of discretization choices, and exist in any dimension. To the best of the author's knowledge, this is the first intrinsically dimension-independent method, and can be used to relax curves on 2-manifolds, surfaces in R^3 or even hyper-surfaces in R^n. We demonstrate our method on a variety of discrete entities, spanning from triangular, quadrilateral and polygonal surfaces, to solid tetrahedral meshes.