A heat flow based relaxation scheme for n dimensional discrete hyper surfaces. (April 2018) - Livesu, Marco
Highlights: A diffusion based operator for discrete hyper-surfaces. Independent on the specific discretization choice. Independent from the dimensionality of the dataset. Fast and easy to use (no complex parameter tuning). Graphical abstract: Abstract: 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.