Local enhancement and denoising algorithms on arbitrary mesh surfaces

In the process of analyzing the surfaces of 3d scanned objects, it is desirable to perform per-vertex calculations on a region of connected vertices, much in the same way that 2d image filters perform per-pixel calculations on a window of adjacent pixels. Operations such as blurring, averaging, and noise reduction would be useful for these applications, and are already well-established in 2d image enhancement. In this paper, we present a method for adapting simple windowed 2d image processing operations to the problem domain of 3d mesh surfaces. The primary obstacle is that mesh surfaces are usually not flat, and their vertices are usually not arranged in a grid, so adapting the 2d algorithms requires a change of analytical models. First we characterize 2d rectangular arrays as a special case of a graph, with edges between adjacent pixels. Next we treat filter windows as a limitation on the walks from a given source node to every other reachable node in the graph. We tested the common windowed average, weighted average, and median operations. We used 3d meshes comprised of sets of vertices and polygons, modeled as weighted undirected graphs. The edge weights are taken as the Euclidean distance between two vertices, calculated from their XYZ coordinates in the usual way. Our method successfully provides a new way to utilize these existing 2d filters. In addition, further generalizations and applications are discussed, including potential applications in any field that uses graph theory, such as social networking, marketing, telecom networks, epidemiology, and others.