Digital Geometry Processing

Digital Geometry, large polygonal meshes coming from digitizing of complex geometry, is the fourth wave of multimedia after sound, images, and video. The basic idea behind digital geometry processing is to bring the entire suite of standard signal processing algorithms such as editing, filtering, and compression to digital geometry. This is challenging because unlike sound, images, and video, geometry is not defined on a Euclidean space and traditional Fourier based techniques no longer apply. Instead we propose a new paradigm based on so-called semi-regular meshes formed by recursive subdivision and local displacements. We show how parameterizations can be used to build semi-regular meshes, how they are almost perfectly suited for compression, and how they can be used in editing and filtering operations. We discuss extensions which deal with sets of meshes and with topology changes. Finally we show some reent theoretical results on regularity, stability, and approximation quality.