Discrete surface signal processing: The polygon as the surface element

Abstract

Smooth surfaces are approximated by polyhedral surfaces both for display, and for other computational purposes. This is probably the most common surface representation in computer graphics, because polygons are what the current generation of graphics hardware can display efficiently. In computer vision this representation has not been very popular for a long time, probably because it was not seen as very appropriate for object recognition and positioning applications. But it is now being used in more applications, particularly in the medical domain, where polyhedral surface approximations of unknown surfaces are routinely generated with iso-surface construction algorithms. We look at polyhedral surfaces as a generalization of digital images, where the polygon takes the place of the pixel. The polygon is the fundamental surface element. Digital images are functions defined at the nodes of a regular rectangular grid. Discrete surface signals are functions defined at the vertices of a polyhedral surface of arbitrary topology. Signal processing operations, and Fourier analysis in particular, are the fundamental tools of low level computer vision. The lack of regularity and self-similarity of arbitrary polyhedral surface meshes, as opposed to the regular rectangular grids of digital images, complicates the analysis and processing of digital surface signals. Nevertheless, in this paper we generalize Fourier analysis to discrete surface signals. As a first application of this theory we consider the problem of surface smoothing, which corresponds to low-pass filtering within this framework. As in the classical cases of one-dimensional signals, and of digital images, the analysis of discrete surface signals is reduced to matrix analysis, and matrix computation techniques are used to achieve fast discrete surface signal processing operations. We intend to continue along this line of research in the near future, extending higher level computer vision operations to arbitrary polyhedral surfaces.