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Unformatted text preview: Encoding Meshes in Differential Coordinates Daniel Cohen-Or Olga Sorkine School of Computer Science Tel Aviv University Abstract Representing surfaces in local, rather than global, coordinate sys- tems proves to be useful for various geometry processing applica- tions. In particular, we have been investigating surface representa- tions based on differential coordinates, constructed using the Lapla- cian operator and discrete forms. Unlike global Cartesian coordi- nates, that only represent the spatial locations of points on the sur- face, differential coordinates capture the local surface details which greatly affect the shading of the surface and thus its visual appear- ance. On polygonal meshes, differential coordinates and the dis- crete mesh Laplacian operator provide an efficient linear surface re- construction framework suitable for various mesh processing tasks. In this paper we discuss the important properties of differential co- ordinates and show their applications for surface reconstruction. In particular, we discuss quantization of the differential coordinates, Least-squares meshes and mesh editing. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object ModelingCurve, surface, solid, and object representations Keywords: differential coordinates, Laplacian, least-squares meshes, mesh editing 1 Introduction Differential encoding for mesh deformation has been an active re- search area in recent years. Early forms of differential encoding of meshes have been used for mesh compression. Using predictive schemes for encoding mesh geometry, such as the parallelogram rule, a vertex is encoded as the difference between the predicted position and its actual position. Better schemes are multi-way, that is, the prediction is based on an average of several predictions from different directions. The prediction based on the Laplacian opera- tor takes into account all possible directions and thus yields better prediction than schemes based on a single way or just a few (see Table 2 in [Sorkine et al. 2003]). The Laplacian operator will be described below. As we shall see, it has been proved to be more than a prediction scheme per se. The Laplacian operator has a nice, well studied mathematical formula- tion, for which we can apply spectral analysis and understand its be- havior. With certain extension, the Laplacian operator allows to de- fine discrete bases for surface geometry that are tailored to represent specific geometric features [Sorkine et al. 2005]. The Laplacian op- erator has lead to various developments in several applications, like mesh encoding, mesh deformation and mesh manipulation. 2 Mesh encoding Polygonal meshes and triangular meshes in particular are common- place and the de-facto standard for representing surfaces in com- puter graphics. The representation of a mesh M consists of its geometry and its topology. The geometry is a list of n Cartesian coordinates encoding the position of n vertices. The topology en-vertices....
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This note was uploaded on 05/20/2011 for the course CAP 6701 taught by Professor Staff during the Spring '08 term at University of Florida.
- Spring '08