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Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

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Discrete Differential-Geometry Operators for Triangulated 2-Manifolds Mark Meyer 1 , Mathieu Desbrun 1 , 2 , Peter Schr¨ oder 1 , and Alan H. Barr 1 1 Caltech 2 USC Summary. This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and cur- vatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing for- mulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in ac- curacy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhance- ment, and quality checking, and show results of denoising in higher dimensions, such as for tensor images. 1 Introduction Despite extensive use of triangle meshes in Computer Graphics, there is no consensus on the most appropriate way to estimate simple geometric at- tributes such as normal vectors and curvatures on discrete surfaces. Many surface-oriented applications require an approximation of the first and sec- ond order properties with as much accuracy as possible. This could be done by polynomial reconstruction and analytical evaluation, but this often intro- duces overshooting or unexpected surface behavior between sample points. The triangle mesh is therefore often the only “reliable” approximation of the continuous surface at hand. Unfortunately, since meshes are piecewise linear surfaces, the notion of continuous normal vectors or curvatures is non trivial. It is fundamental to guarantee accuracy in the treatment of discrete sur- faces in many applications. For example, robust curvature estimates are im- portant in the context of mesh simplification to guarantee optimal triangu- lations [HG99]. Even if the quadric error defined in [GH97] measures the Gaussian curvature on an infinitely subdivided mesh, the approximation be- comes rapidly unreliable for sparse sampling. In surface modeling, a number of other techniques are designed to create very smooth surfaces from coarse meshes, and use discrete curvature approximations to measure the quality of
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2 Mark Meyer, Mathieu Desbrun, Peter Schr¨oder, and Alan H. Barr the current approximation (for example, see [MS92]). Accurate curvature nor- mals are also essential to the problem of surface denoising [DMSB99, GSS99] where good estimates of mean curvatures and normals are the key to undis- torted smoothing. More generally, discrete operators satisfying appropriate discrete versions of continuous properties would guarantee reliable numerical behavior for many applications using meshes.
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