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Chapter 5: Analysis and Results 61Gauss-Bonnet approach [Lin, 1982] makes use of the angle α i at v and two successiveedges. The reduced form of Gauss-Bonnet theorem for polygonal meshes is given interms of the loss of angle as Equation 5.5. − KdA = π −i=An 12 α0i(5.5)Assuming that K is a constant in that neighborhood, Gaussian and mean curvature iscomputed asn−1n−112π−αi||ei|| βii= 0 4 i=0κ = ;H =AA33(5.6)where A is the accumulated area of all the triangles that contain vertex v and || e || iβiisthe measure of angle deviation between the normal at vertex v and its neighbor.Desbrun et al. [Desbrun et al., 1999] reduces the normal deviation as the sum of thecotangents of the angle formed at the neighbor vertex. Taubin [Taubin, 1995] defines asymmetric matrix using the integral formula involving the normal curvature.Assuming that vertex normals at each vertex have been computed, a matrix M v isapproximated with a weighted sum over the neighbor vertices v where T i is the unitlength normalized projection of vector (v i –v) onto the tangent plane at v.n 1= − M v wiκn(Ti)TiTii=0Ti=||t[ I − NvNv] [ v − vi]t[ I − NvN ] [ v − v ]||vit(5.7)(5.8)The weights w i in Equation 5.7 are selected proportional to the sum of surface areas ofthe triangles incident to both vertices v and v i . The matrix M v is restricted to the tangentplane and its Eigen values correspond to the principal values of curvature.We compare different approaches to choose one of these for our CVM algorithm. Wemake our decision based on a few experiments. We have chosen a saddle surface forwhich we can compute the analytical curvature. We show the surface Gaussiancurvature also as a 3D mesh in Figure 5.2 because the variation of curvature along thesurface can be visualized better. We have also presented a simple multi-resolutionexperiment in Figure 5.2. We have sampled the saddle surface so that each surface meshmodel is made up of 161 vertices, 961 vertices and 10000 vertices respectively. Wehave computed curvature based on each of the methods discussed in the previousparagraphs. We observe that the curvature estimate of the Gauss-Bonnets approach and
Chapter 5: Analysis and Results 62Saddle SurfaceAnalyticSurfaceCurvatureGauss- BonnetApproachParaboloidFittingLoss of angle ascurvatureTaubin’sMethodN =121N =961N = 10000Figure 5.2: Curvature analysis – Multi-resolution error analysis experiment withfour different approaches to curvature estimation on triangle meshes.
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To the Graduate Council:I am submit
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Acknowledgementsii“Experience is
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Contents1 INTRODUCTION.............
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List of TablesviTable 2.1:Qualitati
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viiiFigure 5.2: Curvature analysis
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Chapter 1: Introduction 21.1 Motiva
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Chapter 1: Introduction 41.2 Propos
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Chapter 1: Introduction 6We then pe
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Chapter 2: Literature Review 8Shape
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Page 91 and 92: Chapter 6: Conclusions 82complexity
Page 93 and 94: Bibliography 84BIBLIOGRAPHY
Page 95 and 96: Bibliography 86[Biermann, 2001][Bel
Page 97 and 98: Bibliography 88[Cyr and Kimia, 2001
Page 99 and 100: Bibliography 90[Gourley, 1998][Gosh
Page 101 and 102: Bibliography 92[Joshi and Chang, 19
Page 103 and 104: Bibliography 94[Meyer, 2002][Morse,
Page 105 and 106: Bibliography 96[Safar et al., 2000]
Page 107 and 108: Bibliography 98parts,” IEEE Trans
Page 109: Vita 100VITASreenivas Rangan Sukuma
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