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Chapter 5: Analysis and Results 63loss of angle approach are two methods that give us good estimates of Gaussiancurvature in comparison with the paraboloid and Taubin methods. We also observe thatas the resolution of the data increases Taubin’s method also improves drastically. Theparaboloid fitting method appears to have the sampled version of the analyticalcurvature. Since we have fit an analytical surface at each vertex to compute curvaturearound it, the error in this method seems to have accumulated throughout the mesh.We next performed the curvature analysis on the unit sphere whose Gaussian curvatureestimate should be equal to the reciprocal of the radius. We show how each one of thesemethods has behaved with the sphere at different resolutions in Figure 5.3. We wouldlike to reiterate the large error in the paraboloid fitting methods at low resolutions.Since we are interested in a scheme that is consistent at all resolutions we need tomake a choice between Gauss-Bonnet, loss of angle and Taubin’s methods.We have created synthetic surfaces such as the spherical cup, saddle and a monkeysaddle. Visually and analytically the monkey saddle surface has the maximum variationin curvature. We decided to choose the method that categorically shows the variation.We call the variation as the span for curvature and plot it against each surface for thefour methods in Figure 5.4. We conclude that Taubin and Gauss-Bonnet’s approach forcurvature estimation yields accurate results. We have used Taubin’s method to computeprincipal curvatures and the Gauss-Bonnet approach for the Gaussian curvature for ourimplementation.We have combined the simplicity of the Harvard mesh library (written by X. Gu) andspeed of Triangle mesh library (written by Michael Roy) for our triangle meshprocessing. Both the libraries are open source implementations of the half-edge datastructure in C++. We have used the Microsoft Developer Environment (MicrosoftVisual C++7.0) as our programming platform. For graphs and plots however we haveused MATLAB.5.1.2 Density Estimation for Information MeasureWe would like to document our experience with the bandwidth optimization methods.Before incorporating it into our algorithm we have used the MATLAB(implementation of Christian Beardah’s) toolbox on kernel density estimation. Withground truth normal density, we have concluded that cross validation methods give usaccurate results. We have compared least squares cross validation, smoothed crossvalidation, likelihood cross validation, biased cross validation, distribution scalemethods and the plug-in method. With large data cross validation though accurate wasthe most time consuming. Cross validation is a O (N 2 ) complex algorithm in the worstcase and had convergence problems with our real data. Sometimes cross validationmethods result in monotonic cost functions that output the lower limit as the optimalbandwidth. We use the plug-in method. The plug-in method is a multi pass paradigm
Chapter 5: Analysis and Results 64Figure 5.3: Curvature analysis – Error in curvature of a sphere at multipleresolutions.
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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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Chapter 2: Literature Review 102.2.
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Page 59 and 60: Chapter 4: Algorithm Overview 50∞
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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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