This Article 
 Bibliographic References 
 Add to: 
Spherical DCB-Spline Surfaces with Hierarchical and Adaptive Knot Insertion
Aug. 2012 (vol. 18 no. 8)
pp. 1290-1303
Xin Li, Dept. of Electr. & Comput. Eng., Louisiana State Univ., Baton Rouge, LA, USA
Juan Cao, Sch. of Math. Sci., Xiamen Univ., Xiamen, China
Zhonggui Chen, Dept. of Comput. Sci., Xiamen Univ., Xiamen, China
Hong Qin, Dept. of Comput. Sci., Stony Brook Univ., Stony Brook, NY, USA
This paper develops a novel surface fitting scheme for automatically reconstructing a genus-0 object into a continuous parametric spline surface. A key contribution for making such a fitting method both practical and accurate is our spherical generalization of the Delaunay configuration B-spline (DCB-spline), a new non-tensor-product spline. In this framework, we efficiently compute Delaunay configurations on sphere by the union of two planar Delaunay configurations. Also, we develop a hierarchical and adaptive method that progressively improves the fitting quality by new knot-insertion strategies guided by surface geometry and fitting error. Within our framework, a genus-0 model can be converted to a single spherical spline representation whose root mean square error is tightly bounded within a user-specified tolerance. The reconstructed continuous representation has many attractive properties such as global smoothness and no auxiliary knots. We conduct several experiments to demonstrate the efficacy of our new approach for reverse engineering and shape modeling.

[1] J. Hoschek and D. Lasser, Fundamefntals of Computer Aided Geometric Design. A.K. Peters, 1993.
[2] M. Neamtu, "Splines on Surfaces," Handbook of Computer Aided Geometric Design, Chapter 9, pp. 229-253, Elsevier, 2002.
[3] G.E. Fasshauer, "Adaptive Least Squares Fitting with Radial Basis Functions on the Sphere," Mathematical Methods for Curves and Surfaces, pp. 141-150, Vanderbilt Univ. Press, 1995.
[4] T. Lyche and L.L. Schumaker, "A Multiresolution Tensor Spline Method for Fitting Functions on the Sphere," SIAM J. Scientific Computing, vol. 22, no. 2, pp. 724-746, 2000.
[5] R.E. Barnhill, K. Opitz, and H. Pottmann, "Fat Surfaces: A Trivariate Approach to Triangle-Based Interpolation on Surfaces," Computer Aided Geometric Design, vol. 9, no. 5, pp. 365-378, 1992.
[6] W. Freeden, M. Schreiner, and R. Franke, "A Survey on Spherical Spline Approximation," Surveys Math. Industry, vol. 7, pp. 29-85, 1997.
[7] H. Wang, Y. He, X. Li, X. Gu, and H. Qin, "Polycube Splines," Proc. ACM Symp. Solid and Physical Modeling, pp. 241-251, 2007.
[8] X. Gu, Y. He, and H. Qin, "Manifold Splines," Proc. ACM Symp. Solid and Physical Modeling, pp. 27-38, 2005.
[9] S.R. Buss and J.P. Fillmore, "Spherical Averages and Applications to Spherical Splines and Interpolation," ACM Trans. Graphics, vol. 20, no. 2, pp. 95-126, 2001.
[10] G.E. Fasshauer and L.L. Schumaker, "Scattered Data Fitting on the Sphere," Proc. Int'l Conf. Math. Methods for Curves and Surfaces II, pp. 117-166, 1998.
[11] V. Baramidze, M.J. Lai, and C.K. Shum, "Spherical Splines for Data Interpolation and Fitting," SIAM J. Scientific Computing, vol. 28, pp. 241-259, 2005.
[12] P. Alfeld, M. Neamtu, and L.L. Schumaker, "Bernstein-Bézier Polynomials on Spheres and Sphere-Like Surfaces," Computer Aided Geometric Design, vol. 13, pp. 333-349, 1996.
[13] P. Alfeld, M. Neamtu, and L.L. Schumaker, "Fitting Scattered Data on Sphere-Like Surfaces Using Spherical Splines," J. Computational and Applied Math., vol. 73, pp. 5-43, 1996.
[14] M. Neamtu, "Homogeneous Simplex Splines," J. Computational and Applied Math., vol. 73, pp. 1-2, 1996.
[15] M.-J. Lai and L.L. Schumaker, Spline Functions on Triangulations. Cambridge Univ. Press, 2007.
[16] G. Greiner and H.-P. Seidel, "Modeling with Triangular B-Splines," IEEE Computer Graphics and Applications, vol. 14, pp. 211-220, 1993.
[17] M.G.J. Franssen, "Evaluation of DMS-Splines," master's thesis, Eindhoven Univ. of Tech nology, 1995.
[18] C. de Boor, "On Calculating with B-Splines," J. Approximation Theory, vol. 6, pp. 50-62, 1972.
[19] R. Pfeifle and H.-P. Seidel, "Fitting Triangular B-Splines to Functional Scattered Data," Proc. Graphics Interface, pp. 26-33, 1995.
[20] H. Qin and D. Terzopoulos, "Triangular NURBS and Their Dynamic Generalizations," Computer Aided Geometric Design, vol. 14, no. 4, pp. 325-347, 1997.
[21] S. Han and G. Medioni, "Triangular NURBS Surface Modeling of Scattered Data," Proc. Seventh Conf. Visualization, pp. 295-302, 1996.
[22] Y. He and H. Qin, "Surface Reconstruction with Triangular B-Splines," Proc. Geometric Modeling and Processing, pp. 279-290, 2004.
[23] Y. He, X. Gu, and H. Qin, "Automatic Shape Control of Triangular B-Splines of Arbitrary Topology," J. Computer Science and Technology, vol. 21, pp. 232-237, 2006.
[24] R. Pfeifle and H.-P. Seidel, "Spherical Triangular B-Splines with Application to Data Fitting," Proc. Eurographics, pp. 89-96, 1995.
[25] Y. He, X. Gu, and H. Qin, "Rational Spherical Splines for Genus Zero Shape Modeling," Proc. Int'l Conf. Shape Modeling and Applications, pp. 82-91, 2005.
[26] R. Gormaz, "B-Spline Knot-Line Elimination and Bézier Continuity Conditions," Proc. Int'l Conf. Curves and Surfaces in Geometric Design, pp. 209-216, 1994.
[27] Y. He, X. Gu, and H. Qin, "Fairing Triangular B-Splines of Arbitrary Topology," Proc. Pacific Graphics, pp. 153-156, 2005.
[28] M. Neamtu, "What Is the Natural Generalization of Univariate Splines to Higher Dimensions?," Mathematical Methods for Curves and Surfaces, Vanderbilt Univ., 2001.
[29] M. Neamtu, "Bivariate Simplex B-Splines: A New Paradigm," SCCG '01: Proc. 17th Spring Conf. Computer Graphics, pp. 71-78, 2001.
[30] C. de Boor, "The Way Things Were in Multivariate Splines: A Personal View," Mulitscale, Nonlinear and Adaptive Approximation, pp. 10-37, Springer, 2009.
[31] J. Cao, X. Li, G. Wang, and H. Qin, "Surface Reconstruction Using Bivariate Simplex Splines on Delaunay Configurations," Computer and Graphics, vol. 33, no. 3, pp. 341-350, 2009.
[32] B. Dembart, D. Gonsor, and M. Neamtu, "Bivariate Quadratic B-Splines Used as Basis Functions for Data Fitting," Mathematics for Industry: Challenges and Frontiers 2003. A Process View: Practice and Theory, SIAM, pp. 178-198, 2005.
[33] J.-D. Boissonnat, O. Devillers, and M. Teillaud, "A Semidynamic Construction of Higher-Order Voronoi Diagrams and Its Randomized Analysis," Algorithmica, vol. 9, pp. 329-356, 1993.
[34] A. Okabe, B. Boots, K. Sugihara, and S.N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, second ed. Wiley, 2000.
[35] F.P. Preparata and M.I. Shamos, Computational Geometry: An Introduction. Springer-Verlag, 1985.
[36] H.-S. Na, C.-N. Lee, and O. Cheong, "Voronoi Diagrams on the Sphere," Computational Geometry, vol. 23, no. 2, pp. 183-194, 2002.
[37] J.L. Brown and A.J. Worsey, "Problems with Defining Barycentric Coordinates for the Sphere," Math. Modelling and Numerical Analysis, vol. 26, pp. 37-49, 1992.
[38] W. Li, S. Xu, G. Zhao, and L.P. Goh, "Adaptive Knot Placement in B-Spline Curve Approximation," Computer-Aided Design, vol. 37, no. 8, pp. 791-797, 2005.
[39] C. Gotsman, X. Gu, and A. Sheffer, "Fundamentals of Spherical Parameterization for 3D Meshes," Proc. ACM SIGGRAPH, pp. 358-363, 2003.
[40] X. Gu and S.-T. Yau, "Global Conformal Surface Parameterization," Proc. Symp. Geometry Processing, pp. 127-137, 2003.
[41] M. Quicken, C. Brechbuhler, J. Hug, H. Blattmann, and G. Szekely, "Parameterization of Closed Surfaces for Parametric Surface Description," Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 354-360, 2000.
[42] E. Praun and H. Hoppe, "Spherical Parametrization and Remeshing," Proc. ACM SIGGRAPH '03, pp. 340-349, 2003.
[43] R. Zayer, C. Rössl, and H.-P. Seidel, "Curvilinear Spherical Parameterization," Proc. Int'l Conf. Shape Modeling and Applications, pp. 57-64, 2006.
[44] P. Degener, J. Meseth, and R. Klein, "An Adaptable Surface Parameterization Method," Proc. Int'l Meshing Roundtable, pp. 201-213, 2003.
[45] K. Hormann and G. Greiner, "Mips: An Efficient Global Parametrization Method." Curve and Surface Design, pp. 153-162, Vanderbilt Univ. Press, 2000.
[46] H. Hoppe, "Progressive Meshes," Proc. ACM SIGGRAPH, pp. 99-108, 1996.
[47] P.V. Sander, J. Snyder, S.J. Gortler, and H. Hoppe, "Texture Mapping Progressive Meshes," Proc. ACM SIGGRAPH, pp. 409-416, 2001.
[48] Y. Lai, S.-H. Hu, and H. Pottmann, "Surface Fitting Based on a Feature Sensitive Spatial Tessellations: Concepts and Applications Parametrization," Computer-Aided Design, vol. 38, no. 7, pp. 800-807, 2006.
[49] D. Cohen-Steiner and J.-M. Morvan, "Restricted Delaunay Triangulations and Normal Cycle," Proc. Symp. Computational Geometry, pp. 312-321, 2003.
[50] G. Taubin, "Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation," Proc. Int'l Conf. Computer Vision (ICCV), pp. 902-907, 1995.
[51] M. Meyer, M. Desbrun, P. Schröder, and A.H. Barr, "Discrete Differential-Geometry Operators for Triangulated 2-Manifolds," Visualization and Mathematics III, H.-C. Hege and K. Polthier, eds., pp. 35-57, Springer-Verlag, 2003.
[52] Q. Du, M.D. Gunzburger, and L. Ju, "Constrained Centroidal Voronoi Tessellations for Surfaces," SIAM J. Scientific Computing, vol. 24, no. 5, pp. 1488-1506, 2002.
[53] S. Lloyd, "Least Squares Quantization in PCM," IEEE Trans. Information Theory, vol. 28, no. 2, pp. 129-137, Jan. 1982.
[54] Y. Liu, W. Wang, B. Lévy, F. Sun, D.-M. Yan, L. Lu, and C. Yang, "On Centroidal Voronoi Tessellation—Energy Smoothness and Fast Computation," ACM Trans. Graphics, vol. 28, no. 4, pp. 1-17, 2009.
[55] D.-M. Yan, B. Lévy, Y. Liu, F. Sun, and W. Wang, "Isotropic Remeshing with Fast and Exact Computation of Restricted Voronoi Diagram," Computer Graphic Forum, vol. 28, no. 5, pp. 1445-1454, 2009.
[56] A. Gersho, "Asymptotically Optimal Block Quantization," IEEE Trans. Information Theory, vol. IT-25, no. 4, pp. 373-380, July 1979.

Index Terms:
tensors,computational geometry,least mean squares methods,mesh generation,splines (mathematics),surface fitting,hierarchical knot insertion,spherical DCB-spline surfaces,adaptive knot insertion,novel surface fitting scheme,continuous parametric spline surface,Delaunay configuration B-spline,nontensor-product spline,surface geometry,genus-0 model,spherical spline representation,reconstructed continuous representation,reverse engineering,shape modeling,root mean square error,Splines (mathematics),Surface reconstruction,Polynomials,Approximation methods,Surface treatment,Electronic mail,Image reconstruction,non-tensor-product B-splines.,Delaunay configurations,spherical splines,knot placement,knot insertion
Xin Li, Juan Cao, Zhonggui Chen, Hong Qin, "Spherical DCB-Spline Surfaces with Hierarchical and Adaptive Knot Insertion," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 8, pp. 1290-1303, Aug. 2012, doi:10.1109/TVCG.2011.156
Usage of this product signifies your acceptance of the Terms of Use.