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Approximating Digital 3D Shapes by Rational Gaussian Surfaces
January-March 2003 (vol. 9 no. 1)
pp. 56-69

Abstract—A method for approximating spherical topology digital shapes by rational Gaussian (RaG) surfaces is presented. Points in a shape are parametrized by approximating the shape with a triangular mesh, determining parameter coordinates at mesh vertices, and finding parameter coordinates at shape points from interpolation of parameter coordinates at mesh vertices. Knowing the locations and parameter coordinates of the shape points, the control points of a RaG surface are determined to approximate the shape with a required accuracy. The process starts from a small set of control points and gradually increases the control points until the error between the surface and the digital shape reduces to a required tolerance. Both triangulation and surface approximation proceed from coarse to fine. Therefore, the method is particularly suitable for multiresolution creation and transmission of digital shapes over the Internet. Application of the proposed method in editing of 3D shapes is demonstrated.

[1] E. Bardinet, L.D. Cohen, and N. Ayache, “A Parametric Deformable Model to Fit Unstructured 3D Data,” Computer Vision and Image Understanding, vol. 71, no. 1, pp. 39-54, 1998.
[2] J. Barhak and A. Fischer, “Parametrization and Reconstruction from 3D Points Based on Neural Network and PDE Techniques,” IEEE Trans. Visualization and Computer Graphics, vol. 7, no. 1, pp. 1-16, Jan.-Mar. 2001.
[3] J. Bezdek, L. Hall, and L. Clarke, “Review of MR Image Segmentation Techniques Using Pattern Recognition,” Med. Phys., vol. 20, pp. 233-260, 1993.
[4] W. Boehm and G. Farin, “Concerning Subdivision of Bézier Triangles,” Computer Aided Design, vol. 15, no. 5, pp. 260-261, 1983.
[5] M. Bomans, K.-H. Höhne, U. Tiede, and M. Riemer, “3D Segmentation of MR Images of the Head for 3D Display,” IEEE Trans. Medical Imaging, vol. 9, no. 2, pp. 177-183, 1990.
[6] C. Brechbühler, G. Gerig, and O. Kübler, “Parametrization of Closed Surfaces for 3D Shape Description,” Computer Vision and Image Understanding, vol. 61, no. 2, pp. 154-170, 1995.
[7] M. Brejl and M. Sonka, “Directional 3D Edge Detection in Anisotropic Data: Detector Design and Performance Assessment,” Computer Vision and Image Understanding, vol. 77, no. 2, pp. 84-110, 2000.
[8] J.C. Carr, R.K. Beatson, J.B. Cherrie, T.J. Mitchell, W.R. Fright, B.C. McCallum, and T.R. Evans, “Reconstruction and Representation of 3D Objects with Radial Basis Functions,” Proc. SIGGRAPGH, pp. 67-76, 2001.
[9] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer Aided Design, vol. 10, pp. 350-355, 1978.
[10] I. Cohen and L.D. Cohen, “A Hybrid Hyperquadric Model for 2-D and 3-D Data Fitting,” Computer Vision and Image Understanding, vol. 63, no. 3, pp. 527-541, 1996.
[11] D. Cohen-Or, D. Levin, and O. Remez, “Progressive Compression of Arbitrary Triangular Meshes,” Proc. IEEE Conf. Visualization, pp. 67-72, Oct. 1999.
[12] M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, “Multiresolution Analysis of Arbitrary Meshes,” Proc. SIGGRAPH, pp. 173-182, 1995.
[13] G. Farin, “Triangular Bernstein-Bézier Patches,” Computer Aided Geometric Design, vol. 3, no. 2, pp. 83-128, 1986.
[14] M.S. Floater, “Parametrization and Smooth Approximation of Surface Triangulations,” Computer Aided Geometric Design, vol. 14, pp. 231-250, 1997.
[15] M.S. Floater and M. Reimers, “Meshless Parametrization and Surface Reconstruction,” Computer Aided Geometric Design, vol. 18, pp. 77-92, 2001.
[16] J.M. Galvez and M. Canton, “Normalization and Shape Recognition of Three-Dimensional Objects by 3D Moments,” Pattern Recognition, vol. 26, no. 5, pp. 667-682, 1993.
[17] A. Goshtasby, D. Turner, and L. Ackerman, “Matching of Tomographic Image Slices for Interpolation,” IEEE Trans. Medical Imaging, vol. 11, no. 4, pp. 507-516, 1992.
[18] A. Goshtasby, “Design and Recovery of 2-D and 3-D Shapes Using Rational Gaussian Curves and Surfaces,” Int'l J. Computer Vision, vol. 10, no. 3, pp. 233-256, 1993.
[19] A. Goshtasby, “Geometric Modeling Using Rational Gaussian Curves and Surfaces,” Computer Aided Design, vol. 27, no. 5, pp. 363-375, 1995.
[20] J. Gregory and P. Charrot, “A $\big. C^1\bigr.$ Triangular Interpolation Patch for Computer-Aided Geometric Design,” Computer Graphics and Image Processing, vol. 13, no. 1, pp. 80-87, 1980.
[21] I. Guskov, K. Vidimce, W. Sweldens, and P. Schroder, “Normal Meshes,” Proc. SIGGRAPH, pp. 95-102, 2000.
[22] A.J. Hanson, “Hyperquadrics: Smooth Deformable Shapes with Convex Polyhedral Bounds,” Computer Vision, Graphics, and Image Processing, vol. 44, pp. 191-210,. 1988.
[23] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Surface Reconstruction from Unorganized Points,” Computer Graphics, vol. 26, no. 2, pp. 71-78, 1992.
[24] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Mesh Optimization,” Proc. SIGGRAPH, pp. 19-26, 1993.
[25] H. Hoppe, “Progressive Meshes,” Proc. SIGGRAPH, pp. 99-108, 1996.
[26] H. Hoppe, “Efficient Implementation of Progressive Meshes,” Computer & Graphics, vol. 22, no. 1, pp. 27-36, 1998.
[27] J. Hoschek and D. Lasser, Computer Aided Geometric Design, pp. 289-291. A.K. Peters, 1989.
[28] A. Khodakovsky, P. Shröder, and W. Sweldens, “Progressive Geometry Compression,” Proc. SIGGRAPH, pp. 271-278, 2000.
[29] L.P. Kobbelt, J. Vorsatz, U. Labsik, H.-P. Seidel, “A Shrink Wrapping Approach to Remeshing Polygonal Surfaces,” EUROGRAPHICS, vol. 18, no. 3, 1999.
[30] A.W.F. Lee, W. Sweldens, P. Schröder, L. Cowsar, and D. Dobkin, “MAPS: Multiresolution Adaptive Parametrization of Surfaces,” Computer Graphics Proc., pp. 95-104, 1998.
[31] A. Lee, H. Moreton, and H. Hoppe, “Displaced Subdivision Surfaces,” Proc. SIGGRAPH, pp. 85-94, 2000.
[32] C. Loop and T. DeRose, “Generalized B-Spline Surfaces of Arbitrary Topology,” Computer Graphics, vol. 24, no. 4, pp. 347-356, 1990.
[33] A.M. López, D. Lloret, J. Serrat, and J.J. Villanueva, “Multilevel Creases Based on the Level-Set Extrinsic Curvature,” Computer and Image Understanding, vol. 77, no. 2, pp. 111-144, 2000.
[34] W. Lorensen and H. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Computer Graphics, vol. 21, pp. 163-169, 1987.
[35] M. Lounsbery, T.D. DeRose, and J. Warren, “Multiresolution Analysis for Surfaces of Arbitrary Topological Type,” ACM Trans. Graphics, vol. 16, no. 1, pp. 34-73, 1997.
[36] T. McInerney and D. Terzopoulos, “A Dynamic Finite Element Surface Model for Segmentation and Tracking in Multidimensional Medical Images with Application to Cardiac 4D Image Analysis,” Computerized Medical Imaging and Graphics, vol. 19, no. 1, pp. 69-83, 1995.
[37] B.S. Morse, T.S. Yoo, P. Rheingans, D.T. Chen, and K.R. Subramanian, Shape Modeling International, pp. 1-12, 2001.
[38] R. Pajarola and J. Rossignac, “Compressed Progressive Meshes,” IEEE Trans. Visualization and Computer Graphics, vol. 6, no. 1, pp. 79-93, 2000.
[39] A. Pentland, “Automatic Extraction of Deformable Parts Models,” Int'l J. Computer Vision, pp. 107-126, 1990.
[40] J. Peters, “Constructing $\big. {\rm C}^1\bigr.$ Surfaces or Arbitrary Topology Using Biquadratic and Bicubic Splines,” Designing Fair Curves and Surfaces, N. Sapidis, ed., pp. 277-293, 1994.
[41] M. Powell and M. Sabin, “Piecewise Quadratic Approximation on Triangles,” ACM Trans. Math. Software, vol. 3, pp. 316-325, 1997.
[42] S. Punam, J.K. Udupa, and O. Dewey, “Scale-Based Fuzzy Connected Image Segmentation: Theory, Algorithms, and Validation,” Computer Vision and Image Understanding, vol. 77, no. 2, pp. 145-174, 2000.
[43] H. Qin and D. Terzopoulos, “Triangular NURBS and Their Dynamic Generalization,” Computer Aided Geometric Design, vol. 14, pp. 325-347, 1997.
[44] S.P. Raya, “Low-Level Segmentation of 3D Magnetic Resonance Brain ImagesA Rule-Based System,” IEEE Trans. Medical Imaging, vol. 9, no. 3, pp. 327-337, 1990.
[45] F. Solina and R. Bajcsy, “Recovery of Parametric Models from Range Images: The Case of Superquadrics with Global Deformations,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 2, pp. 131-147, Feb. 1990.
[46] L.H. Staib and J.S. Duncan, “Deformable Fourier Models for Surface Finding in 3D Images,” Proc. SPIE, vol. 1808, pp. 90-104, 1992.
[47] H. Tek and B. Kimia, “Volumetric Segmentation of Medical Images by Three-Dimensional Bubbles,” Computer Vision and Image Understanding, vol. 65, no. 2, pp. 246-258, 1997.
[48] D. Terzopoulos and D. Metaxas, “Dynamic 3D Models with Local and Global Deformations: Deformable Superquadrics,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 7, pp. 703-713, 1991.
[49] G. Turk and J.F. O'Brien, “Variational Implicit Surfaces,” Georgia Technical Report No. GIT-GVU-99-15, 1999.
[50] G. Turk and J.F. O'Brien, “Shape Transformation Using Variational Implicit Functions,” Proc. SIGGRAPH, pp. 335-342, 1999.
[51] Z.J. Wood, M. Desburn, P. Schröder, and D. Breen, “Semi-Regular Mesh Extraction from Volumes,” Proc. IEEE Visualization, pp. 275-282, 2000.
[52] D. Zorin, P. Schroder, and W. Sweldens, “Interactive Multiresolution Mesh Editing,” Proc. SIGGRAPH, pp. 259-268, 1997.

Index Terms:
Digital 3D shape, triangular mesh, multiresolution representation, rational Gaussian (RaG) surface, shape editing.
Citation:
Marcel Jackowski, Martin Satter, Ardeshir Goshtasby, "Approximating Digital 3D Shapes by Rational Gaussian Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 9, no. 1, pp. 56-69, Jan.-March 2003, doi:10.1109/TVCG.2003.1175097
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