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Dynamic Modeling of Butterfly Subdivision Surfaces
July-September 2000 (vol. 6 no. 3)
pp. 265-287

Abstract—In this paper, we develop integrated techniques that unify physics-based modeling with geometric subdivision methodology and present a scheme for dynamic manipulation of the smooth limit surface generated by the (modified) butterfly scheme using physics-based “force” tools. This procedure-based surface model obtained through butterfly subdivision does not have a closed-form analytic formulation (unlike other well-known spline-based models) and, hence, poses challenging problems to incorporate mass and damping distributions, internal deformation energy, forces, and other physical quantities required to develop a physics-based model. Our primary contributions to computer graphics and geometric modeling include: 1) a new hierarchical formulation for locally parameterizing the butterfly subdivision surface over its initial control polyhedron, 2) formulation of dynamic butterfly subdivision surface as a set of novel finite elements, and 3) approximation of this new type of finite elements by a collection of existing finite elements subject to implicit geometric constraints. Our new physics-based model can be sculpted directly by applying synthesized forces and its equilibrium is characterized by the minimum of a deformation energy subject to the imposed constraints. We demonstrate that this novel dynamic framework not only provides a direct and natural means of manipulating geometric shapes, but also facilitates hierarchical shape and nonrigid motion estimation from large range and volumetric data sets using very few degrees of freedom (control vertices that define the initial polyhedron).

[1] G.M. Chaikin, “An Algorithm for High Speed Curve Generation,” Computer Vision, Graphics, and Image Processing, vol. 3, no. 4, pp. 346-349, 1974.
[2] D. Doo, “A Subdivision Algorithm for Smoothing Down Irregularly Shaped Polyhedrons,” Proc. Interactive Techniques in Computer Aided Design, pp. 157-165, 1978.
[3] M. Sabin, “The Use of Piecewise Forms for the Numerical Representation of Shape” PhD thesis, Hungarian Academy of Sciences, Budapest, 1976.
[4] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer Aided Design, vol. 10, no. 6, pp. 350-355, 1978.
[5] C. Loop, “Smooth Subdivision Surfaces Based on Triangles,” MS thesis, Dept. of Math., Univ. of Utah, 1987.
[6] H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, and W. Stuetzle, “Piecewise Smooth Surface Reconstruction,” Proc., ACM SIGGRAPH, pp. 295-302, July 1994.
[7] M. Halstead, M. Kass, and T. DeRose, “Efficient, Fair Interpolation Using Catmull-Clark Surfaces,” Proc., ACM SIGGRAPH, pp. 35-44, Aug. 1993.
[8] J. Peters and U. Reif, “The Simplest Subdivision Scheme for Smoothing Polyhedra,” ACM Trans. Graphics, vol. 16, no. 4, pp. 420-431, Oct. 1997.
[9] T.W. Sederberg, J. Zheng, D. Sewell, and M. Sabin, “Non-Uniform Recursive Subdivision Surfaces,” Proc., ACM SIGGRAPH, pp. 387-394, July 1998.
[10] T. DeRose, M. Kass, and T. Truong, “Subdivision Surfaces in Character Animation,” Proc., ACM SIGGRAPH, pp. 85-94, July 1998.
[11] N. Dyn, D. Levin, and J.A. Gregory, “A Butterfly Subdivision Scheme for Surface Interpolation with Tension Control,” ACM Transactions on Graphics, vol. 9, no. 2, pp. 160-169, Apr. 1990.
[12] N. Dyn, S. Hed, and D. Levin, “Subdivision Schemes for Surface Interpolation,” Proc. Workshop Computational Geometry, pp. 97-118, 1993.
[13] D. Zorin, P. Schröder, and W. Sweldens, “Interpolating Subdivision for Meshes with Arbitrary Topology,” Proc., ACM SIGGRAPH, pp. 189-192, Aug. 1996.
[14] D. Doo and M. Sabin, “Analysis of the Behavior of Recursive Division Surfaces Near Extraordinary Points,” Computer Aided Design, vol. 10, no. 6, pp. 356-360, 1978.
[15] A.A. Ball and D.J.T. Storry, “Conditions for Tangent Plane Continuity over Recursively Generated B-Spline Surfaces,” ACM Trans. Graphics, vol. 7, no. 2, pp. 83-102, 1988.
[16] A.A. Ball and D.J.T. Storry, “An Investigation of Curvature Variations over Recursively Generated B-Spline Surfaces,” ACM Trans. Graphics, vol. 9, no. 4, pp. 424-437, 1990.
[17] U. Reif, “A Unified Approach to Subdivision Algorithms Near Extraordinary Points,” Computer Aided Geometric Design, vol. 12, no. 2, pp. 153-174, 1995.
[18] C.A. Micchelli and H. Prautzsch, “Uniform Refinement of Curves,” Linear Algebra and Its Applications, vols. 114/115, pp. 841-870, 1989.
[19] A.S. Cavaretta, W. Dahmen, and C.A. Micchelli, “Stationary Subdivision,” Memoirs of the Am. Math. Soc., vol. 93, no. 453, Sept. 1991.
[20] J.E. Schweitzer, “Analysis and Application of Subdivision Surfaces,” PhD thesis, Univ. of Washington, Seattle, 1996.
[21] A. Habib and J. Warren, “Edge and Vertex Insertion for a Class of$C^1$Subdivision Surfaces,” Computer Aided Geometric Design, to appear.
[22] J. Peters and U. Reif, “Analysis of Generalized B-Spline Subdivision Algorithms,” SIAM J. Numerical Analysis, to appear, available at / tr/ pub/
[23] D. Zorin, “Smoothness of Stationary Subdivision on Irregular Meshes,” Constructive Approximation, submitted, available as Stanford Computer Science Lab Technical Report CSL-TR-98-752.
[24] J. Stam, “Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values,” Proc. ACM SIGGRAPH, pp. 395-404, July 1998.
[25] L. Kobbelt, “Interpolatory Refinement by Variational Methods,” Approximation Theory VIII, C. Chui and L. Schumaker, eds., pp. 217-224, World Scientific, 1995.
[26] L. Kobbelt, “A Variational Approach to Subdivision,” Computer-Aided Geometric Design, vol. 13, pp. 743-761, 1996.
[27] L. Kobbelt and P. Schröder, “Constructing Variationally Optimal Curves through Subdivision,” Technical Report CS-TR-97-05, California Inst. of Technology Computer Science Dept., 1997.
[28] H. Weimer and J. Warren, “Subdivision Schemes for Thin Plate Splines,” Computer Graphics Forum, vol. 17, no. 3, pp. 303-313, 1998.
[29] T. Kurihara, “Interactive Surface Design Using Recursive Subdivision,” Communicating with Virtual Worlds, N.M. Thalmann and D. Thalmann, eds., pp. 228-243, Springer-Verlag, 1993.
[30] K. Pulli and M. Lounsbery, “Hierarchical Editing of Subdivision Surfaces,” Technical Report UW-CSE-97-04-07, Univ. of Washington Computer Science Dept., 1997.
[31] D. Zorin, P. Schröder, and W. Sweldens, “Interactive Multiresolution Mesh Editing,” Proc. ACM SIGGRAPH, pp. 259-268, Aug. 1997.
[32] L. Kobbelt, S. Campagna, J. Vorsatz, and H.P. Seidel, “Interactive Multiresolution Modeling on Arbitrary Meshes,” Proc. ACM SIGGRAPH, pp. 105-114, July 1998.
[33] D. Terzopoulos, J. Platt, A. Barr, and K. Fleischer, “Elastically Deformable Models,” Proc. ACM SIGGRAPH, pp. 205-214, 1987.
[34] D. Terzopoulos and K. Fleischer, “Deformable Models,” The Visual Computer, vol. 4, no. 6, pp. 306-331, 1988.
[35] A. Pentland and J. Williams, “Good Vibrations: Modal Dynamics for Graphics and Animation,” Proc. ACM SIGGRAPH, pp. 215-222, 1989.
[36] D. Metaxas and D. Terzopoulos, “Dynamic Deformation of Solid Primitives with Constraints,” Proc. ACM SIGGRAPH, pp. 309-312, July 1992.
[37] B.C. Vemuri and A. Radisavljevic, “Multiresolution Stochastic Hybrid Shape Models with Fractal Priors,” ACM Trans. Graphics, vol. 13, no. 2, pp. 177-207, Apr. 1994.
[38] G. Celniker and D. Gossard, “Deformable Curve and Surface Finite Elements for Free-Form Shape Design,” Proc. ACM SIGGRAPH, pp. 257-266, July 1991.
[39] M.I.J. Bloor and M.J. Wilson, “Representing PDE Surfaces in Terms of B-Splines,” Computer Aided Design, vol. 22, no. 6, pp. 324-331, 1990.
[40] M.I.J. Bloor and M.J. Wilson, “Using Partial Differential Equations to Generate Free-Form Surfaces,” Computer Aided Design, vol. 22, no. 4, pp. 202-212, 1990.
[41] G. Celniker and W. Welch, “Linear Constraints for Deformable B-Spline Surfaces,” Proc. Symp. Interactive 3D Graphics, pp. 165-170, 1992.
[42] W. Welch and A. Witkin, “Variational Surface Modeling,” Proc. ACM SIGGRAPH, pp. 157-166, July 1992.
[43] H. Qin and D. Terzopoulos, “Dynamic NURBS Swung Surfaces for Physics-Based Shape Design,” Computer-Aided Design, vol. 27, no. 2, pp. 111-127, 1995.
[44] H. Qin and D. Terzopoulos, “D-NURBS: A Physics-Based Framework for Geometric Design,” IEEE Trans. Visualization and Computer Graphics, vol. 2, no. 1, pp. 85-96, Mar. 1996.
[45] D. Terzopoulos and H. Qin, “Dynamic NURBS with Geometric Constraints for Interactive Sculpting,” ACM Trans. Graphics, vol. 13, no. 2, pp. 103-136, Apr. 1994.
[46] C. Mandal, H. Qin, and B.C. Vemuri, “Dynamic Smooth Subdivision Surfaces for Data Visualization,” Proc. IEEE Visualization '97, pp. 371-377, Oct. 1997.
[47] H. Qin, C. Mandal, and B.C. Vemuri, “Dynamic Catmull-Clark Subdivision Surfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 4, no. 3, Sept. 1998.
[48] C. Mandal, B.C. Vemuri, and H. Qin, “Shape Recovery Using Dynamic Subdivision Surfaces,” Proc. Int'l Conf. Computer Vision, pp. 805-810, Jan. 1998.
[49] E. Bardinet, L.D. Cohen, and N. Ayache, “Superquadrics and Free-Form Deformations: A Global Model to Fit and Track 3D Medical Data,” Proc. Computer Visualization and Pattern Recognition in Medicine, N. Ayache, ed., pp. 319-326, 1995.
[50] L.D. Cohen and I. Cohen, “Finite-Element Methods for Active Contour Models and Balloons for 2D and 3D Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 11, pp. 1,131-1,147, Nov. 1993.
[51] E. Koh, D. Metaxas, and N. Badler, “Hierarchical Shape Representation Using Locally Adaptive Finite Elements,” Proc. European Conf. Computer Vision, J.O. Eklundh, ed., pp. 441-446, 1994.
[52] F. Leitner and P. Cinquin, “Complex Topology 3D Objects Segmentation,” Model-Based Vision Development and Tools, SPIE Proc., pp. 16-26, 1991.
[53] 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.
[54] L.H. Staib and J.S. Duncan, “Boundary Finding with Parametrically Deformable Models,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 11, pp. 1,061-1,075, Nov. 1992.
[55] 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. 703-714, July 1991.
[56] D. Metaxas and D. Terzopoulos, “Shape and Non-Rigid Motion Estimation through Physics-Based Synthesis,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 6, pp. 580-591, June 1993.
[57] A. Pentland and B. Horowitz, “Recovery of Non-Rigid Motion and Structure,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 7, pp. 730-742, July 1991.
[58] Y. Chen and G. Medioni, “Surface Description of Complex Objects from Multiple Range Images,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 153-158, June 1994.
[59] W.C. Huang and D. Goldgof, “Adaptive-Size Physically-Based Models for Non-rigid Motion Analysis,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 833-835, June 1992.
[60] H. Tanaka and F. Kishino, “Adaptive Mesh Generation for Surface Reconstruction: Parallel Hierarchical Triangulation without Discontinuities,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 88-94, June 1993.
[61] M. Vasilescu and D. Terzopoulos, “Adaptive Meshes and Shells: Irregular Triangulation, Discontinuities and Hierarchical Subdivision,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 829-832, June 1992.
[62] J.M. Lounsbery, T. DeRose, and J. Warren, “Multiresolution Analysis for Surfaces of Arbitrary Topological Type,” ACM Trans. Graphics, vol. 16, no. 1, pp. 34-73, Jan. 1997.
[63] B.R. Gossick, Hamilton's Principle and Physical Systems. New York: Academic Press, 1967.
[64] H. Kardestuncer, The Finite Element Handbook. New York: McGraw-Hill, 1987.
[65] O. Monga and R. Deriche, “3D Edge Detection Using Recursive Filtering,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 28-35, June 1989.
[66] G.H. Golub and V.H. Van Loan, Matrix Computations. The Johns Hopkins Univ. Press, 1989.
[67] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C. Cambridge Univ. Press, 1992.

Index Terms:
Dynamic modeling, physics-based geometric design, geometric modeling, CAGD, subdivision surfaces, deformable models, finite elements, interactive techniques.
Chhandomay Mandal, Hong Qin, Baba C. Vemuri, "Dynamic Modeling of Butterfly Subdivision Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 6, no. 3, pp. 265-287, July-Sept. 2000, doi:10.1109/2945.879787
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