Dynamic Catmull-Clark Subdivision Surfaces

Recursive subdivision schemes have been extensively used in computer graphics, computer-aided geometric design, and scientific visualization for modeling smooth surfaces of arbitrary topology. Recursive subdivision generates a visually pleasing smooth surface in the limit from an initial user-specified polygonal mesh through the repeated application of a fixed set of subdivision rules. We present a new dynamic surface model based on the Catmull-Clark subdivision scheme, a popular technique for modeling complicated objects of arbitrary genus. Our new dynamic surface model inherits the attractive properties of the Catmull-Clark subdivision scheme, as well as those of the physics-based models. This new model provides a direct and intuitive means of manipulating geometric shapes, and an efficient hierarchical approach for recovering complex shapes from large range and volume data sets using very few degrees of freedom (control vertices). We provide an analytic formulation and introduce the "physical" quantities required to develop the dynamic subdivision surface model which can be interactively deformed by applying synthesized forces. The governing dynamic differential equation is derived using Lagrangian mechanics and the finite element method. Our experiments demonstrate that this new dynamic model has a promising future in computer graphics, geometric shape design, and scientific visualization.

[1]  Jörg Peters,et al.  The simplest subdivision scheme for smoothing polyhedra , 1997, TOGS.

[2]  Ulrich Reif,et al.  A unified approach to subdivision algorithms near extraordinary vertices , 1995, Comput. Aided Geom. Des..

[3]  Peter Schröder,et al.  Interpolating Subdivision for meshes with arbitrary topology , 1996, SIGGRAPH.

[4]  Hiromi T. Tanaka,et al.  Adaptive mesh generation for surface reconstruction: parallel hierarchical triangulation without discontinuities , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[5]  Rachid Deriche,et al.  3D edge detection using recursive filtering: Application to scanner images , 1991, CVGIP Image Underst..

[6]  A. A. Ball,et al.  An investigation of curvature variations over recursively generated B-spline surfaces , 1990, TOGS.

[7]  M. Sabin.  The use of piecewise forms for the numerical representation of shape , 1976 .

[8]  Dimitris N. Metaxas,et al.  Shape and Nonrigid Motion Estimation Through Physics-Based Synthesis , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  John C. Platt,et al.  Elastically deformable models , 1987, SIGGRAPH.

[10]  Andrew P. Witkin,et al.  Variational surface modeling , 1992, SIGGRAPH.

[11]  Malcolm I. G. Bloor,et al.  Using partial differential equations to generate free-form surfaces , 1990, Comput. Aided Des..

[12]  Charles T. Loop,et al.  Smooth Subdivision Surfaces Based on Triangles , 1987 .

[13]  George Celniker,et al.  Linear constraints for deformable non-uniform B-spline surfaces , 1992, I3D '92.

[14]  M. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1978 .

[15]  Demetri Terzopoulos,et al.  Adaptive meshes and shells: irregular triangulation, discontinuities, and hierarchical subdivision , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[16]  Gabriel Taubin,et al.  A signal processing approach to fair surface design , 1995, SIGGRAPH.

[17]  Demetri Terzopoulos,et al.  Regularization of Inverse Visual Problems Involving Discontinuities , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[18]  Hong Qin,et al.  Dynamic NURBS with geometric constraints for interactive sculpting , 1994, TOGS.

[19]  Alex Pentland,et al.  Recovery of Nonrigid Motion and Structure , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  Hong Qin,et al.  D-NURBS: A Physics-Based Framework for Geometric Design , 1996, IEEE Trans. Vis. Comput. Graph..

[21]  Tony DeRose,et al.  Piecewise smooth surface reconstruction , 1994, SIGGRAPH.

[22]  Malcolm I. G. Bloor,et al.  Representing PDE surfaces in terms of B-splines , 1990, Comput. Aided Des..

[23]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[24]  Dmitry B. Goldgof,et al.  Adaptive-size physically-based models for nonrigid motion analysis , 1992, Proceedings 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[25]  A. A. Ball,et al.  Conditions for tangent plane continuity over recursively generated B-spline surfaces , 1988, TOGS.

[26]  Dimitris N. Metaxas,et al.  Dynamic deformation of solid primitives with constraints , 1992, SIGGRAPH.

[27]  Ayman Habib,et al.  Edge and vertex insertion for a class of C1 subdivision surfaces , 1999, Comput. Aided Geom. Des..

[28]  Gene H. Golub,et al.  Matrix computations , 1983 .

[29]  Rachid Deriche,et al.  3D edge detection using recursive filtering: application to scanner images , 1989, Proceedings CVPR '89: IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[30]  George Merrill Chaikin,et al.  An algorithm for high-speed curve generation , 1974, Comput. Graph. Image Process..

[31]  B. Gossick.  Hamilton's principle and physical systems , 1967 .

[32]  Gérard G. Medioni,et al.  Surface description of complex objects from multiple range images , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[33]  J. Peters,et al.  Analysis of Algorithms Generalizing B-Spline Subdivision , 1998 .

[34]  Dimitris N. Metaxas,et al.  Dynamic 3D models with local and global deformations: deformable superquadrics , 1990, [1990] Proceedings Third International Conference on Computer Vision.

[35]  Alex Pentland,et al.  Good vibrations: modal dynamics for graphics and animation , 1989, SIGGRAPH.

[36]  Leif Kobbelt,et al.  A variational approach to subdivision , 1996, Comput. Aided Geom. Des..

[37]  Demetri Terzopoulos,et al.  A finite element model for 3D shape reconstruction and nonrigid motion tracking , 1993, 1993 (4th) International Conference on Computer Vision.

[38]  Baba C. Vemuri,et al.  Multiresolution stochastic hybrid shape models with fractal priors , 1994, TOGS.

[39]  Jean Schweitzer,et al.  Analysis and application of subdivision surfaces , 1996 .

[40]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[41]  Laurent D. Cohen,et al.  Finite-Element Methods for Active Contour Models and Balloons for 2-D and 3-D Images , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[42]  Alex Pentland,et al.  Recovery of non-rigid motion and structure , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[43]  Demetri Terzopoulos,et al.  Dynamic swung surfaces for physics-based shape design , 1995, Comput. Aided Des..

[44]  George Celniker,et al.  Deformable curve and surface finite-elements for free-form shape design , 1991, SIGGRAPH.

[45]  Norman I. Badler,et al.  Hierarchical Shape Representation Using Locally Adaptive Finite Elements , 1994, ECCV.

[46]  Tony DeRose,et al.  Efficient, fair interpolation using Catmull-Clark surfaces , 1993, SIGGRAPH.

[47]  Denis Z orin.  Smoothness of Stationary Subdivision on Irregular Meshes , 1998 .

[48]  H. Kardestuncer,et al.  Finite element handbook , 1987 .