Fast variational design of multiresolution curves and surfaces with B-spline wavelets

Abstract Multiresolution modeling provides a powerful tool for complex shape editing. To achieve a better control of deformations and a more intuitive interface, variational principles have been used in such multiresolution models. However, when handling multiresolution constraints, the existing methods often result in solving large optimization systems. Hence, the computational time may become too excessive to satisfy the requirements for interactive design in CAD. In this paper, we present a fast approach for interactive variational design of multiresolution models. By converting all constraints at different levels to a target level, the optimization problem is formulated and solved at the lower level. Thus, the unknown coefficients of the optimization system are significantly reduced. This improves the efficiency of variational design. Meanwhile, to avoid smoothing out the details of the shape in variational modeling, we optimize the change in the deformation energy instead of the total energy of the deformed shape. Several examples and the experimental results are given to demonstrate the effectiveness and efficiency of this approach.

[1]  David Salesin,et al.  Multiresolution curves , 1994, SIGGRAPH.

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

[3]  Tosiyasu L. Kunii,et al.  Continuous-resolution-level constraints in variational design of multiresolution shapes , 1998, The Visual Computer.

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

[5]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[6]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[7]  Tom Lyche,et al.  Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics , 1980 .

[8]  L. Piegl.  Modifying the shape of rational B-splines. part2: surfaces , 1989 .

[9]  Rémi Ronfard,et al.  Detail-Preserving Variational Surface Design With Multiresolution Constraints , 2005, J. Comput. Inf. Sci. Eng..

[10]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[11]  Günther Greiner,et al.  Variational Design and Fairing of Spline Surfaces , 1994, Comput. Graph. Forum.

[12]  Richard Szeliski,et al.  Fast Surface Interpolation Using Hierarchical Basis Functions , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[13]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[14]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[15]  Shigeo Takahashi,et al.  Variational design of curves and surfaces using multiresolution constraints , 1998, The Visual Computer.

[16]  Steven J. Gortler,et al.  Wavelet Methods For Computer Graphics , 1995 .

[17]  W. Press,et al.  Numerical Recipes in C++: The Art of Scientific Computing (2nd edn)1 Numerical Recipes Example Book (C++) (2nd edn)2 Numerical Recipes Multi-Language Code CD ROM with LINUX or UNIX Single-Screen License Revised Version3 , 2003 .

[18]  Gershon Elber.  Multiresolution Curve Editing With Linear Constraints , 2001 .

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

[20]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[21]  Rémi Ronfard,et al.  Detail-preserving variational surface design with multiresolution constraints , 2004, Proceedings Shape Modeling Applications, 2004..

[22]  Michael F. Cohen,et al.  Hierarchical and variational geometric modeling with wavelets , 1995, I3D '95.