Multiresolution B-Splines Based On Wavelet Constraints

We present a novel method for determining local multiresolution filters for B-spline subdivision curves of any order. Our approach is based on constraining the wavelet coefficients such that the coefficients at even vertices can be computed from the coefficients of neighboring odd vertices. This constraint leads to an initial set of decomposition filters. To increase the quality of these initial filters, we use a line search optimization that reduces the size of the wavelet coefficients. The resulting multiresolution filters are a biorthogonal wavelet system whose construction is similar to the lifting scheme. This approach is demonstrated in depth for cubic B-spline curves. Our filters are shown to perform comparably with established filters.

[1]  Martin Bertram,et al.  Biorthogonal Loop-Subdivision Wavelets , 2004, Computing.

[2]  R. Bartels,et al.  Reversing subdivision rules: local linear conditions and observations on inner products , 2000 .

[3]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[4]  Wim Sweldens,et al.  The lifting scheme: a construction of second generation wavelets , 1998 .

[5]  Malcolm A. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1998 .

[6]  Wim Sweldens,et al.  Building your own wavelets at home , 2000 .

[7]  David Salesin,et al.  Wavelets for computer graphics: theory and applications , 1996 .

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

[9]  Richard F. Riesenfeld,et al.  A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  Wim Sweldens,et al.  An Overview of Wavelet Based Multiresolution Analyses , 1994, SIAM Rev..

[11]  E. J. Stollnitz,et al.  Wavelets for Computer Graphics: A Primer Part 2 , 1995 .

[12]  Richard H. Bartels,et al.  Multiresolution Surfaces having Arbitrary Topologies by a Reverse Doo Subdivision Method , 2002, Comput. Graph. Forum.

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

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