Interpolatory wavelets for manifold-valued data

Abstract Geometric wavelet-like transforms for univariate and multivariate manifold-valued data can be constructed by means of nonlinear stationary subdivision rules which are intrinsic to the geometry under consideration. We show that in an appropriate vector bundle setting for a general class of interpolatory wavelet transforms, which applies to Riemannian geometry, Lie groups and other geometries, Holder smoothness of functions is characterized by decay rates of their wavelet coefficients.

[1]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[2]  Philipp Grohs,et al.  Smoothness of interpolatory multivariate subdivision in Lie groups , 2009 .

[3]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[4]  Rong-Qing Jia,et al.  Vector subdivision schemes and multiple wavelets , 1998, Math. Comput..

[5]  Mats Holmström,et al.  Solving Hyperbolic PDEs Using Interpolating Wavelets , 1999, SIAM J. Sci. Comput..

[6]  Gilles Deslauriers,et al.  Symmetric iterative interpolation processes , 1989 .

[7]  Gang Xie,et al.  Smoothness Equivalence Properties of General Manifold-Valued Data Subdivision Schemes , 2008, Multiscale Model. Simul..

[8]  Johannes Wallner.  Smoothness Analysis of Subdivision Schemes by Proximity , 2006 .

[9]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[10]  T. Yu,et al.  Smoothness equivalence properties of interpolatory Lie group subdivision schemes , 2010 .

[11]  I. Daubechies,et al.  Normal Multiresolution Approximation of Curves , 2004 .

[12]  Philipp Grohs,et al.  Smoothness equivalence properties of univariate subdivision schemes and their projection analogues , 2009, Numerische Mathematik.

[13]  D. Levin,et al.  Subdivision schemes in geometric modelling , 2002, Acta Numerica.

[14]  Gang Xie,et al.  Smoothness Equivalence Properties of Manifold-Valued Data Subdivision Schemes Based on the Projection Approach , 2007, SIAM J. Numer. Anal..

[15]  Gitta Kutyniok,et al.  Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis , 2007, SIAM J. Math. Anal..

[16]  Nira Dyn,et al.  Approximation order of interpolatory nonlinear subdivision schemes , 2010, J. Comput. Appl. Math..

[17]  Johannes Wallner,et al.  Smoothness Properties of Lie Group Subdivision Schemes , 2007, Multiscale Model. Simul..

[18]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[19]  Karl Scherer,et al.  Approximationsprozesse und Interpolationsmethoden , 1968 .

[20]  Philipp Grohs,et al.  Smoothness Analysis of Subdivision Schemes on Regular Grids by Proximity , 2008, SIAM J. Numer. Anal..

[21]  Nira Dyn,et al.  Convergence and C1 analysis of subdivision schemes on manifolds by proximity , 2005, Comput. Aided Geom. Des..

[22]  C. Micchelli,et al.  On vector subdivision , 1998 .

[23]  H. Johnen,et al.  On the equivalence of the K-functional and moduli of continuity and some applications , 1976, Constructive Theory of Functions of Several Variables.

[24]  A. Harten.  Multiresolution representation of data: a general framework , 1996 .

[25]  David L. Donoho,et al.  Interpolating Wavelet Transforms , 1992 .

[26]  Peter Schröder,et al.  Multiscale Representations for Manifold-Valued Data , 2005, Multiscale Model. Simul..

[27]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[28]  C. Micchelli,et al.  Stationary Subdivision , 1991 .