Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets

Abstract In this paper, we build a multidimensional wavelet decomposition based on polyharmonic B-splines. The pre-wavelets are polyharmonic splines and so not tensor products of univariate wavelets. Explicit construction of the filters specified by the classical dyadic scaling relations is given and the decay of the functions and the filters is shown. We then design the decomposition/recomposition algorithm by means of downsampling/upsampling and convolution products.

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

[2]  Zvi Ziegler,et al.  Approximation theory and applications , 1983 .

[3]  Charles A. Micchelli,et al.  Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two , 1991, Curves and Surfaces.

[4]  R. DeVore,et al.  On the construction of multivariate (pre)wavelets , 1993 .

[5]  P. Lemarié.  Wavelets, Spline Interpolation and Lie Groups , 1991 .

[6]  Charles A. Micchelli,et al.  Using the refinement equation for the construction of pre-wavelets , 1991, Numerical Algorithms.

[7]  Christophe Rabut,et al.  A fast wavelet algorithm for multidimensional signals using polyharmonic splines , 2003 .

[8]  Y. Meyer.  Wavelets and Operators , 1993 .

[9]  S. Rippa,et al.  Numerical Procedures for Surface Fitting of Scattered Data by Radial Functions , 1986 .

[10]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[11]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[12]  D. Levin,et al.  Iterative Solution of Systems Originating from Integral Equations and Surface Interpolation , 1983 .

[13]  W. Madych.  Some elementary properties of multiresolution analyses of L 2 (R n ) , 1993 .

[14]  Thierry Blu,et al.  Isotropic polyharmonic B-splines: scaling functions and wavelets , 2005, IEEE Transactions on Image Processing.

[15]  W. R. Madych.  Polyharmonic Splines, Multiscale Analysis, and Entire Functions , 1990 .

[16]  Milvia Rossini,et al.  On the Errors of Multidimensional MRA Based on Non-Separable Scaling Functions , 2006, Int. J. Wavelets Multiresolution Inf. Process..

[17]  P. Lemarié,et al.  Base d'ondelettes sur les groupes de Lie stratifiés , 1989 .

[18]  W. Madych.  Spline type summability for multivariate sampling , 1999 .

[19]  Jean Duchon,et al.  Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces , 1976 .

[20]  Christophe Rabut,et al.  Using the refinement equation for the construction of pre-wavelets III: Elliptic splines , 2005, Numerical Algorithms.

[21]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[22]  Christophe Rabut,et al.  Elementarym-harmonic cardinal B-splines , 1992, Numerical Algorithms.

[23]  Charles K. Chui,et al.  Cardinal interpolation with differences of tempered functions , 1992 .

[24]  C. Rabut.  B-splines polyharmoniques cardinales : interpolation, quasi-interpolation, filtrage , 1990 .

[25]  Stéphane Jaffard.  Propriétés des matrices « bien localisées » près de leur diagonale et quelques applications , 1990 .

[26]  K. K. Vo,et al.  Distributions, analyse de Fourier, opérateurs aux dérivées partielles , 1972 .