Multiwavelet Frames from Refinable Function Vectors

Starting from any two compactly supported d-refinable function vectors in (L2(R))r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L2(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.

[1]  Qiyu Sun,et al.  Algorithm for the construction of symmetric and antisymmetric M-band wavelets , 2000, SPIE Optics + Photonics.

[2]  C. Heil,et al.  Accuracy of Lattice Translates of Several Multidimensional Refinable Functions , 1998 .

[3]  Ding-Xuan Zhou,et al.  Approximation by Multiple Refinable Functions , 1997, Canadian Journal of Mathematics.

[4]  Gerlind Plonka-Hoch,et al.  A new factorization technique of the matrix mask of univariate refinable functions , 2001, Numerische Mathematik.

[5]  J. Benedetto,et al.  The Theory of Multiresolution Analysis Frames and Applications to Filter Banks , 1998 .

[6]  I. Selesnick.  Smooth Wavelet Tight Frames with Zero Moments , 2001 .

[7]  George C. Donovan,et al.  Construction of Orthogonal Wavelets Using Fractal Interpolation Functions , 1996 .

[8]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[9]  Zuowei Shen.  Affine systems in L 2 ( IR d ) : the analysis of the analysis operator , 1995 .

[10]  C. Chui,et al.  Compactly supported tight and sibling frames with maximum vanishing moments , 2001 .

[11]  B. Han.  Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix , 2003 .

[12]  Charles K. Chui,et al.  Affine frames, quasi-affine frames, and their duals , 1998, Adv. Comput. Math..

[13]  Bin Han,et al.  Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets , 2001 .

[14]  Qingtang Jiang,et al.  Parameterizations of Masks for Tight Affine Frames with Two Symmetric/Antisymmetric Generators , 2003, Adv. Comput. Math..

[15]  Charles K. Chui,et al.  Tight frame oversampling and its equivalence to shift-invariance of affine frame operators , 2002 .

[16]  I. Daubechies,et al.  Framelets: MRA-based constructions of wavelet frames☆☆☆ , 2003 .

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

[18]  C. Chui,et al.  Compactly supported tight frames associated with refinable functions , 2000 .

[19]  G. Weiss,et al.  A First Course on Wavelets , 1996 .

[20]  George C. Donovan,et al.  Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets , 1996 .

[21]  Zuowei Shen.  Refinable function vectors , 1998 .

[22]  A. Ron,et al.  Affine Systems inL2(Rd): The Analysis of the Analysis Operator , 1997 .

[23]  A. Ron,et al.  Affine systems inL2 (ℝd) II: Dual systems , 1997 .

[24]  Bin Han,et al.  Tight wavelet frames generated by three symmetric B-spline functions with high vanishing moments , 2003 .

[25]  G. Strang,et al.  Approximation by translates of refinable functions , 1996 .

[26]  B. Han,et al.  Pairs of Dual Wavelet Frames from Any Two Refinable Functions , 2004 .

[27]  Charles K. Chui,et al.  Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments , 2003, Adv. Comput. Math..

[28]  C. Micchelli,et al.  Biorthogonal Wavelet Expansions , 1997 .

[29]  G. Plonka.  Approximation order provided by refinable function vectors , 1997 .

[30]  A. Aldroubi.  Portraits of frames , 1995 .

[31]  B. Han.  On Dual Wavelet Tight Frames , 1997 .

[32]  A. Petukhov.  Explicit Construction of Framelets , 2001 .