The Theory of Wavelets with Composite Dilations

A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L 2(ℝn) under the action of lattice translations and dilations by products of elements drawn from non-commuting sets of matrices A and B. Typically, the members of B are matrices whose eigenvalues have magnitude one, while the members of A are matrices expanding on a proper subspace of ℝn. The theory of these systems generalizes the classical theory of wavelets and provides a simple and flexible framework for the construction of orthonormal bases and related systems that exhibit a number of geometric features of great potential in applications. For example, composite wavelets have the ability to produce “long and narrow” window functions, with various orientations, well-suited to applications in image processing.

[1]  Akram Aldroubi,et al.  Wavelets, Multiwavelets, and Their Applications , 1998 .

[2]  Y. Zeevi,et al.  Analysis of Multiwindow Gabor-Type Schemes by Frame Methods☆ , 1997 .

[3]  Wang-Q Lim,et al.  Wavelets with composite dilations , 2004 .

[4]  D. Walnut,et al.  Differentiation and the Balian-Low Theorem , 1994 .

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

[6]  E. Candès,et al.  New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities , 2004 .

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

[8]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[9]  Demetrio Labate,et al.  A unified characterization of reproducing systems generated by a finite family, II , 2002 .

[10]  Christopher Heil,et al.  Continuous and Discrete Wavelet Transforms , 1989, SIAM Rev..

[11]  E. Candès.  New tight frames of curvelets and optimal representations of objects with C² singularities , 2002 .

[12]  James S. Byrnes,et al.  Twentieth century harmonic analysis : a celebration , 2001 .

[13]  Yang Wang,et al.  Wavelets, tiling, and spectral sets , 2002 .

[14]  Yang Wang,et al.  Lattice tiling and the Weyl—Heisenberg frames , 2001 .

[15]  J. Seo,et al.  Unimodular wavelets for $L^2$ and the Hardy space $H^2$. , 1994 .

[16]  G. Kutyniok,et al.  Density of weighted wavelet frames , 2003 .

[17]  Guido Weiss,et al.  The Mathematical Theory of Wavelets , 2001 .

[18]  Wang-Q Lim,et al.  Wavelets with composite dilations and their MRA properties , 2006 .

[19]  Demetrio Labate,et al.  A unified characterization of reproducing systems generated by a finite family , 2002 .

[20]  O. Christensen,et al.  Density of Gabor Frames , 1999 .

[21]  D. Speegle.  On the existence of wavelets for non-expansive dilation matrices , 2003 .