A wavelet Plancherel theory with application to sparse continuous wavelet transform

We introduce a framework for calculating sparse approximations to signals based on elements of continuous wavelet systems. The method is based on an extension of the continuous wavelet theory. In the new theory, the signal space is embedded in larger "abstract" signal space, which we call the window-signal space. There is a canonical extension of the wavelet transform on the window-signal space, which is an isometric isomorphism from the window-signal space to a space of functions on phase space. Hence, the new framework is called a wavelet-Plancherel theory, and the extended wavelet transform is called the wavelet-Plancherel transform. Since the wavelet-Plancherel transform is an isometric isomorphism, any operation on phase space can be pulled-back to an operation in the window-signal space. Using this pull back property, it is possible to pull back a search for big wavelet coefficients to the window-signal space. We can thus avoid inefficient calculations on phase space, performing all calculations entirely in the window-signal space. We consider in this paper a matching pursuit algorithm based on this coefficient search approach. Our method has lower computational complexity than matching pursuit algorithms based on a naive coefficient search.

[1]  Olivier Rioul,et al.  Fast algorithms for discrete and continuous wavelet transforms , 1992, IEEE Trans. Inf. Theory.

[2]  Demetrio Labate,et al.  Optimally Sparse Multidimensional Representation Using Shearlets , 2007, SIAM J. Math. Anal..

[3]  V. Jones Von Neumann Algebras , 2020, Lectures on von Neumann Algebras.

[4]  Y. C. Pati,et al.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[5]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[6]  Ron Levie,et al.  Randomized Signal Processing with Continuous Frames , 2018, Journal of Fourier Analysis and Applications.

[7]  Ron Levie,et al.  Uncertainty principles and optimally sparse wavelet transforms , 2017, Applied and Computational Harmonic Analysis.

[8]  David Bernier,et al.  Wavelets from square-integrable representations , 1996 .

[9]  A. Aldroubi,et al.  Polynomial splines and wavelets: a signal processing perspective , 1993 .

[10]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .

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

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

[13]  Bernhard Laback,et al.  Time–Frequency Sparsity by Removing Perceptually Irrelevant Components Using a Simple Model of Simultaneous Masking , 2010, IEEE Transactions on Audio, Speech, and Language Processing.

[14]  Gitta Kutyniok,et al.  The Uncertainty Principle Associated with the Continuous Shearlet Transform , 2008, Int. J. Wavelets Multiresolution Inf. Process..

[15]  H. Führ Abstract Harmonic Analysis of Continuous Wavelet Transforms , 2005 .

[16]  Calvin C. Moore,et al.  On the regular representation of a nonunimodular locally compact group , 1976 .

[17]  Alexander G. Sklar A Wavelet-based Pitch-shifting Method , 2006 .

[18]  R. Lipsman,et al.  The plancherel formula for group extensions , 1972 .

[19]  D. Labate,et al.  Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators , 2006 .

[20]  Jean Laroche,et al.  Improved phase vocoder time-scale modification of audio , 1999, IEEE Trans. Speech Audio Process..

[21]  Gilbert Helmberg,et al.  Introduction to Spectral Theory in Hilbert Space , 1970 .

[22]  Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions , 2001, math-ph/0106014.

[23]  Anders Vretblad,et al.  Fourier Analysis and Its Applications , 2005 .

[24]  Wigner functions for a class of semi-direct product groups , 2002, math-ph/0201007.

[25]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[26]  Richard Kronland-Martinet,et al.  A Class of Algorithms for Time-Frequency Multiplier Estimation , 2013, IEEE Transactions on Audio, Speech, and Language Processing.

[27]  H. Führ,et al.  A classification of continuous wavelet transforms in dimension three , 2016, Applied and Computational Harmonic Analysis.

[28]  I. Daubechies,et al.  PAINLESS NONORTHOGONAL EXPANSIONS , 1986 .

[29]  Piotr Majdak,et al.  A time-frequency method for increasing the signal-to-noise ratio in system identification with exponential sweeps , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[30]  The Weak Paley-Wiener Property for Group Extensions , 2004, math/0401436.

[31]  C. Torrence,et al.  A Practical Guide to Wavelet Analysis. , 1998 .

[32]  Boualem Boashash,et al.  Time-Frequency Signal Analysis and Processing: A Comprehensive Reference , 2015 .

[33]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[34]  A. Grossmann,et al.  Transforms associated to square integrable group representations. I. General results , 1985 .

[35]  Christina Gloeckner Foundations Of Time Frequency Analysis , 2016 .

[36]  B. Pettis On integration in vector spaces , 1938 .

[37]  H. Führ Continuous wavelets transforms from semidirect products , 2000 .