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A wavelet Plancherel theory with application to multipliers and sparse approximations. (English) Zbl 1505.42040

The objects of study in this paper are continuous wavelet transforms based on square integrable unitary representations of locally compact groups. In classical Fourier analysis, every operation in the time domain is equivalent to an operation in the frequency domain and conversely (e.g. ‘convolution in the time domain is multiplication in the frequency domain’). This interchangeability of time and frequency representations of the signal is very useful. Such an equivalence does not hold in wavelet analysis – the analysis operator is not invertible. This paper seeks to ameliorate this deficiency by extending the signal space to the ‘window-signal space’ and develops a theory of ‘wavelet-Plancherel transform’ as the authors call it. The wavelet transform extends to a unitary, the wavelet-Plancherel transform, from the window-signal space to the coefficient space. Thus any operation on the coefficient space is equivalent to a corresponding operation on the window-signal space and vice versa. As an illustration of the efficacy of this analysis, it is shown how continuous wavelet multipliers with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. A framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems is also developed.
The authors, in an earlier paper, had developed a theory of transforms called the semi-direct product wavelet transforms that includes short-term Fourier transforms, 1D wavelet transforms and shearlet transforms as special cases. The wavelet-Plancherel theory of the present paper is derived from the semi-direct product wavelet transforms. Helpfully, first, the authors present the special case of the wavelet-Plancherel theory of 1D wavelet transforms to fix the ideas of the general theory.
The main contributions of the paper are as follows. (1) Developing what the authors call the wavelet-Plancherel theory for generic wavelet transforms based on square integrable representations of a locally compact group. In this theory, the wavelet transform is canonically extended to an isometric isomorphism from the window-signal space to a space of functions on phase space. (2) Obtaining closed-form formulas for the pull-back of phase space multiplicative operators for semi-direct product wavelet transforms, a large class of generic wavelet transforms. (3) Showing that carrying all computations in the window-signal space leads to fast multiplicative operator computations for a large class of functions in the case of 1D continuous wavelet transforms. For instance, for polynomial or characteristic functions, computational complexity is reduced to \(O(N)\) from \(O(N^2)\). (4) Utilising the fast implementation of multiplication by characteristic functions in a coefficient search method – the method takes \(O(N\log N)\) operations, instead of the usual \(O(N^2 \log N)\). As an example, this search method is used in a matching pursuit algorithm.
Here is an outline of the various sections of this long paper. After the introduction, section 2 gives an outline of the wavelet-Plancherel theory in dimension one. Representation theory and functional calculus preliminaries are presented in the next section. Section 4 is devoted to a summary of classical wavelet transforms and the recently developed theory of semi-direct product wavelet transforms (due to the authors) which form the basis of the wavelet-Plancherel theory of the present paper. The STFT, 1D wavelet transforms, the rotation-dilation wavelet transform, and the shearlet transform are shown to be particular cases. The next two sections form the core of the paper. The wavelet-Plancherel theory is developed in section 5. Section 6 is on Wavelet-Plancherel phase space filtering where it is shown how to calculate the pull-back of phase space multiplicative operators to the window-signal space. The range of the wavelet-Plancherel transform is characterised. A detailed discussion of greedy approximation methods formulating a greedy algorithm for extracting sparse approximations to signals using atoms from the 1D wavelet system is presented in the last section. Discretisations of the wavelet Plancherel method are given and the complexity of the algorithm proposed is also discussed. Four appendices on functional calculus, some technical proofs, pull-back formulas, and implementation of the algorithm complete the paper.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
20C35 Applications of group representations to physics and other areas of science
65T60 Numerical methods for wavelets
43A80 Analysis on other specific Lie groups
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
20C99 Representation theory of groups
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References:

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