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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.