TY - JOUR

T1 - A Wavelet Plancherel Theory with Application to Multipliers and Sparse Approximations

AU - Levie, Ron

AU - Sochen, Nir

N1 - Publisher Copyright:
© 2022 Taylor & Francis Group, LLC.

PY - 2022

Y1 - 2022

N2 - We introduce an extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space–the so called window–signal space. There is a canonical extension of the wavelet transform to an isometric isomorphism between the window–signal space and the coefficient 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 in the coefficient space can be pulled-back to an operation in the window–signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window–signal space. As one example application, we show how continuous wavelet multipliers (also called Calderón–Toeplitz operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.

AB - We introduce an extension of continuous wavelet theory that enables an efficient implementation of multiplicative operators in the coefficient space. In the new theory, the signal space is embedded in a larger abstract signal space–the so called window–signal space. There is a canonical extension of the wavelet transform to an isometric isomorphism between the window–signal space and the coefficient 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 in the coefficient space can be pulled-back to an operation in the window–signal space. It is then possible to improve the computational complexity of methods that involve a multiplicative operator in the coefficient space, by performing all computations directly in the window–signal space. As one example application, we show how continuous wavelet multipliers (also called Calderón–Toeplitz operators), with polynomial symbols, can be implemented with linear complexity in the resolution of the 1D signal. As another example, we develop a framework for efficiently computing greedy sparse approximations to signals based on elements of continuous wavelet systems.

KW - Continuous wavelet

KW - Plancherel theorem

KW - matching pursuit

KW - sparse decomposition

KW - wavelet multiplier

UR - http://www.scopus.com/inward/record.url?scp=85134041867&partnerID=8YFLogxK

U2 - 10.1080/01630563.2022.2060253

DO - 10.1080/01630563.2022.2060253

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AN - SCOPUS:85134041867

SN - 0163-0563

VL - 43

SP - 1303

EP - 1400

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

IS - 11

ER -