TY - JOUR

T1 - Uncertainty principles and optimally sparse wavelet transforms

AU - Levie, Ron

AU - Sochen, Nir

N1 - Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2020/5

Y1 - 2020/5

N2 - In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.

AB - In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.

KW - Continuous wavelet transform

KW - Group representation

KW - Sparse representation

KW - Uncertainty principle

UR - http://www.scopus.com/inward/record.url?scp=85054074445&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2018.09.008

DO - 10.1016/j.acha.2018.09.008

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AN - SCOPUS:85054074445

SN - 1063-5203

VL - 48

SP - 811

EP - 867

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 3

ER -