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 -