Uncertainty principles and optimally sparse wavelet transforms

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 based on a window 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.

[1]  Robert L. Grossman,et al.  Wavelet transforms associated with finite cyclic groups , 1992, [1992] Conference Record of the Twenty-Sixth Asilomar Conference on Signals, Systems & Computers.

[2]  A. Grossmann,et al.  Transforms associated to square integrable group representations. I. General results , 1985 .

[3]  William O. Alltop,et al.  Complex sequences with low periodic correlations (Corresp.) , 1980, IEEE Trans. Inf. Theory.

[4]  William O. Alltop,et al.  Complex sequences with low periodic correlations , 1980 .

[5]  G. Teschke.  Construction of Generalized Uncertainty Principles and Wavelets in Bessel Potential Spaces , 2005 .

[6]  Demetrio Labate,et al.  Optimally Sparse Multidimensional Representation Using Shearlets , 2007, SIAM J. Math. Anal..

[7]  B. Hall.  Lie Groups, Lie Algebras, and Representations: An Elementary Introduction , 2004 .

[8]  Thomas Strohmer,et al.  High-Resolution Radar via Compressed Sensing , 2008, IEEE Transactions on Signal Processing.

[9]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[10]  R. Levie,et al.  Adjoint translation, adjoint observable and uncertainty principles , 2014, Adv. Comput. Math..

[11]  George W. Mackey,et al.  A theorem of Stone and von Neumann , 1949 .

[12]  Gabriele Steidl,et al.  Shearlet coorbit spaces and associated Banach frames , 2009 .

[13]  Syed Twareque Ali,et al.  Coherent States, Wavelets, and Their Generalizations , 2013 .

[14]  Calvin C. Moore,et al.  On the regular representation of a nonunimodular locally compact group , 1976 .

[15]  A. Prasad.  An easy proof of the Stone-von Neumann-Mackey Theorem , 2009, 0912.0574.

[16]  Gitta Kutyniok,et al.  The Uncertainty Principle Associated with the Continuous Shearlet Transform , 2008, Int. J. Wavelets Multiresolution Inf. Process..

[17]  D. Labate,et al.  Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators , 2006 .

[18]  Gilbert Helmberg,et al.  Introduction to Spectral Theory in Hilbert Space , 1970 .

[19]  H. Feichtinger,et al.  Banach spaces related to integrable group representations and their atomic decompositions. Part II , 1989 .

[20]  Syed Twareque Ali,et al.  Two-Dimensional Wavelets and their Relatives , 2004 .

[21]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[22]  G. Teschke,et al.  Coorbit Spaces with Voice in a Fréchet Space , 2014, 1402.3917.

[23]  Karlheinz Gröchenig,et al.  Foundations of Time-Frequency Analysis , 2000, Applied and numerical harmonic analysis.

[24]  Nir Sochen,et al.  Do Uncertainty Minimizers Attain Minimal Uncertainty? , 2010 .

[25]  H. Feichtinger,et al.  Banach spaces related to integrable group representations and their atomic decompositions, I , 1989 .

[26]  N. Sochen,et al.  Square Integrable Group Representations and the Uncertainty Principle , 2011 .

[27]  Michael Martin Nieto,et al.  Coherent States , 2009, Compendium of Quantum Physics.

[28]  E. Candès,et al.  Continuous curvelet transform , 2003 .

[29]  E. Candès,et al.  New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities , 2004 .

[30]  R. Carter.  Lie Groups , 1970, Nature.

[31]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[32]  Bruno Torrésani,et al.  Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers , 2008, Structured Decompositions and Efficient Algorithms.

[33]  B. Demange.  Uncertainty Principles for the Ambiguity Function , 2005 .

[34]  Hans G. Feichtinger,et al.  A method for optimizing the ambiguity function concentration , 2012, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO).

[35]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[36]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

[37]  H. Führ.  Abstract Harmonic Analysis of Continuous Wavelet Transforms , 2005 .

[38]  J.P. Costas,et al.  A study of a class of detection waveforms having nearly ideal range—Doppler ambiguity properties , 1983, Proceedings of the IEEE.

[39]  G. Folland.  Harmonic analysis in phase space , 1989 .

[40]  Johannes Grassberger,et al.  A note on representations of the finite Heisenberg group and sums of greatest common divisors , 2001, Discret. Math. Theor. Comput. Sci..

[41]  P. Maass,et al.  The Affine uncertainty principle in one and two dimensions , 1995 .

[42]  H. Feichtinger,et al.  A unified approach to atomic decompositions via integrable group representations , 1988 .

[43]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .