Sparsity Equivalence of Anisotropic Decompositions

Anisotropic decompositions using representation systems such as curvelets, contourlet, or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the concept of sparsity equivalence, we aim to provide a framework which allows categorization of the ability for sparse approximations of representation systems. This framework, in particular, enables transferring results on sparse approximations from one system to another. We demonstrate this concept for the example of curvelets and shearlets, and discuss how this viewpoint immediately leads to novel results for both systems.

[1]  Minh N. Do,et al.  Ieee Transactions on Image Processing the Contourlet Transform: an Efficient Directional Multiresolution Image Representation , 2022 .

[2]  D. Labate,et al.  Resolution of the wavefront set using continuous shearlets , 2006, math/0605375.

[3]  E. Candès,et al.  Continuous Curvelet Transform : I . Resolution of the Wavefront Set , 2003 .

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

[5]  E. Candès,et al.  Continuous curvelet transform: II. Discretization and frames , 2005 .

[6]  Wang-Q Lim,et al.  Compactly Supported Shearlets , 2010, 1009.4359.

[7]  Mark J. T. Smith,et al.  A filter bank for the directional decomposition of images: theory and design , 1992, IEEE Trans. Signal Process..

[8]  D. Donoho.  Wedgelets: nearly minimax estimation of edges , 1999 .

[9]  Stéphane Mallat,et al.  Sparse geometric image representations with bandelets , 2005, IEEE Transactions on Image Processing.

[10]  G. Kutyniok,et al.  Irregular Shearlet Frames: Geometry and Approximation Properties , 2010, 1002.2657.

[11]  Wang-Q Lim,et al.  Compactly supported shearlets are optimally sparse , 2010, J. Approx. Theory.

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

[13]  E. Candès,et al.  Recovering edges in ill-posed inverse problems: optimality of curvelet frames , 2002 .

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

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

[16]  Michael Elad,et al.  From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images , 2009, SIAM Rev..

[17]  Emmanuel J. Candès,et al.  Curvelets and Curvilinear Integrals , 2001, J. Approx. Theory.

[18]  E. Candès.  New tight frames of curvelets and optimal representations of objects with C² singularities , 2002 .

[19]  N. Kingsbury.  Complex Wavelets for Shift Invariant Analysis and Filtering of Signals , 2001 .

[20]  E. Candès,et al.  Ridgelets: a key to higher-dimensional intermittency? , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[21]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[22]  Pierre Vandergheynst,et al.  Directional Wavelets Revisited: Cauchy Wavelets and Symmetry Detection in Patterns , 1999 .

[23]  E. Candès,et al.  The curvelet representation of wave propagators is optimally sparse , 2004, math/0407210.

[24]  M. Nielsen,et al.  Frame Decomposition of Decomposition Spaces , 2007 .

[25]  Gitta Kutyniok,et al.  Development of a digital shearlet transform based on Pseudo-Polar FFT , 2009, Optical Engineering + Applications.

[26]  Gitta Kutyniok,et al.  Microlocal Analysis of the Geometric Separation Problem , 2010, ArXiv.