Compactly supported shearlets are optimally sparse

Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2 discontinuity curve, have by now become a standard model for measuring sparse (nonlinear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit sparse approximations within this model, which are optimal up to a log-factor. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications. In this paper, we present the first complete proof of optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of (non-tight) shearlet frames with shearlet generators having compact support and satisfying some weak directional vanishing moment conditions.

[1]  Steffen Dereich,et al.  Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction , 2011, 1101.1369.

[2]  Steffen Dereich,et al.  A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations , 2011 .

[3]  G. Easley,et al.  Sparse directional image representations using the discrete shearlet transform , 2008 .

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

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

[6]  Wang-Q Lim,et al.  Sparse multidimensional representation using shearlets , 2005, SPIE Optics + Photonics.

[7]  G. Teschke,et al.  Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints , 2010 .

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

[9]  Lutz Kämmerer,et al.  On the stability of the hyperbolic cross discrete Fourier transform , 2011, Numerische Mathematik.

[10]  G. Teschke,et al.  Inversion of the noisy Radon transform on SO(3) by Gabor frames and sparse recovery principles , 2011 .

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

[12]  G. Kutyniok,et al.  Construction of Compactly Supported Shearlet Frames , 2010, 1003.5481.

[13]  Wang-Q Lim.  Discrete Shearlet Transform : New Multiscale Directional Image Representation , 2009 .

[14]  Tobias Jahnke,et al.  Solving chemical master equations by adaptive wavelet compression , 2010, J. Comput. Phys..

[15]  E. Ullmann,et al.  Preconditioning Stochastic Galerkin Saddle Point Problems , 2022 .

[16]  Wang-Q Lim,et al.  Edge analysis and identification using the continuous shearlet transform , 2009 .

[17]  G. Plonka.  The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data , 2009 .

[18]  G. Teschke,et al.  A compressive Landweber iteration for solving ill-posed inverse problems , 2008 .

[19]  Winfried Sickel,et al.  Best m-term approximation and Lizorkin-Triebel spaces , 2011, J. Approx. Theory.

[20]  Fred J. Hickernell,et al.  Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN , 2010, J. Complex..

[21]  K. Bredies,et al.  Regularization with non-convex separable constraints , 2009 .

[22]  Erwan Faou,et al.  Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets , 2009, SIAM J. Sci. Comput..

[23]  Christian Bender,et al.  Dual pricing of multi-exercise options under volume constraints , 2011, Finance Stochastics.

[24]  Stephan Dahlke,et al.  Adaptive wavelet methods and sparsity reconstruction for inverse heat conduction problems , 2010, Adv. Comput. Math..

[25]  Elisabeth Ullmann,et al.  Stochastic Galerkin Matrices , 2010, SIAM J. Matrix Anal. Appl..

[26]  CONSTRUCTION OF REGULAR AND IRREGULAR SHEARLET FRAMES , 2007 .

[27]  I. Daubechies,et al.  Tree Approximation and Optimal Encoding , 2001 .

[28]  Erich Novak,et al.  Optimal approximation of elliptic problems by linear and nonlinear mappings IV: Errors in L2 and other norms , 2004, J. Complex..

[29]  Dajana Conte,et al.  Mathematical Modelling and Numerical Analysis an Error Analysis of the Multi-configuration Time-dependent Hartree Method of Quantum Dynamics , 2022 .

[30]  B. M. Fulk.  MATH , 1992 .

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

[32]  R. Schilling,et al.  Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise , 2009, 0911.4681.

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

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

[35]  Wang-Q Lim,et al.  The Discrete Shearlet Transform: A New Directional Transform and Compactly Supported Shearlet Frames , 2010, IEEE Transactions on Image Processing.

[36]  Justin K. Romberg,et al.  Image compression using an efficient edge cartoon + texture model , 2002, Proceedings DCC 2002. Data Compression Conference.

[37]  Daniela Rosca,et al.  A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform , 2011, IEEE Transactions on Image Processing.

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

[39]  Daniel Rudolf,et al.  Error bounds for computing the expectation by Markov chain Monte Carlo , 2009, Monte Carlo Methods Appl..

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

[41]  Lars Grasedyck,et al.  Hierarchical Singular Value Decomposition of Tensors , 2010, SIAM J. Matrix Anal. Appl..

[42]  M. Fornasier,et al.  Multilevel Preconditioning for Adaptive Sparse Optimization , 2009 .

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

[44]  D. Donoho.  Sparse Components of Images and Optimal Atomic Decompositions , 2001 .