Cartoon Approximation with -Curvelets

It is well-known that curvelets provide optimal approximations for so-called cartoon images which are dened as piecewise C 2 -functions, separated by a C 2 singularity curve. In this paper, we consider the more general case of piecewise C -functions, separated by a C singularity curve for 2 (1; 2]. We rst prove a benchmark result for the possibly achievable best N-term approximation rate for this more general signal model. Then we introduce what we call -curvelets, which are systems that interpolate between wavelet systems on the one hand ( = 1) and curvelet systems on the other hand ( = 1 ). Our main result states that those frames achieve this optimal rate for = 1 , up to log-factors.

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

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

[3]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[4]  T. Berger.  Rate-Distortion Theory , 2003 .

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

[6]  R. Gribonval,et al.  Nonlinear Approximation with Dictionaries I. Direct Estimates , 2004 .

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

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

[9]  Gitta Kutyniok,et al.  Shearlets: Multiscale Analysis for Multivariate Data , 2012 .

[10]  Wang-Q Lim,et al.  Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames , 2011, SIAM J. Math. Anal..

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

[12]  P. Grohs.  Ridgelet-type Frame Decompositions for Sobolev Spaces related to Linear Transport , 2012 .

[13]  Gitta Kutyniok,et al.  α-Molecules : Curvelets , Shearlets , Ridgelets , and Beyond , 2013 .

[14]  Gitta Kutyniok,et al.  Alpha molecules: curvelets, shearlets, ridgelets, and beyond , 2013, Optics & Photonics - Optical Engineering + Applications.

[15]  Gitta Kutyniok,et al.  Parabolic Molecules , 2012, Found. Comput. Math..