A Unitary Extension Principle for Shearlet Systems

In this paper, we first introduce the concept of an adaptive MRA (AMRA) structure which is a variant of the classical MRA structure suited to the main goal of a fast flexible decomposition strategy adapted to the data at each decomposition level. We then study this novel methodology for the general case of affine-like systems, and derive a Unitary Extension Principle (UEP) for filter design. Finally, we apply our results to the directional representation system of shearlets. This leads to a comprehensive theory for fast decomposition algorithms associated with shearlet systems which encompasses tight shearlet frames with spatially compactly supported generators within such an AMRA structure. Also shearlet-like systems associated with parabolic scaling and unimodular matrices optimally close to rotation as well as 3D shearlet systems are studied within this framework.

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

[2]  B. Han.  On Dual Wavelet Tight Frames , 1997 .

[3]  Jian-Feng Cai,et al.  Blind motion deblurring using multiple images , 2009, J. Comput. Phys..

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

[5]  Laurent Demanet,et al.  Fast Discrete Curvelet Transforms , 2006, Multiscale Model. Simul..

[6]  Raymond H. Chan,et al.  Wavelet Algorithms for High-Resolution Image Reconstruction , 2002, SIAM J. Sci. Comput..

[7]  Bin Han,et al.  Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets , 2001 .

[8]  B. Han.  Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space , 2009, 0907.3501.

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

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

[11]  Jian-Feng Cai,et al.  A framelet-based image inpainting algorithm , 2008 .

[12]  I. Daubechies,et al.  Framelets: MRA-based constructions of wavelet frames☆☆☆ , 2003 .

[13]  B. Han.  DUAL MULTIWAVELET FRAMES WITH HIGH BALANCING ORDER AND COMPACT FAST FRAME TRANSFORM , 2008 .

[14]  B. Han.  Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix , 2003 .

[15]  A. Ron,et al.  Affine Systems inL2(Rd): The Analysis of the Analysis Operator , 1997 .

[16]  Bin Han,et al.  Compactly Supported Symmetric C∞ Wavelets with Spectral Approximation Order , 2008, SIAM J. Math. Anal..

[17]  Raymond H. Chan,et al.  Simultaneously inpainting in image and transformed domains , 2009, Numerische Mathematik.

[18]  Gitta Kutyniok,et al.  Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis , 2007, SIAM J. Math. Anal..

[19]  Bin Han,et al.  Vector cascade algorithms and refinable function vectors in Sobolev spaces , 2003, J. Approx. Theory.

[20]  Rong-Qing Jia,et al.  Approximation properties of multivariate wavelets , 1998, Math. Comput..

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

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

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

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

[25]  R. Chan,et al.  A framelet algorithm for enhancing video stills , 2007 .

[26]  Wang-Q Lim,et al.  Wavelets with composite dilations and their MRA properties , 2006 .

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

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

[29]  Zuowei Shen,et al.  Deconvolution: a wavelet frame approach , 2007, Numerische Mathematik.

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

[31]  Gabriele Steidl,et al.  The Continuous Shearlet Transform in Arbitrary Space Dimensions , 2009, Structured Decompositions and Efficient Algorithms.

[32]  Bin Han,et al.  Symmetry property and construction of wavelets with a general dilation matrix , 2002 .

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

[34]  Zuowei Shen,et al.  Dual Wavelet Frames and Riesz Bases in Sobolev Spaces , 2009 .

[35]  Jian-Feng Cai,et al.  Blind motion deblurring from a single image using sparse approximation , 2009, CVPR.

[36]  R. Chan,et al.  Tight frame: an efficient way for high-resolution image reconstruction , 2004 .

[37]  Raymond H. Chan,et al.  Restoration of Chopped and Nodded Images by Framelets , 2008, SIAM J. Sci. Comput..

[38]  Raymond H. Chan,et al.  Convergence analysis of tight framelet approach for missing data recovery , 2009, Adv. Comput. Math..

[39]  Jian-Feng Cai,et al.  Linearized Bregman Iterations for Frame-Based Image Deblurring , 2009, SIAM J. Imaging Sci..

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

[41]  Zuowei Shen.  Affine systems in L 2 ( IR d ) : the analysis of the analysis operator , 1995 .

[42]  Jian-Feng Cai,et al.  Split Bregman Methods and Frame Based Image Restoration , 2009, Multiscale Model. Simul..