A split-augmented Lagrangian algorithm for spectral factorization of a set of 2D directional filters and application to the design of compact shearlet frames

In this paper, we first briefly review the directional properties of the Dual-Tree complex wavelet transform and we investigate how the directional selectivity of the transform can be increased (i.e., to obtain more than 6 orientations per scale). To this end, we describe a new augmented Lagrangian optimization algorithm to jointly perform the 2D spectral factorization of a set of 2D directional filters, with a high numerical accuracy. We demonstrate how this approach can be used to design compactly supported shearlet frames that are tight. Finally, a number of experimental results are given to show the merits of the resulting shearlet frames.

[1]  Benjamin Belzer,et al.  Complex, linear-phase filters for efficient image coding , 1995, IEEE Trans. Signal Process..

[2]  Martin J. Wainwright,et al.  Image denoising using scale mixtures of Gaussians in the wavelet domain , 2003, IEEE Trans. Image Process..

[3]  Baltasar Beferull-Lozano,et al.  Directionlets: anisotropic multidirectional representation with separable filtering , 2006, IEEE Transactions on Image Processing.

[4]  Björn Svensson,et al.  Approximate spectral factorization for design of efficient sub-filter sequences , 2008 .

[5]  T.,et al.  Shiftable Multi-scale TransformsEero , 1992 .

[6]  Minh N. Do,et al.  Multidimensional Directional Filter Banks and Surfacelets , 2007, IEEE Transactions on Image Processing.

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

[8]  W. Lim,et al.  Construction of Compactly Supported Shearlet Frames , 2010, 1003.5481.

[9]  Justin K. Romberg,et al.  Multiscale wedgelet image analysis: fast decompositions and modeling , 2002, Proceedings. International Conference on Image Processing.

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

[11]  Minh N. Do,et al.  The finite ridgelet transform for image representation , 2003, IEEE Trans. Image Process..

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

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

[14]  Gabriel Cristóbal,et al.  Self-Invertible 2D Log-Gabor Wavelets , 2007, International Journal of Computer Vision.

[15]  Hamid Krim,et al.  A Shearlet Approach to Edge Analysis and Detection , 2009, IEEE Transactions on Image Processing.

[16]  Edward H. Adelson,et al.  Shiftable multiscale transforms , 1992, IEEE Trans. Inf. Theory.

[17]  Tai Sing Lee,et al.  Image Representation Using 2D Gabor Wavelets , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

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

[19]  J. Lina,et al.  Complex Daubechies Wavelets , 1995 .

[20]  Wilfried Philips,et al.  Design of a tight frame of 2D shearlets based on a fast non-iterative analysis and synthesis algorithm , 2011, Optical Engineering + Applications.

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

[22]  Richard Baraniuk,et al.  The Dual-tree Complex Wavelet Transform , 2007 .

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

[24]  Jon F. Claerbout,et al.  The Wilson-Burg method of spectral factorization with application to helical filtering , 2003 .

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

[26]  Wilfried Philips,et al.  Efficient multiscale and multidirectional representation of 3D data using the 3D discrete shearlet transform , 2011, Optical Engineering + Applications.

[27]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

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

[29]  Ronald R. Coifman,et al.  Brushlets: A Tool for Directional Image Analysis and Image Compression , 1997 .

[30]  Ramesh A. Gopinath,et al.  The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform , 2003, IEEE Trans. Signal Process..