ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet...

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

[2]  Bin Han,et al.  Adaptive Multiresolution Analysis Structures and Shearlet Systems , 2011, SIAM J. Numer. Anal..

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

[4]  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.

[5]  MalekiArian,et al.  Reproducible Research in Computational Harmonic Analysis , 2009 .

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

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

[8]  Arian Maleki,et al.  Reproducible Research in Computational Harmonic Analysis , 2009, Computing in Science & Engineering.

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

[10]  E. Candès,et al.  Astronomical image representation by the curvelet transform , 2003, Astronomy & Astrophysics.

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

[12]  K. Gröchenig,et al.  Numerical and Theoretical Aspects of Nonuniform Sampling of Band-Limited Images , 2001 .

[13]  R. Lipsman Abstract harmonic analysis , 1968 .

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

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

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

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

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

[19]  Felix J. Herrmann,et al.  Non-parametric seismic data recovery with curvelet frames , 2008 .

[20]  Ronald R. Coifman,et al.  A Framework for Discrete Integral Transformations I-The Pseudopolar Fourier Transform , 2008, SIAM J. Sci. Comput..

[21]  K. Gröchenig A discrete theory of irregular sampling , 1993 .

[22]  Bin Han,et al.  A Unitary Extension Principle for Shearlet Systems , 2009, 0912.4529.

[23]  K. Gröchenig RECONSTRUCTION ALGORITHMS IN IRREGULAR SAMPLING , 1992 .

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

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

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

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

[28]  David H. Bailey,et al.  The Fractional Fourier Transform and Applications , 1991, SIAM Rev..

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

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

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

[32]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

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