A Review of Curvelets and Recent Applications

Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing interest in the community of applied mathematics and signal processing over the past years. In this paper, we present a review on the curvelet transform, including its history beginning from wavelets, its logical relationship to other multiresolution multidirectional methods like contourlets and shearlets, its basic theory and discrete algorithm. Further, we consider recent applications in image/video processing, seismic exploration, fluid mechanics, simulation of partial different equations, and compressed sensing.

[1]  Charles Fefferman,et al.  Wave packets and fourier integral operators , 1978 .

[2]  Edward H. Adelson,et al.  The Design and Use of Steerable Filters , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

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

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

[5]  Stéphane Mallat,et al.  Wavelets for a vision , 1996, Proc. IEEE.

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

[7]  David J. Field,et al.  Emergence of simple-cell receptive field properties by learning a sparse code for natural images , 1996, Nature.

[8]  Hart F. Smith.  A Hardy space for Fourier integral operators , 1998 .

[9]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[10]  E. Candès.  Harmonic Analysis of Neural Networks , 1999 .

[11]  N. Kingsbury.  Image processing with complex wavelets , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  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.

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

[14]  E. Candès,et al.  Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges , 2000 .

[15]  M. Farge,et al.  Coherent vortex extraction in 3D turbulent flows using orthogonal wavelets. , 2001, Physical review letters.

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

[17]  Xiaoming Huo,et al.  Beamlets and Multiscale Image Analysis , 2002 .

[18]  Emmanuel J. Candès,et al.  The curvelet transform for image denoising , 2002, IEEE Trans. Image Process..

[19]  Truong Q. Nguyen,et al.  A study of two-channel complex-valued filterbanks and wavelets with orthogonality and symmetry properties , 2002, IEEE Trans. Signal Process..

[20]  Emmanuel J. Candès,et al.  New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction , 2002, Signal Process..

[21]  E. Candès,et al.  Curvelets and Fourier Integral Operators , 2003 .

[22]  Fionn Murtagh,et al.  Gray and color image contrast enhancement by the curvelet transform , 2003, IEEE Trans. Image Process..

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

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

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

[26]  Robert D. Nowak,et al.  Platelets: a multiscale approach for recovering edges and surfaces in photon-limited medical imaging , 2003, IEEE Transactions on Medical Imaging.

[27]  Bruno A Olshausen,et al.  Sparse coding of sensory inputs , 2004, Current Opinion in Neurobiology.

[28]  E. Candès,et al.  The curvelet representation of wave propagators is optimally sparse , 2004, math/0407210.

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

[30]  Myeong-Ryong Nam,et al.  Fusion of multispectral and panchromatic Satellite images using the curvelet transform , 2005, IEEE Geosci. Remote. Sens. Lett..

[31]  Gabriele Steidl,et al.  Dual-Tree Complex Wavelet Transform in the Frequency Domain and an Application to Signal Classification , 2005, Int. J. Wavelets Multiresolution Inf. Process..

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

[33]  Michael Elad,et al.  Submitted to Ieee Transactions on Image Processing Image Decomposition via the Combination of Sparse Representations and a Variational Approach , 2022 .

[34]  Lexing Ying,et al.  3D discrete curvelet transform , 2005, SPIE Optics + Photonics.

[35]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

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

[37]  Felix J. Herrmann,et al.  Seismic denoising with nonuniformly sampled curvelets , 2006, Computing in Science & Engineering.

[38]  Jianwei Ma,et al.  Combined Complex Ridgelet Shrinkage and Total Variation Minimization , 2006, SIAM J. Sci. Comput..

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

[40]  Stéphane Mallat,et al.  A review of Bandlet methods for geometrical image representation , 2007, Numerical Algorithms.

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

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

[43]  L. Demanet,et al.  Wave atoms and sparsity of oscillatory patterns , 2007 .

[44]  Mohamed-Jalal Fadili,et al.  Morphological Component Analysis: An Adaptive Thresholding Strategy , 2007, IEEE Transactions on Image Processing.

[45]  Jianwei Ma,et al.  Curvelets for surface characterization , 2007 .

[46]  Huub Douma,et al.  Leading-order seismic imaging using curvelets , 2007 .

[47]  Gerlind Plonka-Hoch,et al.  Combined Curvelet Shrinkage and Nonlinear Anisotropic Diffusion , 2007, IEEE Transactions on Image Processing.

[48]  M. Y. Hussaini,et al.  A refining estimation for adaptive solution of wave equation based on curvelets , 2007, SPIE Optical Engineering + Applications.

[49]  Jianwei Ma,et al.  Three-dimensional curvelets for coherent vortex analysis of turbulence , 2007 .

[50]  Laurent Demanet,et al.  Curvelets and wave atoms for mirror-extended images , 2007, SPIE Optical Engineering + Applications.

[51]  Mohamed-Jalal Fadili,et al.  Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal , 2008, IEEE Transactions on Image Processing.

[52]  R. Neelamani,et al.  Coherent and random noise attenuation using the curvelet transform , 2008 .

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

[54]  Aleksandra Pizurica,et al.  Context adaptive image denoising through modeling of curvelet domain statistics , 2008, J. Electronic Imaging.

[55]  D. J. Verschuur,et al.  Adaptive curvelet-domain primary-multiple separation , 2008 .

[56]  Ivan Bermejo-Moreno,et al.  On the non-local geometry of turbulence , 2008, Journal of Fluid Mechanics.

[57]  Hervé Chauris,et al.  Seismic demigration/migration in the curvelet domain , 2008 .

[58]  Xiangchu Feng,et al.  Structure and Texture Image Inpainting Using Sparse Representations and an Iterative Curvelet Thresholding Approach , 2008, Int. J. Wavelets Multiresolution Inf. Process..

[59]  Zhengding Qiu,et al.  Multipurpose Watermarking Based on Multiscale Curvelet Transform , 2008, IEEE Transactions on Information Forensics and Security.

[60]  Gerlind Plonka-Hoch,et al.  The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data , 2008, Multiscale Model. Simul..

[61]  Gerlind Plonka-Hoch,et al.  Nonlinear Regularized Reaction-Diffusion Filters for Denoising of Images With Textures , 2008, IEEE Transactions on Image Processing.

[62]  Fredrik Andersson,et al.  A Multi-Scale Approach to Hyperbolic Evolution Equations with Limited Smoothness , 2008 .

[63]  F. Herrmann,et al.  Sparsity- and continuity-promoting seismic image recovery with curvelet frames , 2008 .

[64]  Petros Koumoutsakos,et al.  Edge detection in microscopy images using curvelets , 2009, BMC Bioinformatics.

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

[66]  Seismic imaging in the curvelet domain: achievements and perspectives , 2009 .

[67]  F. Herrmann,et al.  Optimized Compressed Sensing for Curvelet-based Seismic Data Reconstruction , 2009 .

[68]  François-Xavier Le Dimet,et al.  Multiscale geometric analysis of turbulence by curvelets , 2009 .

[69]  François-Xavier Le Dimet,et al.  Deblurring From Highly Incomplete Measurements for Remote Sensing , 2009, IEEE Transactions on Geoscience and Remote Sensing.

[70]  W. Hackbusch,et al.  Black Box Low Tensor-Rank Approximation Using Fiber-Crosses , 2009 .

[71]  Henryk Wozniakowski,et al.  Approximation of infinitely differentiable multivariate functions is intractable , 2009, J. Complex..

[72]  E. Novak,et al.  Optimal Order of Convergence and (In)Tractability of Multivariate Approximation of Smooth Functions , 2009 .

[73]  Jianwei Ma,et al.  Comparisons of wavelets, contourlets and curvelets in seismic denoising , 2009 .

[74]  Jianwei Ma,et al.  Single-Pixel Remote Sensing , 2009, IEEE Geoscience and Remote Sensing Letters.

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

[76]  Gerlind Plonka-Hoch,et al.  Curvelet-Wavelet Regularized Split Bregman Iteration for Compressed Sensing , 2011, Int. J. Wavelets Multiresolution Inf. Process..

[77]  G. Easley,et al.  Shearlet Based Total Variation for Denoising , 2022 .