Fast Discrete Curvelet Transforms

This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.

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

[2]  David L. Donoho,et al.  Digital curvelet transform: strategy, implementation, and experiments , 2000, SPIE Defense + Commercial Sensing.

[3]  Wang-Q Lim,et al.  Wavelets with composite dilations , 2004 .

[4]  Chris Anderson,et al.  Rapid Computation of the Discrete Fourier Transform , 1996, SIAM J. Sci. Comput..

[5]  Raymond H. Chan,et al.  Conjugate Gradient Methods for Toeplitz Systems , 1996, SIAM Rev..

[6]  D. J. Verschuur,et al.  Separation of Primaries and Multiples by Non-Linear Estimation in the Curvelet Domain , 2004, 66th EAGE Conference & Exhibition.

[7]  Hart F. Smith.  A parametrix construction for wave equations with $C^{1,1}$ coefficients , 1998 .

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

[9]  G. Beylkin.  On the Fast Fourier Transform of Functions with Singularities , 1995 .

[10]  Felix J. Herrmann,et al.  Sparseness‐ and continuity‐constrained seismic imaging , 2005 .

[11]  N. Aghanim,et al.  Detection and discrimination of cosmological non-Gaussian signatures by multi-scale methods , 2003, astro-ph/0311577.

[12]  D. Donoho,et al.  Redundant Multiscale Transforms and Their Application for Morphological Component Separation , 2004 .

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

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

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

[16]  M. Do.  Directional multiresolution image representations , 2002 .

[17]  Huub Douma,et al.  Wave-character Preserving Pre-stack Map Migration Using Curvelets , 2004 .

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

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

[20]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[21]  E. Candès,et al.  Recovering edges in ill-posed inverse problems: optimality of curvelet frames , 2002 .

[22]  Eero P. Simoncelli.  Shiftable multi-scale transforms [or "What's wrong with orthonormal wavelets"] , 1992 .

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

[24]  A. G. Flesia,et al.  Digital Implementation of Ridgelet Packets , 2003 .

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

[26]  P. Heywood.  Trigonometric Series , 1968, Nature.

[27]  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.

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

[29]  Vladimir Rokhlin,et al.  Fast Fourier Transforms for Nonequispaced Data , 1993, SIAM J. Sci. Comput..

[30]  A. Duijndam,et al.  Nonuniform Fast Fourier Transform , 1997 .

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

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

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

[34]  William T. Freeman,et al.  Presented at: 2nd Annual IEEE International Conference on Image , 1995 .

[35]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[36]  Pierre Vandergheynst,et al.  Directional dyadic wavelet transforms: design and algorithms , 2002, IEEE Trans. Image Process..

[37]  Emmanuel J. Candès,et al.  The curvelet transform for image denoising , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[38]  V. Rokhlin,et al.  Fast Fourier Transforms for Nonequispaced Data, II , 1995 .

[39]  G. Strang.  A proposal for toeplitz matrix calculations , 1986 .