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Zbl 1122.65134
Candès, Emmanuel; Demanet, Laurent; Donoho, David; Ying, Lexing
Fast discrete curvelet transforms. (English)
[J] Multiscale Model. Simul. 5, No. 3, 861-899 (2006). ISSN 1540-3459; ISSN 1540-3467/e

Summary: 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.

MSC 2000:: *65T60 Wavelets
94A08 Image processing
68U10 Image processing
42C40 Wavelets

Keywords: two-dimensional and three-dimensional curvelet transforms; fast Fourier transforms; unequally spaced fast Fourier transforms; smooth partitioning; interpolation; digital shear; filtering; wrapping

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