Fast Discrete Curvelet Transforms

TLDR

The paper presents two digital implementations of the second‑generation curvelet transform in two and three dimensions. The first implementation uses unequally spaced fast Fourier transforms, while the second employs wrapping of selected Fourier samples, differing mainly in the spatial grid used to translate curvelets at each scale and angle. Both implementations produce a table of curvelet coefficients indexed by scale, orientation, and location, run in O(n² log n) flops, are invertible with similar complexity, and outperform earlier first‑generation curvelets by being conceptually simpler, faster, and less redundant. The software CurveLab, implementing both transforms, is available at http://www.curvelet.org.

Abstract

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.