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.

This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [9,5] in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width ≈ length 2 at fine scales. We prove that for a wide class of linear hyperbolic differential equations, the curvelet representation of the solution operator is both optimally sparse and well organized.• It is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e. faster than any negative polynomial),• and well-organized in the sense that the very few nonnegligible entries occur near a shifted diagonal.Indeed, we actually show that the action of the wave-group on a curvelet is wellapproximated by simply translating the center of the curvelet along the Hamiltonian flow-hence the diagonal shift in the curvelet representation. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.

This paper is concerned with the fast computation of Fourier integral operators of the general form R d e 2πıΦ(x,k) f (k)dk, where k is a frequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. In two dimensions, when the input and output are sampled on N × N Cartesian grids, a direct evaluation requires O(N 4 ) operations, which is often times prohibitively expensive.This paper introduces a novel algorithm running in O(N 2 log N ) time, i. e. with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm [30]. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel e 2πıΦ(x,k) to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel has approximately low-rank; we propose constructing such low-rank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part and, lastly, remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.

We introduce "wave atoms" as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2 . We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the "wrapping" technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shift-invariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.

We address the problem of recovering an n-vector from m linear measurements lacking sign or phase information. We show that lifting and semidefinite relaxation suffice by themselves for stable recovery in the setting of m = O(n log n) random sensing vectors, with high probability. The recovery method is optimizationless in the sense that trace minimization in the PhaseLift procedure is unnecessary. That is, PhaseLift reduces to a feasibility problem. The optimizationless perspective allows for a Douglas-Rachford numerical algorithm that is unavailable for PhaseLift. This method exhibits linear convergence with a favorable convergence rate and without any parameter tuning.Acknowledgements. The authors acknowledge generous funding from the National Science Foundation, the Alfred P. Sloan Foundation, TOTAL S.A., and the Air Force Office of Scientific Research. The authors would also like to thank Xiangxiong Zhang for helpful discussions.