Higher - Dimensional Discrete Smith Curvelet Transform : a Parallel Algorithm

We revisit Smith’s frame of curvelets and discretize it making use of the USFFT as developed by Dutt and Roklin, and Beylkin. In the discretization, we directly approximate the underlying dyadic parabolic decomposition, its (rotational) symmetry, and approximate all the necessary decay estimates in phase space with arbitrary accuracy. Numerically, our transform is unitary. Moreover, if we apply the inverse transform after the forward transform, we approximate the identity matrix, as it should, and if we apply the forward transform after the inverse transform we recover the necessary decay estimates (of the matrix representing the identity operator). Another relevant aspect of our discretization is the appearance of parameters that control the tiling of phase space corresponding with the dyadic parabolic decomposition, preserving the relative parabolic scaling, while adapting to the physical problem at hand. We consider applications in exploration seismology and global seismology. For these, we need transforms in higher dimensions, that is, in dimension n = 3, 4, 5, while the data sets and images are inherently very large. We propose a parallel algorithm suited for such large-scale problems in the realm of high performance computing.