The curvelet representation of wave propagators is optimally sparse

Abstract

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.

Microlocal Analysis of the Geometric Separation Problem

“…A suitable distance has first been introduced by Hart Smith in [38]. Our analysis will employ results obtained by Candès and Demanet in their work on the curvelet representation of wave propagators [5], in which they use the following variation of Hart Smith's distance:…”

“…In [5], this distance was then extended to derive a distance adapted to discrete curvelet indices, which means, in particular, including the scaling component. For a pair of curvelet indices η = (j, k, ℓ) and η ′ = (j ′ , k ′ , ℓ), this so-called dyadic-parabolic pseudo-distance is defined by…”

The Construction of Smooth Parseval Frames of Shearlets

“…Notice that this definition of shearlet molecule is similar to the curvelet molecules introduced by Candès and Demanet [1]. Both definitions adapt the notion of vaguelettes of Coifman and Meyer [32].…”

“…[14,31]). For an appropriate γ ∈ L 2 (R 2 ), a system of shearlets is defined as a collection of functions of the form Notice that the matrix A (1) is a dilation matrix whose integer powers produce anisotropic dilations and, more specifically, parabolic scaling dilations similar to those used in the curvelet construction; the shearing matrix B (1) is non-expanding and its integer powers control the directional features of the shearlet system. The generator function γ is defined in the frequency domain aŝ…”

“…Recall that the elements of the system (2.6) are of the form ψ (1) j,ℓ,k (x) = | det A (1) Next, a direct calculation show that, for all M ∈ N, there is a constant C M such that…”