We construct frames of wavepackets produced by parabolic dilation, rotation and translation of (a finite sum of) Gaussians and give asymptotics on the analogue of Daubechies frame criterion. We show that the coefficients in the corresponding approximate expansion decay fast away from the wavefront set of the original data.
This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 -j , each element has an envelope that is aligned along a ridge of length 2 -j 2 and width 2 -j . We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ∥f - f C n ∥ 2 L2 ≤ C . n -2 . (log n) 3 , n → ∞. This rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior. We introduce 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 signicantly sparser expansion in wave atoms than in other xed 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 " 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 ngerprint images demonstrate the potential of the tool introduced. 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 [7, 9] in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width length2 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 few shifted diagonals. Indeed, we show that the wave group maps each curvelet onto a sum of curveletlike waveforms whose locations and orientations are obtained by following the different Hamiltonian flows - hence the diagonal shifts 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. Curvelets are plausible candidates for simultaneous compressionofseismicdata,theirimages,andtheimagingoperator itself. We show that with curvelets, the leading-order approximation in angular frequency, horizontal wavenumber, and migrated location to common-offset CO Kirchhoff depth migration becomes a simple transformation of coordinates of curvelets in the data, combined with amplitude scaling. This transformation is calculated using map migration, whichemploysthelocalslopesfromthecurveletdecomposition of the data. Because the data can be compressed using curvelets, the transformation needs to be calculated for relatively few curvelets only. Numerical examples for homogeneous media show that using the leading-order approximationonlyprovidesagoodapproximationtoCOmigrationfor moderate propagation times.As the traveltime increases and raysdivergebeyondthespatialsupportofacurvelet;however, the leading-order approximation is no longer accurate enough. This shows the need for correction beyond leading order,evenforhomogeneousmedia.