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.
While the differences between the various parabolic wavepackets are very relevant in practice, their asymptotic properties are similar. The Gaussian wavepackets in are an example of from @cite_22 and therefore they provide the same sparse approximation properties as curvelets for functions with discontinuities along smooth edges @cite_5 . The notion of curvelet molecules and the related notion of in @cite_4 are in turn particular cases of the notion of , as recently introduced in @cite_10 . In that article it is shown that all expansions with adequate systems of parabolic molecules share similar asymptotic properties. As a consequence, our results on the decay of frame coefficients away from the wavefront set can be transferred to other parabolic systems.
@cite_5 @cite_4 @cite_22 @cite_10
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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. 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. We discuss a Continuous Curvelet Transform (CCT), a transform f → Γf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b ∈ R^2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) = f, γabθ where the inner products project f onto analyzing elements called curvelets γ_(abθ) which are smooth and of rapid decay away from an a by √a rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to ‘track’ the behavior of singularities along curves. They are continuum scale space orientation analogs of the discrete frame of curvelets discussed in Candes and Donoho (2002). We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Γf (a, x0, θ0) decays rapidly as a → 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of Γf (a, x0, θ0) for fixed (x0, θ0), as a → 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Γf (a, x, θ) is not of rapid decay as a → 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the ‘directional parabolic square function’ S^m(x, θ) = ( ʃ|Γf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1 2) is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set. It is well documented that natural images are compressible in wavelet bases and tend to exhibit fractal properties. In this paper, we investigate statistical models that mimic these behaviors. We then use our models to make predictions on the statistics of the wavelet coefficients. Following an innovation modeling approach, we identify a general class of finite-variance self-similar sparse random processes. We first prove that spatially dilated versions of self-similar sparse processes are asymptotically Gaussian as the dilation factor increases. Based on this fundamental result, we show that the coarse-scale wavelet coefficients of these processes are also asymptotically Gaussian, provided the wavelet has enough vanishing moments. Moreover, we quantify the degree of Gaussianity by deriving the theoretical evolution of the kurtosis of the wavelet coefficients across scales. Finally, we apply our analysis to one- and two-dimensional signals, including natural images, and show that the wavelet coefficients ...