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.
The Gaussian wavepackets that we treat in this article share the geometric features of other parabolic wavepackets including curvelets and shearlets. Specifically, the atoms are concentrated on elongated regions obeying the relation: @math @math and oscillate across their ridge. The packets in are produced exactly by applying a parabolic dilation, rotation and translation to a single window function. In constrast, the classic Gaussian wavepackets from @cite_16 @cite_39 @cite_6 use an additional frequency cut-off to enforce vanishing moments. The curvelets from @cite_5 are produced by parabolic dilation and rotation of a slightly different window at each scale. In the case of shearlets, rotations are replaced by the shear maps @math . Wave packets based on tilings of frequency space are discussed in @cite_35 @cite_20 @cite_41 @cite_11 .
@cite_35 @cite_41 @cite_6 @cite_39 @cite_5 @cite_16 @cite_20 @cite_11
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Abstract In this article, we develop a general method for constructing wavelets | det A j | 1 2 ψ ( A j x − x j , k ) : j ∈ J , k ∈ K on irregular lattices of the form X = x j , k ∈ R d : j ∈ J , k ∈ K , and with an arbitrary countable family of invertible d × d matrices A j ∈ G L d ( R ) : j ∈ J that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L 2 ( R d ) developed in this article, that allow other time frequency decompositions such as nonharmonic Gabor frames with nonuniform covering of the Euclidean space R d . Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology. Abstract In this article we construct affine systems that provide a simultaneous atomic decomposition for a wide class of functional spaces including the Lebesgue spaces L p ( R d ) , 1 p + ∞ . The novelty and difficulty of this construction is that we allow for non-lattice translations. We prove that for an arbitrary expansive matrix A and any set Λ —satisfying a certain spreadness condition but otherwise irregular—there exists a smooth window whose translations along the elements of Λ and dilations by powers of A provide an atomic decomposition for the whole range of the anisotropic Triebel–Lizorkin spaces. The generating window can be either chosen to be bandlimited or to have compact support. To derive these results we start with a known general “painless” construction that has recently appeared in the literature. We show that this construction extends to Besov and Triebel–Lizorkin spaces by providing adequate dual systems. We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Holder regularity s ≥ 2, and (iii) suggests a novel computational algorithm that requires solving for the mentioned geometry on the one hand and solving a matrix Volterra integral equation of the second kind on the other hand. The Volterra equation can be solved by recursion—as in the... 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. In this paper we present a construction of frames generated by a single band-limited function for decomposition smoothness spaces on ( R ^d ) of modulation and Triebel–Lizorkin type. A perturbation argument is then used to construct compactly supported frame generators.