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.
Closest to our work is the construction of curvelet-type expansions in @cite_36 . Whereas we construct frames by using an analogue of Daubechies criterion for wavelets, the curvelet-type frames for @math in @cite_36 are obtained by using a perturbation argument. The main focus of that work is the production of compactly-supported atoms, but the results also apply to sums of modulated Gaussians. As mentioned in [Section 6] rani12 , the perturbation argument yields a frame where each element is a linear combination of dilated, rotated and translated Gaussians. However, the coefficients in that linear combination depend on the indices of the particular frame element, but the number of terms is proved to be uniformly bounded. By contrast, the packets in consist exactly of parabolic dilations, rotations and translations of a function, which may be taken to be a fixed linear combination of Gaussians.
@cite_36
2034276301
We study a flexible method for constructing curvelet-type frames. These curvelet-type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to give a constructive description of how to construct curvelet-type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet-type spaces. Such a system of curvelet molecules can then be constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay.