Representation of Fourier Integral Operators Using Shearlets

Abstract

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However, these methods are not equally effective in dealing with Fourier Integral Operators in general. In this article, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an affine-like system of well-localized waveforms at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candès and Demanet using curvelets, which illustrates the benefits of directional multiscale representations (such as curvelets and shearlets) in the study of those functions and operators where traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space.

Shearlet Smoothness Spaces

“…In this section, we show that is possible to characterize the same spaces using frames of shearlet-like systems which are not necessarily band-limited. We recall the notion of shearlet molecules from [19]. Let the matrices A (h) , B (h) , h = 1, 2, be as in Section 3 and…”

“…One of the main tools of the new method consists in extending the machinery of almost diagonal matrices to Triebel-Lizorkin type spaces. Interestingly, in Section 4.6, we use the almost orthogonality of shearlets (from a work about almost diagonal shearlet decomposition of Fourier Integral Operators by one of us in [19]) to characterize the shearlet smoothness spaces using shearlet molecules, a shearlet-like family of functions which are not required to be band-limited. The recent paper by Vera [29] adapts the framwork of Triebel-Lizorking spaces to the shearlet decomposition.…”

The Construction of Smooth Parseval Frames of Shearlets

“…Indeed, one can try to define abstractly the type conditions which are required by a generic shearlet-like system in order to form a sparse representation system and this leads to a notion of shearlet molecule. This type of notion was originally introduced by the authors in [15], in connection with the study of the shearlet representation of Fourier Integral Operators, where the following definition was introduced (in dimension D = 2).…”

“…These properties are particularly useful, as discussed in [15], where it is shown that there is a notion of almost orthogonality associated with the shearlet molecules. Related to this, it is useful to recall that shearlet molecules were recently used to develop a notion of sparsity equivalence in [27], implying, essentially, that all systems satisfying the conditions given in Definition 9 share the same sparsity properties.…”

Discrete shearlet transform on GPU with applications in anomaly detection and denoising

“…The shearlet decomposition has been successfully employed in many problems from applied mathematics and signal processing, including decomposition of operators [11], inverse problems [12,13], edge detection http://asp.eurasipjournals.com/content/2014/1/64 [14][15][16], image separation [17], and image restoration [18][19][20]. However, one major bottleneck to the wider applicability of the shearlet transform is that current discrete implementations tend to be very time consuming, making its use impractical for large data sets and for realtime applications.…”