The curvelet representation of wave propagators is optimally sparse

By E.J. Candes, Laurent Demanet. 2004

322 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geophysics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e74c49438d902547b88e4d3eba7440b17caeeadb

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Citations in the corpus

Block Compressed Sensing of Natural Images (2007, background): “, the curvelet [17]), better results can be expected.”
The Curvelet Transform (2010, background): “Wave fronts of solutions can be also sparsely represented in curvelet domain [6].”
Fast Discrete Curvelet Transforms (2006, background): “4 Higher Dimensions Curvelets exist in any dimension [5].”
Deblending of Simultaneous-source Seismic Data using Fast Iterative Shrinkage-thresholding Algorithm with Firm-thresholding (2016, background): “Moreover, Candes and Tao (2005) and Donoho (2006) have shown that solving Eq.”
Seislet transform and seislet frame (2010, background): “Curvelets are particularly appropriate for seismic data because they provide a provably optimal decomposition of wavepropagation operators (Candés and Demanet, 2005).”
Curvelet-based POCS interpolation of nonuniformly sampled seismic records (2012, background): “• The parabolic constructed curvelet exhibits many intriguing properties, such as fast implementation, multiscale, multidirectional features, and excellent localization in spatial and frequency domain (Candès and Demanet, 2005; Herrmann et al., 2007), as can be shown in Fig.”
Non-linear primary-multiple separation with directional curvelet frames (2007, methodology): “The recently developed curvelet transform (see e.g. Candès & Donoho 2004) compose signals in terms of waveforms that are multiscale and multidirectional.”
Wave atoms and sparsity of oscillatory patterns (2007, background): “Such oscillatory integrals and their generalizations were studied in [5,6], where it was shown (among other results) that they preserve the p sparsity in curvelets.”