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Geometric Separation in \(\mathbb {R}^3\)

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Abstract

The geometric separation problem, initially posed by Donoho and Kutyniok (Commun Pure Appl Math 66:1–47, 2013), aims to separate a distribution containing a non-trivial superposition of point and curvilinear singularities into its distinct geometric constituents. The solution proposed in Donoho and Kutyniok  (2013) considers expansions with respect to a combined wavelet-curvelet dictionary and applies an \(\ell ^1\)-norm minimization over the expansion coefficients to achieve separation asymptotically at fine scales. However, the original proof of this result uses a heavy machinery relying on sparse representations of Fourier integral operators which does not extend directly to the 3D setting. In this paper, we extend the geometric separation result to the 3D setting using a novel and simpler argument which relies in part on techniques developed by the authors for the shearlet-based analysis of curvilinear edges. Our new result also yields a significantly simpler proof of the original 2D geometric separation problem and extends a prior result by the authors which was limited to piecewise linear singularities.

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Notes

  1. Here we ignore that boundary elements corresponding to \(\ell = \pm 2^j\) are slightly modified as it is irrelevant for our arguments.

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Acknowledgements

DL acknowledges support from NSF grant DMS 1720487, GEAR 113491 and by a grant from the Simon Foundation (422488).

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Authors and Affiliations

  1. Department of Mathematics, Missouri State University, Springfield, MO, 65804, USA

    Kanghui Guo

  2. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Demetrio Labate

Authors
  1. Kanghui Guo
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  2. Demetrio Labate
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Correspondence to Demetrio Labate.

Additional information

Communicated by Gitta Kutyniok.

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Guo, K., Labate, D. Geometric Separation in \(\mathbb {R}^3\). J Fourier Anal Appl 25, 108–130 (2019). https://doi.org/10.1007/s00041-017-9569-z

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  • DOI: https://doi.org/10.1007/s00041-017-9569-z

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