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Geometric Separation of Singularities Using Combined Multiscale Dictionaries

Journal of Fourier Analysis and Applications volume 21, pages 667–693 (2015)Cite this article

Abstract

Several empirical results appeared in the literature during the last decade have shown that it is often possible to separate images and other multidimensional data into geometrically distinct constituents. A rigorous mathematical analysis of the geometric separation problem in the two-dimensional setting was recently introduced by Donoho and Kutyniok (Comm Pure Appl Math 66:1–47, 2013), who proposed a mathematical framework to separate point and smooth curve singularities in 2D images using a combined dictionary consisting of curvelets and wavelets. In this paper, we adapt their approach and introduce a novel argument to extend geometric separation to the three-dimensional setting. We show that it is possible to separate point and piecewise linear singularities in 3D using a combined dictionary consisting of shearlets and wavelets. Our new approach takes advantage of the microlocal properties of the shearlet transform and has the ability to handle singularities containing vertices and corner points, which cannot be handled using the original arguments.

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Fig. 1

Notes

  1. Note: as also remarked in [6], the geometric separation we consider here is rather different from the separation of texture from smooth structure where sparsity in the representation does not play a relevant role.

  2. Here we ignore the fact that the boundary elements corresponding to \(\ell = \pm 2^j\) are slightly modified, since this is irrelevant for all our arguments.

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Acknowledgments

The authors acknowledge support from NSF Grant DMS 1008900/1008907; the second author also acknowledges support from NSF Grant DMS 1005799 and DMS 1320910.

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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.

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Communicated by Joel Tropp.

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Guo, K., Labate, D. Geometric Separation of Singularities Using Combined Multiscale Dictionaries. J Fourier Anal Appl 21, 667–693 (2015). https://doi.org/10.1007/s00041-014-9381-y

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  • DOI: https://doi.org/10.1007/s00041-014-9381-y

Keywords

  • Cluster coherence
  • Geometric separation
  • \(\ell ^1\) Minimization
  • Shearlets
  • Sparse representations
  • Wavelets

Mathematics Subject Classification

  • 42C15
  • 42C40
  • 94A11