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Characterization of Piecewise-Smooth Surfaces Using the 3D Continuous Shearlet Transform

Journal of Fourier Analysis and Applications volume 18, pages 488–516 (2012)Cite this article

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Abstract

One of the most striking features of the Continuous Shearlet Transform is its ability to precisely characterize the set of singularities of multivariable functions through its decay at fine scales. In dimension n=2, it was previously shown that the continuous shearlet transform provides a precise geometrical characterization for the boundary curves of very general planar regions, and this property sets the groundwork for several successful image processing applications. The generalization of this result to dimension n=3 is highly nontrivial, and so far it was known only for the special case of 3D bounded regions where the boundary set is a smooth 2-dimensional manifold with everywhere positive Gaussian curvature. In this paper, we extend this result to the general case of 3D bounded regions with piecewise-smooth boundaries, and show that also in this general situation the continuous shearlet transform precisely characterizes the geometry of the boundary set.

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Notes

  1. Notice that the continuous curvelet transform [2] also employs analyzing elements defined at various locations, scales and orientations, and it shares some of the properties of the continuous shearlet transform. However, the shearlet transform has the distinctive feature of being derived from the theory of affine systems, and this provides several advantages in terms of discretization and extensions to higher dimensions [3, 4, 9, 13].

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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, 651 Phillip G Hoffman, Houston, TX, 77204-3008, USA

    Demetrio Labate

Authors
  1. Kanghui Guo
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  2. Demetrio Labate
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Corresponding author

Correspondence to Demetrio Labate.

Additional information

Communicated by Stephan Dahlke.

KG and DL are partially supported by NSF grant DMS 1008900/1008907. DL is partially supported by NSF grant DMS 1005799.

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Guo, K., Labate, D. Characterization of Piecewise-Smooth Surfaces Using the 3D Continuous Shearlet Transform. J Fourier Anal Appl 18, 488–516 (2012). https://doi.org/10.1007/s00041-011-9209-y

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  • DOI: https://doi.org/10.1007/s00041-011-9209-y

Keywords

  • Analysis of singularities
  • Continuous wavelets
  • Curvelets
  • Directional wavelets
  • Edge detection
  • Shearlets
  • Wavelets

Mathematics Subject Classification (2000)

  • 42C15
  • 42C40