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Irregular Shearlet Frames: Geometry and Approximation Properties

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Abstract

Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2-dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far.

In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L 2(ℝ2), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such as oversampled shearlet systems and co-shearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.

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Authors and Affiliations

  1. Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, 90112, Thailand

    Pisamai Kittipoom

  2. Institute of Mathematics, University of Osnabrück, 49069, Osnabrück, Germany

    Gitta Kutyniok & Wang-Q Lim

Authors
  1. Pisamai Kittipoom
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  2. Gitta Kutyniok
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  3. Wang-Q Lim
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Corresponding author

Correspondence to Gitta Kutyniok.

Additional information

Communicated by Karlheinz Gröchenig.

P.K. acknowledges support from the Prince of Songkla University Grant PSU 0802/2548. G.K. would like to thank Stephan Dahlke, Christopher Heil, Demetrio Labate, Gabriele Steidl, and Gerd Teschke for various discussions on related topics. G.K. and W.-Q L. acknowledge support from DFG Grant SPP-1324, KU 1446/13-1. The authors would like to thank the referees for valuable comments and suggestions.

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Kittipoom, P., Kutyniok, G. & Lim, WQ. Irregular Shearlet Frames: Geometry and Approximation Properties. J Fourier Anal Appl 17, 604–639 (2011). https://doi.org/10.1007/s00041-010-9163-0

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  • DOI: https://doi.org/10.1007/s00041-010-9163-0

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