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Shearlets pp 39–67Cite as

Shearlets and Microlocal Analysis

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Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Although wavelets are optimal for describing pointwise smoothness properties of univariate functions, they fail to efficiently characterize the subtle geometric phenomena of multidimensional singularities in high-dimensional functions. Mathematically these phenomena can be captured by the notion of the wavefront set which describes point- and direction-wise smoothness properties of tempered distributions. After familiarizing ourselves with the definition and basic properties of the wavefront set, we show that the shearlet transform offers a simple and convenient way to characterize the wavefront set in terms of the decay properties of the shearlet coefficients.

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Acknowledgements

This work was supported by the European Research Council under grant ERC AdG 247277.

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Authors and Affiliations

  1. ETH Zürich, Rämistraße 101, 8001, Zürich, Switzerland

    Philipp Grohs

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  1. Philipp Grohs
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Correspondence to Philipp Grohs .

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Editors and Affiliations

  1. , Institut für Mathematik, Technische Universität Berlin, Berlin, 10623, Berlin, Germany

    Gitta Kutyniok

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

    Demetrio Labate

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Grohs, P. (2012). Shearlets and Microlocal Analysis. In: Kutyniok, G., Labate, D. (eds) Shearlets. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8316-0_2

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