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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Resolution of the wavefront set using continuous shearlets

Authors: Gitta Kutyniok and Demetrio Labate
Journal: Trans. Amer. Math. Soc. 361 (2009), 2719-2754
MSC (2000): Primary 42C15; Secondary 42C40
Published electronically: October 24, 2008
MathSciNet review: 2471937
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Abstract: It is known that the Continuous Wavelet Transform of a distribution $f$ decays rapidly near the points where $f$ is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of $f$. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of $f$ and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by $\mathcal {SH}_\psi f(a,s,t) = \langle {f}{\psi _{ast}}\rangle$, where the analyzing elements $\psi _{ast}$ are dilated and translated copies of a single generating function $\psi$. The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements $\{\psi _{ast}\}$ form a system of smooth functions at continuous scales $a>0$, locations $t \in \mathbb {R}^2$, and oriented along lines of slope $s \in \mathbb {R}$ in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution $f$.

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Additional Information

Gitta Kutyniok
Affiliation: Department of Statistics, Stanford University, Stanford, California 94305

Demetrio Labate
Affiliation: Department of Mathematics, North Carolina State University, Campus Box 8205, Raleigh, North Carolina 27695

Keywords: Analysis of singularities, continuous wavelets, curvelets, directional wavelets, shearlets, wavefront set, wavelets
Received by editor(s): April 24, 2006
Received by editor(s) in revised form: November 1, 2007
Published electronically: October 24, 2008
Additional Notes: The first author acknowledges support from Deutsche Forschungsgemeinschaft (DFG), Grant KU 1446/5-1
The second author acknowledges support from NSF Grant DMS 0604561
Article copyright: © Copyright 2008 American Mathematical Society