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Continuous Curvelet Transform: I. Resolution of the Wavefront Set

Candès, Emmanuel J. and Donoho, David L. (2005) Continuous Curvelet Transform: I. Resolution of the Wavefront Set. Applied and Computational Harmonic Analysis, 19 (2). pp. 162-197. ISSN 1063-5203.

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We discuss a Continuous Curvelet Transform (CCT), a transform f → Γf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b ∈ R^2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) = {f, γabθ} where the inner products project f onto analyzing elements called curvelets γ_(abθ) which are smooth and of rapid decay away from an a by √a rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to ‘track’ the behavior of singularities along curves. They are continuum scale/space/orientation analogs of the discrete frame of curvelets discussed in Candès and Donoho (2002). We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Γf (a, x0, θ0) decays rapidly as a → 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of Γf (a, x0, θ0) for fixed (x0, θ0), as a → 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Γf (a, x, θ) is not of rapid decay as a → 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the ‘directional parabolic square function’ S^m(x, θ) = ( ʃ|Γf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1/2) is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set.

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Additional Information:© 2005 Elsevier Inc. Received 5 November 2003; accepted 16 February 2005. Communicated by Guido L. Weiss. Available online 31 March 2005. Communicated by Guido L. Weiss. This work has been partially supported by DMS 00-77261, DMS 01-40698 (FRG), DMS 98–72890 (KDI), DARPA ACMP and AFOSR MURI. EJC was supported by an Alfred P. Sloan Fellowship and DOE DE-FG03-02ER25529. DLD would like to thank the Mathematical Institute at the University of Leiden for hospitality during Fall 2002.
Funding AgencyGrant Number
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Air Force Office of Scientific Research (AFOSR)UNSPECIFIED
Multidisciplinary University Research Initiative (MURI)UNSPECIFIED
Alfred P. Sloan fellowshipUNSPECIFIED
Department of Energy (DOE)DE-FG03-02ER25529
Subject Keywords:Wavelets; Curvelets; Wave packets; Directional wavelets; Analysis of singularities; Singular support; Wavefront set; Parabolic scaling
Record Number:CaltechAUTHORS:20110602-140244176
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Official Citation:Emmanuel J. Candes, David L. Donoho, Continuous curvelet transform: I. Resolution of the wavefront set, Applied and Computational Harmonic Analysis, Volume 19, Issue 2, September 2005, Pages 162-197, ISSN 1063-5203, DOI: 10.1016/j.acha.2005.02.003. (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:23884
Deposited By: Ruth Sustaita
Deposited On:02 Jun 2011 22:40
Last Modified:26 Dec 2012 13:17

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