A Caltech Library Service

Continuous Curvelet Transform: II. Discretization and Frames

Candès, Emmanuel J. and Donoho, David L. (2005) Continuous Curvelet Transform: II. Discretization and Frames. Applied and Computational Harmonic Analysis, 19 (2). pp. 198-222. ISSN 1063-5203.

PDF - Submitted Version
See Usage Policy.


Use this Persistent URL to link to this item:


We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle length ≈ width^2. Our comparisons allow to extend Theorems known for one decomposition to others. We start from a Continuous Curvelet Transform f → Γ_f (a, b, θ) of functions f(x_1, x_2) on R^2, with parameter space indexed by scale a > 0, location b ∈ R^2, and orientation θ. The transform projects f onto a curvelet γ_(abθ), yielding coefficient Γ_f (a, b, θ) = f, _(γabθ); the corresponding curvelet γ_(abθ) is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform introduced by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on true affine parabolic scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in euclidean coordinates by taking a slightly different mother wavelet at each scale. Smith’s transform, unlike the CCT, does not provide a continuous tight frame. We show that, with the right underlying wavelet in Smith’s transform, the analyzing elements of the two transforms become increasingly similar at increasingly fine scales. We derive a discrete tight frame essentially by sampling the CCT at dyadic intervals in scale a_j = 2^−j, at equispaced intervals in direction, θ_(jℓ), = 2π2^(−j/2)ℓ, and equispaced sampling on a rotated anisotropic grid in space. This frame is a complexification of the ‘Curvelets 2002’ frame constructed by Emmanuel Candès et al. [1, 2, 3]. We compare this discrete frame with a composite system which at coarse scales is the same as this frame but at fine scales is based on sampling Smith’s transform rather than the CCT. We are able to show a very close approximation of the two systems at fine scales, in a strong operator norm sense. Smith’s continuous transform was intended for use in forming molecular decompositions of Fourier Integral Operators (FIO’s). Our results showing close approximation of the curvelet frame by a composite frame using true affine paraboblic scaling at fine scales allow us to cross-apply Smith’s results, proving that the discrete curvelet transform gives sparse representations of FIO’s of order zero. This yields an alternate proof of a recent result of Candès and Demanet about the sparsity of FIO representations in discrete curvelet frames.

Item Type:Article
Related URLs:
URLURL TypeDescription
Additional Information:© 2005 Elsevier Inc. Received 5 November 2003; accepted 17 February 2005. Communicated by Guido L. Weiss. Available online 31 March 2005. This work has been partially supported by DMS 00-77261, DMS 01-40698 (FRG), DMS 98-72890 (KDI), DARPA and AFOSR. D.L.D. thanks the Mathematical Institute of the University of Leiden for hospitality during Fall 2002. Thanks to the referee for helpful comments.
Funding AgencyGrant Number
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Air Force Office of Scientific Research (AFOSR)UNSPECIFIED
Subject Keywords:Curvelets, Parabolic Scaling, Fourier Integral Operator, Tight Frame
Record Number:CaltechAUTHORS:20110602-152619943
Persistent URL:
Official Citation:Emmanuel J. Candes, David L. Donoho, Continuous curvelet transform: II. Discretization and frames, Applied and Computational Harmonic Analysis, Volume 19, Issue 2, September 2005, Pages 198-222, ISSN 1063-5203, DOI: 10.1016/j.acha.2005.02.004. (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:23885
Deposited By: Ruth Sustaita
Deposited On:03 Jun 2011 16:03
Last Modified:26 Dec 2012 13:17

Repository Staff Only: item control page