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The curvelet transform for image denoising

Starck, Jean-Luc and Candès, Emmanuel J. and Donoho, David L. (2002) The curvelet transform for image denoising. IEEE Transactions on Image Processing, 11 (6). pp. 670-684. ISSN 1057-7149. https://resolver.caltech.edu/CaltechAUTHORS:STAieetip02

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Abstract

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.


Item Type:Article
Additional Information:© Copyright 2002 IEEE. Reprinted with permission. Manuscript received January 19, 2001; revised November 21, 2001. Posted online: 2002-08-07. This work was supported by the National Science Foundation under Grants DMS 98-72890 (KDI) and DMS 95-05151 and also by AFOSR MURI-95-P49620-96-1-0028. The authors would like to thank one referee for some very helpful comments on the original version of the manuscript.
Subject Keywords:Curvelets, discrete wavelet transform, FFT, filtering, FWT, radon transform, ridgelets, thresholding rules, wavelets
Issue or Number:6
Record Number:CaltechAUTHORS:STAieetip02
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:STAieetip02
Alternative URL:http://dx.doi.org/10.1109/TIP.2002.1014998
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1381
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:13 Jan 2006
Last Modified:02 Oct 2019 22:42

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