Open Access
June 2002 Recovering edges in ill-posed inverse problems: optimality of curvelet frames
Emmanuel J. Candès, David L. Donoho
Ann. Statist. 30(3): 784-842 (June 2002). DOI: 10.1214/aos/1028674842


We consider a model problem of recovering a function $f(x_1,x_2)$ from noisy Radon data. The function $f$ to be recovered is assumed smooth apart from a discontinuity along a $C^2$ curve, that is, an edge. We use the continuum white-noise model, with noise level $\varepsilon$.

Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level $\varepsilon$ only as $O(\varepsilon^{1/2})$ as $\varepsilon\to 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(\varepsilon^{2/3})$. However, as we show here, this rate is not optimal.

In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.

We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(\varepsilon^{4/5})$ as noise level $\varepsilon\to 0$. This rate of convergence holds uniformly over a class of functions which are $C^2$ except for discontinuities along $C^2$ curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example.


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Emmanuel J. Candès. David L. Donoho. "Recovering edges in ill-posed inverse problems: optimality of curvelet frames." Ann. Statist. 30 (3) 784 - 842, June 2002.


Published: June 2002
First available in Project Euclid: 6 August 2002

zbMATH: 1101.62335
MathSciNet: MR1922542
Digital Object Identifier: 10.1214/aos/1028674842

Primary: 62C20 , 62G20
Secondary: 41A30 , ‎42C40 , 94A08 , 94A12

Keywords: curvelets , Deconvolution , Edge , edge-preserving regularization , Ill-posed inverse problems , minimax estimation , Optimal rates of convergence , Radon transform , regularization , ridgelets , Singular value decomposition , wavelet shrinkage , Wavelets , wavelet-vaguelette decomposition , white-noise model

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.30 • No. 3 • June 2002
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