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- W2029026810 abstract "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 $varepsilonto 0$. A recent innovation--nonlinear 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 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 can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(varepsilon^{4/5})$ as noise level $varepsilonto 0$. 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." @default.
- W2029026810 created "2016-06-24" @default.
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- W2029026810 date "2002-06-01" @default.
- W2029026810 modified "2023-09-26" @default.
- W2029026810 title "Recovering edges in ill-posed inverse problems: optimality of curvelet frames" @default.
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- W2029026810 doi "https://doi.org/10.1214/aos/1028674842" @default.
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