Abstract

We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. 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 ɛ only as O(ɛ1/2)asɛ → 0. A recent innovation – nonlinear shrinkage in the wavelet domain – visually improves edge sharpness and improves MSE convergence to O(ɛ2/3). However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recentlyintroduced 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

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