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 $arepsilon$.ududTraditional 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 $arepsilon$ only as $O(arepsilon^{1/2})$ as $arepsilono 0$. A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to $O(arepsilon^{2/3})$. However, as we show here, this rate is not optimal.ududIn 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.ududWe prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE $O(arepsilon^{4/5})$ as noise level $arepsilono 0$. 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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机译：我们考虑从嘈杂的Radon数据中恢复函数$ f（x_1，x_2）$的模型问题。假定要恢复的函数$ f $除了沿$ C ^ 2 $曲线（即边沿）的不连续性之外，都是平滑的。我们使用连续白噪声模型，其噪声水平为 varepsilon $。 ud ud使用传统的线性方法来解决此类反问题，在存在边的情况下，其性能较差。定性地，重建在边缘附近模糊。从数量上讲，它们在模型中给出了均方误差（MSE），该均方根误差趋于零，而噪声水平$ varepsilon $仅作为$ O（ varepsilon ^ {1/2}）$作为$ varepsilon 至0 $。最近的创新-小波域中的非线性收缩-在视觉上提高了边缘清晰度，并改善了MSE收敛到$ O（ varepsilon ^ {2/3}）$。但是，正如我们在此处显示的那样，该速率不是最佳的。 ud ud实际上，通过在此设置中部署最近引入的Curvelet紧帧，可以获得实质上最佳的性能。 Curvelet是光滑，高度各向异性的元素，非常适合检测和合成弯曲边缘。要在Radon设置中部署它们，我们构造Radon算子的基于Curvelet的双正交分解，并基于嘈杂的Curvelet系数的阈值构建“曲线收缩”估计量。实际上，估算器会检测Radon域中某些位置和方向的边缘，并自动合成原始域中相应位置和方向的边缘。 ud ud我们证明可以调整Curvelet的收缩率，以便估算器可以在对数因子，MSE $ O（ varepsilon ^ {4/5}）$作为噪声水平$ varepsilon 到0 $。该收敛速度在$ C ^ 2 $的函数类别上保持一致，除了$ C ^ 2 $曲线上的不连续性外，（对数项除外）是该类别的函数的极大值。我们的方法是一般策略的一个实例，该策略应适用于其他反问题。我们画一个反卷积的例子。
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