带噪声 Radon 逆问题的点态估计

已知函数f的 Radon 投影数据，便可利用 Radon 变换的逆重构原来的函数。Radon 逆问题是图像重建的数学基础，已被广泛应用于医学、工程等领域。由于自然图像中包含大量曲线和边缘信息，我们假定f除一条光滑曲线外处处二次可微，用它来模拟具有奇异性曲线边缘的二维图像。小波变换提供了 Besov 空间中函数的良好估计，但因其不具有方向性，不能很好地估计具有奇异性曲线边缘的函数。鉴于切波的多方向性、各向异性及对奇异性曲线边缘函数的最优表示，Colona 等人利用切波变换研究带噪声的 Radon 逆问题，构造了奇异性曲线边缘函数的最优切波估计。切波方法在带噪声 Radon 逆问题研究中取得了显著成果，但其结果基于 L_{2} 范数意义下，关于点态意义下切波估计的收敛性尚待研究。. 本项目尝试利用切波方法构造带噪声的 Radon 逆问题的点态切波估计，研究其点态收敛阶及最优性。

It is well known that we can use Radon inversion to reconstruct a function f from its Radon projection data. Radon inversion is the mathematical foundation of image reconstruction, which is applied widely in medicine, engineering and etc. Noticing that natural images contain a large number of curves and edges, we assume that f is two times continuous differentiable except a smooth curve, which is used to simulate a two-dimensional image with singular curve edge. The wavelet transform provides good estimation over Besov spaces, but it is not well adapted for analyzing the functions with discontinuities along cure edge due to the lack of direction. In view that shearlets have multi-direction、anisotropy and optimal representation ability for the functions containing singularity curve edges, Colona and etc use shearlet method to study noisy Radon inversion, construct the optimal shearlet estimates for the functions containing singularity curve edges. The shearlet method has made remarkable achievements in noisy Radon model in the sense of L_{2} norm. However, the pointwise convergence of shearlet estimators has not been reported. . The project aims to construct shearlet estimates of noisy Radon inversion in pointwise sense using shearlet transforms, and attemps to study the pointwise convergence rate and optimality.