The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (na\"{\i}ve) solution does not depend on the measured data continuously, regularization is needed to re-establish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: One generic linear regularization that learns how to manipulate the singular values of the linear operator in an extension of [1], and one tailored approach in the Fourier domain that is specific to CT-reconstruction. We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typically much smoother than the training data they were trained on. Finally, we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically, discuss their advantages and disadvantages and investigate the effect of discretization errors at different resolutions.

翻译：从相应的吵闹的雷达变换中重建图像,是计算机断层摄影(CT)应用中出现的错误的线性反问题的一个典型例子。由于(na\"{i}ve)解决方案并不取决于持续测量的数据,因此需要正规化以重新建立持续依赖性。在这项工作中,我们调查从数据中学习线性整顿方法的简单但仍然可以比较的一致方法。更具体地说,我们分析两种方法:一种通用线性正规化方法,在[1]的延伸中学习如何操纵线性操作员的单值,一种针对CT重建的Fourier域的定制方法。我们证明,这种方法已成为趋同性整顿方法,而且它们所提供的重建通常比培训数据更加顺利。最后,我们比较光谱以及基于四面的CT重建方法,讨论其利弊,并在不同的决议中调查离散错误的影响。