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 (naive) 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 our previous work, 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：从其对应噪声Radon变换重建图像是计算机断层成像（CT）应用中出现的不适定线性逆问题的典型示例。由于（朴素）解不连续依赖于测量数据，因此需要正则化来重新建立连续依赖性。本文研究简单但可证明收敛的学习线性正则化方法的数据驱动途径。具体而言，我们分析两种方法：一种是我们先前工作扩展中通过学习操纵线性算子奇异值实现的通用线性正则化方法，另一种是针对CT重建的傅里叶域定制方法。我们证明这类方法不仅成为收敛的正则化方法，而且其提供的重建结果通常比训练数据更平滑。最后，我们通过数值实验比较CT重建的谱方法与傅里叶方法，讨论其优缺点，并考察不同分辨率下离散化误差的影响。