Convergent Data-driven Regularizations for CT Reconstruction

TL;DR

This paper introduces and analyzes two data-driven regularization methods for CT image reconstruction, demonstrating their convergence and smoother results compared to training data, with a focus on spectral and Fourier domain approaches.

Contribution

The paper proposes two novel, provably convergent linear regularization techniques for CT reconstruction, extending previous spectral manipulation methods and tailoring approaches to the Fourier domain.

Findings

Both methods are proven to be convergent regularization techniques.

Reconstructed images are generally smoother than the training data.

Numerical comparisons highlight advantages and disadvantages of spectral and Fourier approaches.

Abstract

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.