Iterative Reconstruction Methods for Cosmological X-Ray Tomography

TL;DR

This paper develops and compares iterative reconstruction algorithms, including Landweber, Tikhonov, and sparsity-promoting regularizers, for solving the ill-posed inverse problem of imaging cosmic strings from CMB data via the light ray transform.

Contribution

It introduces a microlocal stability analysis and applies advanced iterative methods to improve high-resolution cosmic string imaging from CMB data.

Findings

Landweber iteration provides stable inversion under certain conditions

Tikhonov regularization enhances reconstruction stability

Sparsity-promoting regularizers improve image quality for singular features

Abstract

We consider the imaging of cosmic strings by using Cosmic Microwave Background (CMB) data. Mathematically, we study the inversion of an X-ray transform in Lorentzian geometry, called the light ray transform. The inverse problem is highly ill-posed, with additional complexities of being large-scale and dynamic, with unknown parameters that represent multidimensional objects. This presents significant computational challenges for the numerical reconstruction of images that have high spatial and temporal resolution. In this paper, we begin with a microlocal stability analysis for inverting the light ray transform using the Landweber iteration. Next, we discretize the spatiotemporal object and light ray transform and consider iterative computational methods for solving the resulting inverse problem. We provide a numerical investigation and comparison of some advanced iterative methods for regularization including Tikhonov and sparsity-promoting regularizers for various example scalar functions with conormal type singularities.