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.

翻译：我们考虑利用宇宙微波背景（CMB）数据对宇宙弦进行成像。在数学上，我们研究洛伦兹几何中X射线变换（称为光线变换）的反演问题。该反问题高度病态，且具有大尺度、动态性以及代表多维目标的未知参数等额外复杂性，这对实现具有高时空分辨率的数值图像重建构成了显著的计算挑战。本文首先利用Landweber迭代对光线变换反演进行微局域稳定性分析；随后，我们对时空目标及光线变换进行离散化，并考虑求解相应反问题的迭代计算方法。针对具有共法型奇点的各类标量函数示例，我们提供了包括Tikhonov和稀疏促进正则化在内的若干先进迭代正则化方法的数值研究及对比分析。