Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Iterative Reconstruction Methods for Cosmological X-Ray Tomography

Julianne Chung, Lucas Onisk, Yiran Wang

TL;DR

This paper tackles the ill-posed inverse problem of recovering cosmic-string-induced gravitational perturbations from CMB data via the light ray transform $L$ in Minkowski space. It combines microlocal stability analysis of the Landweber iteration with a discretized forward model and a comparative study of iterative regularization methods, including Tikhonov and $\ell_1$-based approaches (ISTA/FISTA). The authors reveal that $N=L^*L$ has a paired-Lagrangian structure that stabilizes reconstruction of space-like singularities while time-like singularities and light-like artefacts pose challenges, and they demonstrate that FISTA and generalized Tikhonov with a space-time prior tend to yield superior reconstructions in numerical experiments. The results provide a practical framework for iterative reconstruction in cosmological X-ray tomography and point toward future extensions to 4D problems and partial-data scenarios with structure-exploiting algorithms.

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.

Iterative Reconstruction Methods for Cosmological X-Ray Tomography

TL;DR

This paper tackles the ill-posed inverse problem of recovering cosmic-string-induced gravitational perturbations from CMB data via the light ray transform in Minkowski space. It combines microlocal stability analysis of the Landweber iteration with a discretized forward model and a comparative study of iterative regularization methods, including Tikhonov and -based approaches (ISTA/FISTA). The authors reveal that has a paired-Lagrangian structure that stabilizes reconstruction of space-like singularities while time-like singularities and light-like artefacts pose challenges, and they demonstrate that FISTA and generalized Tikhonov with a space-time prior tend to yield superior reconstructions in numerical experiments. The results provide a practical framework for iterative reconstruction in cosmological X-ray tomography and point toward future extensions to 4D problems and partial-data scenarios with structure-exploiting algorithms.

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.
Paper Structure (9 sections, 3 theorems, 57 equations, 11 figures, 3 tables)

This paper contains 9 sections, 3 theorems, 57 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Overview of contributions
  3. The Landweber Iteration
  4. Analysis of the Instability
  5. Computational Approaches for the Inverse Problem
  6. Discrete Forward Problem
  7. Iterative Methods and Regularization
  8. Numerical Results
  9. Conclusion

Key Result

Proposition 2.1

\newlabelprop-stab0 For any $f\in L^2({\mathbb R}^{n+1}), n\geq 2$, we have where $\mu_n = C_n^\frac{1}{2} \delta^{(n-3)/4}>0$ and $C_n = 2\pi^{(n-1)/2}/\Gamma((n-1)/2)$ is the surface area of ${\mathbb S}^{n-2}, n\geq 2$.

Figures (11)

  • Figure 1: Setup of the inverse problem. Each point at $t = t_0$ plane is regarded as a light source. The light signal from $x$ is recorded at $t = t_1$ in the dashed circle. The light ray $\gamma$ can be parametrized by $x\in {\mathbb R}^n, v\in {\mathbb S}^{n-1}$. The inverse problem is to reconstruct $f$ supported between $t = t_0$ and $t = t_1$ from $Lf$.
  • Figure 1: Reconstruction of the characteristic function of the unit ball. The figure shows the projection to the $t, x$-plane so that each point represents a circle in ${\mathbb R}^2.$ On the circle in the figure, the highlighted arches (in red) represent parts of the sphere that can be stably reconstructed. The four straight lines tangent to the sphere represent possible artefacts in the reconstruction.
  • Figure 1: Discrete problem setup. On the left is an illustration of the geometry of the discrete CMB problem, with the object of interest located between the source grid at the bottom and the detector grid on the top. For each source location $s_i$, detectors (determined by a cone of 45 degrees) receive contributions from event planes $X(t), t=1, \ldots, T$. The weights of contribution from $X(t)$ are determined via bilinear interpolation, as shown in the right figure, where the red cross denotes the ray's intersection with $X(t)$.
  • Figure 1: Isosurface plots of the true strings of Example $\#$1 with $a=0.05$ for three different values of $c$. A clipping plane through the $yt$-plane is utilized to display the inside of the strings. The temporal direction of each of the strings increases from the bottom of figure to the top.
  • Figure 2: RRE and RRN plots for Example $\#1$ with $a=0.05$ and $c=0$. The black dashed line in the RRN plot (right) indicates where the methods would terminate according to the DP. Colored stars and circles in the RRE plot (left) indicate termination according to the DP and the smallest RRE achieved in the allotted iterations, respectively.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Proposition 2.1
  • Proof 1
  • Theorem 3.1: Theorem 1 of AnUh
  • Theorem 3.2
  • Proof 2