Singularly Weighted X-ray Tensor Tomography

TL;DR

This paper studies singularly-weighted X-ray transforms of symmetric tensors on the disk, providing a sharp range decomposition, kernel characterization, and a new tensor decomposition method for reconstruction.

Contribution

It introduces a comprehensive analysis of weighted X-ray transforms for tensors, including range, kernel, and a novel tensor decomposition in weighted spaces.

Findings

Sharp range decomposition for all tensor orders

Complete kernel characterization for the transforms

A new tensor decomposition method for reconstruction

Abstract

If $d$ is a boundary defining function for the Euclidean unit disk and $I$ denotes the geodesic X-ray transform, for $\gamma\in (-1,1)$, we study the singularly-weighted X-ray transforms $I_m d^\gamma$ acting on symmetric $m$-tensors. For any $m$, we provide a sharp range decomposition and characterization in terms of a distinguished Hilbert basis of the data space, that comes from earlier studies of the Singular Value Decomposition for the case $m=0$. Since for $m\ge 1$, the transform considered has an infinite-dimensional kernel, we fully characterize this kernel, and propose a representative for an $m$-tensor to be reconstructed modulo kernel, along with efficient procedures to do so. This representative is based on a new generalization of the potential/conformal/transverse-tracefree decomposition of tensor fields in the context of singularly weighted $L^2$-topologies.