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arXiv · 2511.08871

Singularly Weighted X-ray Tensor Tomography

Jonathan Kay·François Monard
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Abstract

If $d$ is a boundary defining function for the Euclidean unit disk and $I$ denotes the geodesic X-ray transform, for $γ\in (-1,1)$, we study the singularly-weighted X-ray transforms $I_m d^γ$ 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.

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Jonathan Kay, François Monard. 2025-11-12. Singularly Weighted X-ray Tensor Tomography. https://arxiv.org/abs/2511.08871

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