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Singularly Weighted X-ray Tensor Tomography

Jonathan Kay, François Monard

TL;DR

This work analyzes singularly weighted X-ray transforms $I_m d^\gamma$ on symmetric $m$-tensors over the unit disk for $\gamma\in(-1,1)$. It builds a comprehensive framework comprising forward mapping properties, a generalized York/tt tensor decomposition, and a sharp range characterization, together with explicit reconstruction procedures that are robust to the kernel. The authors derive a gauge-representative modulo kernel, establish an elliptic decomposition in the weighted setting, and provide both SVD-based and Pestov-Uhlmann-type inversion techniques applicable across tensor orders. The results combine fiberwise Fourier analysis with Guillemin-Kazhdan operators to enable order-blind reconstruction and rigorous handling of boundary behavior, with potential impact on tensor tomography in bounded domains and weighted boundary data problems.

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.

Singularly Weighted X-ray Tensor Tomography

TL;DR

This work analyzes singularly weighted X-ray transforms on symmetric -tensors over the unit disk for . It builds a comprehensive framework comprising forward mapping properties, a generalized York/tt tensor decomposition, and a sharp range characterization, together with explicit reconstruction procedures that are robust to the kernel. The authors derive a gauge-representative modulo kernel, establish an elliptic decomposition in the weighted setting, and provide both SVD-based and Pestov-Uhlmann-type inversion techniques applicable across tensor orders. The results combine fiberwise Fourier analysis with Guillemin-Kazhdan operators to enable order-blind reconstruction and rigorous handling of boundary behavior, with potential impact on tensor tomography in bounded domains and weighted boundary data problems.

Abstract

If is a boundary defining function for the Euclidean unit disk and denotes the geodesic X-ray transform, for , we study the singularly-weighted X-ray transforms acting on symmetric -tensors. For any , 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 . Since for , the transform considered has an infinite-dimensional kernel, we fully characterize this kernel, and propose a representative for an -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 -topologies.

Paper Structure

This paper contains 27 sections, 20 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Preliminaries and notation
  4. Forward mapping properties
  5. Potential tensors
  6. Tensor field decompositions
  7. Data space decomposition
  8. Range characterization and reconstruction
  9. Forward mapping properties - Proof of Proposition \ref{['prop:Ibounded']}
  10. Tensor decompositions - Proof of Theorems \ref{['thm:Yorkdecomp']}, \ref{['thm:rangedecomp']} and Corollary \ref{['cor:decomposition']}
  11. A singularly-weighted elliptic decomposition
  12. Tensor fields and fiberwise Fourier analysis
  13. Decompositions. Proofs of Lemma \ref{['lem:oneformdecompSD']}, Theorem \ref{['thm:Yorkdecomp']}, and Corollary \ref{['cor:decomposition']}
  14. Range decomposition
  15. Description of the range over tt $m$-tensors
  16. ...and 12 more sections

Key Result

Proposition 1

For every $\gamma>-1$, the operator is bounded (with norm at most $2^{\gamma+1/2} \sqrt{B(\gamma+1,\gamma+1)}$) and surjective.

Figures (2)

  • Figure 1: Decomposition of $L^2_{\gamma,+}(\partial_+ S\mathbb{D})$ induced by Equation \ref{['eq:evendecomposition']}: $\operatorname{Ran}(I_0d^\gamma)$ is spanned by the polynomials in-between the solid and dashed lines. For fixed $j\geq 1$, the diagonals $\left\{\psi_{n,-j}^{\gamma,+}\right\}$ and $\left\{\psi_{n,n+j}^{\gamma,+}\right\}$ capture the range of $L^2_\gamma(\mathbb{D};S_\mathrm{tt}^{2j}(T^*\mathbb{D}))$.
  • Figure 2: Decomposition of $L^2_{\gamma,-}(\partial_+S\mathbb{D})$ induced by Equation \ref{['eq:odddecomposition']}: $\operatorname{Ran}(I_1(\star \mathrm{d}))$ is spanned by the polynomials in between the solid and and dashed lines. For fixed $j\geq 0$, the diagonals $\left\{\psi_{n,-j}^{\gamma,-}\right\}$ and $\left\{\psi_{n,n+j+1}^{\gamma,-}\right\}$ capture the range of $L^2_\gamma(\mathbb{D};S_\mathrm{tt}^{2j+1}(T^*\mathbb{D}))$.

Theorems & Definitions (39)

  • Proposition 1
  • Corollary 2
  • Lemma 3
  • proof : Proof of Lemma \ref{['lem:kernel']}
  • Lemma 4
  • Theorem 5: Tensor field decomposition
  • Corollary 6
  • Definition 1: Hilbert Scales
  • Lemma 7
  • Theorem 8: Orthogonal decomposition of the range
  • ...and 29 more