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Tensor tomography on asymptotically hyperbolic surfaces

Nikolas Eptaminitakis, François Monard, Yuzhou Joey Zou

TL;DR

This work develops tensor tomography on non-trapping asymptotically hyperbolic surfaces by constructing a gauge-representative framework through a tt-potential-conformal decomposition and elliptic decompositions of geodesic transport operators via the 0-calculus. It establishes robust forward-mapping properties of the geodesic X-ray transform on polyhomogeneous and weighted L^2 spaces and provides an iterated-tt gauge for even-rank tensors, with injectivity results modulo gauge. Specializing to the Poincaré disk, the authors obtain a precise L^2 data-space decomposition into orthogonal components, a complete range characterization for even tensors, and explicit, implementable reconstruction formulas from both the X-ray data and the normal operator. Collectively, these results advance exact tensor tomography on AH surfaces, yielding concrete reconstruction procedures and shedding light on the role of gauge choices in inverse problems on non-compact geometries.

Abstract

We initiate a study of the inversion of the geodesic X-ray transform $I_m$ over symmetric $m$-tensor fields on asymptotically hyperbolic surfaces. This operator has a non-trivial kernel whenever $m\ge 1$. To propose a gauge representative to be reconstructed from X-ray data, we first prove a "tt-potential-conformal" decomposition theorem for $m$-tensor fields (where "tt" stands for transverse traceless), previously used in integral geometry on compact Riemannian manifolds with boundary in Sharafutdinov, 2007; Dairbekov and Sharafutdinov, 2011. The proof is based on elliptic decompositions of the Guillemin-Kazhdan operators $η_\pm$ (Guillemin and Kazhdan, 1980) and leverages in the current setting the 0-calculus of Mazzeo-Melrose (Mazzeo and Melrose, 1987; Mazzeo, 1991). Iterating this decomposition gives rise to an "iterated-tt" representative modulo $\ker I_m$ for a tensor field, which is distinct from the often-used solenoidal representative. In the case of the Poincaré disk, we show that the X-ray transform of a tensor in iterated-tt form splits into components that are orthogonal relative to a specific $L^2$ structure in data space. For even tensor fields, we provide a full picture of the data space decomposition, in particular a range characterization of $I_{2n}$ for every $n$ in terms of moment conditions and spectral decay. Finally, we give explicit approaches for the reconstruction of even tensors in iterated-tt form from their X-ray transform or its normal operator, using specific knowledge of geodesically invariant distributions with one-sided Fourier content, whose properties are analyzed in detail.

Tensor tomography on asymptotically hyperbolic surfaces

TL;DR

This work develops tensor tomography on non-trapping asymptotically hyperbolic surfaces by constructing a gauge-representative framework through a tt-potential-conformal decomposition and elliptic decompositions of geodesic transport operators via the 0-calculus. It establishes robust forward-mapping properties of the geodesic X-ray transform on polyhomogeneous and weighted L^2 spaces and provides an iterated-tt gauge for even-rank tensors, with injectivity results modulo gauge. Specializing to the Poincaré disk, the authors obtain a precise L^2 data-space decomposition into orthogonal components, a complete range characterization for even tensors, and explicit, implementable reconstruction formulas from both the X-ray data and the normal operator. Collectively, these results advance exact tensor tomography on AH surfaces, yielding concrete reconstruction procedures and shedding light on the role of gauge choices in inverse problems on non-compact geometries.

Abstract

We initiate a study of the inversion of the geodesic X-ray transform over symmetric -tensor fields on asymptotically hyperbolic surfaces. This operator has a non-trivial kernel whenever . To propose a gauge representative to be reconstructed from X-ray data, we first prove a "tt-potential-conformal" decomposition theorem for -tensor fields (where "tt" stands for transverse traceless), previously used in integral geometry on compact Riemannian manifolds with boundary in Sharafutdinov, 2007; Dairbekov and Sharafutdinov, 2011. The proof is based on elliptic decompositions of the Guillemin-Kazhdan operators (Guillemin and Kazhdan, 1980) and leverages in the current setting the 0-calculus of Mazzeo-Melrose (Mazzeo and Melrose, 1987; Mazzeo, 1991). Iterating this decomposition gives rise to an "iterated-tt" representative modulo for a tensor field, which is distinct from the often-used solenoidal representative. In the case of the Poincaré disk, we show that the X-ray transform of a tensor in iterated-tt form splits into components that are orthogonal relative to a specific structure in data space. For even tensor fields, we provide a full picture of the data space decomposition, in particular a range characterization of for every in terms of moment conditions and spectral decay. Finally, we give explicit approaches for the reconstruction of even tensors in iterated-tt form from their X-ray transform or its normal operator, using specific knowledge of geodesically invariant distributions with one-sided Fourier content, whose properties are analyzed in detail.

Paper Structure

This paper contains 43 sections, 32 theorems, 214 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Results on AH Spaces
  4. The unit sphere and cosphere bundles and their compactifications.
  5. A model for the manifold of geodesics ${\cal G}$.
  6. The 0-sphere bundle.
  7. Mapping properties of the X-ray transform on L 2 and phg spaces
  8. Representatives modulo gauge for symmetric tensor fields
  9. The iterated-tt representative.
  10. Results on the Poincaré disk
  11. Data space decomposition and range characterization
  12. Range characterization.
  13. Reconstruction
  14. Reconstruction from X-Ray Data.
  15. Reconstruction from the normal operator.
  16. ...and 28 more sections

Key Result

Proposition 2.1

Let $(M,g)$ be a non-trapping AH manifold and let $E$ be an index set with $\mathop{\mathrm{Re}}\nolimits(E)>0$. Then the operator eq:IH extends as a map Moreover, for every $\alpha>0$ and bdf $\rho$ on $M$, there exists a function $R_\alpha\in C^\infty(\overline{{\cal G}})$ such that where $B$ is the Beta function. Moreover, there exist $\mu',\alpha'>0$ such that for all $\mu_\rho\leq \mu'$ and

Figures (2)

  • Figure 1: The case $n=2$. On the left, $\overline{{\cal G}}$, whose compactified fibers are parameterized by a point $y_*\in \partial M$. We write $\pi_b$ for the natural projection ${}^b\overline{ S^*M}\to M$. On the right, ${}^b\overline{ S^*M}$ (with the $y$ variable suppressed) sitting over $M$. The gray curves are level sets of $\tilde{x}$. Some integral curves of $\mathcal{Y}$ (defined in \ref{['eq:Y']}) corresponding to initial conditions with $\delta$ small are shown in red.
  • Figure 2: A spectral representation of the data space $L^2_+({\cal G}_{\mathrm h})$. Each "$\bullet$" represents an element $\psi_{n,k}^{0,H}$. The spaces $E_{\pm}$ and $E_r$ are defined in \ref{['eq:Epm']} and \ref{['eq:Er']}, respectively.

Theorems & Definitions (68)

  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 1
  • Corollary 2.5
  • Proposition 2.6: Injectivity
  • Theorem 2
  • Proposition 2.7
  • Corollary 2.8: Range decomposition
  • ...and 58 more