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Tensor field tomography with attenuation and refraction: adjoint operators for the dynamic case and numerical experiments

Lukas Vierus, Thomas Schuster, Bernadette Hahn

TL;DR

This work extends tensor field tomography to time-dependent tensors in media with attenuation and refraction by formulating data as attenuated ray transforms on a Riemannian manifold. It develops two adjoint representations for the forward operator—an integral backprojection and a dual transport PDE—applied to static and dynamic cases, with viscosity regularization for numerics. Numerical experiments in 2D show that integral-based adjoints are more efficient than PDE-based adjoints, and that incorporating refraction via a non-constant index improves reconstruction accuracy, though at higher computational cost. The results highlight the importance of accounting for refraction in dynamic TFT and provide a practical framework for using Landweber iterations with Nesterov acceleration for static fields, while outlining paths for extending to time-dependent tensor fields.

Abstract

This article is concerned with tensor field tomography in a fairly general setting, that takes refraction, attenuation and time-dependence of tensor fields into account. The mathematical model is given by attenuated ray transforms of the fields along geodesic curves corresponding to a Riemannian metric that is defined by the index of refraction. The data are given at the boundary tangent bundle of the domain and it is well-known that they can be characterized as boundary data of a transport equation turning tensor field tomography into an inverse source problem. This way the adjoint of the forward mapping can be computed using the integral representation or, equivalently, associated to a dual transport equation. The article offers and proves two different representations for the adjoint mappings both in the dynamic and static case. The numerical implementation is demonstrated and evaluated for static fields using the damped Landweber method with Nesterov acceleration applied to both, the integral and PDE-based formulations. The transport equations are solved using a viscosity approximation. The error analysis reveals that the integral representation significantly outperforms PDE-based methods in terms of computational efficiency while achieving comparable reconstruction accuracy. The impact of noise and deviations from straight-line trajectories are investigated confirming improved accuracy if refraction is taken into account. We conclude that the inclusion of refraction to the forward model pays in spite of increased numerical cost.

Tensor field tomography with attenuation and refraction: adjoint operators for the dynamic case and numerical experiments

TL;DR

This work extends tensor field tomography to time-dependent tensors in media with attenuation and refraction by formulating data as attenuated ray transforms on a Riemannian manifold. It develops two adjoint representations for the forward operator—an integral backprojection and a dual transport PDE—applied to static and dynamic cases, with viscosity regularization for numerics. Numerical experiments in 2D show that integral-based adjoints are more efficient than PDE-based adjoints, and that incorporating refraction via a non-constant index improves reconstruction accuracy, though at higher computational cost. The results highlight the importance of accounting for refraction in dynamic TFT and provide a practical framework for using Landweber iterations with Nesterov acceleration for static fields, while outlining paths for extending to time-dependent tensor fields.

Abstract

This article is concerned with tensor field tomography in a fairly general setting, that takes refraction, attenuation and time-dependence of tensor fields into account. The mathematical model is given by attenuated ray transforms of the fields along geodesic curves corresponding to a Riemannian metric that is defined by the index of refraction. The data are given at the boundary tangent bundle of the domain and it is well-known that they can be characterized as boundary data of a transport equation turning tensor field tomography into an inverse source problem. This way the adjoint of the forward mapping can be computed using the integral representation or, equivalently, associated to a dual transport equation. The article offers and proves two different representations for the adjoint mappings both in the dynamic and static case. The numerical implementation is demonstrated and evaluated for static fields using the damped Landweber method with Nesterov acceleration applied to both, the integral and PDE-based formulations. The transport equations are solved using a viscosity approximation. The error analysis reveals that the integral representation significantly outperforms PDE-based methods in terms of computational efficiency while achieving comparable reconstruction accuracy. The impact of noise and deviations from straight-line trajectories are investigated confirming improved accuracy if refraction is taken into account. We conclude that the inclusion of refraction to the forward model pays in spite of increased numerical cost.
Paper Structure (10 sections, 4 theorems, 80 equations, 7 figures, 4 tables)

This paper contains 10 sections, 4 theorems, 80 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Attenuated ray transforms on manifolds
  3. Adjoint operators of the ray transform
  4. Numerical experiments
  5. Numerical results for the Euclidean case
  6. Numerical approximation of $\mathcal{I}_{\alpha_0} f$
  7. Numerical computation of $\mathcal{I}_{\alpha_0}^* h$
  8. Numerical computation of $\mathcal{S}_{\alpha_0}^* h$
  9. Numerical results for variable $n(x)$
  10. Conclusions

Key Result

Theorem 2.1

Let $(M,g)$ be a Riemannian manifold with $g_{ij}=n^2(x)\delta_{ij}$ and $n\geq 1$. Further, let $\alpha\in H^k(\Omega M)$ with $\alpha\geq \alpha_0 >0$. Then, $\mathcal{I}_\alpha^d$ from Iadf is bounded, i.e., for all $l\in\mathbb{N}_0$ there is a constant $C=C(T,\alpha,\Omega M)>0$, such that

Figures (7)

  • Figure 1: Sketch of a geodesic curve $\gamma_{x,\xi}$
  • Figure 2: Segment between two concentrical circles
  • Figure 3: Segment inside the inner circle
  • Figure 4: Reconstruction of $f_1^{(2)}$ for varying noise levels $\delta\in\lbrace 0,0.1,0.2\rbrace$
  • Figure 5: Grid points for approximation of $\frac{\partial^2 w}{\partial\mu^2}$ at $p=1$
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • Remark 3.1
  • Corollary 3.2
  • Remark 3.2