Bernadette Hahn | Searcharxiv

arXiv2026

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

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.

math.NA↗

arXiv2025

The method of the approximate inverse for limited-angle CT

Limited-angle computerized tomography stands for one of the most difficult challenges in imaging. Although it opens the way to faster data acquisition in industry and less dangerous scans in medicine, standard approaches, such as the filtered backprojection (FBP) algorithm or the widely used total-variation functional, often produce various artefacts that hinder the diagnosis. With the rise of deep learning, many modern techniques have proven themselves successful in removing such artefacts but at the cost of large datasets. In this paper, we propose a new model-driven approach based on the method of the approximate inverse, which could serve as new starting point for learning strategies in the future. In contrast to FBP-type approaches, our reconstruction step consists in evaluating linear functionals on the measured data using reconstruction kernels that are precomputed as solution of an auxiliary problem. With this problem being uniquely solvable, the derived limited-angle reconstruction kernel (LARK) is able to fully reconstruct the object without the well-known streak artefacts, even for large limited angles. However, it inherits severe ill-conditioning which leads to a different kind of artefacts arising from the singular functions of the limited-angle Radon transform. The problem becomes particularly challenging when working on semi-discrete (real or analytical) measurements. We develop a general regularization strategy, named constrained limited-angle reconstruction kernel (CLARK), by combining spectral filter, the method of the approximate inverse and custom edge-preserving denoising in order to stabilize the whole process. We further derive and interpret error estimates for the application on real, i.e. semi-discrete, data and we validate our approach on synthetic and real data.

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