Refractive dynamic tensor field tomography: towards a holistic approach
Final Report Abstract
Within the framework of this project significant advancements in tensor tomography across various settings have been achieved including static, dynamic, attenuated, and refractive cases. Generalized attenuated ray transforms for higher-order tensor fields were studied, where integral moments were explicitly calculated providing unique solutions to boundary value problems under certain smoothness conditions. Novel numerical methods, particularly using the approximate inverse, demonstrated accurate reconstructions for 2D vector and 2-tensor tomography from limited data. Furthermore, singular value decompositions for the underlying integral transforms have been computed and continuity estimates in Sobolev-Bochner spaces were derived. An alternative approach investigated tensor tomography as an inverse source problem for transport equations that emerge from the geodesic vector field. By adding a viscosity term, weak solutions were analyzed, proving unique existence under mild conditions on refractive indices and absorption coefficients. The adjoint of the dynamic ray transform was deduced leading to two different, equivalent representations. The integral representation proved computationally simpler, albeit requiring geodesic equation solutions for different initial values. As for the numerical implementations, we focused on the stationary case, employing a polar grid. Challenges included optimal grid sampling and handling singularities at the origin. Synthetic data tests validated numerical convergence for decreasing viscosity parameters. The error analysis revealed that the integral representation significantly outperforms PDE-based methods in view of computational efficiency while achieving comparable reconstruction accuracy. The reconstruction was performed using the attenuated Landweber method with Nesterov acceleration applied to both, the integral and PDE-based formulations. The integral operator method proved to be superior regarding efficiency leading to its exclusive use for the non-Euclidean case. Additional experiments analyzed the impact of noise and deviations from straight-line trajectories confirming improved accuracy by using refraction modeling, despite increased computational cost.
Link to the final report
https://oa.tib.eu/renate/handle/123456789/21446
Publications
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Integral and differential operators as the tools of integral geometry and tomography. Numerical Computations: Theory and Algorithms NUMTA 2019
E. Derevtsov, Y. Volkov & T. Schuster
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Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields. Lecture Notes in Computer Science, 97-111. Springer International Publishing.
Derevtsov, Evgeny Yu.; Volkov, Yuriy S. & Schuster, Thomas
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Integral Operators at Settings and Investigations of Tensor Tomography Problems. Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy, 111-117. Springer International Publishing.
Derevtsov, Evgeny; Volkov, Yuriy & Schuster, Thomas
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The Method of Approximate Inverse in Slice-by-Slice Vector Tomography Problems. Lecture Notes in Computer Science, 487-494. Springer International Publishing.
Svetov, Ivan E.; Maltseva, Svetlana V. & Louis, Alfred K.
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An iterative algorithm for reconstructing a 2D vector field by its limited-angle ray transform. Journal of Physics: Conference Series, 1715(1), 012037.
Maltseva, S. V.; Svetov, I. E. & Louis, A. K.
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Generalized attenuated ray transforms and their integral angular moments. Applied Mathematics and Computation, 409, 125494.
Derevtsov, Evgeny Yu.; Volkov, Yuriy S. & Schuster, Thomas
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On solving the slice-by-slice three-dimensional 2-tensor tomography problems using the approximate inverse method. Journal of Physics: Conference Series, 1715(1), 012036.
Louis, A. K.; Maltseva, S. V.; Polyakova, A. P.; Schuster, T. & Svetov, I. E.
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Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations. ETNA - Electronic Transactions on Numerical Analysis, 57, 80–100.
Vierus, Lukas & Schuster, Thomas
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A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality. Inverse Problems, 40(8), 085007.
Louis, Alfred K.
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Dynamic refractive tensor field tomography as an inverse problem for a transport equation. Doctoral thesis, Saarland University Saarbrücken
L. Vierus
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Inversion formulae for ray transforms in vector and tensor tomography. Inverse Problems, 38(6), 065008.
Louis, Alfred K.
