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Inverting the geodesic ray transform with finite measurements: stability and reconstruction

  • Yernat M. Assylbekov
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Abstract

We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.

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Cite

BibTeXRIS
BibTeX
@misc{assylbekov2026inverting,
  title = {{Inverting the geodesic ray transform with finite measurements: stability and reconstruction}},
  author = {Yernat M. Assylbekov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40015},
  eprint = {2609.40015},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40015},
}
RIS
TY  - UNPB
TI  - Inverting the geodesic ray transform with finite measurements: stability and reconstruction
AU  - Yernat M. Assylbekov
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40015
UR  - https://arxiv.org/abs/2609.40015
AB  - We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.
ER  -