Broken ray tomography in the disc

Ilmavirta, Joonas

doi:10.1088/0266-5611/29/3/035008

by Joonas Ilmavirta

Abstract:

Given a bounded $C^1$ domain $\Omega\subset\R^n$ and a nonempty subset $E$ of its boundary (set of tomography), we consider broken rays which start and end at points of $E$. We ask: If the integrals of a function over all such broken rays are known, can the function be reconstructed? We give positive answers when $\Omega$ is a ball and the unknown function is required to be uniformly quasianalytic in the angular variable and the set of tomography is open. We also analyze the situation when the set of tomography is a singleton.

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Reference:

Broken ray tomography in the disc (Joonas Ilmavirta), Inverse Problems, volume 29, number 3, pp. 035008, 2013. [show abstract] [hide abstract] Given a bounded $C^1$ domain $\Omega\subset{\mathbb R}^n$ and a nonempty subset $E$ of its boundary (set of tomography), we consider broken rays which start and end at points of $E$. We ask: If the integrals of a function over all such broken rays are known, can the function be reconstructed? We give positive answers when $\Omega$ is a ball and the unknown function is required to be uniformly quasianalytic in the angular variable and the set of tomography is open. We also analyze the situation when the set of tomography is a singleton. [arXiv] [MathSciNet]

Bibtex Entry:

@article{disk,
  author={Joonas Ilmavirta},
  title={Broken ray tomography in the disc},
  journal={Inverse Problems},
  volume={29},
  number={3},
  pages={035008},
  year={2013},
  month=mar,
  doi={10.1088/0266-5611/29/3/035008},
  url={http://users.jyu.fi/~jojapeil/pub/disk.pdf},
  gsid={5877025966077680622,11789774664916568283},
  arxiv={1210.4354},
  MRNUMBER = {3040563},
  abstract={Given a bounded $C^1$ domain $\Omega\subset\R^n$ and a nonempty subset $E$ of its boundary (set of tomography), we consider broken rays which start and end at points of $E$.
We ask: If the integrals of a function over all such broken rays are known, can the function be reconstructed?
We give positive answers when $\Omega$ is a ball and the unknown function is required to be uniformly quasianalytic in the angular variable and the set of tomography is open.
We also analyze the situation when the set of tomography is a singleton.}
}