Momentum ray transforms

Venkateswaran P. Krishnan, Ramesh Manna, Suman Kumar Sahoo, Vladimir A. Sharafutdinov

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)

Abstract

The momentum ray transform I k integrates a rank m symmetric tensor field f over lines in R n with the weight t k : (I k f)(x,ξ)=∫ -∞ t k 〈 f(x+tξ),ξ m〉 dt. In particular, the ray transform I=I0 was studied by several authors since it had many tomographic applications. We present an algorithm for recovering f from the data (I 0 f,I 1 f,…,I m f). In the cases of m=1 and m=2, we derive the Reshetnyak formula that expresses ∥f∥Hst(ℝn) through some norm of (I 0 f,I 1 f,…,I m f). The Hst-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

Original languageEnglish
Pages (from-to)679-701
Number of pages23
JournalInverse Problems and Imaging
Volume13
Issue number3
DOIs
Publication statusPublished - Jun 2019

Keywords

  • Inverse problems
  • Ray transform
  • Reshetnyak formula
  • Stability estimates
  • Tensor analysis
  • tensor analysis
  • inverse problems
  • stability estimates

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