Limited Data Problems for the Generalized Radon Transform in Rn

Jürgen Frikel, Eric Todd Quinto

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We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in Rn. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in Rn and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in R2 and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to analyze more general imaging situations.
Original languageEnglish
JournalS I A M Journal on Mathematical Analysis
Issue number4
Pages (from-to)2301-2318
Publication statusPublished - 2016


  • Radon transforms
  • Microlocal analysis
  • Computed tomography
  • Lambda tomography
  • Limited angle tomography
  • Fourier integral operators


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