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Integral Geometry and Tomography

  • Victor Isakov
Chapter
  • 1.9k Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 127)

Abstract

The problems of integral geometry are to determine a function given (weighted) integrals of this function over a “rich” family of manifolds. These problems are of importance in medical applications (tomography), and they are quite useful for dealing with inverse problems in hyperbolic differential equations (integrals of unknown coefficients over ellipsoids or lines can be obtained from the first terms of the asymptotic expansion of rapidly oscillating solutions and an information about first-arrival times of a wave). There has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is the unity; there are interesting results in the plane case when a family of curves is regular (resembling locally the family of straight lines) or in case of the family of straight lines with an arbitrary regular attenuation. Still there are many interesting open questions about the problem with local data and simultaneous recovery of density of a source and of attenuation. We give a brief review of this area, referring for more information to the books of Natterer [Nat] and Sharafutdinov [Sh].

Keywords

Attenuation Coefficient Inversion Formula Integral Geometry Collision Kernel Hyperbolic Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Victor Isakov
    • 1
  1. 1.Department of Mathematics and StatisticsWichita State UniversityWichitaUSA

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