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Image reconstruction from radially incomplete spherical Radon data

  • GAIK AMBARTSOUMIAN (a1), RIM GOUIA-ZARRAD (a2), VENKATESWARAN P. KRISHNAN (a3) and SOUVIK ROY (a4)
Abstract

We study inversion of the spherical Radon transform with centres on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result, our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm and demonstrate its accuracy and efficiency on several numerical examples.

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Ambartsoumian was supported in part by US NSF Grants DMS 1109417, DMS 1616564 and Simons Foundation Grant 360357. Gouia-Zarrad was supported in part by the American University of Sharjah (AUS) research grant FRG3. Krishnan was supported in part by NSF grants DMS 1109417 and DMS 1616564. He and Roy benefited from support of the Airbus Group Corporate Foundation Chair “Mathematics of Complex Systems” established at TIFR Centre for Applicable Mathematics and TIFR International Centre for Theoretical Sciences, Bangalore, India.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
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