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Image reconstruction from radially incomplete spherical Radon data

Published online by Cambridge University Press:  11 September 2017

GAIK AMBARTSOUMIAN ,
RIM GOUIA-ZARRAD ,
VENKATESWARAN P. KRISHNAN  and
SOUVIK ROY
GAIK AMBARTSOUMIAN
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, Texas, USA email: gambarts@uta.edu
RIM GOUIA-ZARRAD
Affiliation:
Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE email: rgouia@aus.edu
VENKATESWARAN P. KRISHNAN
Affiliation:
Tata Institute of Fundamental Research – Centre for Applicable Mathematics, Bangalore, India email: vkrishnan@math.tifrbng.res.in
SOUVIK ROY
Affiliation:
Department of Mathematics, University of Würzburg, Würzburg, Germany email: souvik.roy@mathematik.uni-wuerzburg.de
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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.

Keywords

spherical Radon transformmedical imagingspherical harmonicsVolterra integral equationstruncated singular value decomposition
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 3 , June 2018 , pp. 470 - 493
Copyright
Copyright © Cambridge University Press 2017 

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