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Inversion and Symmetries of the Star Transform

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Abstract

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along “star-shaped” trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

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Acknowledgements

We would like to thank the anonymous reviewers for several insightful observations that allowed us to shorten some of the proofs and considerably improve the manuscript.

This work was partially funded by NSF grant DMS 1616564 and Simons Foundation grant 360357. The authors are thankful to Alessandro Conflitti for updates about the current status of his conjecture and useful references.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Arlington, Arlington, USA

    Gaik Ambartsoumian

  2. Department of Mathematics, University of Arizona, Tucson, USA

    Mohammad J. Latifi

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  1. Gaik Ambartsoumian
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  2. Mohammad J. Latifi
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Correspondence to Gaik Ambartsoumian.

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Ambartsoumian, G., Latifi, M.J. Inversion and Symmetries of the Star Transform. J Geom Anal 31, 11270–11291 (2021). https://doi.org/10.1007/s12220-021-00680-7

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  • DOI: https://doi.org/10.1007/s12220-021-00680-7

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