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Invertibility and stability for a generic class of radon transforms with application to dynamic operators

  • Siamak RabieniaHaratbar EMAIL logo
Published/Copyright: December 5, 2018

Abstract

Let X be an open subset of 2. We study the dynamic operator, 𝒜, integrating over a family of level curves in X when the object changes between the measurement. We use analytic microlocal analysis to determine which singularities can be recovered by the data-set. Our results show that not all singularities can be recovered as the object moves with a speed lower than the X-ray source. We establish stability estimates and prove that the injectivity and stability are of a generic set if the dynamic operator satisfies the visibility, no conjugate points, and local Bolker conditions. We also show this results can be implemented to fan beam geometry.

MSC 2010: 44A12; 46F12; 53C65

Award Identifier / Grant number: 1600327

Funding statement: Partly supported by NSF Grant DMS 1600327.

Acknowledgements

The author would like to express his special gratitudes to Professor Plamen D. Stefanov for introducing the problem and his valuable discussions throughout this work. The author thanks Professor Todd Quinto for his helpful comments. The author also thanks the referees for their valuable comments that have helped in improving the manuscript.

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Received: 2018-03-31
Revised: 2018-10-31
Accepted: 2018-11-04
Published Online: 2018-12-05
Published in Print: 2019-08-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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