Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

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• Leonid Kunyansky (University of Arizona)
• Thursday 18 May 2023, 14:30-15:30
• Seminar Room 1, Newton Institute.

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RNTW03 - New tomographic methods using particles

Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). We will discuss the connection between the weighted Radon transforms and MAET , and will demonstrate performance of the new inversion formulas in numerical simulations.

This talk is part of the Isaac Newton Institute Seminar Series series.

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