Spherical Means in Odd Dimensions and EPD equations
Boris Rubin
TL;DR
This paper presents a straightforward proof of the inversion formula for spherical mean transforms in odd dimensions, with applications to thermoacoustic tomography and the Euler-Poisson-Darboux equation.
Contribution
It offers a simple proof of the Finch-Patch-Rakesh inversion formula using analytic continuation and fractional integrals, enhancing understanding of spherical mean transforms in odd dimensions.
Findings
Proof of the inversion formula using analytic continuation
Applications to thermoacoustic tomography
Insights into the Euler-Poisson-Darboux equation
Abstract
The paper contains a simple proof of the Finch-Patch-Rakesh inversion formula for the spherical mean Radon transform in odd dimensions. This transform arises in thermoacoustic tomography. Applications are given to the Cauchy problem for the Euler-Poisson-Darboux equation with initial data on the cylindrical surface. The argument relies on the idea of analytic continuation and known properties of Erdelyi-Kober fractional integrals.
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Taxonomy
TopicsPhotoacoustic and Ultrasonic Imaging · Thermography and Photoacoustic Techniques · Numerical methods in inverse problems