14.12.2019 Geodesic tomography problems on Riemannian manifolds (Railo)

One of the most fundamental inverse problems is to determine an unknown function from its line integrals over all lines. Johann Radon proved that such information determines a function uniquely on the plane in his famous article in 1917. In his dissertation Jesse Railo at the Ä¢¹½Ö±²¥
studies such integral geometric inverse problems on Riemannian manifolds with good results. Geometric inverse problems form an active international research field of mathematics.
Published
14.12.2019

The dissertation of Jesse Railo studies the geodesic ray transform which encodes the line integrals of a function along geodesics. This operator appears in many applications in imaging and partial differential equations. The dissertation establishes many conditions when such information determines a function uniquely and stably. A new numerical model for computed tomography imaging is created as a part of the dissertation.

"The research of inverse problems has been one of the success stories of Finnish mathematics. Inverse problems form a field of mathematics that studies how to determine information about internal structure of an object based on measurements on the boundary or the exterior of the object. Important applications can be found in medical imaging, or for example in the imaging of Earth or even the Universe. The power of inverse problems mathematics is in this general applicability", says Railo.

One of the main results shows that a symmetric solenoidal tensor field can be determined uniquely from its geodesic ray transform on Cartan-Hadamard manifolds, when certain geometric decay conditions are satisfied. The studied integral transforms appear in inverse scattering theory in quantum physics and general relativity.

Another main result shows that a piecewise constant vector-valued function can be determined uniquely from its geodesic ray transform with a continuous and non-singular matrix weight on Riemannian manifolds that admit a strictly convex function and have a strictly convex boundary. These integral transforms can be used to model attenuated ray transforms and inverse problems for connections and Higgs fields. The dissertation also comprehensively studies periodic methods for computed tomography.

The dissertation is published in JYU Dissertations series, number 161, The Ä¢¹½Ö±²¥, Jyväskylä, 2019. ISBN: 978-951-39-7958-4
Link to publication: 

M.Sc. Jesse Railo defends his doctoral dissertation in Mathematics "Geodesic tomography problems on Riemannian manifolds" on Saturday 14th of December 2019 At Mattilanniemi in lecture hall MaA211 at 12 o'clock. Opponent is Assistant Professor François Monard from University of California, Santa Cruz, USA and Custos is Professor Mikko Salo from Ä¢¹½Ö±²¥. The doctoral dissertation is held in English.

For further information:
Jesse Railo, jesse.t.railo@jyu.fi
Communications officer Tanja Heikkinen, tanja.s.heikkinen@jyu.fi, tel.+358 50 581 8351 
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