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MS54 2: The x-ray transform and its generalizations: Theory, methods, and applications
Time:
Monday, 04/Sept/2023:
4:00pm - 6:00pm
Session Chair: Suman Kumar Sahoo
Location:VG1.101
Presentations
Inversion of the momenta X-ray transform of symmetric tensor fields in the plane
Kamran Sadiq
Johann Radon Institute (RICAM), Austria
The X-ray transform of symmetric tensor fields recovers the tensor field only up to a potential field. In 1994, V. Sharafutdinov showed that augmenting the X-ray data with several momentum-ray transforms establishes uniqueness, with a most recent work showing stability of the inversion. In this talk, I will present a different approach to stably reconstruct symmetric tensor fields compactly supported in the plane.
The method is based on the extension of Bukhgeim's theory to a system of $A$-analytic maps.
This is joint work with H. Fujiwara, D. Omogbhe and A. Tamasan.
Simultaneous recovery of attenuation and source density in SPECT and multibang regularisation
Sean Holman
University of Manchester, United Kingdom
I will discuss results about simultaneous recovery of the attenuation and source density in the SPECT inverse problem, which is given mathematically by the attenuated ray transform. Assuming the attenuation is piecewise constant and the source density piecewise smooth we show that, provided certain conditions are satisfied, it is possible to uniquely determine both. I will also discuss a numerical algorithm that allows for determination of both parameters in the case when the range of the piecewise constant attenuation is known and look at some synthetic numerical examples. This is based on joint work with Philip Richardson.
Inversion of a restricted transverse ray transform on symmetric $m$-tensor fields in $\mathbb{R}^3$
Rohit Kumar Mishra, Chandni Thakkar
Indian Institute of Technology Gandhinagar, India
In this work, we study a restricted transverse ray transform on symmetric $m$-tensor fields in $\mathbb{R}^3$ and provide an explicit inversion algorithm to recover the unknown $m$-tensor field. We restrict the transverse ray transform to all lines passing through a fixed curve $\gamma$ satisfying the Kirillov-Tuy condition. This restricted data is used to find the weighted Radon transform of components of the unknown tensor field, which we use to recover components of the tensor field explicitly.
Inverse problems, unique continuation and the fractional Laplacian
Jesse Railo
University of Cambridge, United Kingdom
The Calderón problem is a famous nonlinear model inverse problem: Do voltage and current measurements on the boundary of an object determine its electric conductivity uniquely? X-ray computed tomography is a famous linear model inverse problem studied via Radon transforms. We discuss how the fractional Laplacians pop up in the analysis of Radon transforms. We then discuss recent results on the unique continuation of the fractional Laplacians and the related Caffarelli-Silvestre extension problem for $L^p$ functions. We explain some of the implications to the analysis of Radon transforms with partial data and its further generalizations. Finally, we discuss the role of unique continuation in recent mathematical studies of the Calderón problem to nonlocal equations.
