## The IMI Colloquium Report in January 9, 2024

2024.01.31

Title: Case study of industrial inverse problems in Finland: X-ray tomography

Speaker: Samuli Siltanen (University of Helsinki/The Finnish Center of Excellence in Inverse Problems Research)

Place: IMI Auditorium(W1-D-413)

Attendance: 14 (Students: 5; Staffs: 7; Other: 2)

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X-ray tomography is based on recording X-ray images of a patient or object along multiple projection directions and is mathematically an inverse problem of reconstructing a non-negative function from information about line integrals of the function along a finite number of lines. Since inverse problems are generally ill-posed problems, the stability, or even uniqueness, of the unknown function with respect to the observed data is not expected in X-ray tomography. The difficulties encountered in such ill-posed problems are explained to non-experts using some simple examples. He also carefully explained the basic idea of reconstruction by the regularization method using total variation, using some examples of matrices of 2×2. Finally, as an application, he introduced the results of joint research on limited-angle and cone-beam dental X-ray tomography.

Markus Juvonen, a student of the presenter, also came to the seminar and gave a lecture on new parameter choice method for total variation deblurring of images which is based on a dual-grid computation of the solution.