Linear and Nonlinear Inverse problems in imaging with practical Applications

Samuli Siltanen

Lecture 1: Introduction to X-ray tomography

  • What is an X-ray image? The Beer-Lambert law

  • Slice imaging and history of CT. Inverse problem of tomography

  • Are you a natural tomographer?

  • Filtered back-projection and the Radon transform

  • Applications

Lecture 2: Pixel-based imaging and matrix models

  • Why matrices for tomography instead of filtered back-projection?

  • Singular value decomposition and ill-posedness

  • Naive and regularized reconstructions for 12x12 pixel tomography

  • Review of basic regularization methods: Truncated SVD, Tikhonov regularization, total variation regularization, wavelet sparsity

  • A learning-based approach

Lecture 3:

  • Limited angle data and the Helsinki Tomography Challenge 2022

  • Nonlinear imaging: passive gamma emission tomography of spent nuclear fuel

  • More applications

Exercise/Lab classes:

Siiri Rautio, Salla Latva-Äijö, Elli Karvonen and Elena Morotti will help with the sessions.

  • Exercises on Monday: simple tomographic matrix models.
    https://github.com/ssiltane/RICAM2022tomography

  • Lab class on Tuesday: working with open datasets from Helsinki
    https://www.fips.fi/dataset.php