This repository contains materials for the master's course "Introduction to Inverse Problems and Imaging" at ČVUT. It includes homework assignments, lecture notes, and MATLAB code focused on solving inverse problems using numerical methods and regularization techniques.
This homework explores the inversion of the Abel integral equation, a classic ill-posed problem in inverse problems, using regularization techniques like Truncated SVD and Tikhonov regularization to handle noise and instability.
- Exam Slides: Exam_slides_1_Martin_Kunz.pdf
- Report: HW_1_Martin_Kunz.pdf
Key figures:
Decay of singular values illustrating the ill-posedness of the problem.
Truncated SVD regularization applied to noisy Abel data.
Tikhonov method balancing data fit and smoothness.
This assignment addresses the backwards heat equation, an extremely ill-posed inverse problem where small errors in data measurements lead to large errors in the reconstructed initial condition, demonstrating the need for advanced regularization methods.
- Exam Slides: Exam_slides_2_Martin_Kunz.pdf
- Report: HW_2_Martin_Kunz.pdf
Key figures:
Initial condition for the backwards heat equation problem.
Iterative Landweber method recovering the initial temperature distribution.
Tikhonov vs. Sobolev regularization for stability.
This homework deals with the non-linear inverse problem of autoconvolution, where the forward operator is a convolution of the unknown function with itself, solved using Newton's method with appropriate regularization for convergence.
- Report: HW_3_Martin_Kunz.pdf
Key figures:
Numerical vs. analytical forward operator for autoconvolution.
Convergence of Newton's method for the non-linear inverse problem.
