One-dimensional deconvolution
One-dimensional deconvolution is one of the basic examples in the book Mueller & Siltanen: Linear and nonlinear inverse problems with practical applications (SIAM 2012). It is very educational to study how Tikhonov regularization, Total Variation regularization and wavelet sparsity sharpen blurred signals, each in their own way. The Matlab codes are available here: https://github.com/ssiltane/deconv1D
- Building convolution matrices
- Simulating tomographic data without committing inverse crime
- Calculating singular value decomposition (SVD)
- Reconstruction by truncated SVD
- Reconstruction by Tikhonov regularization
- Reconstruction by Total Variation regularization
- Reconstruction by wavelet sparsity
- Reconstruction by neural networks
D-bar method for Electrical Impedance tomography
Electrical Impedance Tomography (EIT) aims to recover the internal electrical conductivity of a physical body from measurements of voltages and currents at the boundary of the body. EIT has applications in medical imaging, underground prospecting, and nondestructive testing. The image reconstruction problem of EIT is a nonlinear and severely ill-posed inverse problem. The D-bar method is a non-iterative regularized reconstruction method based on low-pass filtering a nonlinear Fourier transform. This repository contains Matlab routines implementing the D-bar method for simulated EIT data:
https://github.com/ssiltane/Dbar_method_for_EIT
Literature: Chapters 12-15 of the book Linear and nonlinear inverse problems with practical applications by Jennifer L Mueller and Samuli Siltanen (SIAM 2012).
Authors of the code: Jennifer Mueller, Samuli Siltanen and Janne Tamminen.
X-ray tomography
Two-dimensional X-ray tomography is one of the basic examples in the book Mueller & Siltanen: Linear and nonlinear inverse problems with practical applications (SIAM 2012). This learning resource offers presentation slides and Matlab codes that Professor Samuli Siltanen used in the winter school Advanced methods for mathematical image analysis in Bologna, Italy, January 2023. Codes are available in this repository:
https://github.com/ssiltane/BolognaWinterSchool2023
- Building tomographic matrices
- Simulating tomographic data without committing inverse crime
- Calculating singular value decomposition (SVD)
- Reconstruction by truncated SVD
- Reconstruction by Tikhonov regularization
- Reconstruction by Total Variation regularization
- Reconstruction by wavelet sparsity
