Author of this post: Samuli Siltanen (samuli.siltanen “at” helsinki.fi)
This blog aims to help everyone interested in inverse problems and their computational solution. We offer open software and links to open datasets so that anyone can easily try different reconstruction methods on both simulated and measured data.
Inverse problems are about interpreting indirect measurements: there is an object we want to see or understand, but we cannot image or measure the object directly. However, we do have available measurement data that is somehow related to the object of interest, but needs further processing for extracting information out. Inverse problems arise for example in medical imaging, underground prospecting, remote sensing and nondestructive testing.
A classical example of an inverse problem is medical X-ray tomography. Several two-dimensional X-ray images are taken of a patient along different projection directions. The inverse problem is to reconstruct the three-dimensional structure inside the patient from all those two-dimensional projection images. For more information about X-ray tomography, see Wikipedia and this page.
Another example is the nonlinear inverse problem of Electrical Impedance Tomography (EIT). There one feeds electric currents into a conductive body using electrodes, measures the resulting voltages, and aims to recover the electrical conductivity distribution inside the body. The EIT image formation problem is very ill-posed, or sensitive to modelling errors and measurement noise. The image above shows a phantom measured at University of Eastern Finland (left), reconstruction using the D-bar method (center, computation by Andreas Hauptmann) and reconstruction using Bayesian inversion (right, computation by Ville Kolehmainen). The EIT data is openly available at the page https://www.fips.fi/EIT_dataset.php, and we will later publish instructions in the FIPS Computational Blog (https://blog.fips.fi/) showing how to reconstruct the phantom. For more information about EIT, see Wikipedia and this page.
We hope you enjoy the blog and find it useful! Please send feedback by either commenting the blog posts or by sending email to the address samuli.siltanen “at” helsinki.fi.
The Finnish Inverse Problems Society (FIPS) wants to increase awareness and technical skills about inverse problems worldwide. This blog is part of the public outreach efforts of FIPS.