Inverse Problems

Roughly spoken, inverse problems are concerned with the determination of the cause of an observed or a desired effect. A typical exampe of an inverse problem is ultrasound imaging, whose application you probably have seen at a doctor. The goal is to estimate/image the so-called acoustic impedance of some interior region of the human body. For that purpose, ultrasound waves are sent into the patient and afterwards the pressure variations are measured outside of the patient. Because the wave propagation inside the patient depends on the acoustic impedance, it is possible to estimate the impedance from the measured pressure data. Ultrasound imaging has the advantage that it is very cheap, fast and mobile.

A drawback of pure ultrasound imaging, however, is that the acoustic impedance of many types of tumors is very similar to that of healthy tissue. An alternative, more recently developed imaging method offering improved contrast for cancer detection is photoacoustic tomography (PAT). The basic principle of PAT can be described as follows: The tissue of interest is illuminated with pulsed light, which becomes partially absorbed inside the tissue. As a consequence of the heating, pressure waves are generated within the tissue, which are measured outside the patient, for example by piezoelectric films or optical fibers. In our research, we work on various theoretical and practical aspects of PAT including image reconstruction using linear, planar or circular detectors, or taking acoustic wave dissipation into account. In our research, we also develop iterative and variational reconstruction methods that can be applied for solving more general types of inverse problems.