Tomographic imaging modalities play a vital role in medicine and engineering, especially whenever the
inaccessible inner part of an object is to be investigated. For example, in medical imaging, it is crucial to have reliable and fast imaging methods for diagnostics. In nondestructive testing, tomographic methods are often used to identify damages or other irregularities to ensure the quality of a product. Mathematically, tomographic imaging requires the solution of an inverse problem.
In this talk, some basic concepts for inverse problems and their solution are introduced. Three different imaging techniques from nondestructive testing and medical imaging are addressed: Terahertz tomography, dynamic computerized tomography, and magnetic particle imaging. In addition, a fast reconstruction method is presented that yields approximate solutions which are stable with respect to noise in the measured data and can be adapted to solving nonlinear problems or to incorporate modelling inexactness.