Efficient algorithms for the regularization of dynamic inverse problems - part I and II

Abstract

Part I: Theory In this paper dynamic inverse problems are studied, where the investigated object is allowed to change during the measurement procedure. In order to achieve reasonable results, temporal a priori information will be considered. Here, "temporal smoothness" is used as a quite general, but for many applications sufficient, a priori information. This is justified in the case of slight movements during a x-ray scan in computerized tomography, or in the field of current density reconstruction, where one wants to conclude from electrical measurements on the heads surface to locations of brain activity. First, the notion of a dynamic inverse problem is introduced, then we describe how temporal smoothness can be incorporated in the regularization of the problem, and finally an efficient solver and some regularization properties of this solver are presented. Part II: Applications In this part the application of the new methods to three practical important problems, namely dynamic computerized tomography, dynamic electrical impedance tomography and spatio-temporal current density reconstructions will be presented. Dynamic reconstructions will be carried out in simulated objects which show the quality of the methods and the efficiency of the solution process. A comparison to a Kalman-smoother approach will be given for dynEIT.