Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization

Abstract

One challenge for modern imaging methods is the investigation of objects which change during the data acquisition. This occurs in non-destructive testing as well as in medical applications, e.g. on account of patient or organ movements.Due to the object's deformations, the respective imaging modality is described by a dynamic inverse problem. In this paper, a classification scheme for linear dynamic problems depending on the object's motion is provided.Based on this scheme, we study the class in detail where the dynamic problem is still related to the operator in the static case, and where we call the deformations moderate. We proof important properties of the dynamic operator, derive a singular value decomposition and develop suitable regularization methods. The application of these methods to specific problems is illustrated at two examples including dynamic computerized tomography.