On variational problems and gradient flows in image
This thesis treats different methods and theoretical aspects of the calculus of variations and their applications in image processing, especially those in which gradient flows for the minimization of the variational problems are used. The focus lies on geometrical methods for image registration, as well as numerical methods for computing the solution for an implicit variant of Willmore flow (level set method). The problem of image registration aims at finding a suitable correspondence map by means of a deformation of one image domain to the other, such that significant structures are overlaid correctly. In practical applications, these images are often aquired by different sensors and hence yield different image modalities. Image registration plays an important role in the clinical evaluation of medical data as for example from computed tomography or magnetic resonance imaging, as well as diagnosis and surgery planning. The proposed method aims at an alignment of geometrical descriptors instead of merely comparing image intensities. In particular, the approach is morphological (contrast invariant). Registration methods are in general ill-posed inverse problems, hence regularization methods are necessary. In order to ensure a robust and stable minimization process, three complementary regularization strategies are incorporated and linked to each other, namely multiscale methods, Tikhonov-regularization and regularized gradient flows. The latter are based on the definition of a regularizing metric, that is used to define the representation of the regularized gradient of the energy functional. Furthermore, a nonlinear polyconvex and hyperelastic energy functional that takes the role of the Tikhonov-regularization functional allows to devise an existence proof of the geometric registration approach. The theoretical justification of the multiscale approach is given by the study of the variational convergence of the regularized functionals by means of Gamma-convergence. In order to further stabilize the registration process, a new Mumford-Shah-based method to align significant edges is proposed. It aims the simultaneous process of registration, segmentation (feature detection) and image restoration, since these processes typically interdepend on each other. Here, two different approaches are proposed, numerically incorporated and compared: a phase-field approach, that is based on the Ambrosio-Tortorelli approximation of the Mumford-Shah functional and a corresponding level set approach. The minimization of the Willmore functional via a semi-implicit numerical scheme is applied to the recontruction of destroyed regions of surfaces that are given by a level set representation. The introduction of an implicit narrow band method allows a significant acceleration of the geometrical evolution.