Keywords: numerical shape sensitivities, 4D segmentation, geometric image registration, semi-smooth Newton methods
Information: Project part FREELEVEL will focus on two research streams: (i) shape and topological sensitivity-based solvers in tomography and (ii) the extension of spatially adapted regularization to more general image restoration problems, e.g., involving blind deconvolution, and non-convex regularization. Concerning tomography problems, a level-set-based algorithm relying on shape and extended topological sensitivities will be realized for FDOT and MIT, respectively. For MIT, first a reduced model leading to an elliptic PDE-system will be studied and, in a next step, the full Maxwell system will be taken into account. For numerical efficiency purposes, a shape-aware adaptive finite element method will be intertwined with the level-set solver (partly with FEMBEM). In the area of image restoration, we are motivated by optical diffusion tomography problems for detecting objects located behind turbid media and by convolution identification in dual-MR techniques (MRI). We formulate these problems in terms of blind deconvolution, preferably with non-convex regularization with respect to the image. The resulting problems will be studied and solved numerically. With respect to the latter - split Bregman - iteratively re-weighted total variation and semismooth Newton solvers will be investigated. Further, motivated by sparse magnetic resonance imaging, problems in compressed sensing with convex (OPTIM) and non-convex relaxation (MRI) of the 0-norm will be treated. The latter is interesting as there is evidence that non-convex regularization goes along with a possible reduction in acquired data. Further, FREELEVEL will focus on topics supporting other projects, such as piecewise polynomial Mumford-Shah based image segmentation using topological sensitivities (INVERSE).
Within the SFB, FREELEVEL acts as a center of expertise for shape and topological sensitivity-based level-set solvers for tomography problems. Moreover, FREELEVEL contributes expertise and software in image restoration to the SFB through various cooperations with practitioners (MRI) as well as the applied mathematics group within the SFB (OPTIM, INVERSE).