In various problems in image processing, the data take values in a nonlinear manifold. Examples are circle-valued data in SAR imaging, special orthogonal group-valued data expressing vehicle headings, aircraft orientations or camera positions, motion group data as well as shape-space data. A further example is the space of positive matrices representing the diffusibility of water molecules in DTI imaging. In this talk we consider various variational approaches for such data including TV and higher order methods.
Many non-convex variational models from image processing can be solved globally using convex relaxation frameworks. A popular framework comes from the context of calibration methods. We show that this framework can be interpreted as a measure-valued variational problem. This perspective can be applied to scalar, vectorial and manifold-valued problems with first- and second-order regularization.
It is known that optimality criteria form the foundations of mathematical programming both theoretically and computationally. In this talk will be approach generalized optimality conditions for weak Pareto-optimal solutions on Riemannian manifolds which allowed to consider the proximal point method for nondifferentiable multiobjective programs without any assumption of convexity over the constraint sets that determine the vectorial improvement steps throughout the iterative process.