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Isometry invariant shape priors for variational image segmentation

Variational methods play a fundamental role in mathematical image analysis as a bridge between models and algorithms. 
A major challenge is to formulate a given model as a feasible optimization problem. There has been a huge leap in that respect concerning local data models in the framework of convex relaxation. But non-local concepts such as the shape of a sought-after object are still difficult to implement. 
In this thesis we study mathematical representations for shapes and develop shape prior functionals for object segmentation based thereon. 
A particular focus is set on the isometry invariance of the functionals and the compatibility with existing convex functionals for image labelling. 
Optimal transport is used as a central modelling and computational tool to compute registrations between different shapes as a basis for a shape similarity measure. This point of view leads to a link between the two somewhat dual representations of a shape by the region it occupies and its outline, allowing to combine their respective strengths. 
Naively the computational complexity implied by the derived functionals is unfeasible. Therefore suitable hierarchical optimization methods are developed.