The CMA-ES on Riemannian Manifolds to Reconstruct Shapes in 3-D Voxel Images

The covariance matrix adaptation evolution strategy (CMA-ES) has been successfully used to minimize functionals on vector spaces. We generalize the concept of the CMA-ES to Riemannian manifolds and evaluate its performance in two experiments. First, we minimize synthetic functionals on the 2-D sphere. Second, we consider the reconstruction of shapes in 3-D voxel data. A novel formulation of this problem leads to the minimization of edge and region-based segmentation functionals on the Riemannian manifold of parametric 3-D medial axis representation. We compare the results to gradient-based methods on manifolds and particle swarm optimization on tangent spaces and differential evolution.