Reconstructing Karcher Means of Shapes on a Riemannian Manifold of Metrics and Curvatures

In a recent paper [1], the authors suggest a novel Riemannian framework for comparing shapes. In this framework, a simple closed surface is represented by a field of metric tensors and curvatures. A product Riemannian metric is developed based on the L norm on symmetric positive definite matrices and scalar fields. Taken as a quotient space under the group of volume-preserving diffeomorphisms, the space becomes a proper metric manifold of shapes. In this work, we simplify this representation, showing that only mean curvature and metric tensor fields are needed for a complete surface representation. In this simplified framework, we develop an algorithm for computing Karcher means, and compare the results to standard Euclidean averages of surface embeddings.