Riemannian methods for optimization in a shape space of triangular meshes

Abstract

In this paper, the applicability of two optimization algorithms in the context of Riemannian manifolds is investigated. More precisely, the steepest descent method and the nonlinear conjugate gradient method are used to solve the Shape-From-Shading problem in a shape space of triangular meshes. For this reason, appropriate Riemannian metrics are introduced in this shape space. Instead of taking steps along straight lines, propagation along geodesics with respect to the Riemannian metric is employed. This requires to formulate the geodesic equation and the equation of parallel translation in the shape space of triangular meshes. In general, the proposed geodesic optimization algorithms outperform the standard steepest descent method concerning various quality measures and the visual impression of the reconstructed surface. In addition, it is shown that an appropriately chosen Riemannian metric can guide the optimization process towards a desired class of surfaces.