Shape matching by variational computation of geodesics on a manifold

Klassen et al. [9] recently developed a theoretical formulation to model shape dissimilarities by means of geodesics on appropriate spaces. They used the local geometry of an infinite dimensional manifold to measure the distance dist(A, B) between two given shapes A and B. A key limitation of their approach is that the computation of distances developed in the above work is inherently unstable, the computed distances are in general not symmetric, and the computation times are typically very large. In this paper, we revisit the shooting method of Klassen et al. for their angle-oriented representation. We revisit explicit expressions for the underlying space and we propose a gradient descent algorithm to compute geodesics. In contrast to the shooting method, the proposed variational method is numerically stable, it is by definition symmetric, and it is up to 1000 times faster.