  Many algorithms for surface registration risk producing significant errors if
surfaces are significantly nonisometric. Manifold learning has been shown to be
effective at improving registration quality, using information from an entire
collection of surfaces to correct issues present in pairwise registrations.
These methods, however, are not robust to changes in the collection of
surfaces, or do not produce accurate registrations at a resolution high enough
for subsequent downstream analysis. We propose a novel algorithm for
efficiently registering such collections given initial correspondences with
varying degrees of accuracy. By combining the initial information with recent
developments in manifold learning, we employ a simple metric condition to
construct a measure on the space of correspondences between any pair of shapes
in our collection, which we then use to distill soft correspondences. We
demonstrate that this measure can improve correspondence accuracy between
feature points compared to currently employed, less robust methods on a diverse
dataset of surfaces from evolutionary biology. We then show how our methods can
be used, in combination with recent sampling and interpolation methods, to
compute accurate and consistent homeomorphisms between surfaces.
