Geodesic Warps by Conformal Mappings

Journal article, 2012

In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications in e.g. medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D’ Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern meth- ods, which are drawn from the full set of diffeomorphisms, he deliberately chose lower-dimensional sets of transforma- tions, such as planar conformal mappings. In this paper we study warps composed of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several exam- ples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations.