International Journal of Computer Vision

, Volume 105, Issue 2, pp 144-154

First online:

Geodesic Warps by Conformal Mappings

  • Stephen MarslandAffiliated withSchool of Engineering and Advanced Technology (SEAT), Massey University
  • , Robert I. McLachlanAffiliated withInstitute of Fundamental Sciences (IFS), Massey University
  • , Klas ModinAffiliated withInstitute of Fundamental Sciences (IFS), Massey University Email author 
  • , Matthew PerlmutterAffiliated withInstitute of Fundamental Sciences (IFS), Massey University

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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 methods, which are drawn from the full set of diffeomorphisms, he deliberately chose lower-dimensional sets of transformations, 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 examples 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.


Image registration Conformal mappings Infinite dimensional manifolds Geodesic warps LDDMM