International Journal of Computer Vision
, Volume 105, Issue 2, pp 144154
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 lowerdimensional 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.
Keywords
Image registration Conformal mappings Infinite dimensional manifolds Geodesic warps LDDMM Title
 Geodesic Warps by Conformal Mappings
 Journal

International Journal of Computer Vision
Volume 105, Issue 2 , pp 144154
 Cover Date
 201311
 DOI
 10.1007/s112630120584x
 Print ISSN
 09205691
 Online ISSN
 15731405
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Image registration
 Conformal mappings
 Infinite dimensional manifolds
 Geodesic warps
 LDDMM
 Industry Sectors
 Authors

 Stephen Marsland ^{(1)}
 Robert I. McLachlan ^{(2)}
 Klas Modin ^{(2)}
 Matthew Perlmutter ^{(2)}
 Author Affiliations

 1. School of Engineering and Advanced Technology (SEAT), Massey University, Palmerston North, New Zealand
 2. Institute of Fundamental Sciences (IFS), Massey University, Private Bag 11222, Palmerston North, New Zealand