In recent years there has been considerable interest in methods fordiffeomorphic warping of images, with applications e.g.\ in medical imaging andevolutionary biology. The original work generally cited is that of theevolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps todeform images of one species into another. However, unlike the deformations inmodern methods, which are drawn from the full set of diffeomorphism, hedeliberately chose lower-dimensional sets of transformations, such as planarconformal mappings. In this paper we study warps of such conformal mappings. The approach is toequip the infinite dimensional manifold of conformal embeddings with aRiemannian metric, and then use the corresponding geodesic equation in order toobtain diffeomorphic warps. After deriving the geodesic equation, a numericaldiscretisation method is developed. Several examples of geodesic warps are thengiven. We also show that the equation admits totally geodesic solutionscorresponding to scaling and translation, but not to affine transformations.
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