Abstract
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.
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Notes
This generalisation of EPDiff has not yet been worked out in detail in the literature. However, it is likely that the approach developed in Gay-Balmaz et al. (2012) for free boundary flow can be used with only minor modifications.
One can also look at the generalisation of EPDiff to embeddings from a Klein geometry perspective. Indeed, let \(\mathrm{Diff}_\mathsf{U }(\mathbb{R }^2)\) denote the diffeomorphisms that leaves the domain \(\mathsf U \) invariant. Then the embeddings \(\mathrm{Emb}(\mathsf U ,\mathbb{R }^2)\) can be identified with the space of co-sets \(\mathrm{Diff}(\mathbb{R }^2)/\mathrm{Diff}_\mathsf{U }(\mathbb{R }^2)\).
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Acknowledgments
This work was funded by the Royal Society of New Zealand Marsden Fund and the Massey University Postdoctoral Fellowship Fund. The authors would like to thank the reviewers for helpful comments and suggestions.
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Marsland, S., McLachlan, R.I., Modin, K. et al. Geodesic Warps by Conformal Mappings. Int J Comput Vis 105, 144–154 (2013). https://doi.org/10.1007/s11263-012-0584-x
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DOI: https://doi.org/10.1007/s11263-012-0584-x