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In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which infinitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocity-based image warping methods, which have proved to yield efficient and robust estimations of diffeomorphisms in the case of large deformation. Here, we provide a rigorous and general construction of this infinite dimensional "shape manifold" on which we place a Riemannian metric. We then obtain the geodesic equations, for which we show the existence and uniqueness of solutions for all times. We finally use this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.
Trouvé et al. (Sat,) studied this question.
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