On Completeness of Groups of Diffeomorphisms

We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or compact without boundary. The main result is that for $s > \dim M/2 + 1$, the group $\mathcal D^s(M)$ is geodesically and metrically complete with a surjective exponential map. We also extend the result to its closed subgroups, in particular the group of volume preserving diffeomorphisms and the group of symplectomorphisms. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching

On Completeness of Groups of Diffeomorphisms

We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or compact without boundary. The main result is that for $s > \dim M/2 + 1$, the group $\mathcal D^s(M)$ is geodesically and metrically complete with a surjective exponential map. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.Comment: 43 pages, revised versio

Multiscale Point Correspondence Using Feature Distribution and Frequency Domain Alignment

In this paper, a hybrid scheme is proposed to find the reliable point-correspondences between two images, which combines the distribution of invariant spatial feature description and frequency domain alignment based on two-stage coarse to fine refinement strategy. Firstly, the source and the target images are both down-sampled by the image pyramid algorithm in a hierarchical multi-scale way. The Fourier-Mellin transform is applied to obtain the transformation parameters at the coarse level between the image pairs; then, the parameters can serve as the initial coarse guess, to guide the following feature matching step at the original scale, where the correspondences are restricted in a search window determined by the deformation between the reference image and the current image; Finally, a novel matching strategy is developed to reject the false matches by validating geometrical relationships between candidate matching points. By doing so, the alignment parameters are refined, which is more accurate and more flexible than a robust fitting technique. This in return can provide a more accurate result for feature correspondence. Experiments on real and synthetic image-pairs show that our approach provides satisfactory feature matching performance