Group Actions, Homeomorphisms, and Matching: A General Framework
International Journal of Computer Vision - Special issue on statistical and computational theories of vision: Part II
Landmark Matching via Large Deformation Diffeomorphisms on the Sphere
Journal of Mathematical Imaging and Vision
Analysis of Planar Shapes Using Geodesic Paths on Shape Spaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
International Journal of Computer Vision
2D-Shape Analysis Using Conformal Mapping
International Journal of Computer Vision
Diffeomorphic Matching of Diffusion Tensor Images
CVPRW '06 Proceedings of the 2006 Conference on Computer Vision and Pattern Recognition Workshop
International Journal of Computer Vision
Large Deformation Diffeomorphic Metric Curve Mapping
International Journal of Computer Vision
The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature
The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature
CVPR'04 Proceedings of the 2004 IEEE computer society conference on Computer vision and pattern recognition
Optimal data-driven sparse parameterization of diffeomorphisms for population analysis
IPMI'11 Proceedings of the 22nd international conference on Information processing in medical imaging
Kernel bundle EPDiff: evolution equations for multi-scale diffeomorphic image registration
SSVM'11 Proceedings of the Third international conference on Scale Space and Variational Methods in Computer Vision
Landmark matching via large deformation diffeomorphisms
IEEE Transactions on Image Processing
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This paper deals with the computation of sectional curvature for the manifolds of $N$ landmarks (or feature points) in $D$ dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e., the cometric), when written in coordinates, is such that each of its elements depends on at most $2D$ of the $ND$ coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, thus suggesting solving the highly nontrivial problem of developing a formula that expresses sectional curvature in terms of the cometric and its first and second partial derivatives (we call this Mario's formula). We apply such a formula to the manifolds of landmarks, and in particular we fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesics. The latter example gives insight into the geometry of the full manifolds of landmarks.