I will talk about recent results from a number of people in my group
on Riemannian manifolds in computer vision. In many Vision problems
Riemannian manifolds come up as a natural model. Data related to
a problem can be naturally represented as a point on a Riemannian manifold.
This talk will give an intuitive introduction to Riemannian manifolds,
and show how they can be applied in many situations.
Manifolds of interest include the manifold of Positive Definite matrices and the Grassman Manifolds,
which have a role in object recognition and classification, and the Kendall
shape manifold, which represents the shape of 2D objects.
Of particular interest is the question of when one can define positive-
definite kernels on Riemannian manifolds. This would allow the
application of kernel techniques of SVMs, Kernel FDA, dictionary
learning etc directly on the manifold.
An algorithm to segment and track the myocardium using the level-set formalism is described. To this end, two priors are proposed: a shape prior based on a geometric model (hyperquadrics) and a motion one expressed as a level conservation constraint on the implicit function associated to the level-set. These priors term are coupled with a local data attachment term and a thickness term that prevents both contours from merging. The algorithm is validated on 20 echocardiographic sequences with manual references of 2 experts.