Some Matrix manifolds (Lie group, Grassmann manifold and Riemannian manifold) for computer vision

Lie group

A Lie group is a set G with two structures: G is a group and G is a (smooth,

real) manifold. These structures agree in the following sense: multiplication and

inversion are smooth maps.

Stiefel manifold

The Stiefel manifold of orthonormal -frames in Rn is the collection of vectors (v1,v2 ...,vn ) where vi is in Rn

for all i, and the k-tuple (v1, v2...,vk ) is orthonormal.

Grassmann manifolds is a certain collection of vector subspaces of a vector space. In particular, gnk is the Grassmann manifold of k-dimensional subspaces of the vector

space.

Riemannian manifolds- In the Riemannian framework, the tangent space TxM at each point x of a manifold M is endowed with a smooth inner product <⋅, ⋅>x.

转自：http://home.iitk.ac.in/~maninder/cs365/hw2/hw2.pdf

some papers in computer vision:

Learning the Irreducible Representations of Commutative Lie Groups