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

本文探讨了Lie群的概念,它结合了群结构和流形结构,并介绍了Stiefel流形作为正交基的集合。进一步讨论了Grassmann流形,即向量子空间的集合。同时,文章还触及了Riemann流形中内积的引入。内容丰富,适合计算机视觉领域的研究者深入理解几何结构在数学和应用中的角色。

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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 

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