In [7], the exponential mapping from the arithmetic mean of points on the Lie algebra (3) to the Lie group SE(3) was constructed to give the Riemannian mean in order selleckchem to get a mean filter.In this paper, the Riemannian means on SE(n) and those on UP(n), which are both important noncompact matrix Lie groups [13, 14], are considered, respectively. Especially, SE(3) is the spacial rigid body motion, and UP(3) is the 3-dimensional Heisenberg group H(3). Based on the left invariant metric on the matrix Lie groups, we get the geodesic distance between any two points and take their sum as a cost function. And the Riemannian mean will minimize it. Furthermore, the Riemannian mean on SE(n) is gotten using the Riemannian gradient algorithm, rather than the approximate mean.

An iterative formula for computing the Riemannian mean on UP(n) is proposed according to the Jacobi field. Finally, we give some numerical simulations on SE(3) and those on H(3) to illustrate our results.2. Overview of Matrix Lie GroupsIn this section, we briefly introduce the Riemannian framework of the matrix Lie groups [15, 16], which forms the foundation of our study of the Riemannian mean on them.2.1. The Riemannian Structures of Matrix Lie GroupsA group G is called a Lie group if it has differentiable structure: the group operators, that is, G �� G �� G, (x, y) x ? y and G �� G, x x?1, are differentiable, x, y G. A matrix Lie group is a Lie group with all elements matrices. The tangent space of G at identity is the Lie algebra , where the Lie bracket is defined.

The exponential map, denoted by exp , is a map from the Lie algebra to the group G. Generally, the exponential map is neither surjective nor injective. Nevertheless, it is a diffeomorphism between a neighborhood of the identity I on G and a neighborhood of the identity 0 on . The (local) inverse of the exponential map is the logarithmic map, denoted by log .The most general matrix Lie group is the general linear group GL(n, ) consisting of the invertible n �� n matrices with real entries. As the inverse image of ? 0 under the continuous map A det (A), GL(n, ) is an open subset of the set of n �� n real matrices, denoted by Mn��n, which is isomorphic to n��n, it has a differentiable manifold structure (submanifold). The group multiplication of GL(n, ) is the usual matrix multiplication, the inverse map takes a matrix A on GL(n, ) to its inverse A?1, and the identity element is the identity matrix I.

The Lie algebra (n, ) of GL(n, ) turns out to be Mn��n ?X,Y��??(n,?).(1)All?with the Lie bracket defined by the matrix commutator[X,Y]=XY?YX, other real matrix Lie groups are subgroups of GL(n, ), and their group operators Carfilzomib are subgroup restrictions of the ones on GL(n, ). The Lie bracket on their Lie algebras is still the matrix commutator.Let S denote a matrix Lie group and its Lie algebra.