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Riemannian Means on Special Euclidean Group and Unipotent Matrices Group Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 292787, 9 pages http://dx.doi.org/10.1155/2013/292787 Research Article Riemannian Means on Special Euclidean Group and Unipotent Matrices Group Xiaomin Duan,1 Huafei Sun,1 and Linyu Peng2 1 School of Mathematics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK Correspondence should be addressed to Huafei Sun; [email protected] Received 1 August 2013; Accepted 16 September 2013 Academic Editors: R. Abu-Saris and P. Bracken Copyright © 2013 Xiaomin Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results. 1. Introduction exponential mapping from the arithmetic mean of points on the Lie algebra se(3) to the Lie group SE(3) was constructed A matrix Lie group, which is also a differentiable manifold to give the Riemannian mean in order to get a mean filter. simultaneously, attracts more and more researchers’ atten- In this paper, the Riemannian means on SE() and those tion from both theoretic interest and its applications [1–5]. on UP(), which are both important noncompact matrix The Riemannian mean on the matrix Lie groups is widely Lie groups [13, 14], are considered, respectively. Especially, studied for its varied applications in biomedicine, signal SE(3) is the spacial rigid body motion, and UP(3) is the processing, and robotics control [6–9]. Fiori and Tanaka 3-dimensional Heisenberg group (3).Basedontheleft [10] suggested a general-purpose algorithm to compute the invariant metric on the matrix Lie groups, we get the geodesic average element of a finite set of matrices belonging to distance between any two points and take their sum as a any matrix Lie group. In [11], the author investigated the cost function. And the Riemannian mean will minimize Riemannian mean on the compact Lie groups and proposed a it. Furthermore, the Riemannian mean on SE() is gotten globally convergent Riemannian gradient descent algorithm. using the Riemannian gradient algorithm, rather than the Different invariant notions of mean and average rotations on approximate mean. An iterative formula for computing the SO(3) (it is compact) are given in [9]. Recently, Fiori [12]dealt Riemannian mean on UP() is proposed according to the with computing averages over the group of real symplectic Jacobi field. Finally, we give some numerical simulations on matrices, which found applications in diverse areas such as SE(3) and those on (3) to illustrate our results. optics and particle physics. However, the Riemannian mean on the special Euclidean group SE() and the unipotent matrix group UP(),whichare 2. Overview of Matrix Lie Groups the noncompact matrix Lie groups, has not been well studied. Fletcher et al. [6]proposedaniterativealgorithmtoobtain In this section, we briefly introduce the Riemannian frame- the approximate solution of the Riemannian mean on SE(3) work of the matrix Lie groups [15, 16], which forms the by use of the Baker-Cambell-Hausdorff formula. In [7], the foundation of our study of the Riemannian mean on them. 2 The Scientific World Journal 2.1. The Riemannian Structures of Matrix Lie Groups. Agroup where denotes the left translation, denotes the right is called a Lie group if it has differentiable structure: the translation, and ()∗ and (−1 )∗ are the tangent mappings group operators, that is, × → , (,) →⋅and associated with and −1 , respectively. The adjoint action −1 →,→ , are differentiable, , .AmatrixLie ∈ Ad : s → s is group is a Lie group with all elements matrices. The tangent −1 space of at identity is the Lie algebra g,wheretheLiebracket Ad= . (5) is defined. The exponential map, denoted by exp, is a map from It is also easy to see the formula that the Lie algebra g to the group . Generally, the exponential = . map is neither surjective nor injective. Nevertheless, it is Ad (6) a diffeomorphism between a neighborhood of the identity on and a neighborhood of the identity 0 on g.The Then, the left invariant metric on is given by (local) inverse of the exponential map is the logarithmic map, ⟨, ⟩ =⟨( −1 ) , ( −1 ) ⟩ denoted by log. ∗ ∗ The most general matrix Lie group is the general linear (7) (, R) × =⟨−1, −1⟩ := ((−1) −1) group GL consisting of the invertible matrices tr with real entries. As the inverse image of R −{0}under the → () (, R) continuous map det ,GL is an open subset with , ∈ and tr denoting the trace of the matrix. of the set of ×real matrices, denoted by ×,whichis × Similarly, we can define the right invariant metric on as well. isomorphic to R , it has a differentiable manifold structure It has been shown that there exist the left invariant metrics on (submanifold). The group multiplication of GL(, R) is the all matrix Lie groups. usual matrix multiplication, the inverse map takes a matrix (, R) −1 on GL to its inverse , and the identity element is the 2.2. Compact Matrix Lie Group. A Lie group is compact if identity matrix .TheLiealgebragl(, R) of GL(, R) turns () its differential structure is compact. The unitary group , out to be × with the Lie bracket defined by the matrix the special unitary group SU(), the orthogonal group (), commutator the special orthogonal group SO(),andthesymplecticgroup Sp() are the examples of the compact matrix Lie groups [17]. [, ] =−, gl∀,∈ (, R) . (1) Denote a compact Lie group by 1 anditsLiealgebrabys1. ⟨⋅, ⋅⟩ All other real matrix Lie groups are subgroups of There exists an adjoint invariant metric on 1 such that GL(, R), and their group operators are subgroup restrictions ⟨Ad, Ad⟩ = ⟨, ⟩ (8) of the ones on GL(, R). The Lie bracket on their Lie algebras is still the matrix commutator. with , ∈ s1. Notice the fact that the left invariant metric Let denote a matrix Lie group and s its Lie algebra. of any adjoint invariant metric is also right invariant; namely, The exponential map for turns out to be just the matrix it is a bi-invariant metric; so all compact Lie groups have exponential; that is, given an element ∈s, the exponential bi-invariant metrics. Furthermore, if the left invariant and map is the adjoint invariant metrics on 1 deduce a Riemannian ∇ ∞ connection , then the following properties are valid: exp () = ∑ . (2) ! 1 =0 ∇ = [, ] , 2 (9) Theinversemap,thatis,thelogarithmicmap,isdefinedas 1 follows: ⟨R (, ) , ⟩ =− ⟨[, ] , [, ]⟩ , 4 ∞ +1 (−) log () = ∑ (−1) , (3) where R(, ) is a curvature operator about the smooth =1 tangent vector field on the Riemannian manifold (1,∇). Therefore, the section curvature K is given by for in a neighborhood of the identity of . The exponential ofamatrixplaysacrucialroleinthetheoryoftheLiegroups, ⟨[, ] , [, ]⟩ which can be used to obtain the Lie algebra of a matrix Lie K (, ) = ≥0, 4(⟨, ⟩⟨, ⟩ − ⟨, ⟩2) (10) group, and it transfers information from the Lie algebra to the Lie group. K The matrix Lie group also has the structure of a Rieman- which means that is nonnegative on the compact Lie nian manifold. For any , ∈ and ∈,thetangent group. space of at ,wehavethemapsthat In addition, according to the Hopf-Rinow theorem, a compact connected Lie group is geodesically complete. It = , ()∗=, means that, for any given two points, there exists a geodesic curve connecting them and the geodesic curve can extend −1 (4) = ,(−1 )∗=, infinitely. The Scientific World Journal 3 3 2.3. The Riemannian Mean on Matrix Lie Group. Let : with =(,,)∈R . ‖‖ gives the amount of [0, 1] → be a sufficiently smooth curve on . We define rotation with respect to the unit vector along ,where‖⋅‖ the length of () by denotes the Frobenius norm. Physically, represents the 1 angularvelocityoftherigidbody,whereasV corresponds to ̇ ̇ ℓ():=∫ √⟨ () , ()⟩() the linear velocity [19]. In [18], the author presents a closed- 0 form expression of the exponential map se(3) → SE(3) by (11) 1 1− − = ∫ √ {(()−1̇ ()) ()−1̇ ()}, cos ( ) 2 sin ( ) 3 tr exp () =4 ++ + (18) 0 2 3 where denotes the transpose of the matrix. The geodesic 2 2 2 2 with ∈se(3) and = + +.Notethatitcan distance between two matrices and on considered as be regarded as an extension of the well-known Rodrigues a differentiable manifold is the infimum of the lengths ofthe formula on SO(3).ThelogarithmicmapSE(3) → se(3) is curves connecting them; that is, yielded as (,) := inf {ℓ()|:[0, 1] = ( − + 2 − 3), (12) log ( ) 1 2 4 3 4 5 (19) → with (0) =,(1) =}. where According to the Euclidean analogue (mean on Euclidean 1 3 space), a definition of the mean of matrices 1,..., is = ( ) ( ), 1 8csc 2 sec 2 the minimizer of the sum of the squared distances from any matrix to the given matrices 1,..., on . Now, we define 2 =cos (2) − sin () , theRiemannianmeanbasedonthegeodesicdistance(12). (20) 3 =cos () +2cos (2) − sin () − sin (2) , Definition 1. The mean of given matrices 1,..., on in the Riemannian sense corresponding to the metric (7)is 4 =2cos () +cos (2) − sin () − sin (2) , defined as = () − () , 5 cos sin 1 2 =arg min ∑(,) .
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