Furthermore,(EF0H)?1=(E?1?E?1FH?10H?1).(26)Now, we give the geodesic distance on SE(n) as follows.Lemma 3 ��The geodesic distance between two points P selleckchem and Q on SE(n) induced by the scale-dependent left invariant metric (7) is given byd(P,Q)=(||log?(A1TA2)||F2+||b2?b1||F2)1/2.(27)Proof ��As mentioned above, the geodesic distance between two matrices P and Q on SE(n) is achieved by the length of geodesics connecting them; thus, we will compute it through substituting (22) into (11).From Lemma 2, we =((A1TA2)?tA1T?(A1TA2)?tA1T(b1+(b2?b1)t)01).(28)Then, according?get��P,Q?1(t) to the principle about the derivatives of the matrix-valued functions, the following formula is valid:�èBP,Q(t)=(A1(A1TA2)tlog?(A1TA2)b2?b100).(29)Moreover, we have =?log?2(A1TA2)+(b2?b1)T(b2?b1).

(30)Therefore,??thattr?((��P,Q?1(t)�èBP,Q(t))T��P,Q?1(t)�èBP,Q(t)) the geodesic distance on SE(n) between P and Q is given byd(P,Q)=��01(?log?2(A1TA2)+(b2?b1)2)1/2dt=(||log?(A1TA2)||F2+||b2?b1||F2)1/2.(31)This completes the proof of Lemma 2.In addition, it is valuable to mention that the distance ||log (A1TA2)||F, induced by the standard bi-invariant metric on SO(n), stands for the rotation motion from the point P to Q and the distance ||b2 ? b1||F stands for the translation motion on n. Therefore, considering an object undergoing a rigid body Euclidean motion, then, this motion can be decomposed into a rotation with respect to the center of mass of the object and a translation of the center of mass of the object.Theorem 4 ��For N given points on SE(n)Pk=(Akbk01),(32)where Ak SO(n)andbk n, k = 1,2,��, N, if the Riemannian mean of A1, A2,��, AN and the Riemannian mean ofb1, b2,��, bN (i.

e., arithmetic mean) are denoted by A�� and b��, respectively, then, one has the Riemannian mean P�� of P1, P2,��, PN SE(n) byP��=(A��b��01).(33)Proof ��In the Riemannian sense, by (13), the mean P�� is defined asP��=arg?min?P��SE(n)12N��k=1Nd(Pk,P)2=arg?min?P��SE(n)12N��k=1N(||log?(AkTA)||F2+||bk?b||F2)=arg?min?A��SO(n)12N��k=1N||log?(AkTA)||F2+argmin?b��?n12N��k=1N||bk?b||F2.(34)From [9], the geodesic distance between Ak and A on SO(n) is given byd(Ak,A)2=||log?(AkTA)||F2,(35)so we have thatargmin?A��SO(n)12N��k=1N||log?(AkTA)||F2=argmin?A��SO(n)d(Ak,A)2=A��.(36)On the other hand, for bk n, k = 1,2,��, N, it is easy to see thatargmin?b��?n12N��k=1N||bk?b||F2=1N��k=1Nbk=b��.

(37)Therefore, the fact is shown that the Riemannian Batimastat mean b�� of bk is equivalent to the arithmetic mean.Consequently, we prove that equality (33) is valid. In addition, let L denote the cost function of the minimization problem (34) on SE(n); that is,L(P)=Lrota(A)+Ltrans?(b)=12N��k=1N||log?(AkTA)||F2+12N��k=1N||bk?b||F2,(38)where LrotaandLtrans stand for the rotation and the translation components of the cost function L, respectively. We have the gradient of Lrota(A) for A SO(n) as follows [21]:grad(Lrota)=?A��k=1Nlog?(ATAk).