62 D.-M. Chen and W.D. Zhu Substituting Eq. (6.10) into Eq. (6.7) yields ı2 Dmin NX iD1 .pi p/ R p0i p0 2 (6.11) Let xi Dpi p; yi Dp0i p0 (6.12) Substituting Eq. (6.12) into Eq. (6.11) yields ı2 Dmin NX iD1 kxi Ryik 2 Dmin tr .X RY/ T .X RY/ Dmin tr XTXCYTRTRY YTRTX XTRY (6.13) where XD[x1, x2, : : : , xN] is a 3 by N matrix that consists of coordinates of Nscan points with respect to the SCS, YD[y1, y2, : : : , yN] is a 3 by N matrix that consists of coordinates of N scan points with respect to the SMCS, and tr denotes the trace of a matrix. The rotation matrix Ris an orthogonal matrix with RTRDI, where I is an identity matrix. The first two terms in Eq. (6.13) do not depend onR; excluding them would not affect the optimization problem. Hence, Eq. (6.13) becomes min tr YTRTX XTRY Dmin tr XTRY XTRY Dmin tr 2RYXT (6.14) The minimization problem in Eq. (6.14) is equivalent to a maximization problem: min tr 2RYXT Dmax tr 2RYXT Dmax tr RYXT (6.15) Let ADYXT, which is called a covariance matrix. By SVD, the covariance matrixAis decomposed as ADUDVT (6.16) where Uis a 3 by 3 orthogonal matrix, Dis a 3 by 3 diagonal matrix with non-negative real values in the descending order on the diagonal, and Vis a 3 by 3 orthogonal matrix. Substituting Eq. (6.16) into Eq. (6.15) yields max tr RYXT Dmax tr RUDVT Dmax tr DVTRU (6.17) Note that V, RandUare all orthogonal matrices; MDVTRUis also an orthogonal matrix. This means that columns of M are orthonormal vectors and the magnitude of each column is 1. Hence, the domain of entryMij inMsatisfies 1 Mij 1. Equation (6.17) is expressed as max tr DVTRU Dmax tr ŒDM Dmax 3X iD1 iMii 3X iD1 i (6.18) where i is the ith non-negative singular value of A. The optimization problem in Eq. (6.18) attains a maximum whenMii D1. Because Mis an orthogonal matrix, this means that Mhas to be an identity matrix. The rotation matrix Ris obtained by RDVIUT DVUT (6.19) Since Ris a rotation matrix, the determinant of Ris 1. If the determinant of Rderived from Eq. (6.19) is 1, Ris a reflection matrix rather than a rotation matrix [10]. If Ris a reflection matrix when Mii D1, one needs to find the next maximization solution to get a rotation matrix. Equation (6.18) can be expanded as
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