where S11S1251 S21S22S23 The eigenvector corresponding to the maximum positive eigenvalue is a quaternion which minimizes the equation(21.6). An orthonormal rotation matrix R can be recovered from a quaternion q=[90, 91, 92, 93 by G+q2-9-932(qg-993)2(q1g3+9g2) R 2(qy2+g3)+q2-qi-932(q93-9q) 2(q93-992)2(g2q3+q)9+g2-q2-9 The procedure described above can also be applied to the case where more than three corre- sponding point pairs are provided 21.6 Matching problems IG WOS, IPWOS, IGWS or IPWS 21.6.1 Iterative Closest Point (ICP) algorithm [1 for IG WOS or IPWOS algorithm The point set P with Np points iFil from the data shape and the model shape X (with Nr supporting geometric primitives: points, lines, or triangles) are given The iteration is initialized by setting Po= P, go=[1,0, 0,0,0,0, 0 and k=0. The reg istration vectors are defined relative to the initial data set Po so that the final registration represents the complete transformation. Steps 1, 2, 3, and 4 are applied until convergence within a tolerance t a. Compute the closest points: Yk= C(Pk, X) b. Compute the registration:(k, dk)=Q(Po,Yk) C. Apply the registration: Pk+1=gi (Po) d. Terminate the iteration when the change in mean-square error falls below a preset threshold T>0 specifying the desired precision of the registration: Idk -dk+1 <T 21.6.2 ICP algorithm for scaling effects When initial information on transformation is provided, the ICP method can be extended to resolve scaling effects in the matching problem. A scaling factor o is included in the objective function(21.4). The scaling transformation is performed at step c in the iCP algorithm. In this case we have to provide seven initial values(three for translation, three for rotation and lingwhere M = X 3 i=1 nimT i =    s11 s12 s13 s21 s22 s23 s31 s32 s33    . (21.8) The eigenvector corresponding to the maximum positive eigenvalue is a quaternion which minimizes the equation (21.6). An orthonormal rotation matrix R can be recovered from a quaternion qˇ = [q0, q1, q2, q3] by R =    q 2 0 + q 2 1 − q 2 2 − q 2 3 2(q1q2 − q0q3) 2(q1q3 + q0q2) 2(q1q2 + q0q3) q 2 0 + q 2 2 − q 2 1 − q 2 3 2(q2q3 − q0q1) 2(q1q3 − q0q2) 2(q2q3 + q0q1) q 2 0 + q 2 3 − q 2 1 − q 2 2    . (21.9) The procedure described above can also be applied to the case where more than three corre￾sponding point pairs are provided. 21.6 Matching problems : IGWOS, IPWOS, IGWS or IPWS 21.6.1 Iterative Closest Point (ICP) algorithm [1] for IGWOS or IPWOS Algorithm • The point set P with Np points {p~i} from the data shape and the model shape X (with Nx supporting geometric primitives: points, lines, or triangles) are given. • The iteration is initialized by setting P0 = P, q~0 = [1, 0, 0, 0, 0, 0, 0]T and k = 0. The reg￾istration vectors are defined relative to the initial data set P0 so that the final registration represents the complete transformation. Steps 1,2,3, and 4 are applied until convergence within a tolerance τ . a. Compute the closest points: Yk = C(Pk, X). b. Compute the registration: (~qk, dk) = Q(P0, Yk). c. Apply the registration: Pk+1 = ~qk(P0). d. Terminate the iteration when the change in mean-square error falls below a preset threshold τ > 0 specifying the desired precision of the registration: |dk − dk+1| < τ . 21.6.2 ICP algorithm for scaling effects When initial information on transformation is provided, the ICP method can be extended to resolve scaling effects in the matching problem. A scaling factor σ is included in the objective function (21.4). The scaling transformation is performed at step c in the ICP algorithm. In this case we have to provide seven initial values (three for translation, three for rotation and one for scaling). 6