(A) 铺镇铺 Figure 21. 3: Matching via integral properties Method 1 Let us assume that we have two surfaces r1 and r2, where rI is an approximated surface of input data points. The overall procedure is shown in Figure 21.4 In step 100, all generic umbilical points are located on both surfaces ri and r2 using the IPP algorithm [9, 10. Non-generic umbilical points are not used in this process. If a generic umbilical point does not exist, this procedure cannot be applied In step 102, the correspondence search is performed. The value w in the complex plane is scale-independent so that qualitative correspondences can be built from this step. Suppose that matched pairs are denoted as mk,(k= 1, . nk), where nk is the number of matched pairs Then when at least one pair is found, the next step 104 is performed. If no correspondence is established, then the algorithm stops, implying that the umbilical point method cannot used in this case Step 104 resolves the scaling transformation. To calculate a scaling factor, the normal curvatures are evaluated at the corresponding umbilical points on both surfaces r1 and r2 Then the ratio between them is obtained as a scaling factor. Suppose that a surface r is scaled with a scaling factor o, denoted as r, Then the normal curvature k on r is scaled to be k on ra. Therefore using this relation, the scaling factor can be recovered In step 106, after sorting out candidate points, a rigid body transformation is estimated by using the unit quaternion method 3]. Since the number of matched pairs is more than three and if at least three pairs survive the selection process, the problem reduces to finding a rigid body transformation with three known corresponding pairs. Using the methods in 3 a rotation matrix and a translation vector can be calculated. If the matched pairs fail in the(B) (C) (D) (A) Figure 21.3: Matching via integral properties Method 1 Let us assume that we have two surfaces r1 and r2, where r1 is an approximated surface of input data points. The overall procedure is shown in Figure 21.4. In step 100, all generic umbilical points are located on both surfaces r1 and r2 using the IPP algorithm [9, 10]. Non-generic umbilical points are not used in this process. If a generic umbilical point does not exist, this procedure cannot be applied. In step 102, the correspondence search is performed. The value ω in the complex plane is scale-independent so that qualitative correspondences can be built from this step. Suppose that matched pairs are denoted as mk, (k = 1, · · · , nk), where nk is the number of matched pairs. Then when at least one pair is found, the next step 104 is performed. If no correspondence is established, then the algorithm stops, implying that the umbilical point method cannot be used in this case. Step 104 resolves the scaling transformation. To calculate a scaling factor, the normal curvatures are evaluated at the corresponding umbilical points on both surfaces r1 and r2. Then the ratio between them is obtained as a scaling factor. Suppose that a surface r is scaled with a scaling factor σ, denoted as rσ. Then the normal curvature κ on r is scaled to be κ σ on rσ. Therefore using this relation, the scaling factor can be recovered. In step 106, after sorting out candidate points, a rigid body transformation is estimated by using the unit quaternion method [3]. Since the number of matched pairs is more than three and if at least three pairs survive the selection process, the problem reduces to finding a rigid body transformation with three known corresponding pairs. Using the methods in [3] a rotation matrix and a translation vector can be calculated. If the matched pairs fail in the 10