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Overlapping calculation subproblemThe overlapping calculation subproblem is solved in the global frame with fixed parameters {c_{q}}, {r_{i}}, and {g_{i}}. It comprises two steps: recognizing the overlapping point pairs and then calculating the overlapping deviations. Mathematically, the corresponding point of the measuring point in subaperture k is the projection on the surface represented by the discrete measuring point set in subaperture i. It typically implies a surface is first fitted to the point set and then the measuring point is projected to this surface. It is time consuming and not suitable for subaperture stitching because a large number of points are treated simultaneously. Actually, the nominal surface model can be utilized to simplify the problem. First of all, measuring points in subaperture k and subaperture i are projected to the nominal surface, yielding projections {hj,k} and {h,,,}. At the same time, the signed distance to the nominal surface is calculated along with the normal vector n according to Eq. (42). The point in subaperture k is said to lie in the overlapping region if its projection hj_{o},_{k} on the OXY plane lies in the convex hull of projections of {hji} on the OXY plane. Consequently, it is simplified as a 2D computational geometry problem. Figure 20 2D case of overlapping calculation. Figure 20 shows the 2D case of the overlapping calculation subproblem. Because the projection lies within the segment PjJpjij, we say point {gJ^{1}Wj _{j}} is an overlapping point. For simple surfaces such as a plane, sphere, and so on, it is straightforward to calculate the projection of a point to the surface. For conic surfaces, it involves solving a cubic or quartic equation. The most complex case is to calculate the projection on a freeform surface, which requires nonlinear optimization techniques. Based on Voronoi triangularization, the computational complexity is approximately linear for finding all points in a 2D set enclosed in the other 2D set. 
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