 # Overlapping calculation subproblem

The overlapping calculation subproblem is solved in the global frame with fixed parameters {cq}, {ri}, and {gi}. 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 hjo,k on the OXY plane lies in the convex hull of projections of {hji} on the OXY plane. Consequently, it is simplified as a 2-D computational geometry problem. Figure 20 2-D case of overlapping calculation.

Figure 20 shows the 2-D case of the overlapping calculation subproblem. Because the projection lies within the segment PjJpjij, we say point {gJ1Wj 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 2-D set enclosed in the other 2-D set.