Polynomial optimization of the 7-parameter datum transformation problem when only three stations in both systems are given

摘要

We present here the Gauss-Jacobi combinatorial algorithm to solve in a closed form the overdetermined problem of 7-parameter datum transformation with only bare minimum number of points (i.e. three points in both coordinate systems). From the nine 7-parameter datum transformation equations, 36 minimum combinatorial subsets, each comprising seven equations are formed and solved using the Groebner basis algorithm in the first step. With each minimal combinatorial subset yielding seven elements of the solution set, a total of {7 * 36 = 252} solutions are formed. The 252 minimum combinatorial solutions are reduced to their final adjusted values in step two by means of their weighted mean via the nonlinear error/variance-covariance propagation. The advantage is that the Groebner basis algorithm (the computing engine of the Gauss-Jacobi combinatorial algorithm), which is already implemented in algebraic software such as Mathematica and Maple, does not require approximate starting values, as is always the case with traditional procedures (iterative/linearization). The procedure makes it possible for the stochasticity of both coordinate systems involved to be taken into account and becomes handy in a situation where only minimum points are given with no knowledge of the initial approximate values.