Procrustean solution of the 9-parameter transformation problem

The Procrustean matching bed is employed here to provide direct solution to the 9-parameter transformation problem inherent in geodesy, navigation, computer vision and medicine. By computing the centre of mass coordinates of two given systems; scale, translation and rotation parameters are optimised using the Frobenius norm. To demonstrate the Procrustean approach, three simulated and one real geodetic network are tested. In the first case, a minimum three point network is simulated. The second and third cases consider the over-determined eight- and I million-point networks, respectively. The I million point simulated network mimics the case of an air-borne laser scanner, which does not require an isotropic scale since scale varies in the X, Y, Z directions. A real network is then finally considered by computing both the 7 and 9 transformation parameters, which transform the Australian Geodetic Datum (AGD 84) to Geocentric Datum Australia (GDA 94). The results indicate the effectiveness of the Procrustean method in solving the 9-parameter transformation problem; with case I giving the square root of the trace of the error matrix and the mean square root of the trace of the error matrix as 0.039 m and 0.013 m, respectively. Case 2 gives 1. 13 x 10(-12) m and 2.31 X 10(-13) m, while case 3 gives 2.00 x 10(-4) in and 1.20 x 10(-5) m, which is acceptable from a laser scanning point of view since the acceptable error limit is below I m. For the real network, the values 6.789 m and 0.432 m were obtained for the 9-parameter transformation problem and 6.867 m and 0.438 m for the 7-parameter transformation problem, a marginal improvement by 1.14%.