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Procrustean solution of the 9-parameter transformation problem

Abstract

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 1 million-point networks, respectively. The 1 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 1 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 1.13×10−12 m and 2.31×10−13 m, while case 3 gives 2.00×10−4 m and 1.20 × 10−5 m, which is acceptable from a laser scanning point of view since the acceptable error limit is below 1 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%.

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Correspondence to J. L. Awange.

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Key words

  • Procrustes
  • 9-parameter transformation
  • least squares solution
  • Frobenius
  • singular value decomposition (SVD)