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标题：Procrustean solution of the 9-parameter transformation problem
其他标题：Procrustean solution of the 9-parameter transformation problem
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作者：J. L. Awange ; K. -H. Bae ; S. J. Claessens 等
期刊名称：Earth, Planets and Space
电子版ISSN：1880-5981
出版年度：2008
卷号：60
期号：6
页码：529-537
DOI：10.1186/BF03353115
出版社：Springer Verlag
摘要：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%.
关键词：Procrustes ;9-parameter transformation ;least squares solution ;Frobenius ;singular value decomposition (SVD)