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Computational Study of the 3D Affine Transformation Part II. N-point Problem

Bela Palancz
Organization: Budapest University of Technology and Economics
Department: Photogrammetry and Geoinformatics
Piroska Zaletnyik
Organization: Budapest University of Technology and Economics
Department: Department of Geodesy and Surveying
Robert H. Lewis
Organization: Fordham University
Department: Department of Mathematics
Joseph Awange
Organization: Curtin University of Technology
Department: Spatial Sciences, Division of Resource and Environmental
Revision date


In case of considerable nonlinearity. i.e. in geodesy, photogrammetry, robotics, it is difficult to find proper initial values to solve the parameter estimation problem of 3D affine transformation with 9 parameters via linearization and/or iteration. In this paper we developed a symbolic - numeric method to achieve the solution without initial guess. Our method employs explicit analytical expressions developed by computer algebra technique via Dixon resultant as well as reduced Groebner basis for solving 3 points problem, see in Part I. Then this solution can be used as initial value for a Newton -Raphson- Krylov numerical method to solve the N points problem in a form of determined system of 9 polynomials developed by symbolic computation.

The suggested method is fast, robust and it has low complexity comparing with other global as well as local numerical methods, like direct minimization via genetic algorithm, linear homotopy continuation method, Newton-Raphson method with deflation employing SVD algorithm. Numerical illustration is presented with real world geodesical data representing 1138 points of Hungarian Datum, OGPSH network.

*Applied Mathematics
*Wolfram Technology > Application Packages > Applications from Independent Developers > GeometricalGeodesy

9-parameter 3D affine transformation, solution of polynomial system, local numerical methods, Newton-Raphson method, deflation, global minimization, genetic algorithm.
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3Daffine_FullPart_2modified05.nb (245.1 KB) - Mathematica Notebook [for Mathematica 6.0]