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Evaluation of the Newton-Raphson Method for Three-Point Resection in Photogrammetry

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Three-point resection in photogrammetry (based on measured distances between three object points and the corresponding image coordinates) has been a viable approach because it does not require data on the object coordinates. Many iterative and closed-form solutions of the problem have been developed. The problem may have up to four solutions for the distances between the exposure station and the objects. Due to the complexity of the closed-form solutions, a simple method involving Newton-Raphson (N-R) search has recently been suggested for determining the unique (correct) solution for the Grunert's quartic polynomial. The N-R method, however, may not converge to the unique solution. This paper first describes the geometry of the three-point resection and the difficulties with the N-R method. A new Excel-based method that identifies all four solutions of the quartic polynomial is then presented. The method does not require initial estimates of the roots. Situations in which the N-R method is useful are highlighted. Application examples are used to illustrate the issues and concepts addressed. The proposed method, which provides insights into space resection, should be of interest to both researchers and educators.
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Keywords: IMAGE; NEWTON-RAPHSON; OBJECTS; PHOTOGRAMMETRY; POLYNOMIAL; THREE-POINT RESECTION; UNIT VECTORS

Document Type: Research Article

Publication date: March 1, 2007

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