The Sequential Givens method for adjustment computations in photogrammetry

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Virginia Tech


The Givens orthogonalization algorithm is an efficient alternative to the normal equations method for solving many adjustment problems in photogrammetry. The Givens method is one of a class of methods for solving linear systems known generally as orthogonalization or QR methods. It allows for sequential processing and greatly simplifies the computation of statistics on the observations and residuals. The underlying reason for these advantages is the immediate availability of the orthogonal Q matrix, which is computed as the data are processed and is intimately related to the statistics needed for blunder detection. One of these statistics, the F statistic computed from externally studentized residuals, is both easily obtained and well-suited for blunder detection.

The Givens method requires nearly four times the number of computations as compared to the normal equations approach in order to reach a solution. However, depending on the size of the problem, blunder detection through the normal equations requires far more computer time than is required when starting with a Givens decomposition.

The method allows a user to review intermediate results, test residuals and modify the solution without having to compute a full solution. Adjustments of a level net and a single-photo resection are used to demonstrate the method.

Because of the advantage in computational time, the Givens method is superior to the normal equations approach when rigorous blunder detection is required.