Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth

• Published in: Linear algebra and its applications Vol. 432; no. 8; pp. 2061 - 2076
• Main Authors: Schaffrin, Burkhard, Snow, Kyle
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier Inc 01.04.2010; Elsevier
• Subjects: Algebra; Circle fitting; Exact sciences and technology; Gauss–Helmert Model; Iterative linearization; Linear and multilinear algebra, matrix theory; Mathematics; Sciences and techniques of general use; Total Least-Squares; Tykhonov regularization; 65F10; Total Least-Squares; Circle fitting; Iterative linearization; Tykhonov regularization; 49M05; Gauss–Helmert Model; 47A52; 62H12; Circle; Parallel algorithm; Iterative method; Modeling; Errors-in-Variables Model; Gaussian process; Gauss-Helmert Model; Model matching; Lagrange interpolation; Least squares method; Non linear effect; Regularization; Linearization
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2009.09.014

In this contribution a variation of Golub/Hansen/O’Leary’s Total Least-Squares (TLS) regularization technique is introduced, based on the Hybrid APproximation Solution (HAPS) within a nonlinear Gauss–Helmert Model. By applying a traditional Lagrange approach to a series of iteratively linearized Gauss–Helmert Models, a new iterative scheme has been found that, in practice, can generate the Tykhonov regularized TLS solution, provided that some care is taken to do the updates properly. The algorithm actually parallels the standard TLS approach as recommended in some of the geodetic literature, but unfortunately all too often in combination with erroneous updates that would still show convergence, although not necessarily to the (unregularized) TLS solution. Here, a key feature is that both standard and regularized TLS solutions result from the same computational framework, unlike the existing algorithms for Tykhonov-type TLS regularization. The new algorithm is then applied to a problem from archeology. There, both the radius and the center-point coordinates of a circle have to be determined, of which only a small part of the arc had been surveyed in-situ, thereby giving rise to an ill-conditioned set of equations. According to the archaeologists involved, this circular arc served as the starting line of a racetrack in the ancient Greek stadium of Corinth, ca. 500 BC. The present study compares previous estimates of the circle parameters with the newly developed “Regularized TLS Solution of Tykhonov type.”