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

Linear Algebra and Its Applications

Volumn 432, Issue 8, 2010, Pages 2061-2076

  Schaffrin, Burkhard ^a   Snow, Kyle ^b  

^a OHIO STATE UNIVERSITY   (United States)

^b TOPCON CORPORATION   (Japan)

Author keywords

Circle fitting; Gauss Helmert Model; Iterative linearization; Total Least Squares; Tykhonov regularization

Indexed keywords

ANCIENT GREEKS; APPROXIMATION SOLUTION; CIRCLE FITTING; CIRCLE PARAMETER; CIRCULAR ARC; COMPUTATIONAL FRAMEWORK; GAUSS-HELMERT MODEL; ILL-CONDITIONED; IN-SITU; ITERATIVE SCHEMES; KEY FEATURE; LAGRANGE APPROACH; POINT COORDINATES; REGULARIZATION TECHNIQUE; TOTAL LEAST SQUARES;

CAVITY RESONATORS; CONVERGENCE OF NUMERICAL METHODS; ELECTRIC COILS; LINEARIZATION; SURVEYING INSTRUMENTS;

SEEBECK EFFECT;

EID: 75749109318     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2009.09.014     Document Type: Article

References (36)

1. Beck A., and Ben-Tal A. On the solution of the Tikhonov regularization of the TLS problem. SIAM J. Optim. 17 (2006) 98-118
2. Beck A., Ben-Tal A., and Teboulle M. Finding a global optimal solution for a quadratically constrained fractional quadratic problem with applications to the regularized total least squares. SIAM J. Matrix Anal. Appl. 28 2 (2006) 425-445
3. Bookstein F.L. Fitting conic sections to scattered data. Comput. Graph. Image Process. 9 (1979) 59-71
4. Coope D. Circle fitting by linear and nonlinear least-squares. J. Optimiz. Theory Appl. 76 (1993) 381-388
5. Davis T.G. Total least-squares spiral curve fitting. J. Surveying Eng. 125 4 (1999) 159-176
6. Gander W., Golub G.H., and Strebel R. Least-squares fitting of circles and ellipses. BIT 34 (1994) 558-578
7. Golub G.H., and Van Loan C.F. An analysis of the total least-squares problem. SIAM J. Numer. Anal. 17 6 (1980) 883-893
8. Golub G.H., Hansen P.C., and O'Leary D.P. Tikhonov regularization and Total Least Squares. SIAM J. Matrix Anal. Appl. 21 (1999) 185-194
9. E. Grafarend, B. Schaffrin, Adjustment Computations in Linear Models (in German), Bibliograph. Inst., Mannheim, Germany, 1993.
10. H. Guo, R. Renaut, A regularized TLS algorithm, in: Van Huffel/Lemmerling [35], (2002), pp. 57-66.
11. Harville D.A. Maximum Likelihood approaches to variance component estimation and to related problems. J. Amer. Statist. Assoc. 72 358 (1977) 320-338
12. Helmert F.R. Adjustment Computations by the Method of Least Squares (in German). second ed. (1907), Teubner, Leipzig, Germany
13. Kilmer M.E., Hansen P.C., and Espanol M.I. A projection-based approach to general-form Tikhonov regularization. SIAM J. Sci. Comput. 29 (2007) 315-330
14. Kilmer M.E., and O'Leary D.P. Choosing regularization parameters in iterative methods for ill-posed problems. SIAM J. Matrix Anal. Appl. 22 (2001) 1204-1221
15. Koch K.R. Parameter Estimation and Hypothesis Testing in Linear Models. second ed. (1999), Springer, Berlin
16. S. Kupferer, Application of the TLS Technique to Geodetic Problems (in German), PhD dissertation, Geodetic Inst., Univ. of Karlsruhe (TH), Germany, 2005.
17. Lampe J., and Voss H. On a quadratic eigenproblem occurring in regularized total least-squares. Comput. Statist. Data Anal. 52 2 (2007) 1090-1102
18. Malenko A.L., and Kukush A.G. Optimal joint estimator of regression parameter and dispersion in nonlinear Errors-in-Variables models. Probab. Theory Math. Statist. N78 (2008) 142-151
19. Markovsky I., Kukush A., and Van Huffel S. Consistent least-squares fitting of ellipsoids. Numer. Math. 98 1 (2004) 177-194
20. Neitzel F., and Petrovic S. Total Least- Squares in the context of least-squares adjustment of a straight line (in German). Z. für Vermessungswesen 133 (2008) 141-148
21. Nievergelt Y. Hyperspheres and hyperplanes fitting seamlessly by algebraic constrained total least-squares. Linear Algebra Appl. 331 (2001) 145-155
22. A.J. Pope, Some pitfalls to be avoided in the iterative adjustment of nonlinear problems, in: Proceedings of the 38th Annual Meeting of the ASPRS, Falls Church/VA, 1972, pp. 449-477.
23. Pruessner A., and O'Leary D.P. Blind deconvolution using a regularized structured Total Least-Norm algorithm. SIAM J. Matrix Anal. Appl. 24 4 (2003) 1018-1037
24. Renaut R.A., and Guo H. Efficient algorithms for the solution of regularized TLS. SIAM J. Matrix Anal. Appl. 26 (2005) 457-476
25. Rorres C., and Romano D.G. Finding the center of a circular starting line in an ancient Greek stadium. SIAM Rev. 39 4 (1997) 745-754
26. Schaffrin B. Connecting the dots: The straight-line case revisited. Z. für Vermessungswesen 132 (2007) 385-394
27. Schaffrin B. Minimum MSE adjustment and the optimal Tykhonov-Phillips regularization parameter via reproBIQUUE. J. Geodesy 82 (2008) 113-121
28. B. Schaffrin, On penalized least- squares: its mean square error and a quasi-optimal weight ratio, in: Shalabh, Ch. Heumann (Eds.), Recent Advances in Linear Models and Related Areas, Physica-Verlag, Heidelberg, etc., 2008, pp. 313-322.
29. Schaffrin B., and Felus Y. On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms. J. Geodesy 82 (2008) 373-383
30. Schaffrin B., and Felus Y. An algorithmic approach to the Total Least-Squares problem with linear and quadratic constraints. Studia Geophys. Geodaet. 53 (2009) 1-16
31. Sima D., Van Huffel S., and Golub G.H. Regularized Total Least-Squares based on quadratic eigenvalue problem solvers. BIT Numer. Math. 44 (2004) 793-812
32. D. Sima, Regularization Techniques in Model Fitting And Parameter Estimation, Ph.D. Dissertation, Department of Electrical Engineering, Kath. Univ. Leuven, Belgium, 2006, 179 pages.
33. H. Späth, Orthogonal least-squares fitting by conic sections, in: Van Huffel [34], 1997, pp. 259-264.
34. Van Huffel S. Recent Advances in Total Least-Squares Techniques and Errors-In-Variables Modelling (1997), SIAM, Philadelphia/PA
35. Van Huffel S., and Lemmerling P. Total Least-Squares and Errors-In-Variables Modeling: Analysis, Algorithms and Applications (2002), Kluwer, Dordrecht/ Boston/London
36. Van Huffel S., and Vandewalle J. The Total Least-Squares Problem: Computational Aspects and Analysis (1991), SIAM, Philadelphia/PA