Studia Geophysica et Geodaetica

, Volume 59, Issue 3, pp 366–379 | Cite as

On total least squares for quadratic form estimation

  • Xing Fang
  • Jin WangEmail author
  • Bofeng Li
  • Wenxian Zeng
  • Yibin Yao


The mathematical approximation of scanned data by continuous functions like quadratic forms is a common task for monitoring the deformations of artificial and natural objects in geodesy. We model the quadratic form by using a high power structured errors-in-variables homogeneous equation. In terms of Euler-Lagrange theorem, a total least squares algorithm is designed for iteratively adjusting the quadratic form model. This algorithm is proven as a universal formula for the quadratic form determination in 2D and 3D space, in contrast to the existing methods. Finally, we show the applicability of the algorithm in a deformation monitoring.


total least squares quadratic forms high power structured errors-invariables homogeneous equation deformation monitoring 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amiri-Simkooei A.R., 2007. Least-Squares Variance Component Estimation: Theory and GPS Applications. Ph.D. Thesis, Delft University of Technology. Publication on Geodesy, 64, Netherlands Geodetic Commission, Delft, The Netherlands.Google Scholar
  2. Amiri-Simkooei A.R. and Jazaeri S., 2012. Weighted total least squares formulated by standard least squares theory. J. Geod. Sci., 2(2), 113–124.Google Scholar
  3. Amiri-Simkooei A.R., 2013. Application of least squares variance component estimation to errorsin-variables models. J. Geodesy, 87, 935–944.CrossRefGoogle Scholar
  4. Amiri-Simkooei A.R. and Jazaeri S., 2013. Data-snooping procedure applied to errors-in-variables models. Stud. Geophys. Geod., 57, 426–441.CrossRefGoogle Scholar
  5. Bookstein F.L., 1979. Fitting conic sections to scattered data. Comput. Graph. Image Process., 9, 59–71.CrossRefGoogle Scholar
  6. Coope D., 1993. Circle fitting by linear and nonlinear least-squares. J. Optim. Theory Appl., 76, 381–388.CrossRefGoogle Scholar
  7. Davis T.G., 1999. Total least-squares spiral curve fitting. J. Surv. Eng.-ASCE, 125, 159–176.CrossRefGoogle Scholar
  8. Drixler E., 1993. Analyse der Lage und Form von Objekten im Raum. Dissertation. Deutsche Geodätische Kommission, Reihe C, Heft Nr. 409, München, Germany (in German).Google Scholar
  9. Eling D., 2009. Terrestrisches Laserscanning für die Bauwerksüberwachung. Dissertation. Deutsche Geodätische Kommission, Reihe C, Nr. 641, München, Germany (in German, /c-641.pdf).Google Scholar
  10. Fang X., 2011. Weighted Total Least Squares Solution for Application in Geodesy. PhD Thesis, No. 294, Leibniz University, Hannover, Germany.Google Scholar
  11. Fang X., 2013. Weighted total least squares: necessary and sufficient conditions, fixed and random parameters. J. Geodesy, 87, 733–749. DOI:  10.1007/s00190-013-0643-2.CrossRefGoogle Scholar
  12. Fang X., 2014a. A structured and constrained total least-squares solution with cross-covariances. Stud. Geophys. Geod., 58, 1–16, DOI:  10.1007/s11200-012-0671-z.CrossRefGoogle Scholar
  13. Fang X., 2014b. On non-combinatorial weighted total least squares with inequality constraints. J. Geodesy, 88, 805–816.CrossRefGoogle Scholar
  14. Fang X., 2014c. A total least squares solution for geodetic datum transformations. Acta Geod. Geophys., 49, 189–207.CrossRefGoogle Scholar
  15. Fang X., 2015. Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J. Geodesy, DOI:  10.1007/s00190-015-0790-8 (in print).Google Scholar
  16. Fitzgibbon A., Pilu M. and Fisher R., 1999. Direct least-squares fitting of ellipses. IEEE Trans. Pattern Anal. Machine Intell., 21, 476–480.CrossRefGoogle Scholar
  17. Gander W., Golub G.H. and Strebel R., 1994. Least-squares fitting of circles and ellipses. BIT, 34, 558–578.CrossRefGoogle Scholar
  18. Golub G. and Van Loan C., 1980. An analysis of the total least-squares problem. SIAM J. Numer. Anal., 17, 883–893, DOI:  10.1137/0717073.CrossRefGoogle Scholar
  19. Grafarend E. and Awange J.L., 2012. Applications of Linear and Nonlinear Models. Fixed Effects, Random Effects, and Total Least Squares. Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
  20. Hesse C., 2007. Hochauflösende kinematische Objekterfassung mit terrestrischen Laserscannern. Dissertation. Deutsche Geodätische Kommission, Reihe C, Nr. 608, München, Germany (in German).Google Scholar
  21. Hesse C. and Kutterer H., 2006. Automated form recognition of laser scanned deformable objects. In: ei]Sansò F. and Gil A. (Eds), Geodetic Deformation Monitoring: From Geophysical to Engineering Roles. International Association of Geodesy Symposia, 131, Springer-Verlag, Heidelberg, Germany.Google Scholar
  22. Kanatani K., 1994. Statistical bias of conic fitting and renormalization. IEEE Trans. Pattern Anal. Machine Intell., 16, 320–326.CrossRefGoogle Scholar
  23. Kupferer S., 2005. Application of the TLS Technique to Geodetic Problems. PhD Thesis. Geodetic Institute, University of Karlsruhe, Karlsruhe, Germany.Google Scholar
  24. Kutterer H. and Schön S., 1999. Statistische Analyse quadratischer Formen-der Determinantenansatz. AVN, 10/1999, 322-330 (in German, http://www.wichmannverlag. de/images/stories/avn/artikelarchiv/1999/10/5f134c46d35.pdf).Google Scholar
  25. Li B., Shen Y. and Lou L., 2013. Seamless multivariate affine error-in-variables transformation and its application to map rectification. Int. J. GIS, 27, 1572–1592.Google Scholar
  26. Markovsky I., Kukush A. and Van Huffel S., 2004. Consistent least-squares fitting of ellipsoids. Numer. Math., 98, 177–194.CrossRefGoogle Scholar
  27. Nievergelt Y., 2001. Hyperspheres and hyperplanes fitting seamlessly by algebraic constrained total least-squares. Linear Alg. Appl., 331, 145–155.CrossRefGoogle Scholar
  28. Paffenholz J.A., 2012. Direct Geo-Referencing of 3D Point Clouds with 3D Positioning Sensors. Dissertation. Deutsche Geodätische Kommission, Reihe C, Nr. 689, München, Germany (in German)Google Scholar
  29. Pope A.J., 1972. Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. In: Proceedings of the 38th Annual Meeting, American Society of Photogrammetry, Washington, D.C., Mar. 12-17, 1972. American Society of Photogrammetry, Bethesda, MD, 449–477.Google Scholar
  30. Schaffrin B. and Wieser A., 2008. On weighted total least-squares adjustment for linear regression. J. Geodesy, 82, 415–421, DOI: 10.1007/s00190-007-0190-9.CrossRefGoogle Scholar
  31. Schaffrin B. and Snow K., 2010. Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth. Linear Alg. Appl., 432, 2061–2076.CrossRefGoogle Scholar
  32. Shen Y., Li B. and Chen Y., 2011. An iterative solution of weighted total least-squares adjustment. J. Geodesy, 85, 229–238, DOI: 10.1007/s00190-010-0431-1.CrossRefGoogle Scholar
  33. Späth H., 1996. Orthogonal squared distance fitting with parabolas. In: ei]Alefeld G. and Herzberger J. (Eds),Proceedings of the IMACS-GAMM International Symposium on Numerical Methods and Error-Bounds. Akademie-Verlag, Berlin, Germany, 261–269.Google Scholar
  34. Späth H., 1997a. Least-squares fitting of ellipses and hyperbolas. Comput. Stat., 12, 329–341.Google Scholar
  35. Späth H., 1997b. Orthogonal distance fitting by circles and ellipses with given area. Comput. Stat., 12, 343–354.Google Scholar
  36. Teunissen P.J.G., 1985. The Geometry of Geodetic Inverse Linear Mapping and Nonlinear Adjustment. Netherlands Geodetic Commission, Publications on Geodesy, New Series, 8(1), 186 pp.Google Scholar
  37. Teunissen PJG (1988) The non-linear 2D symmetric Helmert transformation: an exact non-linear least-squares solution. Journal of Geodesy, 62 (1):1–15Google Scholar
  38. Teunissen P.J.G., 1989a. A note on the bias in the symmetric Helmert transformation, Festschrift Krarup, 1–8.Google Scholar
  39. Teunissen P.J.G., 1989b. First and second moments of nonlinear least-squares estimators. J. Geodesy, 63, 253–262.Google Scholar
  40. Teunissen P.J.G., 1990. Nonlinear least squares. Manuscripta Geodaetica, 15, 137–150.Google Scholar
  41. Teunissen P.J.G. and Amiri-Simkooei A.R., 2008. Least-squares variance component estimation. J. Geodesy, 82, 65–82.CrossRefGoogle Scholar
  42. Wang J., Kutterer H. and Fang X., 2012. On the detection of systematic errors in terrestrial laser scanning data. J. Appl. Geod., 6, 187–192.Google Scholar
  43. Wang J., 2013a. Block-to-point fine registration in terrestrial laser scanning. Remote Sens., 5, 6921–6937, DOI:  10.3390/rs5126921.CrossRefGoogle Scholar
  44. Wang J., 2013b. Towards Deformation Monitoring with Terrestrial Laser Scanning Based on External Calibration and Feature Matching Methods. PhD Thesis, No. 308, Leibniz University, Hannover, Germany.Google Scholar
  45. Xu P.L., Liu J.N. and Shi C., 2012. Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J. Geodesy, 86, 661–675, DOI:  10.1007/s00190-012-0552-9.CrossRefGoogle Scholar
  46. Xu P.L. and Liu J.N., 2014. Variance components in errors-in-variables models: estimability, stability and bias analysis. J. Geodesy, 88, 719–734, DOI:  10.1007/s00190-014-0717-9.CrossRefGoogle Scholar

Copyright information

© Institute of Geophysics of the ASCR, v.v.i 2015

Authors and Affiliations

  • Xing Fang
    • 1
  • Jin Wang
    • 2
    Email author
  • Bofeng Li
    • 3
  • Wenxian Zeng
    • 1
  • Yibin Yao
    • 1
  1. 1.School of Geodesy & GeomaticsWuhan UniversityWuhanChina
  2. 2.Beijing Key Laboratory of Traffic EngineeringBeijing University of TechnologyBeijingChina
  3. 3.College of Surveying and Geo-InformaticsTongji UniversityShanghaiChina

Personalised recommendations