Direct and Indirect Estimation of the Variance–Covariance Matrix of the Parameters of a Fitted Ellipse and a Triaxial Ellipsoid
Publication: Journal of Surveying Engineering
Volume 147, Issue 1
Abstract
This work deals with the estimation of the variance–covariance matrix of the parameters of a fitted ellipse and an ellipsoid by a direct and an indirect procedure. In the direct approach, the Cartesian equation of an ellipsoid was expressed in terms of the coordinates of the ellipsoid center, the three ellipsoid semiaxes, and the three rotation angles. The general least-squares method was applied to estimate these parameters and their variance–covariance matrix. In the indirect approach, the Cartesian equation of an ellipsoid was expressed as a polynomial. The coefficients of this polynomial equation and their variance–covariance matrix were estimated using the general least-squares method. Then these coefficients were transformed into the parameters of the ellipsoid through an analytical diagonalization of a suitable matrix. The variance–covariance matrix of these parameters was estimated applying the law of propagation of variances. Both approaches are applied to the special case of an ellipse. The numerical examples in both cases indicated that the two procedures produce almost identical results.
Get full access to this article
View all available purchase options and get full access to this article.
Data Availability Statement
The code (MATLAB scripts) that supports the findings of this study is available from the corresponding author upon reasonable request. This code also is available at https://www.researchgate.net/publication/342523137_Panou_Agatza_V-C_Matrix.
References
Aguirre, G. K. 2019. “A model of the entrance pupil of the human eye.” Sci. Rep. 9 (1): 9360. https://doi.org/10.1038/s41598-019-45827-3.
Ahn, S. J., W. Rauh, and H. J. Warnecke. 2001. “Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola.” Pattern Recognit. 34 (12): 2283–2303. https://doi.org/10.1016/S0031-3203(00)00152-7.
Barbero-García, I., J. L. Lerma, Á. Marqués-Mateu, and P. Miranda. 2017. “Low-cost smartphone-based photogrammetry for the analysis of cranial deformation in infants.” World Neurosurg. 102 (Jun): 545–554. https://doi.org/10.1016/j.wneu.2017.03.015.
Bektas, S. 2014. “Orthogonal distance from an ellipsoid.” Bol. Ciênc. Geod. 20 (4): 970–983. https://doi.org/10.1590/S1982-21702014000400053.
Bektas, S. 2015. “Least squares fitting of ellipsoid using orthogonal distances.” Bol. Ciênc. Geod. 21 (2): 329–339. https://doi.org/10.1590/S1982-21702015000200019.
Deakin, R. E. 1998. “3D coordinate transformations.” Surv. Land Inf. Syst. 58 (4): 223–234.
Deakin, R. E. 2005. Notes on least squares. Melbourne, Australia: RMIT Univ.
Dermanis, A. 2017. “Fitting analytical surfaces to points: General approaches and applications to ellipsoid fitting.” In Living with GIS: In honour of the memory of Professor Ioannis Paraschakis, edited by O. Georgoula, M. Papadopoulou, D. Rossikopoulos, S. Spatalas, and A. Fotiou, 81–106. Thessaloniki, Greece: Aristotle Univ. of Thessaloniki.
Ghilani, C., and P. Wolf. 2006. Adjustment computations: Spatial data analysis. 4th ed. New York: Wiley.
Hirvonen, R. A. 1971. Adjustment by least squares in geodesy and photogrammetry. New York: Ungar.
Kopp, J. 2008. “Efficient numerical diagonalization of hermitian 3×3 matrices.” Int. J. Mod. Phys. C 19 (3): 523–548. https://doi.org/10.1142/S0129183108012303.
Malyugina, A., K. Igudesman, and D. Chickrin. 2014. “Least-squares fitting of a three-dimensional ellipsoid to noisy data.” Appl. Math. Sci. 8 (149): 7409–7421. https://doi.org/10.12988/ams.2014.49733.
Markovsky, I., A. Kukush, and S. Van Huffel. 2004. “Consistent least squares fitting of ellipsoids.” Numerische Mathematik 98 (1): 177–194. https://doi.org/10.1007/s00211-004-0526-9.
Mikhail, E. M., and F. Ackermann. 1976. Observations and least squares. New York: IEP–A Dun–Donnelley.
Panou, G., R. Korakitis, and G. Pantazis. 2020. “Fitting a triaxial ellipsoid to a geoid model.” J. Geod. Sci. 10 (1): 69–82. https://doi.org/10.1515/jogs-2020-0105.
Sanghani, P., B. T. Ang, N. K. K. King, and H. Ren. 2018. “Overall survival prediction in glioblastoma multiforme patients from volumetric, shape and texture features using machine learning.” Surg. Oncol. 27 (4): 709–714. https://doi.org/10.1016/j.suronc.2018.09.002.
Soler, T., J. Y. Han, and C. J. Huang. 2020. “Estimating variance–covariance matrix of the parameters of a fitted triaxial ellipsoid.” J. Surv. Eng. 146 (2): 04020003. https://doi.org/10.1061/(ASCE)SU.1943-5428.0000308.
Späth, H. 2001. “Least squares fitting of spheres and ellipsoids using not orthogonal distances.” Math. Commun. 6 (1): 89–96.
Yu, J., S. R. Kulkarni, and H. V. Poor. 2012. “Robust ellipse and spheroid fitting.” Pattern Recognit. Lett. 33 (5): 492–499. https://doi.org/10.1016/j.patrec.2011.11.025.
Information & Authors
Information
Published In
Journal of Surveying Engineering
Volume 147 • Issue 1 • February 2021
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Jun 30, 2020
Accepted: Oct 6, 2020
Published online: Dec 12, 2020
Published in print: Feb 1, 2021
Discussion open until: May 12, 2021
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.