@techreport{
 title = {Direct and Indirect Estimation of the Variance-Covariance Matrix of the Parameters of a Fitted Ellipse and a Triaxial Ellipsoid},
 type = {techreport},
 keywords = {Author keywords: ellipse fitting,ellipsoid fitting,least squares method,singular value decomposition},
 websites = {https://www.researchgate.net/publication/342529585},
 id = {e49f4db9-233e-3f1a-8861-cd4163976a77},
 created = {2024-12-03T08:21:14.898Z},
 file_attached = {true},
 profile_id = {f1f70cad-e32d-3de2-a3c0-be1736cb88be},
 group_id = {5ec9cc91-a5d6-3de5-82f3-3ef3d98a89c1},
 last_modified = {2024-12-03T08:21:23.308Z},
 read = {true},
 starred = {false},
 authored = {false},
 confirmed = {false},
 hidden = {false},
 folder_uuids = {df28411a-ed7f-4991-8358-d39685eb4bf0},
 private_publication = {false},
 abstract = {The purpose of this work is 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 is expressed in terms of the coordinates of the ellipsoid center, of the three ellipsoid semi-axes and the three Euler angles. The general least squares method is applied to estimate these parameters and their variance-covariance matrix. In the indirect approach, the Cartesian equation of an ellipsoid is expressed as a polynomial. The coefficients of this polynomial equation and their variance-covariance matrix are estimated using the general least squares method. Then, these coefficients are transformed into the parameters of the ellipsoid through an analytical diagonalization of a suitable matrix. The variance-covariance matrix of these parameters is estimated applying the law of propagation of variances. Furthermore, both approaches are applied in the special case of an ellipse. Finally, the numerical examples in both cases indicate that the two procedures produce almost identical results.},
 bibtype = {techreport},
 author = {Panou, Georgios and Agatza-Balodimou, Amalia-Maria and Panou, G and Agatza-Balodimou, A.-M}
}

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In the direct approach, the Cartesian equation of an ellipsoid is expressed in terms of the coordinates of the ellipsoid center, of the three ellipsoid semi-axes and the three Euler angles. The general least squares method is applied to estimate these parameters and their variance-covariance matrix. In the indirect approach, the Cartesian equation of an ellipsoid is expressed as a polynomial. The coefficients of this polynomial equation and their variance-covariance matrix are estimated using the general least squares method. Then, these coefficients are transformed into the parameters of the ellipsoid through an analytical diagonalization of a suitable matrix. The variance-covariance matrix of these parameters is estimated applying the law of propagation of variances. Furthermore, both approaches are applied in the special case of an ellipse. 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