Acta Univ. Agric. Silvic. Mendelianae Brun. 2009, 57(6), 163-168 | DOI: 10.11118/actaun200957060163
Parametrizace ortonormálních matic třetího řádu pro lineární kalibraci
- Ústav statistiky a operačního výzkumu, Mendelova zemědělská a lesnická univerzita v Brně, Zemědělská 1, 613 00 Brno, Česká republika
Práce řeší potřebu parametrizace trojrozměrných ortonormálních matic z hlediska lineární kalibrace. Množina ortonormálních matic je parametrizována třemi spojitými a jedním diskrétním parametrem. Smysl jednotlivých spojitých parametrů je zřejmý ze zavedeného pojmu matice otočení a význam diskrétního parametru vysvětluje matice otočení s případnou změnou polarity osy.
ortonormální matice, parametrizace, matice otočení, matice otočení s případnou změnou polarity osy
Parameterization of orthonormal third-order matrices for linear calibration
The paper derives a parametric definition of the set of third-order orthonormal real matrices.
The derivation is done in several partial steps. First a generalized unit matrix is introduced as the simplest case of an orthonormal matrix along with some of its properties and, subsequently, the properties of orthonormal matrices are proved that will be needed.
The derivation itself of a parametric definition of third-order orthonormal matrices is based on the numbers of zero entries that are theoretically possible. Therefore, it is first proved that a third-order square matrix with the number of non-zero entries different from nine, eight, five, or three cannot be orthonormal.
The number of different ways in which the set of third-order orthonormal matrices can be parameterized is greater than one. The concepts of a rotation matrix and a flop-enabling rotation matrix are introduced to motivate the parameterization chosen.
Given the product of two rotation matrices and one flop-enabling rotation matrix, it is first proved that it is a third-order orthonormal matrix. In the last part of the paper, it is then proved that such a product already includes, as special cases, all the third-order orthonormal matrices. It is thus a parametric definition of all third-order orthonormal matrices.
Keywords: orthonormal matrix, parameterization, rotation matrix, rotation matrix with possible axis polarity change
Grants and funding:
The paper has been written with the support of the GAČR 103/07/0183 grant.
Received: July 17, 2009; Published: October 7, 2014 Show citation
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References
- Moll, I., Myšková, K., 2007: Calibration of Identical Objects. Proceedings of the Summer School DATASTAT 06, Masaryk University, pp. 195-201, ISBN 978-80-210-4493-7.
- Moll, I., Myšková, K., 2007: Three Approaches To Linear Calibration [in Czech]. Proceedings of the 6th Mathematical Workshop, VUT Faculty of Civil Engineering, Brno, pp. 41-42, ISBN 80-214-2741-8.
- Myšková, K., 2007: Parameter estimators in a multidimensional calibration model. Proceedings of the Summer School DATASTAT 06, Masaryk University, pp. 203-208, ISBN 978-80-210-4493-7.
- Myšková, K., 2006: 3D Calibration with Identical Diagonal Covariance Matrices [in Czech]. Proceedings of the 17th Summer School of Biometrics, Lednice, pp. 245-252, ISBN 80-86548-89-9.
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