Analysis of Measurements Based on the Singular Value Decomposition

Hanson, Richard J.; Norris, Michael

doi:10.1137/0902029

Cited by 80 publications

(50 citation statements)

References 4 publications

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“…(Hanson and Norris, 1981) and (Gander, 1997). By minimizing Q with respect to t we obtain the necessary equation (note that A(:, 1) indicates the first column of the matrix A): t = r(:, 1) -R r'(:, 1) (11) which means, that after computing R, we can always choose t such that the first column of the matrix Rr' -r* is zero.…”

Section: Mean Square Distance Of Surfaces Definition 2 the Mean Squamentioning

confidence: 99%

Best-Fit of Sculptured Surfaces

Gander

Sourlier²

1999

Machining Impossible Shapes

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The parametric representation of surfaces is often avoided in best-fit computations because of the large number of unknowns. By exploiting the matrix structure, D. Sourlier has developed an effective best-fit software 'FUNKE', which is based on parametric representation. FUNKE is perfectly applicable for sculptured surfaces. We report on the best-fit of a turbine blade.In the second part we propose a new shape distance measure to compare two CAD-described surfaces. We give algorithms in Matlab to compute this shape distance and the corresponding transformation.

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Section: Mean Square Distance Of Surfaces Definition 2 the Mean Squamentioning

confidence: 99%

Best-Fit of Sculptured Surfaces

Gander

Sourlier²

1999

Machining Impossible Shapes

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“…The proof of Corollary 4.5 shows that if a ij has a zero expectation, mapping the noise matrix Ψ λ N ij = exp a ij /λ υ ij , which is already a matrix in its Cartan form, is not too far away from the Cartan form, in expectation, of W ij of (25). For this case, which is a first order perturbation in λ to W ij , we can adapt conditions like (23) from rotation synchronization to synchronization over the Euclidean group.…”

Section: One Interpretation As a Noisy Data Matrix For Synchronizatiomentioning

confidence: 99%

Synchronization over Cartan Motion Groups via Contraction

Özyeşil

Sharon²,

Singer³

2018

SIAM J. Appl. Algebra Geometry

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Abstract. Group contraction is an algebraic map that relates two classes of Lie groups by a limiting process.We utilize this notion for the compactification of the class of Cartan motion groups, which includes the important special case of rigid motions. The compactification process is then applied to reduce a noncompact synchronization problem to a problem where the solution can be obtained by means of a unitary, faithful representation. We describe this method of synchronization via contraction in detail and analyze several important aspects of this application. We then show numerically the advantages of our approach compared to some current state-of-the-art synchronization methods on both synthetic and real data.

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“…At least eight solutions of more recent date (from 1981 up to 2006) can be found in literature. They are the solutions of Hanson and Norris (1981), Bakker et al (1989, p.55), Awange et al (2004), Teunissen (1985, p.148), Arun et al (1987), Hinsken (1987), Horn et al (1988), and Krarup (2006).…”

Section: Direct Solutionsmentioning

confidence: 99%

“…Hanson and Norris consider in their paper (Hanson and Norris, 1981) the estimation of the transformation parameters of the congruence and similarity transformations. Their application is quality control of manufactured parts, where points on parts are matched with points on a drawing.…”

Section: Direct Solutionsmentioning

confidence: 99%

Extendable linearised adjustment model for deformation analysis

Velsink

2014

Survey Review

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This paper gives a linearised adjustment model for the affine, similarity and congruence transformations in 3D that is easily extendable with other parameters to describe deformations. The model considers all coordinates stochastic. Full positive semi-definite covariance matrices and correlation between epochs can be handled. The determination of transformation parameters between two or more coordinate sets, determined by geodetic monitoring measurements, can be handled as a least squares adjustment problem. It can be solved without linearisation of the functional model, if it concerns an affine, similarity or congruence transformation in one-, two-or three-dimensional space. If the functional model describes more than such a transformation, it is hardly ever possible to find a direct solution for the transformation parameters. Linearisation of the functional model and applying least squares formulas is then an appropriate mode of working. The adjustment model is given as a model of observation equations with constraints on the parameters. The starting point is the affine transformation, whose parameters are constrained to get the parameters of the similarity or congruence transformation. In this way the use of Euler angles is avoided. Because the model is linearised, iteration is necessary to get the final solution. In each iteration step approximate coordinates are necessary that fulfil the constraints. For the affine transformation it is easy to get approximate coordinates. For the similarity and congruence transformation the approximate coordinates have to comply to constraints. To achieve this, use is made of the singular value decomposition of the rotation matrix. To show the effectiveness of the proposed adjustment model total station measurements in two epochs of monitored buildings are analysed. Coordinate sets with full, rank deficient covariance matrices are determined from the measurements and adjusted with the proposed model. Testing the adjustment for deformations results in detection of the simulated deformations.

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Analysis of Measurements Based on the Singular Value Decomposition

Cited by 80 publications

References 4 publications

Best-Fit of Sculptured Surfaces

Best-Fit of Sculptured Surfaces

Synchronization over Cartan Motion Groups via Contraction

Extendable linearised adjustment model for deformation analysis

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