The Procrustes Problem for Orthogonal Stiefel Matrices

Bojańczyk, Adam W.; Lutoborski, Adam

doi:10.1137/s106482759630992x

Cited by 21 publications

(21 citation statements)

References 13 publications

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“…The results in [71] indicate that the algorithms proposed by Park [57] and Bojanczyk and Lutoborski [7] are significantly slower than the algorithm of Green and Gower. Furthermore, the algorithm proposed by Zhang and Du [71] is also slightly slower than the algorithm of Green and Gower.…”

Section: Algorithms For Solving the Onp Problem For Non-coplanar Pointsmentioning

confidence: 95%

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Algorithms for the Orthographic-n-Point Problem

Steger

2017

J Math Imaging Vis

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We examine the orthographic-n-point problem (OnP), which extends the perspective-n-point problem to telecentric cameras. Given a set of 3D points and their corresponding 2D points under orthographic projection, the OnP problem is the determination of the pose of the 3D point cloud with respect to the telecentric camera. We show that the OnP problem is equivalent to the unbalanced orthogonal Procrustes problem for non-coplanar 3D points and to the sub-Stiefel Procrustes problem for coplanar 3D points. To solve the OnP problem, we apply existing algorithms for the respective Procrustes problems and also propose novel algorithms. Furthermore, we evaluate the algorithms to determine their robustness and speed and conclude which algorithms are preferable in real applications. Finally, we evaluate which algorithm is most suitable as a minimal solver in a RANSAC scheme.

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Section: Algorithms For Solving the Onp Problem For Non-coplanar Pointsmentioning

confidence: 95%

“…There are further algorithms that have been proposed [7,53,57,71]. The results in [71] indicate that the algorithms proposed by Park [57] and Bojanczyk and Lutoborski [7] are significantly slower than the algorithm of Green and Gower.…”

Section: Algorithms For Solving the Onp Problem For Non-coplanar Pointsmentioning

confidence: 99%

Algorithms for the Orthographic-n-Point Problem

Steger

2017

J Math Imaging Vis

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“…(Mark − in the table II and the table III represents that it takes too much time to record in the tables.) 1) From the figure 2, we notice that the existing methods as EB [7], RSR [8], LSR [9], SP [10] LR [11] and the proposed GPI method converge to the same objective value under the same input data. Besides, our proposed GPI method converges faster than other approaches during iteration.…”

Section: Resultsmentioning

confidence: 96%

“…On the other hand, the experimental results in Section 5 represent that the proposed GPI method uniformly converges Comparisons of the convergence rate are performed for 6 approaches including EB [7], RSR [8], LSR [9], SP [10] LR [11] and our proposed GPI method under 3 different data matrices.…”

Section: Theorem 32mentioning

confidence: 99%

A generalized power iteration method for solving quadratic problem on the Stiefel manifold

Nie

Zhang

2017

Sci. China Inf. Sci.

153

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Abstract-In this paper, we first propose a novel generalized power iteration method (GPI) to solve the quadratic problem on the Stiefel manifold (QPSM) as minT B) along with the theoretical analysis. Accordingly, its special case known as the orthogonal least square regression (OLSR) is under further investigation. Based on the aforementioned studies, we then cast major focus on solving the unbalanced orthogonal procrustes problem (UOPP). As a result, not only a general convergent algorithm is derived theoretically but the efficiency of the proposed approach is verified empirically as well.Index Terms-quadratic problem, Stiefel manifold, power iteration, procrustes problem, orthogonal least square regression.

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“…Schonemann (1966); Bojanczyk and Lutoborski (1999); Golub and Van Loan (1989) ;Edelman, Arias, and Smith (1998). The orthogonal Procrustes problem as stated has an analytic solution in terms of singular value decompositions.…”

Section: An Optimization Problemmentioning

confidence: 99%

On the Aggregation of Local Risk Models for Global Risk Management

et al. 2003

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Given a collection of single-market covariance matrix forecasts for different markets, we describe how to embed them into a global forecast of total risk.We do this by starting with any global covariance matrix forecast that contains information about cross-market correlations, and revising it to agree with the pre-specified submarket matrices, preserving the requirement that a covariance matrix be positive semi-definite.We characterize the ways this can be done and address the resulting numerical optimization problem.

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The Procrustes Problem for Orthogonal Stiefel Matrices

Cited by 21 publications

References 13 publications

Algorithms for the Orthographic-n-Point Problem

Algorithms for the Orthographic-n-Point Problem

A generalized power iteration method for solving quadratic problem on the Stiefel manifold

On the Aggregation of Local Risk Models for Global Risk Management

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