On a sub‐Stiefel Procrustes problem arising in computer vision

Abstract

A sub-Stiefel matrix is a matrix that results from deleting simultaneously the last row and the last column of an orthogonal matrix. In this paper, we consider a Procrustes problem on the set of sub-Stiefel matrices of order n. For n D 2, this problem has arisen in computer vision to solve the surface unfolding problem considered by R. Fereirra, J. Xavier and J. Costeira. An iterative algorithm for computing the solution of the sub-Stiefel Procrustes problem for an arbitrary n is proposed, and some numerical experiments are carried out to illustrate its performance. For these purposes, we investigate the properties of sub-Stiefel matrices. In particular, we derive two necessary and sufficient conditions for a matrix to be sub-Stiefel. We also relate the sub-Stiefel Procrustes problem with the Stiefel Procrustes problem and compare it with the orthogonal Procrustes problem.

Algorithms for the Orthographic-n-Point Problem

“…Remark 4 Theorems 4.2 and 4.4 of [10] give two additional characterizations of sub-Stiefel matrices. We will not make use of these characterizations in this paper.…”

“…2.4, the OnP problem for coplanar points is equivalent to the sub-Stiefel Procrustes problem. The only algorithm for solving this problem that we could find is the algorithm of Cardoso and Ziȩtak [10], which is discussed in Sect. 4.1.…”

“…It is only for n ≤ 200 that the Levenberg-Marquardt algorithm is slightly faster than some of the polynomial system solvers. 10 For n = 100, the polynomial system solvers and the LevenbergMarquardt algorithm are faster by a factor of more than 10 than the algorithm of Green and Gower and by a factor of more than 100 than the algorithm of Koschat and Swayne (note the logarithmic scale of the runtime graphs). Asymptotically, the polynomial system solvers are faster by a factor of more than 4 than all other algorithms.…”

“…The algorithm of Cardoso and Ziȩtak is described in [10,Section 7]. It uses a similar idea as the algorithm of Green and Gower: The sub-Stiefel Procrustes problem is expanded to a 3D orthogonal Procrustes problem.…”