Verifying global minima for L2 minimization problems in multiple view geometry
Date
2013
Authors
Hartley, Richard
Kahl, Fredrik
Olsson, Carl
Seo, Yongduek
Journal Title
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Volume Title
Publisher
Springer
Abstract
We consider the least-squares (L2) minimization problems in multiple view geometry for triangulation, homography, camera resectioning and structure-and-motion with known rotation, or known plane. Although optimal algorithms have been given for these problems under an L-infinity cost function, finding optimal least-squares solutions to these problems is difficult, since the cost functions are not convex, and in the worst case may have multiple minima. Iterative methods can be used to find a good solution, but this may be a local minimum. This paper provides a method for verifying whether a local-minimum solution is globally optimal, by providing a simple and rapid test involving the Hessian of the cost function. The basic idea is that by showing that the cost function is convex in a restricted but large enough neighbourhood, a sufficient condition for global optimality is obtained. The method is tested on numerous problem instances of real data sets. In the vast majority of cases we are able to verify that the solutions are optimal, in particular, for small to medium-scale problems.
Description
Keywords
Keywords: Geometric optimization; Global minima; Global optimality; Homographies; L-infinity; Least Square; Least squares solutions; Local minimums; Minimization problems; Multiple view geometry; Neighbourhood; Optimal algorithm; Problem instances; Rapid test; Real Convex programming; Geometric optimization; Reconstruction
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Source
International Journal of Computer Vision
Type
Journal article
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2037-12-31