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Verifying Global Minima for L Minimization Problems in Multiple View Geometry.

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Title: Verifying Global Minima for L Minimization Problems in Multiple View Geometry.
Authors: Hartley, Richard1 Richard.Hartley@anu.edu.au, Kahl, Fredrik2, Olsson, Carl2, Seo, Yongduek3
Source: International Journal of Computer Vision. Jan2013, Vol. 101 Issue 2, p288-304. 17p. 1 Color Photograph, 3 Diagrams, 6 Graphs.
Subjects: Geometry problems & exercises, Least squares, Triangulation, Algorithms, Cost functions, Iterative methods (Mathematics), Hessian matrices
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. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Computer Vision is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Computer+Vision%22">International Journal of Computer Vision</searchLink>. Jan2013, Vol. 101 Issue 2, p288-304. 17p. 1 Color Photograph, 3 Diagrams, 6 Graphs.
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  Data: <searchLink fieldCode="DE" term="%22Geometry+problems+%26+exercises%22">Geometry problems & exercises</searchLink><br /><searchLink fieldCode="DE" term="%22Least+squares%22">Least squares</searchLink><br /><searchLink fieldCode="DE" term="%22Triangulation%22">Triangulation</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Cost+functions%22">Cost functions</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Hessian+matrices%22">Hessian matrices</searchLink>
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  Data: 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. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of International Journal of Computer Vision is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s11263-012-0569-9
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        Text: English
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      – SubjectFull: Cost functions
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      – SubjectFull: Hessian matrices
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