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# Point triangulation through polyhedron collapse using the l∞ norm

Simon Donné (UGent) , Bart Goossens (UGent) and Wilfried Philips (UGent)
Author
Organization
Abstract
Multi-camera triangulation of feature points based on a minimisation of the overall l(2) reprojection error can get stuck in suboptimal local minima or require slow global optimisation. For this reason, researchers have proposed optimising the l(infinity) norm of the l(2) single view reprojection errors, which avoids the problem of local minima entirely. In this paper we present a novel method for l(infinity) triangulation that minimizes the l(infinity) norm of the l(infinity) reprojection errors: this apparently small difference leads to a much faster but equally accurate solution which is related to the MLE under the assumption of uniform noise. The proposed method adopts a new optimisation strategy based on solving simple quadratic equations. This stands in contrast with the fastest existing methods, which solve a sequence of more complex auxiliary Linear Programming or Second Order Cone Problems. The proposed algorithm performs well: for triangulation, it achieves the same accuracy as existing techniques while executing faster and being straightforward to implement.
Keywords
Optimization, Max-norm, l∞ norm, multi-view geometry, triangulation, MULTIVIEW GEOMETRY, OPTIMIZATION

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## Citation

Chicago
Donné, Simon, Bart Goossens, and Wilfried Philips. 2015. “Point Triangulation Through Polyhedron Collapse Using the L∞ Norm.” In IEEE International Conference on Computer Vision, 792–800.
APA
Donné, S., Goossens, B., & Philips, W. (2015). Point triangulation through polyhedron collapse using the l∞ norm. IEEE International Conference on Computer Vision (pp. 792–800). Presented at the IEEE International Conference on Computer Vision.
Vancouver
1.
Donné S, Goossens B, Philips W. Point triangulation through polyhedron collapse using the l∞ norm. IEEE International Conference on Computer Vision. 2015. p. 792–800.
MLA
Donné, Simon, Bart Goossens, and Wilfried Philips. “Point Triangulation Through Polyhedron Collapse Using the L∞ Norm.” IEEE International Conference on Computer Vision. 2015. 792–800. Print.
@inproceedings{7084788,
abstract     = {Multi-camera triangulation of feature points based on a minimisation of the overall l(2) reprojection error can get stuck in suboptimal local minima or require slow global optimisation. For this reason, researchers have proposed optimising the l(infinity) norm of the l(2) single view reprojection errors, which avoids the problem of local minima entirely. In this paper we present a novel method for l(infinity) triangulation that minimizes the l(infinity) norm of the l(infinity) reprojection errors: this apparently small difference leads to a much faster but equally accurate solution which is related to the MLE under the assumption of uniform noise. The proposed method adopts a new optimisation strategy based on solving simple quadratic equations. This stands in contrast with the fastest existing methods, which solve a sequence of more complex auxiliary Linear Programming or Second Order Cone Problems. The proposed algorithm performs well: for triangulation, it achieves the same accuracy as existing techniques while executing faster and being straightforward to implement.},
author       = {Donn{\'e}, Simon and Goossens, Bart and Philips, Wilfried},
booktitle    = {IEEE International Conference on Computer Vision},
isbn         = {978-1-4673-8390-5},
issn         = {1550-5499},
language     = {eng},
location     = {Santiago, Chile},
pages        = {792--800},
title        = {Point triangulation through polyhedron collapse using the l\ensuremath{\infty} norm},
url          = {http://dx.doi.org/10.1109/ICCV.2015.97},
year         = {2015},
}


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