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International Journal of Computer Vision

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01-01-2014

Closed-Form Solution of Visual-Inertial Structure from Motion

Author: Agostino Martinelli

Published in: International Journal of Computer Vision | Issue 2/2014

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Abstract

This paper investigates the visual-inertial structure from motion problem. A simple closed form solution to this problem is introduced. Special attention is devoted to identify the conditions under which the problem has a finite number of solutions. Specifically, it is shown that the problem can have a unique solution, two distinct solutions and infinite solutions depending on the trajectory, on the number of point-features and on their layout and on the number of camera images. The investigation is also performed in the case when the inertial data are biased, showing that, in this latter case, more images and more restrictive conditions on the trajectory are required for the problem resolvability.

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Actually, the accelerometer bias slightly changes with time, i.e., it would be more appropriate to write \(\varvec{B}(\tau )\). However, as we will show in the next section, few camera images allow us to determine this component and we can assume that the bias is constant during the time interval needed to collect few camera images.

This is equivalent to say that the position of any point-feature and the position of the platform at any time \(t_j\) (\(j=1,\ldots ,n_i\)), are coplanar.

Note that it is not possible to proceed as in the unbiased case since it is not possible to separate the linear system in (13) in two parts because of the bias.

Note that we are not considering the space of the functions \(\theta (\tau )\) but the space of the functions \(\cos \theta (\tau )\).

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Title: Closed-Form Solution of Visual-Inertial Structure from Motion
Author: Agostino Martinelli
Publication date: 01-01-2014
Publisher: Springer US
Published in: International Journal of Computer Vision / Issue 2/2014
Print ISSN: 0920-5691
Electronic ISSN: 1573-1405
DOI: https://doi.org/10.1007/s11263-013-0647-7

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