Structure from Motion: A New Look from the Point of View of Invariant Theory

We present a novel simple formulation of the problem of 3D object reconstruction from images. In this formulation, the object is seen as lying at the intersection of the projection of orbits of custom built Lie group actions. The group parameters correspond to unknown irrelevant quantities such as t...

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Bibliographic Details
Published in:SIAM journal on applied mathematics 2004-01, Vol.64 (4), p.1156-1174
Main Authors: Boutin, Mireille, Bazin, Pierre-Louis
Format: Article
Language:English
Subjects:
Applied mathematics
Cameras
Computer vision
Coordinate systems
Equivalence relation
Geometric shapes
Kinetics
Lie groups
Mathematical functions
Mathematical sets
Mathematics
Optimization techniques
Orbits
Polls
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Summary:We present a novel simple formulation of the problem of 3D object reconstruction from images. In this formulation, the object is seen as lying at the intersection of the projection of orbits of custom built Lie group actions. The group parameters correspond to unknown irrelevant quantities such as the camera orientation, the depth parameters of the object with respect to the camera, and the focal length. We then use an algorithmic method based on moving frames à la Fels-Olver to obtain a fundamental set of invariants of these group actions. The invariants are used to define a set of equations determining the 3D object, thus providing a mathematical formulation of the problem where the irrelevant parameters do not appear.
ISSN:0036-1399
1095-712X
DOI:10.1137/S003613990340246X