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a joint work with Mireille Boutin and Ronan Fablet
A central problem in computer vision is to deal with the transformations that generate a view of the 3D world. To that purpose, geometric invariants provide interesting tools, as they measure intrinsic properties the system studied independently from the geometric transformation itself.
Invariants of projective and affine transformations are generally used, although invariants could be built for almost any Lie group action. For instance, the perspective projection in an Euclidean space can be expressed as a group action, acting on a set of 3D points and the camera optical center. With the moving frame normalisation technique, we have derived a complete fundamental set of invariants for such transformation. These invariants describe the object-camera system independently of gauge settings, and can be derived from image points, for known internal camera parameters.
With this new type of invariants, we can restate the structure and motion problem as the intersection of orbits characterized with the invariants, that determine unique 3D points. The Lie group action involved integrate rotations but not translations: we can therefore detect pure rotations as they are cancelled by the invariant description. Moreover, we have derived invariant measures from the fundamental set that give a direct estimate of the translation direction for each moving point, and can be used for motion segmentation and depth recovery.
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A 3D reconstruction example: the MOVI house reconstructed from the intersection of invariants on six images. Note that most Euclidean properties of the reconstructed objects are respected.
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An example of motion segmentation: the moving car and the stop signal are segmented using only the directions given at any point by the invariants.
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Two examples of depth maps obtained from the invariants: first with a translation along Y, second with a translation along Z.
Acknowledgements
This work has been initiated in the SHAPE
Lab at Brown University, under the NSF grants ITR # 0205477 and KDI #BCS-9980091.
I would like to thank David Cooper, Joseph Mundy and Michael Black for their
helpful discussions and comments on this work.
Further details
Structure from Motion: a new look from the point of view of invariant
theory - P. L. Bazin and M. Boutin, SIAM Journal on Applied Mathematics,
vol
64, 4, 2004 (pdf).
