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Structure from motion: a new look from the point of view of invariant theory

By Pierre-louis Bazin and Mireille Boutin


Abstract We present a novel, simple formulation of the problem of 3D object reconstruc-tion from images. In this formulation, the object is seen as lying at the intersection of the projection of orbits of custom built Lie groups actions. The group param-eters 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 `a la Fels-Olver to obtain a fundamental set of invariants of these groups actions. The invariants areused 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. 1 Introduction This paper has two goals. Its first goal is to illustrate the potential of using the formal-ism of invariant theory in certain applications. This potential is, at this point, rather unexploited and we hope to set a trend with these results. Its second goal is to providenew insights on the problem of structure from motion through a novel formulation in terms of group actions.The problem of structure from motion is rather old and well-studied. It consists in reconstructing an object from a set of pictures of this object (e.g. a movie). In thispaper, we consider the case of objects represented by an ordered set of points in R3and assume that the camera parameters (position and orientation of the camera, foca

Year: 2004
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