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 groups 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 groups 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.