Algebraic and Geometric Tools to Compute Projective and Permutation Invariants

This paper studies the computation of projective invariants in pairs of images from uncalibrated cameras, and presents a detailed study of the projective and permutation invariants for configurations of points and/or lines. We give two basic computational approaches, one algebraic and one geometric, and also the relations between the invariants computed by different approaches. In each case, we show how to compute invariants in projective space assuming that the points and lines have already been reconstructed in an arbitrary projective basis, and also, how to compute them directly from image coordinates in a pair of views using only point and line correspondences and the fundamental matrix. Finally, we develop combinations of those projective invariants which are insensitive to permutations of the geometric primitives of each of the basic configurations.

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