On Reconstructing Configurations of Points in P from a Joint Distribution of Invariants

Consider the diagonal action of the projective group PGL3 on n copies of P . In addition, consider the action of the symmetric group Σn by permuting the copies. In this paper we find a set of generators for the invariant field of the combined group Σn × PGL3. As the main application, we obtain a reconstruction principle for point configurations in P from their sub-configurations of five points. Finally, we address the question of how such reconstruction principles pass down to subgroups. Introduction Consider the problem of recognizing a flat object from its shadow. This is a common problem in computer vision where one often represents objects by the boundary of their image on a picture. For simplicity, assume that the flat object is represented by a finite set of points p1, . . . , pn ∈ R . Rotations and translations of such a flat object in R (almost always) induce a transformation of the image points P1, . . . , Pn ∈ R 2 which can be written as Pi 7→ ( a11 a12 a21 a22 ) Pi + ( a13 a23 ) (a31, a32)Pi + a33 , for all i = 1, . . . , n, with   a11 a12 a13 a21 a22 a23 a31 a32 a33   ∈ GL(3) (0.1) (where of course we have to assume that the above denominator does not vanish). In the computer vision community, this group action is called the projective group action (PGL3(R) = GL3(R)/R ) and plays an important role in many applications. In order to be able to recognize a flat object from its shadow, we thus need to be able to determine whether two sets of n points in the plane lie in the same orbit under the simultaneous action of the projective group on each of the points. More precisely, given P1, . . . , Pn ∈ R 2 and Q1, . . . , Qn ∈ R , we need to be able to determine whether there exists a projective transformation g ∈ PGL3(R) such that g(Pi) = Qi, for all i = 1, . . . , n. However, in many applications, the point correspondence between the two objects is unknown: a priori, we ignore which point is going to be mapped to which. So, more generally, given any P1, . . . , Pn and Q1, . . . , Qn ∈ R , we need to be able to determine whether there exists a permutation π ∈ Σn and a projective transformation g ∈ PGL3(R) such that g(Pi) = Qπ(i), for all i = 1, . . . , n. In an earlier publication [2], we considered the analogue problem with the Euclidean group AO(2), which is a subgroup of the projective group. More precisely, we considered those projective transformations whose matrix is given by   a11 a12 a13 a21 a22 a23 a31 a32 a33   =   a11 a12 a13 a21 a22 a23 0 0 1   , with ( a11 a12 a21 a22 ) ∈ O(2),