Recovering an Algebraic Curve Using its Projections From Different Points. Applications to Static and Dynamic Computational Vision

Abstract

We study how an irreducible closed algebraic curve X embedded in CP^3 can be recovered using its projections from points onto embedded projective planes. The different embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in CP^3 can be recovered modulo some actions of the group of projective transformations of CP^3. For two projections, we show how in a generic situation, a characteristic matrix of the two embeddings can be recovered. We also establish the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degree and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. Then we consider another problem. N projections, with known projections operators and N >> 1, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in CP^3. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve, is projected onto a plane from a different point. The projections operators are known. We show when and how the recovery of the algebraic curve is possible, in function of the degree of the curve of minimal degree generated by the centers of projection. A second part is devoted to applications to computer vision. The results in this paper solve a long standing problem in computer vision that could not have been solved without algebraic-geometric methods.