Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision

We study how an irreducible closed algebraic curve X embedded in CP^3 can berecovered using its projections from points onto embedded projective planes.The different embeddings are unknown. The only input is the defining equationof each projected curve. We show how both the embeddings and the curve in CP^3can be recovered modulo some actions of the group of projective transformationsof CP^3. For two projections, we show how in a generic situation, acharacteristic matrix of the two embeddings can be recovered. We also establishthe minimal number of irreducible algebraic curves required to compute thischaracteristic matrix up to a finite-fold ambiguity, as a function of theirdegree and genus. Then we use this matrix to recover the class of the couple ofmaps and as a consequence to recover the curve. Then we consider anotherproblem. N projections, with known projections operators and N >> 1, areconsidered as an input and we want to recover the curve. The recovery can bedone by linear computations in the dual space and in the Grassmannian of linesin CP^3. A closely related question is also considered. Each point of a finiteclosed subset of an irreducible algebraic curve, is projected onto a plane froma different point. The projections operators are known. We show when and howthe recovery of the algebraic curve is possible, in function of the degree ofthe curve of minimal degree generated by the centers of projection. A secondpart is devoted to applications to computer vision. The results in this papersolve a long standing problem in computer vision that could not have beensolved without algebraic-geometric methods.