Some Applications of Algebraic Curves to Computational Vision

We introduce a new formalism and a number of new results in the context of geometric computational vision. The classical scope of the research in geometric computer vision is essentially limited to static configurations of points and lines in ℙ3. By using some well known material from algebraic geometry, we open new branches to computational vision. We introduce algebraic curves embedded in ℙ3 as the building blocks from which the tensor of a couple of cameras (projections) can be computed. In the process we address dimensional issues and as a result establish the minimal number of algebraic curves required for the tensor variety to be discrete as a function of their degree and genus. We then establish new results on the reconstruction of an algebraic curves in ℙ3 from multiple projections on projective planes embedded in ℙ3. We address three different presentations of the curve: (i) definition by a set of equations, for which we show that for a generic configuration, two projections of a curve of degree d defines a curve in ℙ3 with two irreducible components, one of degree d and the other of degree d(d — 1), (ii) the dual presentation in the dual space ℙ3*, for which we derive a lower bound for the number of projections necessary for linear reconstruction as a function of the degree and the genus, and (iii) the presentation as an hypersurface of ℙ5, defined by the set of lines in ℙ3 meeting the curve, for which we also derive lower bounds for the number of projections necessary for linear reconstruction as a function of the degree (of the curve). Moreover we show that the latter representation yields a new and efficient algorithm for dealing with mixed configurations of static and moving points in ℙ3.