Algebraic Curves in Multiple-View Geometry

We introduce multiple-view geometry for algebraic curveswith applications in both static and dynamic scenes. More precisely we show how the epipolar geometry can be recovered from algebraic curves. For that purpose we introduce a generalization of Kruppa's equations which express the epipolar constraint for algebraic curves. Reconstruction from a single image based on symmetry is also considered and we show how this relates to algebraic curves for a simple example. We also investigate the question of three-dimensional reconstruction of an algebraic curve from two or more views. In the case of two views we show that for a generic situation there are two solutions for the reconstruction which allows extracting the right solution provided the degree of the curve is greater or equal to 3. When more than two views are availablewe show that there construction can be done by linear computations using either the dual curve or the variety of intersecting lines. In both cases no curve ?tting is necessary in the image space. Finally we focus on dynamic scenes and show when and how the trajectory of a moving point can be recovered from a moving camera.