Multiview Differential Geometry of Curves

Trifocal Relative Pose From Lines at Points

We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and the novel case of (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Gröbner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver framework MINUS, which dramatically speeds up previous HC solving by specializing hc methods to generic cases of our problems. We characterize their number of solutions and show with simulated experiments that our solvers are numerically robust and stable under image noise, a key contribution given the borderline intractable degree of nonlinearity of trinocular constraints. We show in real experiments that (i) sift feature location and orientation provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches, where the state of the art structure from motion initialization fails.

Camera Pose Estimation Using First-Order Curve Differential Geometry

This paper considers and solves the problem of estimating camera pose given a pair of point-tangent correspondences between a 3D scene and a projected image. The problem arises when considering curve geometry as the basis of forming correspondences, computation of structure and calibration, which in its simplest form is a point augmented with the curve tangent. We show that while the resectioning problem is solved with a minimum of three points given the intrinsic parameters, when points are augmented with tangent information only two points are required, leading to substantial robustness and computational savings, e.g., as a minimal engine within ransac. In addition, algorithms are developed to find a practical solution shown to effectively recover camera pose using synthetic and real datasets. This technology is intended as a building block of curve-based structure from motion systems, allowing new views to be incrementally registered to a core set of views for which relative pose has been computed.