Applications of Parameter Homotopies

Many problems in applications such as computer vision and engineering can be formulated as solving a parameterized system of polynomial equations for various instances of the parameters. By using homotopy continuation (Paul’s lecture) within numerical algebraic geometry, one can solve these parameterized polynomial systems using parameter homotopies (Sascha’s lecture). In computer vision, parameter homotopies can be naturally overdetermined which can lead to challenges when performing numerical computations. It can also be worthwhile to construct real parameter homotopies in order to more efficiently compute the real solutions for the particular application. This talk will use examples to discuss methods to overcome various challenges as well as discuss methods for constructing parameter homotopies that only compute real solutions.