Parameterized Polynomial Systems and Their Applications

Many problems in computer vision and engineering can be formulated using a parameterized system of polynomials which must be solved for given instances of the parameters. Solutions and behavior over the real numbers are those that provide meaningful information for these applications. By using homotopy continuation within numerical algebraic geometry, one can solve these parameterized polynomial systems. First, we propose a new approach which uses locally adaptive methods and sparse matrix calculations to solve parameterized overdetermined systems in projective space. Examples are provided in 2D image reconstruction to compare the new methods with traditional approaches in numerical algebraic geometry. Second, we discuss new homotopy continuation methods for solving two minimal trifocal calibrated relative pose problems defined by point and line correspondences, which appear together, e.g., in urban scenes or observing curves. Experiments are shown using real and synthetic data to demonstrate that challenging scenes can be reconstructed where standard methods fail. Third, we present a new definition of monodromy action over the real numbers which encodes tiered characteristics regarding real solutions. Examples are given to show the benefits of this definition over a naive extension of the monodromy group (over the complex numbers). In addition, an application in kinematics is discussed to highlight the computational method and impact on calibration.