Optimization of polynomial system solvers with applications to visual odometry

Oleg Naroditsky, University of Pennsylvania


Efficient solutions to polynomial equation systems is an important topic in modern geometric computer vision. The importance stems from the fact that many minimal problems (problems that use the fewest possible number of constraints) have been formulated as polynomial systems in recent years. Minimal problems are extremely important in geometric computer vision because they guarantee the highest probability of rejecting outliers when used with robust estimation frameworks such as RANSAC. However, since many instances of the same minimal problem have to be solved in a typical live vision system, efficient solutions are paramount to reaping full benefit from their solutions. The goal of this work is to solve new and useful minimal geometry problems as well as advance the theory behind the solution methods. New solutions to two such minimal problems are offered and a new development in optimizing solutions for a class of problems using the so-called "action matrix" method is also presented. The first minimal problem that is solved is the structure from motion with a directional correspondence, where image projections of three 3D points in two cameras are combined with a common direction or a point at infinity to solve for camera motion. This algorithm can be solved in closed form or using algebraic geometry and becomes a foundation of a visual odometry algorithm that uses a four-point RANSAC hypothesis to estimate motion instead of the traditional five. The second minimal problem uses 3D point to plane correspondences to establish the motion between two coordinate systems and targets the problem of LIDAR to camera calibration. A set of six correspondence of image line (a plane in 3D) to LIDAR points is sufficient to estimate the relative pose of the devices, including scale. This problem can be solved in closed form using the Macaulay resultant, and becomes the basis for construction of automatic calibration software. The proposed optimization to the action matrix method leads to an improvement in performance and numerical stability of existing algorithms, such as uncalibrated image stitching and three-view triangulation.

Subject Area

Computer science

Recommended Citation

Naroditsky, Oleg, "Optimization of polynomial system solvers with applications to visual odometry" (2012). Dissertations available from ProQuest. AAI3509380.