Algebraic methods for image processing and computer vision

Many important problems in image processing and computer vision can be formulated as the solution of a system of simultaneous polynomial equations. Crucial issues include the uniqueness of solution and the number of solutions (if not unique), and how to find numerically all the solutions. The goal of this paper is to introduce to engineers and scientists some mathematical tools from algebraic geometry which are very useful in resolving these issues. Three-dimensional motion/structure estimation is used as the context. However, these tools should also be helpful in other areas including surface intersection in computer-aided design, and inverse position problems in kinematics/robotics. The tools to be described are Bézout numbers, Gröbner bases, homotopy methods, and a powerful theorem which states that under rather general conditions one can draw general conclusions on the number of solutions of a polynomial system from a single numerical example.