Motion from point matches : multiplicity of solutions
Résumé
In this paper, we study the multiplicity of solutions of the motion problem. Given n point matches between two frames, how many solutions are there to the motion problem ? We show that the maximum number of solutions is 10 when 5 point matches are available. This settles a question which has been around in the computer vision community for a while. We follow two tracks. The first one attempts to recover the motion parameters by studying the essential matrix and has been followed by a number of researchers in the field. A natural extension of this is to use algebraic geometry to characterize the set of possible essential matrices. We present some new results based on this approach. The second one, based on projective geometry, dates from the previous century. We show that the two approaches are compatible and yield the same result. We then describe a computer implementation of the second approach that uses Maple, a language for symbolic computation. The program allows us to compute exactly the solutions for any configuration of 5 points. Some experiments are described.