New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation

We discuss a coordinate-free approach to the geometry of computer vision problems. The technique we use to analyse the three-dimensional transformations involved will be that of geometric algebra: a framework based on the algebras of Clifford and Grassmann. This is not a system designed specifically for the task in hand, but rather a framework for all mathematical physics. Central to the power of this approach is the way in which the formalism deals with rotations; for example, if we have two arbitrary sets of vectors, known to be related via a 3D rotation, the rotation is easily recoverable if the vectors are given. Extracting the rotation by conventional means is not as straightforward. The calculus associated with geometric algebra is particularly powerful, enabling one, in a very natural way, to take derivatives with respect to any multivector (general element of the algebra). What this means in practice is that we can minimize with respect to rotors representing rotations, vectors representing translations, or any other relevant geometric quantity. This has important implications for many of the least-squares problems in computer vision where one attempts to find optimal rotations, translations etc., given observed vector quantities. We will illustrate this by analysing the problem of estimating motion from a pair of images, looking particularly at the more difficult case in which we have available only 2D information and no information on range. While this problem has already been much discussed in the literature, we believe the present formulation to be the only one in which least-squares estimates of the motion and structure are derived simultaneously using analytic derivatives.