NEW GEOMETRIC METHODS FOR COMPUTER VISION - AN APPLICATION TO STRUCTURE AND MOTION ESTIMATION

We discuss a coordinate-free approach to the geometry of computer visi on problems. The technique we use to analyse the three-dimensional tra nsformations involved will be that of geometric algebra: a framework b ased on the algebras of Clifford and Grassmann. This is not a system d esigned 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 h ave two arbitrary sets of vectors, known to be related via a 3D rotati on, the rotation is easily recoverable if the vectors are given. Extra cting the rotation by conventional means is not as straightforward. Th e 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 representin g rotations, vectors representing translations, or any other relevant geometric quantity. This has important implications for many of the le ast-squares problems in computer vision where one attempts to find opt imal rotations, translations etc., given observed vector quantities. W e will illustrate this by analysing the problem of estimating motion f rom a pair of images, looking particularly at the more difficult case in which we have available only 2D information and no information on r ange. While this problem has already been much discussed in the litera ture, we believe the present formulation to be the only one in which l east-squares estimates of the motion and structure are derived simulta neously using analytic derivatives.