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

  • Authors:

  • J. Lasenby;W. J. Fitzgerald;A. N. Lasenby;C. J. L. Doran

  • Affiliations:

  • Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, U.K.;Department of Engineering, Trumpington Street, Cambridge CB2 1PZ, U.K.;MRAO, Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, U.K.;MRAO, Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, U.K.

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 1998

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Abstract

We discuss a coordinate-free approach to the geometry of computervision problems. The technique we use to analyse thethree-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 asstraightforward. The calculus associated with geometric algebra isparticularly powerful, enabling one, in a very natural way, to takederivatives with respect to any multivector (general element of thealgebra). What this means in practice is that we can minimize withrespect to rotors representing rotations, vectors representingtranslations, or any other relevant geometric quantity. This hasimportant implications for many of the least-squares problems incomputer vision where one attempts to find optimal rotations,translations etc., given observed vector quantities. We willillustrate this by analysing the problem of estimating motion from apair of images, looking particularly at the more difficult case inwhich we have available only 2D information and no information onrange. While this problem has already been much discussed in theliterature, we believe the present formulation to be the only one inwhich least-squares estimates of the motion and structure arederived simultaneously using analytic derivatives.