J. Lasenby, W. J. Fitzgerald, C. J. L. Doran and A. N. Lasenby.
New Geometric Methods for Computer Vision
Int. J. Comp. Vision 36(3), p. 191-213 (1998).
Abstract: We discuss a coordinate-free approach to the
geometry of computer vision problems. The technique we use to analyse
the 3-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 3-D 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.