DFG Research project: E.J. Bayro-Corrochano


Clifford algebras are well-known to pure mathematicians. In this
research work we are using an interpretation called geometric algebra
which is a coordinate-free approach to geometry. The elements are
coordinate-independent objects called multivectors which can be
multiplied together using a geometric product.  The system deals
with rotations in n-dimensional space very efficiently.  Since
geometric algebra has already been successfully applied to many areas
of mathematical physics and engineering, we believe that geometric
algebra will throw new lights in modern linear and nonlinear signal
processing. The system appears also as the ideal mathematical
framework for tasks of the perception action cycle systems. 
The topics of my work include:

- Computer Vision:

  binocular and trinocular geometry. Affine and projective
  reconstruction. Trifocal tensor and invariants for matching, 
  object recognition and image coding. 

- Lie Groups and Lie Algebras:

  in the  geometric algebra frame for the computation of  
  differential invariants, affine structure of image sequences and
  visual symmetries for visual guided robot navigation.

- Robotics:

  We use the 4D algebra of the motors for 3D kinematics.
  The motor algebra together with fuzzy logic are being used for
  geometric reasoning useful for object avoidance and navigation.
  In terms of motors we represent points, lines and planes and their motion. 
  These entities and their spatial invariants are being used for manoeuvre.
  Hand-eye calibration using motors for a binocular head on a mobile robot.
  The control of a binocular head is being formulated as a problem of
  multivector control. Related controllers, filters and estimators are
  extended for multidimensional control.

- Neural Computing:

  We believe that neural learning is an issue of geometric learning.
  We have generalized the standard MLP and RBF neural nets
  and the back-propagation training rule in the geometric algebra
  framework. The nets show a much reasonably performance during
  learning and in the generalization due to the geometric product,
  the avoidance of redundant components and the coordinate
  independence  of the data coding. We are improving the learning in
  these architectures using the sub-manifold intrinsic dimensionality.

- Teaching on Applied Clifford or Geometric Algebra:

  We are developing a computer program for teaching applied geometric 
  algebra in schools, colleges, universities and research centers. This
  involves the representation and manipulation of geometric entities in
  different geometric algebras G_{p,q,r}. For the symbolic processing
  and visualization we use Matlab. Our examples involve problems of
  the above cited fields.