Doran.-.Geometric.Algebra.&.Computer.Vision.[].pdf

Geometric Algebra puter Vision
The Auto-calibration Problem
Chris Doran
Department of Physics
Madingley Road, Cambridge
C.******@
.uk/
clifford/
SUMMARY
1. Rotors, bivectors and rotations. Ways of representing
rotations and the group manifold.
2. Extrapolating rotors and rotor calculus. The linear space
associated with rotors, extrapolating and averaging
rotations.
3. The known range data problem. Least squares, its
Bayesian origin and solution for 2 and
cameras.
4. Unknown range data problem. Bayesian analysis again
and its geometric solution. 2 camera data and the

-camera problem.
1
GEOMETRIC ALGEBRA IN 3-D
½









½ ¾ ¿
1 scalar 3 vectors 3 bivectors 1 trivector
NB
for the pseudoscalar. Geometric product has


Æ ·






 
Dot and wedge symbols have usual meaning
½ ½

¡

´
 
µ


´
 
µ
¾ ¾
A rotor is a normalised element of the even subalgebra,
¾ ¾




·


 

½
The tilde is the reverse operation. Rotors generate rotations
via
¼





We can also write
 

¾





R generates a rotation of  in a positive (right handed)
sense in the
plane. In 3-d we can also write
  Á Ò
¾



where
is the unit vector representing the axis.
2
THE GROUP MANIFOLD
Rotors are elements of a 4-d space, normalised to 1. They lie
on a 3-sphere. This is the group manifold. Any paths between
rotors must lie on this manifold.
At any point on the manifold, the tangent space is
3-dimensional. Just like the 2-sphere in 3-d:
Tangent plane
Rotors require 3 parameters, eg. Euler angles
    
¾     
¾     
¾
½ ¾ ¾ ¿ ½ ¾


  
But often more convenient to use the set of bivector
generators with
¾


 
plication: The rotors
and generate the
same rotation. The rotation group manifold is more
 
complicated — it is a projective 3-sphere with
and
identified. Usually easier to work with rotors.