Conformal Geometric Algebra in Stochastic Optimization Problems ...

136 CHAPTER 4. A PRIMER ON POSE ESTIMATION WITH CGA
If the projection rays, and thus the whole imaging system, are to be expressed with
respect to an external world coordinate system, six additional extrinsic parameters
have to be introduced. They reflect a rigid body motion (RBM), consisting of a
rotation and a translation, so as to allow for an arbitrary position and orientation
of the camera in respect to the external coordinate system. So in short, determining
the RBM is pose estimation.
4.1 The overall Principle
The general approach to the pose estimation problem in the text bases on the
following key assumptions:
1. a 3D-point model of the pictured object, given in respect to an external world
coordinate system, is available
2. for each point in the object model the corresponding image point (object
pixel) can be determined if it exists (view dependent) 1
3. for each object pixel a projection ray, expressed in terms of the camera coordinate
system, can be computed
Fig. 4.2: Pose estimation: fitting the red wolf WC to the computed red projection
rays, the black wolf WE is obtained being the connection to the external coordinate
system OE. The inverse RBM (a translation only) then reflects the camera pose.
The principle shall be demonstrated with the help of figure 4.2. There the ‘object
model’ is given by the wolf WE. It is defined with respect to the external coordinate
system at OE. The camera with its optical center OC is assumed to move off the
coordinate system at OE in a fronto-parallel manner, i.e. sideways on, with respect
to the flank of the wolf.
1 The correspondence problem is referred to as ‘matching’.