Conformal Geometric Algebra in Stochastic Optimization Problems ...

200 CHAPTER 7. APPLICATIONS IN COMPUTER VISION
Multiple synthetic experiments are conducted: in the beginning, a cloud of N
Gaussian distributed points {�x ′ 1...N } with a standard deviation of √ 2 and a bias
of (7, 0, 0) is generated directly in front of the camera. Given the ground truth
motor, denoted by M0, these points are displaced yielding the points {a1...N } :=
MK({�x ′ 1...N }) � M, which are intended to represent the object model. A set of N
3 × 3-covariance matrices {Σr1r1 , Σr2r2 , . . ., ΣrNrN } is generated at random so as to
account for image noise. None of those introduces an uncertainty parallel to the
optical axis (the uncertainty that is to be attributed to the image points is always
bounded to the image plane).
For each experimental run the following procedure applied: for each �x ′ i
, 1 ≤ i ≤
N, the corresponding Σriri is used to generate a Gaussian distributed error vector
�ri ∈ �3 , which is in turn used to translate �x ′ i so as to obtain the (noisy) point
xi = K(�x ′ i + �ri). The associated covariance matrix Σxixi is evaluated by means of
the elucidations in section 6.2.1. Next the projection rays {B1...N } passing through
the 3D-point cloud {x1...N } are calculated5 via Bi = (e ∧ eo ∧ xi)I, where error
propagation must again be obeyed when evaluating the covariance matrices {Σb1b1 ,
Σb2b2 , . .., ΣbNbN } ⊂ �6×6 . Hence the best motor � M is estimated, which fits the
{a 1...N } to the corresponding uncertain {B 1...N }. Notice that the ground truth
motor M0 is not necessarily the optimal solution for a single run.
Each experiment - involving 100 runs with N = 15 points - is characterized by three
values: the rotation angle of M0, denoted ω, the angle between the rotation axis
and the optical axis, denoted φ, and the noise level µr, being the arithmetic mean
of the set {��r� 1...N }.
Three motors are compared: the motor M0 (TRUE) and the motor estimated by
the GH-method (GH). This time no SVD-motor is available to serve as an initial
estimate. Hence the geometric method (GEM) as introduced in chapter 4 represents
the third motor, which at the same time plays the role of the initial estimate for the
GH-method. The quality of an estimated motor, here denoted by M, is assessed by
applying it to the actual problem setup, i.e. by transforming the model {a 1...N } into
the point set { ˆ b 1...N } := M{a 1...N } � M. Next the distances between the { ˆ b 1...N }
and their respective projection rays {B 1...N } is calculated, for example with the
help of equation (3.55). The N distances of every single run are averaged, whence
the RMS distance over all 100 runs, denoted by µ, is computed. The standard
deviation is given by σ.
Angle ω 10 ◦ 40 ◦ 70 ◦ 100 ◦
TRUE 0.223 0.230 0.229 0.226
Method GEM 0.229 0.237 0.235 0.230
GH 0.215 0.219 0.215 0.213
Table 7.2: Pose estimation: means µ for varying rotation angles (µr = 0.2).
5 Image plane and image points are thus only fictive entities in this experiment.