l 12mm 
l 20mm 
l 36mm 
  
— are pro- 
oncept on 
iated acti- 
e detected 
xt section. 
struction 
1 
m in com- 
ifferent as- 
t the miss- 
odel-based 
ric object 
is a quan- 
ly the best 
ose param- 
eters by fitting the projection of a three-dimensional 
model to two-dimensional features detected in the 
image(s). We represent the 3D model information 
in the reconstruction level of the hybrid knowledge 
base (see Fig. 7). 
For each object in the domain, there is one 
concept (e.g. RC.3HOLED.BAR) in the knowledge 
base where the necessary geometric information 
is stored. These concepts are linked by spe- 
cialization links to the generic object concept 
RC.OBJECT. The same specialization hierarchy ex- 
ists in the PE-concrete level. So, direct links con- 
nect the 3D object models and the reconstructed 
objects to the recognized objects with all their de- 
tected image features. While the concept RC_VIEW 
collects the reconstructed objects per camera view, 
the concept RC_SCENE establishes the connection 
between all camera views (e.g. stereo images) 
and stands for a 3D representation of the observed 
scene. The concept RC_CAM_PARAM is a context- 
dependent part of each camera view. This concept 
models the external camera parameters and the fo- 
cal length. Our method holds for one ore more views 
of the scene. All concepts in the reconstruction level 
are associated with a numerical model-fitting proce- 
dure which minimizes a multi-variate cost functions 
measuring all differences between projected model 
and detected image features as a function of the ob- 
jects’ pose and the camera parameters®. Common 
features in the scenes we are dealing with are points 
and circles. 
5.1 Projection of model points 
The projection of a model point is the transforma- 
tion of the point x, from model coordinates o to 
the camera coordinate system / and the subsequent 
projection onto the image plane bj. This can be ex- 
pressed in homogeneous coordinates? as 
br = Pi (a) = $1 (Tou “Tio : $(z,)) 
T 0 © 0 
Y = f . 
= ® Za 9 
Y10(9 
c(0)c(4) c(0)s(Q) —s(0) tz 
s(v)s(0)c(9) — c(V)s(9) s(v)s(0)s(0) t c(v)c(6) s(b)e(0) ty |. 
c(v)s(0)c(6) 4 s()s(6) c(y)s(0)s(4) — s(w)e(4) Gi À 
0 0 
e 
o oé- 
© 
Me.) with s(z)=sin(z) and c(x)=cos(z) (16) 
® is a function for the transformation from affine 
to homogeneous coordinates. The projection of a 
model point in a second image plane b, needs one 
3The specializations of RC_OBJECT inherit this feature. 
‘Homogeneous transformations are denoted by 7 with sub- 
scripts indicating destination and source coordinate frame of the 
transformation. 
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additional transformation 7,, from the reference co- 
ordinate system which we place in the first camera 
coordinate system [ to the second camera coordinate 
system r, 
Es, =P} (20) = 97" (0. T4: T5:9(z,) (17) 
5.2 Projection of model circles 
The perspective projection of circles which are pla- 
nar figures can be understood as a collineation in 
the projective plane IP?. The quadratic form of a 
projected model circle is easily computed using four 
projected points on the circle and the corresponding 
cross ratio (see [26] for further details). 
The projection of a model circle to the first and to 
the second image plane are denoted by 
Tp, == T£ te) == I (Tool t Tio 1 Tv.) (18) 
Zp, = Te ota) — D, (Tor “Tri Tio "T. Gro) (19) 
I'; is the function realizing the transformation of the 
projected model circle in homogeneous coordinates 
to the ellipse representation as center point, radii 
and orientation. À model circle x, is characterized 
by its center point, the radius and a normal vector in 
model coordinates o. 'T'he function L', calculates the 
four points that are projected and their cross ratio 
in homogeneous coordinates. This formulation of 
the perspective projection of a model circle allows 
us to measure easily the deviation of projected and 
detected ellipses comparing five parameters. 
5.3 Model-fitting 
The pose of an object is well estimated from the im- 
age data if the value of the non-linear multi-variate 
cost function 
N T 
Go - S SS (20, = Piola, 0.) qe 
i=1 jeB 
Cr Dead) 00 
is minimal. The cost function C measures the de- 
viation of projected model features x,, — these can 
be points or circles — from the corresponding image 
features. The vector a contains all unknown param- 
eters. B is the set of images of a scene. N is the 
number of corresponding model and image feature 
pairs. Depending on the feature, the vectors Tp, 
and zo, contain different representations and the 
projection functions Pi. are the respective trans- 
formations. K is a covariance matrix which is used 
to model the admissible tolerance with respect to 
deviation from projected model to detected image 
features.