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Fig. 2.19
(
a
) Rotating ellipse with
reference points
marked on its boundary. (
b
) Constraint lines
resulting from the rotation of the ellipse. (
c
) Typical distribution of constraint
line intersections
in
UV
space for a real-world image from the first test sequence (cf. Fig.
2.17
). The mean of the
distribution has been subtracted from all points; the principal components are drawn as
solid black
lines
(U
1
,V
1
)
and
(U
2
,V
2
)
. The information about the rotational motion of the object
is thus contained in the range
v
covered by the projections of the intersection
points on the principal component of the distribution which is oriented perpen-
dicular to the longitudinal axis of the object (the second principal component in
Fig.
2.19
c). The value of
v
then corresponds to the velocity dispersion across
the object caused by rotational motion in the image plane. In our system, we ro-
bustly estimate
v
based on the 10 % and 90 % quantiles of the distribution of
the projection values. The angular velocity
ω
p
of the object rotation parallel to
the image plane is then obtained by
ω
p
=
v/l
with
l
as the length interval
parallel to the longitudinal object axis covered by the assigned three-dimensional
points.
The rotation orthogonal to the image plane is determined based on the values
of
∂z/∂t
determined in Sect.
1.5.2.5
for the extracted three-dimensional points.
For each model part, the projections
p
(i)
of the assigned three-dimensional points
on the longitudinal object axis are computed, and a regression line is fitted to the
(p
(i)
,∂z/∂t
(i)
)
data points. The slope of the regression line directly yields the ve-
locity dispersion
w
in the
z
direction and thus the angular velocity
ω
o
of the
object rotation orthogonal to the image plane. Due to the rotational symmetry of
the object models regarded in this study, the rotational motion of the object is al-
ready fully determined by the two components
ω
p
and
ω
o
of the angular veloc-
ity.
The technique described in this section allows us to extend the determination of
the vector
T
of pose parameters by a direct estimation of the temporal pose deriva-
tive
T
without the need for an object tracking stage.
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