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e
and
u
righ
e
computed according to (
1.120
) for the left and the right
interest pixel, respectively:
sections
u
left
u
right
u
left
e
d
=
(v
c
,t
c
)
−
(v
c
,t
c
).
(1.124)
e
To increase the accuracy of the determined disparity values, it is advantageous to
establish the correspondences based on the spacetime approach by searching for the
minimum value of the similarity measure along the epipolar line but to compute the
corresponding disparities without utilising temporal information. This prevents the
disparity value from becoming inaccurate when the true motion behaviour is not
closely approximated by the model function.
Given the optical and geometrical parameters of the camera system, the velocity
component
μ
parallel to the epipolar lines (in pixels per time step) amounts to
¯
2
μ
left
μ
right
.
1
μ
¯
=
+
(1.125)
In metric units, the epipolar velocity
U
=
∂x/∂t
is given by
u
right
1
2
(u
left
(v
c
,t
c
))
∂
∂t
¯
μd
−
(v
c
,t
c
)
+
∂x
∂t
=
e
e
=
U
t
.
(1.126)
d
2
The vertical velocity component
V
∂y/∂t
cannot be inferred pointwise from the
spacetime stereo data due to the aperture problem. The velocity component
∂z/∂t
along the depth axis depends on the first temporal derivative of the disparity, which
is obtained according to
=
∂d
∂t
=
μ
left
μ
right
−
(1.127)
(cf. (
1.122
) and (
1.124
)). Inserting (
1.127
)into(
1.101
) yields for the velocity
W
along the
z
axis
b
0
(μ
left
μ
right
)
∂z
∂t
=−
t
−
W
=
.
(1.128)
d
p
d
2
Note, however, that small relative errors of
μ
left
and
μ
right
may lead to large relative
errors of
∂d/∂t
and
∂z/∂t
. The relative difference between the temporal derivative
∂d/∂t
of the disparity computed according to (
1.127
) and the manually measured
average disparity variation per time step is of the order of 10 % or less (cf. Gövert,
2006
for further details).
Experimental results of the local spatio-temporal intensity modelling approach
by Schmidt et al. (
2007
) outlined in this section are described in Chap.
7
in the
context of three-dimensional scene segmentation and object tracking.
Like the approach described in this section, the spacetime stereo method by
Zhang et al. (
2003
) estimates the temporal derivative
∂d/∂t
of the disparity for each
established point correspondence. However, no quantitative evaluation but merely a
qualitative discussion is given in their study; thus a direct comparison is not possi-
ble. The spacetime stereo approach by Davis et al. (
2005
) does not take into account
the temporal derivative of the disparity.
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