Graphics Reference
In-Depth Information
Substituting the B-spine derivative in the above equation,
a
(
t
)=
s
N
s
o
(
V
x
i
,V
y
j
)
f
i
f
j
ds
i
(
V
x
i
,V
y
j
)
s
N
s
o
j
(5.63)
=
i
f
i
f
j
ds.
j
The integrals can be computed in closed form. For a cubic B-spline, we need
to use ten possible values due to symmetry. In the worst case, we need sixteen
coecient values. At each time instant, multiplication with the control point
gives the area of the contour.
5.8 Recovery of Time to Contact and Surface
Orientation
Cipolla and Blake [38] presented preliminary implementation of their theory.
The examples are based on a camera mounted on a robot arm whose transla-
tions are deliberate while rotations around the camera center are performed to
keep the target of interest in the center of its field of view. The camera intrinsic
parameters (image center, scaling factors, and focal length) and orientation
are unknown. The direction of translation is assumed known and expressed
with bounds. Nelson and Aloimonos [127] demonstrated a robotics system that
computed divergence using spatio-temporal techniques from images of highly
textured visible surfaces, while Cipolla and Blake [38] used image contours for
a real time implementation. The closed contour is localized automatically by
initializing a closed loop B-spline snake in the center of the image. The snake
explodes outwards and deforms under the influence of image forces that cause
it to be attracted to high contrast edges. The robot manipulator then makes
a deliberate motion towards the target. Tracking the area of the contour and
computing its rate of change allows us to estimate the divergence. For motion
along the visual ray, this provides sucient information to estimate the time
to contact. The manipulator, in fact, travels blindly after its sensing actions
and at a uniform speed for the time before contact. In repeated trials, image
divergences measured at distances of 0.5m to 1.0m were estimated accurately
to the nearest half of a time unit. This corresponds to a positional accuracy of
20mm for a manipulator translational velocity of 40mm/s. The ane trans-
formation approximation breaks down at close proximity to the target. This
may lead to a degradation in the estimate of time to contact.
5.8.1 Braking and Object Manipulation
The experiment of Cipolla and Blake shows a sequence of images taken from a
moving car approaching the windshield of a stationary car in front. In the first
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