Three-Dimensional Pose Estimation and Segmentation Methods - 3D Computer Vision: Efficient Methods and Applications - page 130

camera through a static scene or the motion of a rigid object relative to the cam-
era ('rigid motion'). Of special importance is the normal flow, i.e. the component
of the optical flow perpendicular to the edge in the image at which it is measured.
It is shown by Fermüller and Aloimonos ( 1994 ) that an image displacement field
displays a structure which does not depend on the scene itself, where vectors with
specific values of their norm and direction are located on curves of certain shapes
in the image plane. Specifically, sets of identical optical flow (also normal flow) or
disparity vectors correspond to conic curves, termed 'iso-motion contours' by Fer-
müller and Aloimonos ( 1994 ). The properties of these curves only depend on the
parameters of the three-dimensional rigid motion.
In the spatio-temporal pose estimation scenario regarded in this section, for an
object performing a purely translational motion parallel to the image plane all con-
straint lines belonging to pixels on the object intersect in a single point in UV space.
Both components of the translational motion are thus uniquely recovered. For ob-
jects with a rotational motion component in the image plane, a case which is not
addressed by Schunck ( 1989 ), the intersection points between constraint lines are
distributed across an extended region in UV space. This situation is illustrated in
Fig. 2.19 a for an ellipse rotating counterclockwise. The constraint lines belonging
to the indicated contour points are shown in Fig. 2.19 b. In this example, the U
coordinates of the constraint line intersection points are a measure for the mean
horizontal velocities of the corresponding pairs of image points. The V coordinates
have no such physical meaning. The distribution of intersection points is elongated
in vertical direction due to the fact that a vertical edge detector is used for interest
pixel extraction and because only image points with associated values of

<δ max
with δ max typically chosen between 1 and 2 (cf. Sect. 7.3 ) are selected by the space-
time stereo approach. Hence, constraint lines running at small angles to the U axis
do not exist.
Figure 2.19 c shows a distribution of constraint line intersection points obtained
from the scene shown in Fig. 2.17 a, typical of a rotationally symmetric and elon-
gated object like the forearm partial model used in this study. The points in UV
space are weighted according to the spatial density of the corresponding three-
dimensional points along the longitudinal object axis. The mean (U obj ,V obj ) of the
intersection point distribution, corresponding to the translational motion component
of the object, has already been subtracted from the intersection points in Fig. 2.19 c.
The translational motion component W obj parallel to the z axis is given by the me-
dian of the (fairly noisy) values of ∂z/∂t for all three-dimensional points assigned
to the object or object part.
In the example regarded in Fig. 2.19 c, scene points near the wrist are moving
faster in the image plane than scene points near the elbow. The resulting intersec-
tion points are strongly concentrated near the points (U 1 ,V 1 ) and (U 2 ,V 2 ) depicted
in Fig. 2.19 c, which represent the motion of the scene points near the elbow and
near the wrist. In this scenario, two circular markers attached to the upper and the
lower end of the forearm, respectively, would yield two narrow clusters of inter-
section points in UV space at (U 1 ,V 1 ) and (U 2 ,V 2 ) . Regarding scene points at
arbitrary positions on the forearm instead of well-localised markers yields a dis-
tribution which is largely symmetric with respect to the line connecting the points

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