Image Processing Reference
In-Depth Information
Fig. 12.3. (
Left
) The translational motion of a linearly symmetric patch. (
Right
) The trans-
lational motion of a patch containing two distinct directions. The actual and the observable
translations are drawn as
magenta
and
black
vectors, respectively
is perceived. This situation is shown in Fig. 12.2, left, where the normal vector of the
motion plane (magenta) is drawn in blue and the (normal) motion vector of the line
is drawn in black.
By contrast, the motion of a point generates a line in the 3D continuous image.
This situation is shown in Fig. 12.2, right, where dots translate upwards in the image
plane, as represented by the black arrow. The result is many parallel lines, remini-
scient of a bundle of spaghetti.
The two fundamental translation types are not new types of phenomena. They
are not even specific to the physics of motion. Indeed, these occur as a consequence
of physical quantities that happen to be equivalent to or deduced from the structure
tensor. In particular, the motion vectors in the image plane are uniquely determined
by the normal of the plane,
k
,ifthe2D image patch is linearly symmetric (a 2D
property). Likewise, the 2D optical flow is uniquely determined by the direction of
the generated 3D line,
k
,ifthe2D structure tensor of the image patch has a balanced
direction tensor component that is nonzero.
5
In both cases the vector
k
is a 3D vector
and the optical flow is insensitive to its direction, i.e.,
k
is as good as
k
when it
comes to represent, the velocity. Accordingly, the same tensor,
−
kk
T
(12.47)
can represent both types of translational motion conveniently.
5
This is equivalent to saying that the image patch is nontrivial (nonconstant) and it lacks
linear symmetry.
