B.K.P.
HORN
AND
B.G.
SCHUNCK
FIG.
I.
The basic rate of change of image brightness equation constrains the optical flow velocity.
The velocity
(u.
v)
has to lie along a line perpendicular to the brightness gradient vector
(Ex, E,).
The distance of this line from the origin equals
E,
divided by the magnitude of
(E,, E,).
We cannot, however,
determine the component of the movement in the
direction
of
the iso-brightness contours,
at right angles to the brightness
gradient. As a consequence, the flow velocity
(u,
v)
cannot be computed locally
without introducing additional constraints.
5.
The
Smoothness Constraint
If every point
of
the brightness pattern can move independently, there is little
hope of recovering the velocities. More commonly we view opaque objects
of
finite size undergoing rigid motion or deformation. In this case neighboring
points on the objects have similar velocities and the velocity field
of
the
brightness patterns in the image varies smoothly almost everywhere. Dis-
continuities in flow can be expected where one object occludes another. An
algorithm based on a smoothness constraint is likely to have difficulties with
occluding edges as a result.
One way to express the additional constraint is to minimize the square of the
magnitude of the gradient of the optical flow velocity:
(g)'
+
($)'
and
(g
))
+
(E))
Another measure
of
the smoothness of the optical flow field is the sum of the
squares of the Laplacians of the
x-
and y-components of the flow. The
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