Question 9 – Motion
(Optic Flow, Motion Field, Horn and Schunck)
A great deal of useful information can be extracted from time varying images.
Optic Flow  The apparent motion of brightness patterns observed when a camera is moving relative
to the objects being images is called the optical flow.
Motion Field  When objects move in front of a camera, or when a camera moves through a fixed
environment, there are corresponding changes in the image. These changes can be used to recover the
relative motions as well as the shapes of the objects.
We first define the motion field, which assigns a velocity vector to each point in the image. At a
particular instant in time, a point Pi in the image corresponds to some point Po on the surface of an
object. The two are connected by the projection equation. In the case of perspective projection, a ray
from the image point through the center of the lens can be extended until it strikes an opaque surface.
voδt
Po
ri
ri
Pi
voδt
Figure 1. Displacement of a point in the environment causes a displacement of the corresponding
image point. The relationship between the velocities can be found by differentiating the perspective
projection equation.
Let the object point Po have velocity vo relative to the camera. This includes a motion vi in the
corresponding image point Pi. The point Po moves voδt in a time interval δt, and its image Pi moves vi
δt. The velocities are given by:
vo = dro/dt and vi = dri/dt
where ro and ri are related by
1
1
ri 
ro
f'
ro .z
(f is the focal length and z is ??)
Differentiation of the above perspective projection equation yields:
(r .z )vo  (vo .z )ro (ro  vo )  z
1
vi  o

f'
(ro .z ) 2
(ro .z ) 2
A vector can be assigned in this way to every image point. These vectors constitute the motion field.
Optical Flow  is the apparent motion of brightness patterns in the image. Generally, optical flow
corresponds to the motion field, but not always. For example, the motion field and optical flow of a
rotating barber's pole are different, as illustrated in figure 2. In general, such cases are unusual, and for
this lecture we will assume that optical flow corresponds to the motion field.
Figure 2: The motion field and optical flow of a barber's pole.
One problem we do have to worry about, however, is that we are only able to measure the component
of optical flow that is in the direction of the intensity gradient. We are unable to measure the
component tangential to the intensity gradient. This problem is illustrated in figure 3, and further
developed below.
Figure 3: The aperture problem. We can only measure the component b.
Denote the intensity by I(x,y,t). This is a function of three variables as we now have spatiotemporal
variation in our signal. To see how I changes in time, we differentiate with respect to t:
Now, we assume that the image intensity of each visible scene point is unchanging over time (for
example, shadows and illuminations are not changing due to any motion), so we have
which implies
Ixu + Iyv + It = 0,
where the partial derivatives of I are denoted by subscripts, and u and v are the x and y components of
the optical flow vector.
This last equation is called the optical flow constraint equation since it expresses a constraint on the
components u and v of the optical flow.
The optical flow constraint equation can be rewritten as
Thus, the component of the image velocity in the direction of the image intensity gradient at the image
of a scene point is
We cannot, however, determine the component of the optical flow at right angles to this direction. This
ambiguity is known as the aperture problem.
There are two main approaches to reconstructing three dimensional motion from motion in the image
plane:


Convert the motion problem to a stereo problem and find the correspondence between a number
of points in the image at time t to the image at time t +δt.
Computer the optical flow and use its geometrical properties to deduce three dimensional
information about the scene and the motion.