EXISTING TECHNIQUES OF VISUAL PROCESSING - Video Object Extraction and Representation: Theory and Applications - page 34

THE ASSUMPTION OF OPTICAL FLOW
The movement of a physical object through time can be embodied
in the assumption of optical flow. Many motion estimation techniques
use this implicit assumption within their calculations. This assump-
tion is described mathematically by the optical flow equation [Horn and
Schunck, 1981], shown in Eq. 2.1. Given a pixel, ( x ( 0 ), y ( 0 ), 0 ), we can
associate a path through time that describes its projected motion:

I ( x ( t ), y ( t ) ,t ) = c

(2.1)

→

where c is a constant and f = 〈 x ( t ), y ( t ), t 〉 is the path of the pixel

parametrized with respect to time (see Figure 2.4). Eq. 2.1 also has
another form from the Taylor's series expansion and the application of
the chain rule:

∂ I dx ∂ I dy ∂ I

+ + = 0

(2.2)

∂ xdt ∂ y dt ∂ t

where second order terms are considered negligible. Eq. 2.2 relates the

〈 〉 to the partial derivatives

dy
dt
of I(x,y,t). In this topic, when we mention object motion, we refer to
the projected object motion onto the frame,
The optical flow assumption also leads to an alternate representation
of the video sequence where pixels are grouped with respect to the pro-
jected motion. Assuming the motion between frames is correct, we can
group sets of pixels by their time parametrized paths of constant inten-
sity. For a video sequence, we can group these pixels by their paths that
best satisfy the optical flow constraints; these paths are called motion
paths. This grouping of pixels by motion path is a convenient form of
the video sequence.
The issues with motion estimation algorithms based upon the optical
flow are:

dx ,

projected motion of the video object,

dt

dx , dy
dt

〉 .

〈

dt

1. when using Eq. 2.2 to find the projected motion, the problem becomes
a minimization problem with two unknowns and only one constraint.

2. the optical flow equation assumes that the intensity of a point is
constant over time. But there are many counterexamples: a rotating
reflective surface, changing lighting conditions, shading, etc.

When we use optical flow in the system, we need to address the issues
mentioned above. We usually add another constraint, such as motion