Image Processing Reference
In-Depth Information
Now by Equation 4.71 we can substitute for
P
(
t
+ δ
t
)
x
+δ
y
to give
x
,
y
+δ
P
()
t
x
P
()
t
y
P
()
t
xy
,
xy
,
xy
,
P
()
t
=
P
()
t
+
x
+
y
+
t
(4.77)
xy
,
xy
,
t
which with some rearrangement gives the motion constraint equation
x
t
y
t
P
P
P
+
= -
(4.78)
x
y
t
We can recognise some terms in this equation. ∂
P
/∂
x
and ∂
P
/∂
y
are the first-order differentials
of the image intensity along the two image axes. ∂
P
/∂
t
is the rate of change of image
intensity with time. The other two factors are the ones concerned with optical flow, as they
describe movement along the two image axes. Let us call
x
t
y
t
v
u
=
and =
These are the optical flow components:
u
is the
horizontal optical flow
and
v
is the
vertical
optical flow
. We can write these into our equation to give
P
P
P
v
u
+
= -
(4.79)
x
y
t
This equation suggests that the optical flow and the spatial rate of intensity change together
describe how an image changes with time. The equation can actually be expressed more
simply in vector form in terms of the intensity change
∇
P
= [
∇
x
∇
y
] = [
P
/
x
P
/
y
] and
the optical flow
v
= [
u
v
]
T
, as the dot product
∇
P
P
·
v
= -
(4.80)
We already have operators that can estimate the spatial intensity change,
∇
x
=
P
/
x
and
∇
y
, by using one of the edge detection operators described earlier. We also have
an operator which can estimate the rate of change of image intensity,
y
=
P
/
t
, as given
by Equation 4.69. Unfortunately, we cannot determine the optical flow components from
Equation 4.79 since we have one equation in two unknowns (there are many possible pairs
of values for
u
and
v
that satisfy the equation). This is actually called the
aperture problem
and makes the problem
ill-posed.
Essentially, we seek estimates of
u
and
v
that minimise
error in Equation 4.86 over the entire image. By expressing Equation 4.79 as,
u
∇
t
=
P
/
t
= 0 (4.81)
we then seek estimates of
u
and
v
that minimise the error
ec
for all the pixels in an image
∇
x
+
v
∇
y
+
∇
∫∫
v
)
2
ec
=
(
u
∇
x
+
∇
y
+
∇
t
dx dy
(4.82)
We can approach the solution (equations to determine
u
and
v
) by considering the second
assumption we made earlier, namely that neighbouring points move with similar velocity.
This is actually called the
smoothness constraint
as it suggests that the velocity field of the
brightness varies in a smooth manner without abrupt change (or discontinuity). If we add
this in to the formulation, we turn a problem that is ill-posed, without unique solution, to
one that is well posed. Properly, we define the smoothness constraint as an integral over the
