Image Processing Reference
In-Depth Information
area of interest, as in Equation 4.82. Since we want to maximise smoothness, we seek to
minimise the rate of change of the optical flow. Accordingly, we seek to minimise an
integral of the rate of change of flow along both axes. This is an error
es
as
2
2
2
2
v
v
u
x
u
y
∫∫
es
=
+
+
+
dx dy
(4.83)
x
y
The total error is the compromise between the importance of the assumption of constant
brightness and the assumption of smooth velocity. If this compromise is controlled by a
regularisation
parameter
λ
then the total error
e
is
e
=
λ
×
ec
+
es
2
2
2
2
2
v
v
P
P
P
u
x
u
y
∫∫
v
=
λ
×
u
+
+
u
+
+
+
+
dx dy
x
y
t
x
y
(4.84)
There is a number of ways to approach the solution (Horn, 1986), but the most appealing
is perhaps also the most direct. We are concerned with providing estimates of optical flow
at image points. So we are actually interested in computing the values for
u
x,y
and
v
x,y
. We
can form the error at image points, like
es
x,y
. Since we are concerned with image points,
then we can form
es
x,y
by using first-order differences, just like Equation 4.1 at the start of
this chapter. Equation 4.83 can be implemented in discrete form as
=
1
Σ Σ
2
2
v
v
2
v
v
2
es
4
((
u
-
u
)+(
u
-
u
)+(
-
)+ (
-
))
xy
,
xy
+1,
xy
,
xy
,+1
xy
,
xy
+1,
xy
,
xy
,+1
xy
,
xy
(4.85)
The discrete form of the smoothness constraint is then that the average rate of change of
flow should be minimised. To obtain the discrete form of Equation 4.84 we then add in the
discrete form of
ec
(the discrete form of Equation 4.82) to give
Σ Σ
v
2
ec
= (
u
∇
x
+
∇
y
∇
+
t
)
(4.86)
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
xy
where ∇
x
x
,
y
= ∂
P
x
,
y
/∂
x
, ∇
y
x
,
y
= ∂
P
x
,
y
/∂
y
and ∇
t
x
,
y
= ∂
P
x
,
y
/∂
t
are local estimates, at the point
with co-ordinates
x
,
y
,
of the rate of change of the picture with horizontal direction, vertical
direction and time, respectively. Accordingly, we seek values for
u
x
,
y
and
v
x
,
y
that minimise
the total error
e
as given by
Σ Σ
e
= (
λ
×
ec
+
es
)
xy
,
xy
,
xy
,
xy
∇
v
2
λ
×
(
ux
∇
+
∇
y
+
t
)
xy
,
xy
,
xy
,
xy
,
xy
,
Σ Σ
=
+
1
xy
2
2
v
v
2
v
v
2
4
((
u
-
u
) + (
u
-
u
) +(
-
)
+ (
-
) )
xy
+1,
xy
,
xy
,+1
xy
,
xy
+1,
xy
,
xy
,+1
xy
,
(4.87)
Since we seek to minimise this equation with respect to
u
x
,
y
and
v
x
,
y
then we differentiate
it separately, with respect to the two parameters of interest, and the resulting equations
when equated to zero should yield the equations we seek. As such
