Image Processing Reference
In-Depth Information
e
u
xy
xy
,
,
= (
λ
×
2(
ux
∇
+
∇
v
∇ ∇
y
+
t
)
x
+ 2(
u u
-
)) = 0
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
(4.88)
and
e
xy
xy
,
,
v
v
v
= (
λ
×
2(
ux
∇
+
∇
∇ ∇
y
+
t
)
y
+ 2(
-
)) = 0
(4.89)
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
v
This gives a pair of equations in
u
x
,
y
and
v
x
,
y
(1 + (
λ
∇
x
) )
2
u
+
∇
λ
x
∇
y
v
=
u
-
∇
λ
∇
x
t
xy
,
xy
,
xy
,
xy xy
,
,
xy
,
xy
,
xy
,
(4.90)
2
v
v
λ
∇
x
∇
y
u
+ (1 + (
∇
λ
y
) )
=
-
∇
λ
∇
x
t
xy
,
xy xy
,
,
xy
,
xy
,
xy
,
xy
,
xy
,
This is a pair of equations in
u
and
v
with solution
2
2
2
u
v
(1 +
λ
(
(
∇
x
) + (
∇
y
)
))
u
= (1 +
∇
λ
(
y
)
)
∇
-
λ
∇
x
y
∇
-
∇
λ
x
t
xy
,
xy
,
xy
,
xy
,
xy
,
xy
,
xy xy
,
,
xy
,
xy
,
v
v
(1 +
λ
(
(
∇
x
) + (
2
∇
y
)
2
))
=-
∇
λ
x
∇
y
u
+ (1 +
λ
(
∇
x
) )
2
-
∇
λ
y
∇
t
xy
,
xy
,
xy
,
xy
,
xy
,
x
,
y
x y
,
x y
,
x y
,
x y
,
(4.91)
The solution to these equations is in iterative form where we shall denote the estimate of
u
at iteration
n
as
u
<
n
>
so each iteration calculates new values for the flow at each point
according to
v
∇
xu
+
∇
y
+
∇
t
xy xy
,
,
xy xy
,
,
xy
,
〈〉
n
+1
n
〈〉
u
=
u
-
λ
(
∇
x
)
xy
,
xy
,
xy
,
2
2
(1 + (
λ
∇
x
+
∇
y
))
xy
,
xy
,
(4.92)
v
∇
xu
+
∇
y
+
∇
t
xy
,
xxy
,
xy xy
,
,
xy
,
〈〉
n
+1
n
〈〉
v
=
v
-
λ
(
∇
y
)
xy
,
xy
,
xy
,
2
2
(1 + (
λ
∇
x
+
∇
y
))
xy
,
xy
,
Now we have it, the pair of equations gives iterative means for calculating the images of
optical flow based on differentials. In order to estimate the first-order differentials, rather
than use our earlier equations, we can consider neighbouring points in quadrants in successive
images. This gives approximate estimates of the gradient based on the two frames. That is,
(
P
(0)
+
P
(1)
+
P
(0)
+
P
(1)
) -
xy
+1,
xy
+1,
x y
+1,
+1
xy
+1,
+
1
(
P
(0)
+
P
(1)
+
P
(0)
+
P
(1)
)
xy
,
xy
,
xy
,+1
xy
,+1
∇
x
=
xy
,
8
(4.93)
(
P
(0)
+
P
(1)
+
P
(0)
+
P
(1)
) -
xy
,+1
xy
,+1
x
+1,
y
+1
x
+1,
y
+1
(
P
(0)
+
P
(1)
+
P
(0)
+
P
(1)
)
xy
,
xy
,
xy
+1,
xy
+1,
∇
y
=
xy
,
8
In fact, in a later reflection (Horn, 1993) on the earlier presentation, Horn noted with
rancour that some difficulty experienced with the original technique had actually been
caused by use of simpler methods of edge detection which are not appropriate here, as the
simpler versions do not deliver a correctly positioned result between two images. The time
