Image Processing Reference
In-Depth Information
The combination of Eqs. (
7.5
) and (
7.8
), gives the following cost functional,
s
T
w
2
log
s
T
w
2
0
−
λ
v
s
T
w
2
s
T
w
2
d
x
d
y
2
1
XY
J
(
w
)
=
−
.
50 e
x
y
x
y
1
d
x
d
y
+
λ
s
2
+
μ
w
T
w
2
w
x
+
w
y
−
.
(7.11)
Here
is the Lagrangianmultiplier
for the unity norm constraint. The arguments
x
and
y
of the functions are omitted
for the purpose of brevity (
s
λ
v
and
λ
s
are the regularization parameters, while
μ
, etc.). The solution of Eq. (
7.11
) is obtained
using the corresponding Euler-Lagrange equation,
≡
s
(
x
,
y
)
∂
∂
∂
J
I
∂
∂
J
I
∂
∂
J
I
w
−
−
=
0
,
(7.12)
∂
x
∂
w
x
y
∂
w
y
where
J
I
(
w
,
w
x
,
w
y
)
is the integrand in Eq. (
7.11
). On simplification, Eq. (
7.12
)
becomes,
s
T
w
2
x
y
s
T
w
2
d
x
d
y
XY
)
1
log
s
T
w
2
−
)
−
(
s
◦
w
+
log
(
0
.
5e
2
λ
v
−
s
x
y
s
w
d
x
d
y
XY
◦
2
w
◦
w
−
−
λ
s
∇
+
μ
w
=
0
,
(7.13)
2
represent the element-wise product operator, and the Laplacian
operator, respectively. For the RHS of the equation,
0
indicates a zero vector. The
Laplacian operator for a 2-D function
w
is given by Eq. (
7.14
).
where
◦
and
∇
2
w
2
w
∂
=
∂
+
∂
2
w
∇
y
2
.
(7.14)
x
2
∂
A discrete approximation of the Laplacian operator is given as [76]
4
δ
2
w
∇
(
x
,
y
)
≈
2
(
¯
w
(
x
,
y
)
−
w
(
x
,
y
)),
where
represents the local average of the weight vectors in the
X
and
Y
dimensions, and
w
¯
(
x
,
y
)
is the distance between adjacent pixels, trivially set to 1. After
discretization, the iterative solution for the weight vector
w
can be obtained by
re-arranging the terms as given by Eq. (
7.15
).
δ
