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