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

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