Image Processing Reference
In-Depth Information
6.4.5 C
ONJUGATE
G
RADIENT
A
PPROACH
The CG technique is an iterative technique used in solving constrained optimization
problems and systems of linear equations of the form Ax
¼
b, where matrix A is
symmetric and positive de
nite [9]. The CG algorithm can also be used to solve
unconstrained optimization problems such as
finding the inverse printer model
described in the previous section. Assume a forward printer map z
¼
P(y), where
y
2
R
3
is a CMY color patch and z
2
R
3
is its corresponding L*a*b* value. The goal
is to
D
E
a
*
error between x and P(y), that is,
find y for a given z
¼
x by minimizing the
2
1
min
y
E
(
y
) ¼ mi
y
D
E
[
P
(
y
)
, x
] ¼ min
y
2
k
P
(
y
)
x
k
2
[
P
(
y
)
x
]
T
[
P
(
y
)
x
]
1
¼ min
y
(
6
:
52
)
The iteration to obtain the solution using CG method is given by
y
(
k
þ
1
) ¼
y
(
k
) þ a
k
d
k
(
6
:
53
)
where d
k
and
a
k
are the search direction vectors and the step size at the kth iteration,
respectively. The initialization algorithm to obtain an initial estimate y(0) is similar to
the technique used for ICI. The initial search direction is given by
d
k
¼
g
k
¼
e
(
k
)
J
k
(
6
:
54
)
where
e(k)
¼
P(y(k))
x(k)
J
k
is the Jacobian matrix de
ned by Equation 6.39
The updating equations for the search direction and the step size are given as
d
k
J
k
e
T
(
k
)
d
k
J
k
J
k
d
k
a
k
¼
(
6
:
55
)
g
k
þ
1
¼
e
(
k
þ
1
)
J
k
þ
1
(
6
:
56
)
g
k
þ
1
g
k
þ
1
g
k
ð
Þ
b
k
¼
(
6
:
57
)
d
k
g
k
þ
1
g
k
ð
Þ
d
k
þ
1
¼
g
k
þ
1
þ b
k
d
k
(
6
:
58
)
The criteria for stopping the iterations are given by the threshold parameter
e
and the
D
E
a
*
,
maximum number of iterations K
max
. The iteration stops when the MSE,
is less than
or the number of iterations exceeds K
max
. The accuracy of the CG
is similar to the ICI algorithm; however,
e
the computational complexity of the
CG method far exceeds the ICI approach.
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