Image Processing Reference
InDepth Information
7.4.1.1 A Linear in the Parameters Model
A simple
ne, linear, quadratic, or cubic
terms is introduced below. For a scalar case, if y is the output data from the sensor,
and u is the input, then an input
linear in the parameters
model with af
''
''

output model can be written as
y
¼ u
1
f
1
(
u
) þ u
2
f
2
(
u
) þþu
n
f
n
(
u
)
(
7
:
1
)
where
y is the output (sensor) data
u is the input (e.g., CMYK values for a printer) data
u
i
is the parameter that needs
fitting or identi
cation
f
i
(u) is the known form of the model (e.g., af
ne, linear, quadratic, etc.)
In vector and matrix form,
y
¼
A
u
(
:
)
7
2
where the matrix A is named the regression matrix and it contains the form of the
model and
is the matrix containing the parameters of the model.
For example, when modeling a CMYK to L*a*b* input

output data, the
sample output test color data is represented in L*a*b* and the corresponding input
value is represented in CMYK. We can then write parameters of the linear af
u
ne
model (Equation 7.2) as follows:
y
¼
y
0
¼
y
1
½
y
2
y
3
; A
¼
A
0
¼
½
1
u
1
u
2
u
3
u
4
;
2
4
3
5
b
1
b
2
b
3
M
11
M
21
M
31
M
12
M
22
M
32
M
13
M
23
M
33
M
14
M
24
M
34
u ¼ u
0
¼
and
y
0
¼
A
0
u
0
(
:
)
;
7
3
where
u
1
¼
c
u
2
¼
M
u
3
¼
Y
u
4
¼
K
y
1
¼
L*
y
2
¼
a*
y
3
¼
b*
b
i
, M
ij
are constants
to distinguish the variables between initial data
and the updates coming at future time during the adaptation process. If there are N
colors in the training set, then y is a matrix of size N
Note, we used the subscript
0
''
''
3.
A
0
is a matrix with N number of rows. For a linear af
ne model (as shown) A
0
will be of size N
5. Since the A
0
matrix is nonsquare and, therefore, noninvertible,
the parameters contained in
u
0
cannot be solved using simple linear algebra or matrix
methods. First we form a residue equation,
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