Image Processing Reference
In-Depth Information
7.4.1.1 A Linear in the Parameters Model
A simple
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
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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