Image Processing Reference
In-Depth Information
Perform the singular value decomposition (Section 3.10) on the covariance matrix to
get n number of basis functions.
X
¼
X
X
n
i
¼
1
P
2
C
T
¼
2
i
c
i
c
i
SVD
¼ CP
(
7
:
40
)
The vectors,
c
i
, in Equation 7.40, correspond to the ith basis function. They are pair-
wise orthogonal and orthonormal just as the sine and cosine functions in a Fourier
decomposition of the composite function. A linear weighted combination of all these
basis functions represents the complete spectra. That is
X
n
R
(l) ¼
R
0
(l) þ
W
j
c
j
(l)
(
7
:
41
)
j
¼
1
C
T
C¼
I
¼
Note that
identity matrix. Also,
jP
1
j > jP
2
j > > jP
n
j
are rank ordered
singular values contained in the matrix,
P
2
3
P
1
00
:
0
4
5
0
P
2
0
:
0
P ¼
00
P
3
:
0
(
7
:
42
)
:
:
:
:
:
000
: P
n
The parameters, W
j
with
1, 2, . . . , n, are obtained in various ways. One
straightforward way is by using the orthogonality property of the basis vectors
emanating from Equation 7.40 [23]. The PCA method assures that the principal
directions determined by the spectral PCA stage are orthogonal to each other.
Since the basis vectors are also orthonormal, weights are computed from the
following dot product.
j
¼
W
j
¼
r
T
c
j
(
7
:
43
)
If there are only few, K
<
n, dominant eigenvalues, then only few, K, basis vectors
are needed to estimate the spectra as in Equation 7.35. In some situations, it is
dif
cult to know precisely how many dominant eigenvalues (hence eigen or basis
vectors) are required for a good approximation, while the rest are ignored as noise.
In general, if the color gamut of the device is large, as in a 6 or 14 color press, whose
gamuts are typically larger than four-color press,
then more basis vectors are
required for modeling the device. In Figure 7.13,
first four basis vectors
are plotted as a function of wavelength. In Figure 7.14, an example of reconstructed
spectra (Equation 7.35) is shown when up to nine basis vectors are used. The actual
spectra from the training set along with the
the
D
E 2000 and
D
E CIE Lab numbers are
shown in the same
figure to compare with the approximated spectra. In Figure 7.15,
Search WWH ::
Custom Search