Image Processing Reference
In-Depth Information
for the PCA to determine PCAvectors. The results of the PCAvectors are then applied to
the rest of the data set to determine how good a model with PCA can be for the whole set.
This process should be repeated multiple times, by choosing each time a different
random subset for PCA analysis and the rest of the data to
twiththosePCAvectors.
If all of the data come from one distribution, the expected amount of residual energy
(actual variation
modeled variation) is insensitive to the subset chosen for analysis using
PCA. Problem 7.5 illustrates the bootstrapping method for an example dataset.
-
Example 7.3
In a CMYK printing system, input CMYK values are sampled uniformly between 0 and
255 to create 10
4
input
output characterization data set. The spectral data for each
patch is measured at 31 wavelengths from 400 to 700 nm at 10 nm intervals. Use
spectral PCA mathematical techniques to analyze the characterization data and
-
ta
model. Using PCA vectors
find parameters for a least squares and RLS model. Plot the
model accuracy for test colors as a function of the number of basis vectors. Howmany
basis vectors are needed to obtain a reasonably accurate model?
S
OLUTION
We will set P
0
ΒΌ I
(Identity matrix) in the RLS Equations 7.25 and 7.26. The
u
matrix is of size 23
10 (when 10 basis vectors were used) with mean propaga-
tion. All these calculations were done with absolute L*a*b* values. Accuracy for
approx. 2000 prediction colors is shown in Table 7.2 for different number of basis
vectors. RLS algorithm is found to be more accurate than a simple least-squares
regression. Improvement to the prediction accuracy saturates when the number of
basis vectors becomes equal to
five for both algorithms.
TABLE 7.2
Model Accuracy Shown with Respect to Spectral PCA Basis Vectors
D
E
a
*
D
E
2000
No. of
Models
Basis Vectors
Mean
95%
Max
Mean
95%
Max
Least squares
1
20.52
35.50
51.99
31.60
67.34
87.04
2
16.06
32.83
45.31
24.87
60.81
77.89
3
7.74
19.27
41.97
13.46
27.85
56.66
4
7.68
19.25
42.21
13.46
29.16
57.02
5
7.66
19.24
42.02
13.53
29.43
56.72
10
7.65
19.28
41.79
13.52
29.25
57.73
20
7.65
19.28
41.79
13.52
29.25
57.81
RLS
1
17.90
34.99
47.36
27.80
66.30
87.40
2
13.19
32.80
45.50
20.46
60.16
84.97
3
4.89
10.54
25.81
7.74
17.94
41.23
4
4.63
10.41
25.63
7.65
17.79
46.45
5
4.59
10.27
25.37
7.75
19.85
55.22
10
4.58
10.20
25.60
7.71
19.79
57.44
20
4.58
10.20
25.61
7.71
19.78
57.52
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