Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 326

The parameter matrix, u 0 , is a function of CMYK. When it is used in combination

with the 1-D spectral PCA vectors, we have the complete model of the printer.
The training process is summarized in following key steps:

1. Receive spectral vectors, R 1 , R 2 ,...,R N , of the N training set.
2. Determine the sample mean, R 0 , using Equation 7.36.
3. Form covariance matrix (Equation 7.39), perform singular value decom-
position (Equation 7.40), and determine K number of basis vectors c j with

1, 2, . . . , K.
4. Determine zero-mean re

j ¼

ectance spectra for each training spectra,

1, 2, . . . , N.
5. Determine weights W j using Equation 7.43 for j ¼

r i ¼ R i R 0 , i ¼

1, 2, . . . , K basis vectors

1, 2, . . . , N colors. Use the zero-mean spectral r from step 4. For
convenience, the subscript i is not shown in r.
6. Solve the least-square solution for the parameter matrix,

and for i ¼

u 0 , using Equation
7.47, by grouping CMYK values corresponding to each training sample in
A 0 for the appropriate model structure and forming the weight matrix, W T
from step 5.

To use the parameter vector for predicting the spectra for any new CMYK values, at
minimum, run following key process steps:

1. Receive the CMYK values at which spectral re ectance are to be calculated

using the model.
2. Arrange input color CMYK values in the matrix A 0 for the structure chosen
during training step 6. Color numbers are omitted in the matrix A 0 for
simplicity.
3. Determine matrix W T

[W 1 W 2 ...W K ] for all prediction colors (or W j for
j ¼ 1, 2, . . . , K for each color) from Equation 7.46 since the parameter
matrix

¼

u 0 is known from the training process of step 6 above.
4. Obtain predicted re ectance spectra from Equation 7.41 using weight vector

frompredictionprocess step3 and basis vectors fromthe trainingprocess step3.

7.4.2.2 PCA-Based Modeling for Adaptive Estimation

For performing continuous adaptation as fresh samples become available, or to
improve the accuracy of parameters instead of the least squares, an RLS algorithm
(Equation 7.25) can be used with slight modi cations as in Equation 7.48.

u k þ 1 ¼ u k þ P k þ 1 a k þ 1 W k þ 1 a k þ 1 u k ( 7 : 48 )

where the equation for P k þ 1 is shown in Equation 7.26. Adaptive estimation is
initiated by setting P 0 to identity and initial

u 0 to the output of the least-squares

equation (Equation 7.47).
A general warning is required while using PCAmethod for modeling color systems.
In order to gain con dence in the use of experimental data, bootstrapping is often

needed. In bootstrapping, a random subset (1 = 2 in this case) of the data should be used