Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 321

smaller number of basis vectors, reduced computation and measurements for continu-
ous adaptation. We show below how to model the printer using a 1-D PCA approach.
For N number of input colors, given a stream of output spectral training data,
R 1 , R 2 ,...,R N , where each R i is a vector of length n (e.g., n ¼ 31 when there are 31

re 700 nm), then each of

these spectra can be modeled as a random sample from a mixture model in terms of K
basis functions where K < n as

ectance values evenly spaced over the spectrum of 400 -

X

K

R (l) R 0 (l) þ W j c j (l) ( 7 : 35 )

j ¼ 1

where c j ( l ) is the jth basis function derived from PCA analysis. R0(l) 0 ( l ) is the sample

mean, which is computed from the output spectral data, and W j is the jth weight

parameter.

X

N

1
N

R 0 (l) ¼ R i (l) ( 7 : 36 )

i ¼ 1

We can represent, in matrix form, the terms inside the summation of Equation 7.35 as

2
4

3
5

W 1
W 2
.
W K

r ¼ c 1 c 2 ½ c K ( 7 : 37 )

Equation 7.37 can be written in matrix form for each color as

r ¼ BW ( 7 : 38 )

where r is the zero-mean re ectance spectra of size n 1 of the color. If R is the

full re ectance spectra of the color, then the approximated spectral data is given by

R R 0 þ r which represents the full re ectivity vector for that color at speci ed

wavelengths.
The matrix, B ¼ c 1 c 2 c K ], is the mixture matrix of size n K, whose

columns contain the basis vectors with elements at the wavelength intervals used in
the output spectral data. The vector, W ¼

[

[W 1 W 2 W K ] T , is the parameter vector of

size K 1 containing the scalar weights for the color.

The basis vectors are obtained as follows. First compute the covariance matrix
formed by the output spectral data

X X

N

1
N

T ¼ ½ R i R 0 R i R 0 ½ ( 7 : 39 )

i ¼ 1