Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 320

Thus, parameter updates are done adaptively based on sensor measurements to
capture the current state of the device, while at other times models can be used to
estimate the L*a*b* values using the recently updated parameters of the updated
pro

c input CMYK nodes, whose measurements cannot be
obtained easily because the printer may be scheduled for production jobs.

le LUTs, for speci

7.4.2 P RINCIPAL C OMPONENT A NALYSIS -B ASED M ODEL
Though the piecewise linear models based on clustering can produce better models,
the implementation may suffer from a large body of parameters to track, as well as
signi

cant complexity associated with the computation; both time and computing
resources. A much better way to model the input output data empirically is by using

principal component analysis (PCA) (which is also known by many other names,
such as, Karhunen

-

Loeve decomposition, Hotelling transform, singular value
decomposition, empirical orthogonal function decomposition, etc.; Section 3.10).
The PCA approach extracts key information from the data for modeling and throws
away the rest. It decomposes data into linear combinations of mutually orthogonal
and basis vectors of unit length (extracted from a training data set) in such a way that
in that basis, the second order statistics (covariance matrix) of the data is diagona-
lized. This diagonal matrix is the matrix of singular values, all of them nonnegative
by de

-

nition. When sorted from highest singular value to the lowest, the basis
vectors become known as the most signi

cant or principal components. As a result

of this operation on the input output data, it gives less redundancy and as good a

representation of the system as possible with a few principal components.
Many researchers [14

-

18] have attempted to decompose the spectra in terms of
principal basis vectors and use the basis vectors to model the imaging device or use
the PCA to predict the re

-

ectance measurements on multiple substrates (media)
while characterizing PCAs on a reference substrate [17]. This method is widely
used in other fields of engineering [19,20] and may receive more use in color systems

as the demand for high performance systems increase. In PCA-based modeling,
although the representation can be made with any multivariate input output data

samples [21], we show the approach when the sensed output is the re

-

ectance
spectra. A standard spectral sensor is used for measurement. Gretag spectrophotom-
eter, which outputs 36 spectral re ectance values, evenly spaced at 10 nm over the

visible spectrum over the spectrum 380 730 nm, a XRite spectrophotometer [10],

and an in-line spectrophotometer [7], which has 31 outputs evenly spaced at 10 nm
over the wavelength of 400

-

700 nm. In PCA, it is essential that the spectral training
data is mutually correlated. If they are independent, PCA does not help. Also, output
data with a mixture of different wavelengths (i.e., with different vector lengths) are
not considered for PCA.

-

7.4.2.1 PCA-Based Model in Spectral Space

In this section, we first show a one-dimensional (1-D) spectral PCA-based model for a

static printing system. A simple online estimation method is introduced later for a 1-D
PCA to update the parameters during the adaptation process. A more advanced two-
dimensional (2-D) PCA method can be found in Ref. [22] which uses a signi cantly