Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 543

In this section, a theoretically optimal (or minimal) test patch selection algo-
rithm is developed. The method uses techniques outlined in Ref. [53], in which a
measured TRC is approximated in terms of a linear combination of basis functions
obtained from experimental data. It can be used to model the imaging system
with respect to the actuators required for producing compensating TRCs. The basis
vectors which describe the characteristics of the printer are used to determine the
optimum minimal set of gray levels or colors for use in calibration or TRC control
process [54].
For simplicity, let us assume that the test patches are gray levels between 0 and
255 digital count. It is also assumed that the output drift in the system can be
modeled as the weighted sum of K eigenvectors,

X

o ¼ a ( k ) v ( k ) þ n ( 9 : 130 )

k

where
o is the above-mentioned output drift vector with G entries
G being the number of gray levels
v(k) is the kth eigenvector (G

1)
a(k) is the kth model parameter or weight
n is the error vector with independent identical distribution

Equation 9.130 can also be written in matrix form as

o ¼ Va þ n

( : )

9 131

where V is an eigenvector matrix of size G K whose kth column is v(k), and a is
K 1, whose kth entry is a(k). The eigenvector matrix V is constructed via singular
value decomposition (SVD) (Section 3.10) on numerous TRC data samples obtained
from a series of controlled experiments in the laboratory, or obtained from machine
population in the

field. The parameter vector a can be estimated by the least square

method as

1

¼ H T H H T y

a* ( 9 : 132 )

where
superscript T denotes matrix transpose
H is an S K matrix
S is the number of sample patches used in estimation
y is the data vector (S

1) that is obtained from experimental data during the

runtime control = calibration process

The rows of H are selected from the V matrix rows. Speci cally, if the ith patch has a

gray level of g(i), the ith row of H is then a replica of the g(i) th row of matrix V.