Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 343

7.4.4 D EVICE D RIFT M ODEL
A systematic drift in color, which may occur due to changes in humidity and
temperature, can be modeled by sampling a few colors and then updating the
printer model over time. In this section, we examine a method to predict color drift
in digital printers using measurements from a sensor. Two methods are discussed:
(1) channel independent (scalar) autoregressive (AR) model and (2) channel depen-
dent vector autoregressive (VAR) model. The scalar AR model can be used for
predicting single channel drift, that is, drift in the print density on the photoconductor
or on the paper, lightness on the paper, and chroma shift or hue shift. The VAR
model can predict the color drift speci ed in terms of re ectance spectra or the

L*a*b* on the paper.

7.4.4.1 Autoregressive (AR) Model Applied to Printer Drift Prediction

Let time t 0 be the time at which the printer forward model (map) is initially
constructed from measurements. It is easy to build a printer drift model if the
measurements for all the patches used for constructing the forward model are
available. If performed during run time, this would require too many measurements,
which increase cost and decrease productivity. Hence a more preferred method
would be to use a few measurements to sample the color drift and to update the
entire model. Thus a drifted printer model, P(t), is built from the initial forward
printer model, P(t 0 ), at time t 0 þ t based on few measurements of printed color
patches. Printed colors are selected in critical regions of the color space to maximize
the sensitivity to color drift.
We describe the general AR model for signal prediction and explain how it
can be applied to our speci

c problem of printer drift prediction. Consider the
measurable output y(n) from the printing system. The function y(n) could be L*or
a*orb* or chroma or hue or print density for a single color patch. The output of the
system is given by y(n) at time index n. It is assumed that N data samples are
available. We use an AR model of order P of a stationary zero-mean process y(n)of
the form

X

P

y ( n ) ¼ a i y ( n i ) þ e ( n ) ( 7 : 62 )

i ¼ 1

P
i ¼ 1 are the AR parameters and e(n) is the zero-mean white noise process
error with variance

where a fg

2 . In this model, the predicted output is given by

s

X

P

y ( n ) ¼ a i y ( n i ) ( 7 : 63 )

i ¼ 1

The error between the measured output and the predicted output is given by

e ( n ) ¼ y ( n ) y ( n ) ( 7 : 64 )