Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 307

7.4 CHARACTERIZATION OF COLOR SYSTEMS
Conventionally, there are two different approaches to color modeling: (1) empirical
or interpolation-based approaches, which essentially treat the device as a black box
with inputs and outputs and (2) analytical or

first principle approaches which attempt
to characterize the device color response using the fewest number of measurements
to arrive at analytical functions that have a physical connection to the process. Both
kinds of approaches are capable of predicting the color response of the device for a
variety of input images.
The empirical or interpolation-based methods are generally measurement inten-
sive and require the use of a large set of experimentally generated data between
inputs and outputs [2 6]. These models may contain only nonparametric LUTs, or

parameterized analytical functions that

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fit the data. The function passes through the
data points approximately. Thus, although we cannot use an empirical model to
explain a system, we can use such a model to predict the behavior where data does
not exist. One can understand that measurement data is crucial for an empirical
model. Data is used to assess the structure of the functional form and then estimate its
parameters. In this section, we present several empirical models with approaches
to capture both time-zero and time-varying effects. Accurate first principle models
(so-called models) are not available for all kinds of imaging devices.

The complexity of models, errors in capturing the actual physical process in the
presence of device drift over time, light scattering effects, and many other uncer-
tainties associated with the physical device itself make it impossible to perfectly
model the device over a reasonable period of time. In Chapter 10, a more elaborate,
parameterized nonlinear spectral model of the printing system incorporating reason-
able abstractions of the process is described. This type of model can help us to inject
meaningful time-varying effects into the system.

white box

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7.4.1 L EAST -S QUARES E STIMATION
We will consider two parts to the empirical or interpolation-based approach to
printer modeling. The first part is to obtain the model at time t 0 , which is typically

done (at the printer manufacturer) in the factory but could also be performed in

'' '' ''

the 10]. This approach may require

measurements of anywhere from one to thousands of color patches. A mathematical
model is generated and initial parameters are selected from this data using data

eld if an in-line color device is available [7 -

''

fitting techniques such as the least-squares method. Once an optimal parameterized
model is selected, the second part involves an adaptation process. The adaptation
process [11] is implemented using measurements from the sensor to adjust or tune
the model parameters over time. It accounts for variations in printer state over time
t 0 þ t. Using the historical data and the new measurements, an adaptive algorithm
will estimate the model parameters more accurately, and the predicted colors from
the adapted model can better represent the real system output at that time. In
automatic control literature, this kind of empirical modeling of a dynamic system
is called system identi cation.