Internal Process Controls - Control of Color Imaging Systems: Analysis and Design - page 546

Since (H T H) 1 is known from the results of the

final step in the iteration, the above

calculation essentially contains only two matrix - vector multiplications. No matrix

inversion is performed.
In many typical control calibration applications, the eigenvectors are smooth

functions, and the selection process may be conducted in a hierarchical manner to
further reduce computation. Also, intuition can be used in selecting combinations of
gray levels so that the number of combinatorial searches required can be minimized.
Speci

=

first executes the above-described proced-
ural steps a through d with a reduced resolution. The solution for the reduced
resolution is used as the initial condition to run the same procedural steps a through
d again for the full resolution calculation.

cally, the optimization procedure

Example 9.13

As an example, we find the optimal gray levels by solving Equation 9.134 for

spatial TRC data. The quality of TRCs derived from equispaced levels is compared
with the quality derived from levels determined by using the optimal gray level
method. A system is assumed with a small number of gray levels (32, rather than
256) to make the demonstration tractable with a reasonable amount of effort and a
small number of test prints and measurements.
A

first printing is performed using all of the 32 gray levels in a calibration target
where a gray strip for each level spans across the page (Figure 9.53a). Measure-
ments are taken of the spatial nonuniformity using a scanner for each gray level to
obtain spatial TRC data. These first printing and measurement results are used to

1. Derive TRC eigenvector matrix that characterize the system. In our
example, a reduced set of two basis vectors is selected as the system
characterization. Prior experience shows that this number is adequate
when signi cant drift in the print engine response is not present. More

basis vectors can be included depending on the level of accuracy needed.
2. Derive the optimal gray levels for future calibration updates. Optimal gray
levels are derived for three separate cases: 2 levels, 4 levels, and 8 levels.

A second printing of the 32 levels is performed and used for the following steps:

1. Complete print engine state (spatial TRCs) is measured for t > 0. These

measurements serve as a reference for comparison of TRCs derived from
subsets of gray levels.
2. TRCs are generated using several equispaced subsets of gray levels,
namely 2, 4, and 8 levels.
3. TRCs are generated using the optimal levels derived above.
4. Derived TRCs are compared to the fully measured TRCs.

The gray levels used in the procedure are provided in Table 9.3 below.
The error is calculated by first deriving all gray levels (32) from a given

subset through the use of basis vectors. The derived 32-level TRCs are then