Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 312

Substituting Equation 7.24 into Equation 7.22, the following recursive estimation
formula emerges

u k þ 1 ¼ u k þ P k þ 1 a k þ 1 y k þ 1 a k þ 1 u k

( 7 : 25 )

where P k þ 1 is

P k a k þ 1 a k þ 1 P k
1

P k þ 1 ¼ P k ( 7 : 26 )

þ a k þ 1 P k a k þ 1

Equations 7.25 and 7.26 achieve the adaptive estimation by using the previous estimate
and the error between the outputs to converge to a new estimate. The matrix, A 0 ,only
enters into the recursive equation in the

u 0 (Equation 7.6). Hence, the RLS
solution is more robust than the standard least-squares solution (Equation 7.6). In
practice, this recursive estimation can be initiated by simply setting P 0 to a diagonal
matrix, and by letting

first guess for

is obtained
when the error function is driven to zero or to a level below the desired threshold.

u 0 be the best first guess. Convergence to the desired u

7.4.1.3 Piecewise Linear Models

In many practical imaging devices, including printers, the linear model with the RLS
algorithm cannot accurately fit one uniform function to the entire input - output data set

since the functional relationship between input output data may be locally nonlinear

or may be only partially linear. The piecewise linear approach to modeling nonlinear-
ity of many engineering systems is a well-known concept. Hence generating multiple
piecewise linear (or curvilinear due to quadratic, cubic terms) models at various input
points and combining them via interpolation can lead to much more ef

-

cient models
than global linear models, since such models can represent the data more accurately.
To fully exploit the power of piecewise linear models, the color space is partitioned
into a number of regions or clusters. The number of regions, number of training colors
in each region, and the methodology required for partitioning the multidimensional
color space depend on the device under study. K-means algorithm (discussed in
Section 2.4.2.5) based on vector quantization — a classical quantization technique —

widely used in the literature for a variety of classi cation, compression, and signal

partitioning applications [12,13]. It has been found suitable for classifying the multi-
dimensional color space into smaller regions or clusters. It is related to self-organizing
maps, where a large set of input color nodes are divided into groups having approxi-
mately the same number of points closest to them. Linear models are generated for each
region, which is identi

ed by its centroid.
Key steps involved in constructing the initial multiple piecewise linear models based
on clusters are (1) obtain training samples via experiments or historical database, (2) clus-
ter the input or output colors (e.g., L*a*b* values or CMYK values) to obtain partitioned
colors and their centroids, and (3) obtain parameters of the linear model for each cluster
using Equation 7.3. When the adaptation is required, iteratively execute the RLS algo-
rithm using Equation 7.25 for each of the clusters separately. A suitable cluster assign-
ment process is required to assign each color sample to the right cluster. A simple way of
doing this is by comparing the Euclidean distance of the sample output color (assuming