Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 328

7.4.2.3 Log-PCA Model (Log-PCA)

The Log-PCA is sometimes used instead of the linear PCA described in
Section 7.4.2.1. The approach is similar to linear PCA, except that the re

ectance

spectra are replaced by its natural log

R (l) !log ( R (l)) ( 7 : 49 )

A summary of the estimation algorithm for Log-PCA is shown in Figure 7.16a and b.

Example 7.4

Use Example 7.3 input output characterization data and apply log-PCA model to

show the improvements to modeling accuracy.

-

S OLUTION
Table 7.3 shows the results for two different least-squares models. Both algorithms
give similar accuracy when compared to Example 7.3 in which the spectral data
were not converted with log-PCA.

7.4.2.4 Piecewise Linear PCA Model

The main purpose of the PCA approach is to model the printer data set from a
few principal components and a few weights (Equation 7.43), since a few principal
components can capture the essential characteristics of the printer. To improve the
accuracy of the printer model, clustered linearized PCA models can be generated since
the clustering approach can model the nonlinearities well. It requires a reasonably large

Get training data

( CMYK ) 1 R 1 (λ)

( CMYK ) 2
.

R 2 (λ)

( CMYK ) N R N (λ)

Given CMYK, find weight vector
using prediction process steps in
Section 7.4.2.1

Form covariance matrix:
=[-ln( R )] T [-ln( R )]

Σ
where R =[ R 1 R N ] T

R 2 …

Estimate-log of reflectance spectra

8
i= 1

X = -log( R (λ)) = Σ W i ψ i (λ)

Use steps described in Section 7.4.2.1 to estimate the
parameter vector, θ 0 . For RLS algorithm, use Equation 7.48

Obtain predicted reflectance spectra from
equation, R (λ) = exp(- X )

(a) (b)

FIGURE 7.16 (a) Parameter training algorithm shown for log-PCA model. (b) Spectral

prediction algorithm using model parameters and log-PCA.