Interpolation of Multidimensional Functions - Control of Color Imaging Systems: Analysis and Design - page 277

reduced-sized forward LUT, to the full size and then applying an inverse algorithm
on the upsampled LUT to obtain the full-size inverse map.
There are different techniques available in the literature for selecting these
critical colors [3,10,11]. The two most promising techniques for downsampling of
multidimensional LUTs are SLI and dynamic optimization (DO) algorithms. In the
following sections, we consider both SLI and DO algorithms. Numerical examples
are provided for comparison of these two approaches.

6.5.2 D OWNSAMPLING U SING S EQUENTIAL L INEAR I NTERPOLATION
SLI is a numerical technique to optimally downsample a uniformly spaced LUT to
any desired size. Piecewise linear homeomorphism is another technique illustrated
in Ref. [10]. These algorithms are very complex and their derivations are not given in
this topic. Interested readers can refer to Refs. [3,10].

6.5.3 D YNAMIC O PTIMIZATION A LGORITHM
The DO algorithm selects a finite number of colors (or points in L*a*b color space)
by minimizing the MSE (

D E a * ) between the actual printer output and the upsampled
printer output constructed using a

finite number of points as de ned by

D E a *

¼k L*a*b* (out1) L*a*b* (out2)k ( 6 : 60 )

Lab (out1) ¼ P ( CMY ) ( 6 : 61 )

Lab (out2) ¼ P ( CMY ) ( 6 : 62 )

The LUT (P) is obtained by upsampling the smaller LUT containing the

nite
number of critical colors. This is shown in Figure 6.16. Once the upsampled forward
LUT (P) is constructed for a three-to-three map, its inverse LUT (inverse of P) can be
computed using ICI or the other algorithms mentioned above. The DO algorithm is
based on dynamic programming, which uses a multistage decision process and the
performance criteria such as minimization of

D E a * error criteria.

first outline the 1-D case and then extend
the approach to two and three dimensions. The 3-D approach will be used to

To illustrate the algorithm in detail, we

nd the

critical colors for measurement of a three-to-three printer forward map.

6.5.3.1 One-Dimensional DO Algorithm

Consider the 1-D discrete function f(x), where x takes M discrete values x 1 , x 2 ,...,x M .
We would like to choose N < M points x ¼ [ x 1 x 2 x N ] such that x 1 ¼ x 1 , x N ¼ x M ,

and x x while minimizing the MSE resulting from the piecewise linear approxi-

mation de

ned by f (

x) and f(x) over x. The MSE is given by

X

M

fx j L k x ðÞ

1
M

2

E ¼ ( 6 : 63 )

i ¼ 1