Interpolation of Multidimensional Functions - Control of Color Imaging Systems: Analysis and Design - page 248

6 Interpolation of

Multidimensional
Functions

6.1 INTRODUCTION
Multidimensional functions play an important role in digital color imaging systems.
For example, a digital color printer can be modeled as a forward map from four-
dimensional (4-D) device-dependent CMYK color space to three-dimensional (3-D)
device-independent L*a*b* color space. The mapping can be de ned by the follow-

ing three multidimensional functions:

L* ¼ f 1 ( C, M, Y, K )

¼ f 2 ( C, M, Y, K ) ( : )

a* 6 1

¼ f 3 ( C, M, Y, K )

b*

Unfortunately, exact closed-form expressions are not easily available for the above
equations. Therefore, we have two options. The

first option is to have a lookup table
(LUT) of all possible combinations of CMYK; that is, change each separation from
0 to 255 in steps of one, which requires a LUT size of 255 4 entries that is equivalent
to 3 255 4 bytes 12.6856 bytes of CMYK - L*a*b* data. This is not practical.

Another approach would be to have a LUT of size smaller than the full size, for
example, 17 4 or lesser, that is, CMYK - L*a*b* data and use interpolation to

¼

nd
other colors. Multidimensional interpolation is also extensively used in displays,
scanners, and inverse printer maps.
There are two types of interpolation techniques: linear and nonlinear. Generally
speaking, linear interpolation techniques are used to interpolate the data, where the
color space is uniformly sampled (uniform LUT) and is reasonably linear. If the
color space data is nonuniform and is not linear, then nonlinear interpolations are
preferred.
In this chapter, we present different methods of multidimensional interpol-
ation. We first consider uniform LUTs and discuss linear interpolation in one-

dimensional (1-D), two-dimensional (2-D), and 3-D color spaces. An extension to
4-D color space is straightforward. After this, we consider nonuniform LUTs and
describe nonlinear techniques such as the Shepard and moving-matrix approaches.
As an example of application, a numerical computation of inverse printer maps
and techniques for downsampling the color space LUTs are also presented in
some detail.

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