Interpolation of Multidimensional Functions - Control of Color Imaging Systems: Analysis and Design - page 249

6.2 INTERPOLATION OF UNIFORMLY SPACED LOOKUP TABLES
Interpolation of uniformly spaced LUTs can be performed using linear, cubic, and
other nonlinear techniques [1]. We will

first consider linear (1-D) and bilinear (2-D)
interpolation techniques. Trilinear interpolation with application to digital printers is
discussed in the subsequent section.

6.2.1 L INEAR AND B ILINEAR I NTERPOLATIONS
We first consider the linear interpolation of a 1-D LUT. Consider the 1-D function

y ¼ f(x), shown in Figure 6.1, where x is the independent variable and y is the
dependent variable. Assume that a uniform LUT of N data points is available.
The LUT is shown in Table 6.1. Suppose that we would like to interpolate point
b on the curve. This point is located between the two LUT nodes a and c, that is,
x i x x i þ 1 . The interpolated value is located on the straight line connecting the
two nodes a and c, as shown in Figure 6.1. The interpolated value y is given by

y i þ 1 y i
x i þ 1 x i

y ¼ ð x x i Þ y i ( 6 : 2 )

for x i x x i þ 1 .

f ( x )

y 1

y N

y i+ 1

c

y = ?

b

y i

a

x

x 1 x i x i+ 1 x N

x

FIGURE 6.1 One-dimensional function f(x).

TABLE 6.1
Uniform LUT of N Data Points

x y¼ f(x)

x 1 y 1

x 2 y 2

. .

x N y N