Interpolation of Multidimensional Functions - Control of Color Imaging Systems: Analysis and Design - page 254

xyz plane

n 011 n 11

n 111

n 1

t

n 01

n 101 f ( x , y , z )

p ijk

n 001

v

u

n 010 n 10

n 110

n 0

n 000

n 00 n 100

FIGURE 6.7 Interpolation cube and its eight vertices.

The interpolated value of f at location (x, y, z) is computed in three steps as
follows:

(1) Use bilinear interpolation to interpolate the corresponding values of the
function f at nodes ½ n 00 n 01 n 10 n 11 , as shown by the dotted plane in

Figure 6.7:

p 00 ¼ p 000 þ tp 100 p 000 ð Þ

p 01 ¼ p 001 þ tp 101 p 001 ð Þ

( 6 : 6 )

p 10 ¼ p 010 þ tp 110 p 010 ð Þ

p 11 ¼ p 011 þ tp 111 p 011 ð Þ

(2) Use linear interpolation to interpolate the corresponding values of the
function f at nodes [n 0 n 1 ], as shown in Figure 6.7:

p 0 ¼ p 00 þ vp 10 p 00 ð Þ

( 6 : 7 )

p 1 ¼ p 01 þ vp 11 p 01 ð Þ

(3) Use linear interpolation to interpolate the corresponding values of the
function f at nodes [xyz], as shown in Figure 6.7:

p ¼ p 0 þ up 1 p 0

ð Þ ( : )

6 8

The relative distances t, v, and u are given by

x x i
x i þ 1 x i

y y i
y i þ 1 y i

z z i
z i þ 1 z i

t ¼ , v ¼ , and u ¼ ( 6 : 9 )