Interpolation of Multidimensional Functions - Control of Color Imaging Systems: Analysis and Design - page 251

y y

y N

d c

c

y i+ 1
y i

d

a

a

y y

e

e

b

b

y 2

y 1

x x

x i x x i+ 1

x 1 x 2 x x N

FIGURE 6.3 Two-dimensional uniform grid.

p 01

p 11

1- t

1- u

u

t

p 10

p 00

FIGURE 6.4 Relative distances.

The relative distances u and t are given by

x x i
x i þ 1 x i

y y i
y i þ 1 y i

u ¼ , t ¼ ( 6 : 4 )

Substituting u and t in Equation 6.3 results in

x x i
x i þ 1 x i fx i þ 1 , y i

y y i
y i þ 1 y i fx i , y i þ 1

z ¼ f ( x, y ) ¼ fx i , y i ð Þþ ½ ð Þ fx i , y i ð Þ þ ½ ð Þ fx i , y i ð Þ

y y i
y i þ 1 y i

fx i þ 1 , y i þ 1

x x i
x i þ 1 x i

þ ½ ð Þ fx i þ 1 , y i ð Þ fx i , y i þ 1 ð Þ fx i , y i ð Þ

( 6 : 5 )

Example 6.2

Consider a static system with two inputs, x 1 and x 2 , and two outputs, y 1 and y 2 ,
where y 1 ¼ f 1 (x 1 , x 2 ) and y 2 ¼ f 2 (x 1 , x 2 ). The LUT representing these two functions is
given in Table 6.3 and Figure 6.5. Use bilinear interpolation to find the value of

the two functions at point x ¼ [x 1 x 2 ], where x 1 ¼ 1.5 and x 2 ¼ 0.6.