Interpolation of Multidimensional Functions - Control of Color Imaging Systems: Analysis and Design - page 280

y

C

y M

y l j

y j

E i , j , k i , l j

A

B

E

x

x j x k i x M

D

FIGURE 6.17 Grid points in x - y plane.

In the second stage, we will try to select two internal points in the same set. If the
indices corresponding to the

first optimal point are n and m, then the total MSE is

E ¼ E i, ð , j, ðÞ þ E n,m ( 6 : 72 )

where E (i, j),( j,m) is the MSE resulting from interpolating all the grid points between
indices i to n and j to m using all necessary boundary points. We now minimize E
given by Equation 6.72 with respect to i and j. Let the solution be i ¼ n i , j ¼ m j . Then
the indices n i , m j , n, m correspond to the two optimum internal points (total of 14
points including boundary points) with the MSE given by

E 2 ( i, j ) ¼ E n i ð , m j , ðÞ þ E n,m ¼ E n i ð , m j , ðÞ þ E 1 ( n, m ) ( 6 : 73 )

This process is continued until we select N N grid points.

6.5.3.3 Three-Dimensional DO Algorithm

Consider a 3-D discrete function f(X) represented by a set of pairs (X, f(X)), where X
is a 3-D vector. These points are uniformly spaced in the support space of f(X)
forming an M M M LUT. The goal is to downsample this 3-D LUT to a smaller
LUT of size N N N while minimizing the MSE between the original and the
upsampled LUT. The 3-D DO algorithm, which is an extension of the 2-D algorithm,
can be used for optimally selecting these grid points. Similar to 2-D cases, it is a
multistage decision process described by the following steps. In the single-stage
decision process, assume that we start from point {ijk} and form a cube extending
this point to the boundaries of the underlying function. The goal is to find one point

inside the cube with its associated boundary points such that the MSE between the