Interpolation of Multidimensional Functions - Control of Color Imaging Systems: Analysis and Design - page 273

6.4.3.2 Algorithm Initialization

[C(0) M(0) Y(0)] T for the algorithm, we follow the To find an initial estimate y(0) ¼

following steps:

½ T in the forward printer
LUT that is closest to the target color x in an L 2 norm (Euclidean distance)
sense.
(2) Find the point y aux , such that P(y aux )

* * *

(1) Find the auxiliary point z aux ¼ L aux a aux b aux

¼ z aux . Note that y aux can be found
easily, since z aux is a grid point in the forward LUT.
(3) Select N grid points in a neighborhood of y aux ; that is, generate a cluster of
N points y aux1 , y aux2 ,..., y auxN by moving along the C, M, and Y axes
around y aux . Use the LUT P and trilinear interpolation to map these N points
to obtain the corresponding values in L*a*b* color space. Call them

z auxi ¼ P(y auxi ) i ¼

1, 2, . . . , N.
(4) Denote by z 0 the closest point to the set z auxi i ¼ 1, 2, . . . , N. Choose the

initial condition to be y 0 , where z 0 ¼ P(y 0 ).

6.4.4 T ETRAHEDRAL T ECHNIQUE
The tetrahedral interpolation technique covered in Section 6.2.3 can be used to nd

the inverse printer map. Its low computational complexity makes it an affordable
approach for constructing an inverse LUT [2]. For this reason, it is a widely
used methodology for inverse computation. Assume a set of CMY nodes (yi) i ) and
their corresponding L*a*b* nodes (zi) i ) are given in the form of a LUT. To

nd the
inverse of z ¼ L*a*b* using the tetrahedral interpolation method, we perform the
following steps:

Step 1: Partition the CMY color space into tetrahedral segments.

Step 2: Given a target x ¼ L*a*b*, find the tetrahedral that contains the z vector.

Let {z 1 z 2 z 3 z 4 } denote the vertices of the tetrahedron that z belongs to in L*a*b* and
let {y 1 y 2 y 3 y 4 } denote their corresponding vertices in CMY color space.
Step 3: Compute the corresponding CMY value using the following equation:

CMY ¼ y ¼ A CMY A 1 ð z z 1 Þ þ y 1 ( 6 : 49 )

Lab

where

A CMY ¼ y 2 y 1 ½ y 3 y 1 y 4 y 1 ( 6 : 50 )

and

A Lab ¼ z 2 z 1 ½ z 3 z 1 z 4 z 1 ( 6 : 51 )