Use of extrapolation in computing color look-up tables

Colorimetric reproduction requires calibrated color output devices. One way to characterize a color output device is with a three-dimensional look-up table which maps the tristimulus values, t, to the control values, c of the output device. The functional form of the output device can be written in vector notation as t = F(c). The purpose of calibration is to define an inverse mapping from tristimulus values to control values. Since the function F(.) has no closed form, it is defined by interpolation from a table of values. Given a set of control values {c} on a regular grid and the corresponding set of tristimulus values {t} obtained from data collection, we wish to find the {cg} for different {tg} on a grid in the tristimulus space. The grid is obtained from a relatively sparse data set with an appropriately defined interpolation scheme. This interpolation scheme can be complex since it is used only once to compute the grid. The regular finer grid can be used in real-time with simple interpolation. While the functions which represent the device are usually well behaved and smoothly varying, the truncation of the data can cause a problem with interpolations near the edge of the gamut. An approach to solving this problem is to extrapolate the data outside the gamut using bandlimited or linear extrapolation methods. The extrapolated points along with the measured data are used in a single interpolation algorithm over the entire gamut of the device. The results of this method are comparable to other interpolation methods but it is simpler to implement. It has the additional advantage of allowing physical constraints, such as bandlimits, to be easily imposed.