Constrained least-squares regression in color spaces

To characterize color values measured by color devices (e.g., scanners, color copiers, and color cameras) in a device- independent fashion these values must be transformed to colorimet- ric tristimulus values. The measured RGB 3-vectors are not a linear transformation away from such colorimetric vectors, however, but still the best transformation between these two data sets, or be- tween RGB values measured under different illuminants, can easily be determined. Two well-known methods for determining this trans- formation are the simple least-squares fit (LS) procedure and Vrhel's principal component method. The former approach makes no a priori statement about which colors will be mapped well and which will be mapped poorly. Depending on the data set a white reflectance may be mapped accurately or inaccurately. In contrast, the principal component method solves for the transform that ex- actly maps a particular set of basis surfaces between illuminants (where the basis is usually designed to capture the statistics of a set of spectral reflectance data) and hence some statement can be made about which colors will be mapped without error. Unfortu- nately, even if the basis set fits real reflectances well this does not guarantee good color correction. Here we propose a new, compro- mise, constrained regression method based on finding the mapping which maps a single (or possibly two) basis surface(s) without error and, subject to this constraint, also minimizes the sum of squared differences between the mapped RGB data and corresponding XYZ tristimuli values. The constrained regression is particularly useful either when it is crucial to map a particular color with great accuracy or when there is incomplete calibration data. For example, it is gen- erally desirable that the device coordinates for a white reflectance should always map exactly to the XYZ tristimulus white. Surpris- ingly, we show that when no statistics about reflectances are known then a white-point preserving mapping affords much better correc- tion performance compared with the naive least-squares method. Colorimetric results are improved further by guiding the regression using a training set of measured reflectances; a standard data set can be used to fix a white-point-preserving regression that does remarkably well on other data sets. Even when the reflectance sta- tistics are known, we show that correctly mapping white does not incur a large colorimetric overhead; the errors resulting from white- point preserving least-squares fitting and straightforward least- squares are similar. © 1997 SPIE and IS&T. (S1017-9909(97)01204-X)