Mathematical approach for predicting non‐negative tristimulus values using the CAT02 chromatic adaptation transform

Abstract

It was reported that the CAT02 imbedded in the CIECAM02 suffers from predicting the corresponding colours with negative tristimulus values. To overcome this problem, a mathematical approach is proposed for modifying the CAT02. This approach combines the non-negativity constraint for the corresponding colours' tristimulus values with the minimisation of the colour differences between the tristimulus values of the corresponding colours obtained by visual observations and tristimulus values of the corresponding colours predicted by the model, which resulted in a constrained non-linear optimisation problem. A revised matrix is established using the MATLAB routine " fmincon ".The performances of the CAT02 with various matrices including the original CAT02 matrix and the new matrix are tested using the visual data sets and the optimum colours. Test results show that the CAT02 with the new matrix successfully predicted corresponding colours' tristimulus vales without negative values for all optimum colours and colour matching functions of two standard observers under the test illuminants considered. However, accuracy with the new matrix for predicting the visual data becomes about 1 CIELAB colour difference unit worse compared with the original CAT02. It seems that the accuracy has to be sacrificed in order to ensure the non-negativity constraint for the tristimulus values of the corresponding colours.

Different matrices for CIECAM02

“…Li et al . developed a mathematical approach for optimizing the CAT02 matrix M. This approach combines the non‐negativity constraint for the tristimulus values of the corresponding colours with the minimization of the colour differences between the tristimulus values of the corresponding colours obtained by visual observations, and tristimulus values of the corresponding colours predicted by the model.…”

“…developed by Brill and Süsstrunk, the M CAT [Eq. ] given by Li et al , the M HPE [Eq. ] developed by Hunt‐Pointer‐Estévez, and the M CAM [Eq. ]…”

“…Although negative tristimulus values may be useful in some cases such as in extra wide gamut RGB spaces, in reality negative tristimulus values cannot be further used in colour transformations as it is a default assumption for all tristimulus values being non‐negative in all colour transformations up to now. The 82 illuminants used in our previous study for deriving the matrix M CAT defined by Eq. were used here.…”

“…Furthermore, the test for mathematical failure using the CIE 1931 and 1964 colour matching functions was carried out using all the five matrices. The 82 illuminants used in our previous study were used here. Figure shows the chromaticity coordinates of the 82 illuminants.…”

Chromatic adaptation transform by spectral reconstruction

“…The transformation that converts tristimulus values into cone responses is modified through an optimization to yield a CAT that produces corresponding colors that match the empirical data more closely, sometimes including constraints in the optimization formulation to enforce additional desirable behaviors. [7][8][9][10][11][12][13] Examples of these empirically "trained" CATs include Bradford (BFD), CMCCAT97, CMCCAT2000, CAT02, and most recently CAT16. [13][14][15] One characteristic of linear von-Kries-type CATs is that they can sometimes produce tristimulus values with negative components.…”