Optimization of Mixed White Light Color Rendering

A method is developed for optimization of color rendering for a light mixture. No derivatives of the rendering function are required. Constraints of correlated color temperature and approximate white can be incorporated. The method is developed based on the Complex Method with modification. It is first applicable for 3-colors light mixtures and then extended to a hierarchical and iterative structure for higher-order light mixtures. Applicability of the method for a mixture of four colors is demonstrated by simulation. The results show that global and unique convergence to the optimal within required tolerances for CRI and spatial dispersivity is always achieved. Introduction Color rendering ability of mixed white light is an important index to evaluate its illumination quality in applications. However, due to the complexity in measurement of the rendering ability under designated constraints, a method for general mathematical formulation and global optimization of the rendering ability is difficult to develop. It would be very useful for the light emitted diode (LED) die and luminary manufacturers if such a method is available. Zukauskas et al. [1] considered the problem of optimal light mixing as maximization of an objective function. The objective function is formed by linearly interpolating between the luminous efficacy (LE) and the color rendering index (CRI). A boundary finder was developed for the maximization by locating a segment of an outer boundary of the phase distribution (LE, CRI). Zukauskas’ method is full of inspiration. But, as mentioned in Zukauskas’ paper, the method may require large depth and large number of perturbations to access the global maximum. It could be very time-consuming if the dimension of the parameter space is increased. Ries et al. [2] applied a Mathematica routine to search for maximization of the CRI, the LE or the luminous flux (LF) of a light mixture. The maximization is solved for a given mixture color by iteration. During the iteration, three of the relative intensities were computed to meet the requirement of the given color while the remaining intensities are updated by the routine. Ries’ method is computationally efficient. However, the difficulty to access the global optimum increases rapidly with the dimension of the mixture space. Without solving the difficulty, the routine may yield only local maximization. Hsu [3] considered an objective function as a weighting sum of square differences of color properties such as CRI, LE, LF, CCT and color gamut from their set-points. The weighting-sum method is promising to include all the color properties of interest for optimization. However, the approach may achieve only sub-optimal results if precise values for the set-points and the weightings are not available. The CIE CRI is scored in the scale of 0 ~100. In this scale, a CRI of 100 represents 8 sample objects illuminated by a test light may emit zero color differences in the CIE 1964 WUV color space as illuminated by a reference. However, the color space is not of good uniformity at the red region. Currently, the CIE recommends use of the CIE 1976 Lab and 1976 Luv color spaces for calculating the color differences instead. More problems with the CRI may also include inadequate chromatic adaptation correction, limited applicability at extreme CCTs, few sample objects, flawed color fidelity, etc. [4]. Despite the applicability of the CIE 1964 WUV color space for some case may be questionable [5, 6], it is still the one most widely used currently. Therefore, it is used in this study and methods for the CRI optimization developed are also applicable if a substitute of the CRI is given. This study is devoted to working out mathematical formulation and a numerical method for the optimization of CRI of light mixture. Unlike the weighting-sum method, the proposed one requires no preknowledge of the set-points and gives clear-cut restraining border to a mixture space and global optimization. The Complex Method [7] is implemented with modification to determine the relative intensities of light mixture for the optimization. The Complex Method with modification in a hierarchical iterative structure is proposed so that the method becomes generally applicable to this study. Examples for simulation studies are given. Problem Definition In this study, a test illuminator is a mixture of multi-colors. Without changing the chromaticity and CRI of the mixture, the relative intensities of its compound illuminators can be normalized. Moreover, as it will be explained later for a light mixture of higher order (n > 3) for optimization of the CRI, it may be necessary to consider the constraints of the relative intensities as below: