XYZ to ADL: Calculating Logvinenko's Object Color Coordinates

Recently Logvinenko introduced a new objectcolor space, establishing a complete color atlas that is invariant to illumination [2]. However, the existing implementation for calculating the proposed color descriptors is computationally expensive and does not work for all types of illuminants. A new algorithm is presented that allows for an efficient calculation of Logvinenko’s color descriptors for large data sets and a wide variety of illuminants. Introduction In the CIE XYZ space, colors of reflecting objects define a volume called the object-color solid [7, 2], which depends on the spectral power distribution of the illuminant. Figure 1 shows the object color solid for illuminant D65 in the CIE 1931 XYZ tristimulus space. Recently Logvinenko introduced a new objectcolor space, establishing a complete color atlas that is invariant to illumination [2], containing all colors in the object-color solid under any illuminant. However, Logvinenko’s existing implementation for calculating the proposed color descriptors is computationally expensive and does not work for all types of illuminants. Figure 1: The object-color solid for illuminant D65 in the CIE 1931 XYZ space. The points on the surface of the object-color solid are called optimal color stimuli, sometimes described as the object-color stimuli that for a given chromaticity have the greatest luminous reflectance [7]. It is generally accepted that they are generated by reflectance spectra that take values of either zero or one across the visible wavelength range, with no more than two transitions between these values, specified by transition wavelengths λ1 and λ2 [7]. There are two types of optimal reflectance functions, denoted xopt (λ ;λ1,λ2). Type I functions take a value of one for λ1 < λ < λ2, and zero everywhere else, while Type II functions take a value of zero for λ2 < λ < λ1, and one everywhere else. Note that we follow Logvinenko’s notation, where λ1 < λ2 for Type I and λ2 < λ1 for Type II reflectance functions. It is also possible to describe the optimal reflectance functions using central wavelength, λ , and spectral bandwidth δ , to define the center and width of the interval described by the transition wavelengths. The central wavelength λ and spectral bandwidth δ can be calculated from the transition wavelengths as δ = |λ1−λ2| λ = λ1+λ2 2 for Type I optimal reflectances, and for Type II as δ = (λmax −λmin)−|λ1−λ2|