Calculation of number of distinguishable colours by real normal observers

The colour-matching functions of standard observers proposed by the CIE represent the normal colour vision for the worldwide population. But there are deviations in the colourmatching functions for real observers with a normal colour vision, so the observer metamerism index was defined by CIE to evaluate the mismatch between them in colour appearance. In this work, we present an alternative form to evaluate the differences between the CIE standard observer and real observers (Stiles-Burch database) according to the number of distinguishable colours in the colour solid associated to each observer. Unlike the metamerism index defined by CIE, we evaluate globally the colour appearance for a real observer taking into account its colour gamut volume. After analyzing the results, we have seen that the gamut volume of the realobservers is lower than to that associated to the CIE standard observer, even to that associated to the CIE modified observer. Therefore, this work and its methodology could be used to know the ability of the CMF’s of different observers to get the maximum colour gamut under the same illuminant. Furthermore, this methodology could be applicable to study the gamut volume variability of the real observers regarding to age, race, etc, or even with abnormalities in colour vision (dichromacy, pathologies, etc). Introduction The colour-matching functions of standard observers proposed by the CIE (CIE 1931 and CIE 1964) represent the colour vision for the average population. But it is known that there are deviations in the colour-matching functions for real observers with a normal colour vision. For this reason, it was defined the special metamerism index, introduced to describe the mismatch observed among metameric pairs under the CIE standard observer and a standard observer deviated with normal colour vision [1]. A colour appearance model describes the colour perception from the tristimulus values and other parameters relating to the stimulus and the environment. The colour appearance models, for instance, CIELAB and CIECAM02, are based on the descriptors associated to the CIE 1931 standard observer. But an integral model should describe the colour perception for any real observer with normal colour vision. On the other hand, we know that a colour appearance model, for instance CIECAM02, allow us to define the colour solid where all distinguishable colours by the human visual system are enclosed. The colour stimuli shaping the intermediate frontiers of the colour solid, obviously with the maximum colourfulness, are called optimal colours and they were exhaustively studied by MacAdam in 1935 and it is proved that their spectral reflectance or transmittance can be only zero or one. Due to this, the colour solid borders are also known as MacAdam limits. There are two types of optimal colours: type 1, with “mountain”-like spectral profiles, and, type 2, with “valley”-like spectral profiles. Although these colours are not present in nature, they are very important for Colour Science because they constitute the frontier of the human colour solid. In a recently published work [2], a new algorithm have been developed to calculate the optimal colours associated to different illuminants and light sources taking into account the CIE 1931 standard observer. But we can modify this algorithm to obtain the optimal colours for any lightness value and for any illuminant or light source and for any observer. Therefore, the main aim of this work is obtain the colour solid under a fixed illuminant, for instance the illuminant D65, but with different real observers with normal colour vision. After this calculation, we evaluate the number of distinguishable colours according to different packing methods [2] to evaluate the deviations found among these observers and the CIE standard observer. This methodology could be useful to evaluate globally the colour appearance for different observers, that is, this would be an alternative way of evaluating the colour appearance change among observers. Data and Methods In this work, we have studied 10 real observers, observers from Stiles-Burch [3] data and the CIE 1931 modified observer, to analyze the differences among these observers and the CIE 1931 standard observer according to the number of distinguishable colours (Figure 1). These real colour-matching functions, joint to those associated to CIE system, were previously normalized by means the G94 system [4, 5]. Figure 1. Colour matching functions for different real observers and the CIE 1931 observer at the G94 system. CIE 1931 observer CIE 1931 modified observer Observer 2 Observer 4 CIE 1931 observer Observer 6 Observer 8 80 ©2008 Society for Imaging Science and Technology Firstly, we calculated the deviation functions, ( ) ( ) ( ) λ λ λ z y x Δ Δ Δ , , for all observers to know the real deviation among the different real observers as follows: ( ) ( ) ( ) 3 , 2 , 1 1931 , , = − = Δ i x x x CIE i real i i λ λ λ (1) The obtained results can be seen in the Figure 2 and we checked that the deviation functions for the real observers are bigger than the CIE 1931 modified observer. Figure 2. Deviation functions of the 10 real observers and the first deviation functions defined by the CIE (CIE 80:1989) with the CIE 1931 standard observer (black solid line). In spite of the fact all these colour-matching function are not normalised to the same illuminant, we tested if colourmatching functions associated to different real observers fulfil the Luther condition [6]. We found that any colour-matching functions fulfil the Luther condition, even the CIE 1931 modified observer. This preliminary test warns us that there will be observer metamerism, and it is possible that each optimal colour set associated to the real observers may differ from that of the CIE standard observer. Moreover, as the observer metamerism means that color-stimuli encoded equal for the standard colorimetric observer in CIE color space can be encoded different for other real observers, and vice versa, this preliminary result will warranty that we find chromatic regions without perceptual correspondence among observers. Next, we obtained the colour solid to calculate the number of distinguishable colours. The colour solid is obtained following the methodology described in our published work [2]. We need the following inputs: The visible spectrum range, (from 380 to 780 nm). The spectral sampling, N, in this case is equal to 0.1 nm. The spectral power distribution S(λ) of illuminant D65. The lightness value L*, with tolerance ΔL*, to transform it into Y(L*). The colour-matching functions associated to the different real observers and the CIE 1931 standard and modified observer. With these preliminaries, for each fixed Y(L*) value, the routine systematically locates the wavelengths λB1B and λB2B where the sudden change of reflectance or transmittance happens (from 0 to 1 or opposite).With each pair of limiting wavelengths, λ1 and λ2, and the illuminant D65 S(λ) it is very easy to generate the optimal colour stimuli Coptimal(λ) as ρoptimal(λ)*S(λ). Obviously, from here it is almost immediate to compute the XYZ tristimulus values from the colour-matching functions and to encode them into perceptual values in several colour spaces (CIELAB, DIN99d, CIECAM02, etc). Therefore, in this way, changing the colour-matching functions, we can obtain the colour solid associated to different real observer with normal colour vision and the colour solid associated to the CIE 1931 standard observer and the CIE 1931 modified observer. But the real colour-matching functions were previously normalized by means the G94 system. For this reason, we get XYZ values encoded in the G94 system. After this, we transform the XYZ94 values into XYZ values encoded in the CIE 1931 system [4, 5] to obtain the colour solid in different colour spaces. After getting the complete colour solid for each real observer and the CIE 1931 modified observer, we calculate the number of distinguishable colours with the convex hull mathematical technique and the ellipses packing method [2]. In the Figure 3 it can be seen the scheme of the methodology followed in this work: Figure 3. Scheme of the methodology followed in this work. Number of distinguishable colours λ1 λ2