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Abstract
When LEDs are used to mimic daylight, a side-by-side comparison of the chromaticity difference between the LED spectrum and natural daylight will be perceived differently by individual observers. The magnitude of this effect depends on the LED light’s spectral power distribution and can be assessed by using individual observer functions. To minimize the computational effort, an observer metamerism index can be utilized. Here, we compare three methods from the literature to define an observer metamerism index by carrying out a correlation analysis, in which reference spectra of the whole daylight range (1600 K to 88000 K) are used together with an empirical study. The recommended metric is based on a principal component analysis of 1000 individual observers’ color matching functions to define a deviate observer. Using the proposed metamerism index significantly simplifies the calculation of the observer metamerism evaluation. Thus, this metric can be applied in spectral optimization pipelines, which are embedded in smart and adaptive multi-primary LED luminaires.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As formulated in Grassmann's laws [1], every color impression can be described with three parameters. While the underlying spectral distribution is quasi-infinite dimensional, the human visual process transforms it into a three-dimensional color space. This dimensional reduction has the effect that the three color properties of a color stimulus cannot only be generated by a single spectrum but by an infinite number of possible spectral distributions. If two different spectra create the same color impression, these spectra are called a pair of metamers and the effect is called metamerism [2]. This effect is also used to visually match the mixture of three arbitrary primary spectra to different spectral colors, whereby color matching functions (CMFs) are generated. CMFs form the foundation of modern colorimetry and were firstly standardized with the CIE 1931 CMFs for a 2° field size, based on seven observers from Wright [3] and ten observers from Guild [4]. Just like the CIE 1964 CMFs for a 10° field size, which are based on 49 observers from Stiles and Burch [5] and 27 observers from Speranskaya [6], these CMFs are built upon the average of the tested observers and are known to potentially fail in the precise prediction of an individual observer’s color match [7–9]. Due to physiological differences in the retina and aging effects of the lens [10,11], the similarity of a metameric pair can be perceived differently by different observers, which is called observer metamerism [12].
Nimeroff et al. [13] addressed this issue by considering the intra- and intersubject differences in color matching functions. He developed a statistical model in which the variances and covariances could be calculated in addition to the mean of the tristimulus values. Wyszecki and Stiles [14] used the color matching functions from 20 individual observers [15] to define an index for observer metamerism by taking the mean of the individual color differences of a given metameric pair of spectra. A different approach is to define a deviate observer and was firstly proposed by Allen [16]. He used the standard deviation of the 20 observers from Stiles and Burch [15] to calculate the color difference of the standard observer and his deviate observer. Another mathematical approach to derive deviate observer functions was first published by Nayatani et al. [17]. Using single value decomposition on the 20 observer functions from Stiles and Burch, deviation functions were defined, which recreate the 20 observer functions by using linear combinations. As uniform color spaces are highly non-linear, Ohta [18] used a non-linear optimization to define a deviate observer but achieved similar functions like Nayatani.
In 1989, the CIE published with the special metamerism index a standard deviate observer [19]. For a metameric pair, based on either the CIE1931 or the CIE1964 observer, the tristimulus values are also calculated by adding the first deviation functions to the CIE1931 color matching functions. But the resulting color difference, defined as the “special metamerism index for a change in observer” [20], was reported to significantly underestimate color mismatches in metameric color pairs [21–23].
So far, the standardized color calculations and the individual deviations were treated separately. This changed with the CIE Physiological Observer 2006 (CIEPO06) [24], which takes the issue of individual differences into account by including two parameters (age, field of view) in the calculation of CMFs. Based on this, Asano et al. [9] developed an individual colorimetric observer model, which adds eight parameters to the CIEPO06. These additional parameters are lens pigment density, macular pigment density, optical densities of L-, M-, and S-cone photopigments, and λmax shifts of L-, M-, and S-cone photopigments. Using a Monte Carlo simulation, they calculated 10,000 sets of CMFs of hypothetical observers and used a modified k-medoids method for clustering and defining categorical observers [25]. While the number of required categories varies depending on the specific characteristics of the primaries (50 for a laser projector), ten categories are sufficient for most cases, with the first categorical observer being defined by an average of 10,000 observers. This individual observer model provides the basis for the proposed study on defining an observer metamerism index to simplify the calculation when LED spectra are required to mimic daylight for as many individual observers as possible at the same time.
Daylight is used for reference, as its relevance for the physiological and psychological well-being is increasingly supported by scientific evidence [26]. Also, various metrics, for example the color fidelity index Rf [27], use daylight as a reference. Yet, when it comes to color matching artificial daylight to natural daylight, no specific metric addresses observer metamerism. Asano’s categorical observer model simplifies the observer metamerism calculations by using ten representative color matching functions. However, we investigate different approaches of an observer metamerism index, to find the most efficient computational approach by using only one additional set of functions. Also, a single set of deviation functions shows which wavelength sections are the most critical for observer metamerism. For this purpose, we investigate the correlation of the different approaches of an observer metamerism index and the chromaticity differences of metameric spectra for 1000 individual observers.
In order to take into account as many daylight reference spectra as possible, spectra are generated with the use of various daylight models and by measuring with a spectrometer. For each of these reference spectra, eight different metameric spectra are generated by color mixing on eight three-channel LED systems for each of the 1000 individual xyz-observers created with Asano’s Matlab code (according to the US Census 2010 age distribution from 10 to 70 years) [25]. The observer metamerism index with the highest correlation (95.7% for 2° field of view, 98.1% for 10° field of view) is selected for the proposed metric.
In addition, an experiment was carried out in which the observers matched the chromaticity of a multi-channel LED luminaire’s light to natural daylight, aiming to validate the individual observer model and the proposed metric. The observed chromaticity differences are consistent with the prediction of both Asano's observer model and the proposed observer metamerism index.
2. Method
The comprehensive way of creating metameric spectra for multiple observers and comparing the chromaticity differences from the perspective of a reference observer is described in section 2.1.1 to allow comparison with different observer metamerism indexes later. The different approaches for defining an observer metamerism index are presented in section 2.1.2. Various sources of daylight spectra were used to find out which metamerism index achieves the best results when LED spectra are compared to daylight spectra (section 2.1.3). Each daylight reference needed to be matched with multiple LED spectra to have spectra with different magnitudes of the observer metamerism effect. Therefore, different three-channel LED constellations were composed and described in section 2.1.4.
The second part of this investigation consists of a color matching test, in a specially for this purpose constructed mobile observer room (section 2.2.1). This mobile room was used for the comparison of natural daylight to the chromaticity of LED light. The presented individual LED chromaticities were calculated by a custom programmed software which also stored the subject’s input together with spectrometer measurements (section 2.2.2).
2.1.1 Chromaticity differences of individual observer’s metamer spectra
The magnitude of the observer metamerism effect can be calculated by comparing the different individual metameric spectra. For every individual observer, the tristimulus values ${X_i},{Y_i},{Z_i}$ of the reference spectrum ${s_{ref}}(\lambda )\; $ and the LED primaries ${s_{LED,1 - 3}}(\lambda )$ were calculated, using the corresponding set of individual CMFs ${\bar{x}_i}(\lambda ),{\bar{y}_i}(\lambda ),{\bar{z}_i}(\lambda )$:
The normalizing constant K is set for illuminants to Km = 683.002 lm·W-1 for a 2° view and set to Km,10 = 683.599 lm·W-1 for 10° view [20], but is not relevant for the calculation of illuminants chromaticity differences. To separate chromaticity and intensity, xy-coordinates are calculated from the tristimulus values.
Both, the chromaticities ${x_{ref,i}},{y_{ref,i}}$ of the reference spectrum and ${x_{LED,k,i}},{y_{LED,k,i}}$ of three LED primaries, are calculated using Eq. (1)– (6). For the calculation of a metameric spectrum, the LED primaries weights ${Y_{LED,k,i}}$ have to be found, on which the mixed LED spectrum and the reference spectrum are achieving the same tristimulus values (Eq. (7)).
As we are interested in the primaries’ relative ratios, which achieve the reference’s chromaticity, we set YLED,1,i = 1. Then, YLED,2-3,i can be calculated using Eq. (8).
The functions f(D) and g(D) result with $a = x/y$ and $b = z/y$ in
A scaling factor to YLED,1-3,i could be applied, to match the radiance of the two compared spectra, but is not relevant for quantifying the observer metamerism effect. The metameric spectrum ${s_{m,i}}(\lambda )\; $ for the chosen individual observer i is given by
Metameric spectra were calculated for 1000 observers this way, including Asano’s ten categorical observers. The first categorical observer is equivalent to the CIEPO06 of a 38-year-old [25] and was chosen as a reference observer here. The resulting xy coordinates of all these spectra are calculated by Eq. (1)–(6) and transformed into the CIE 1976 uniform chromaticity scale (UCS) [20].
The chromaticity difference is then calculated by
Figure 1 shows an exemplary result of this process. A blackbody spectrum of the temperature 6500 K was used for reference. For a LED system consisting of narrowband R-G-B (red, green, blue) spectra, the variance of the chromaticities from individual metameric spectra, with a mean distance of ${\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime},10}} = 9.35 \times {10^{ - 3}}\; $ to the reference, is significantly higher than for a CW-WW-G (cold white, warm white, green) system. With the broad spectra of phosphor-converted LEDs, a smaller mean of ${\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime},10}} = 1.50 \times {10^{ - 3}}$ was achieved. In the latter case, a group of individual observers would hardly notice any difference, when comparing the LED light to the reference. But using the R-G-B LEDs, many observers would perceive a reddish or greenish-white, while only fewer observers would perceive it as a real match.
Fig. 1. Metameric spectra for 1000 individual observers, including ten categorical observers, for a 6500 K blackbody reference spectrum. When the reference spectrum is matched for all individual observers with R-G-B spectra (a), the mean deviation for the first categorical observer is ${\Delta _{u^{\prime}v^{\prime},10}} = 9.35 \times {10^{ - 3}}$ (c). When using CW-WW-G spectra (b), the mean deviation is ${\Delta _{u^{\prime}v^{\prime},10}} = 1.50 \times {10^{ - 3}}$ (d). The index 10 denotes the 10° field size.
2.1.2 Concept for an observer metamerism index with daylight as reference
The many different observer functions, which need to be taken into account for the previously shown method, make the computational approach relatively effortful. Therefore, various methods from the literature were used and modified to generate an observer metamerism index to allow a simplified calculation of the observer metamerism’s magnitude. Using 1000 individual xyz-CMFs for each field size (2° and 10°) created with Asano’s model [9], as well as his first categorical observer [25] as a reference (cf. Table S1), three different approaches of calculating an observer metamerism index were investigated:
- • The Nimeroff approach: According to Nimeroff’s work [13,28–30], we used the variances $\sigma _X^2(\lambda ),\sigma _Y^2(\lambda ),\sigma _Z^2(\lambda )$ of the 1000 xyz-CMFs (cf. Figure 2 (a),(d) and Table S2) to calculate the between observer covariance matrix ${\mathrm{\Sigma }_{u^{\prime}v^{\prime}}}$ for a given spectral distribution $s(\lambda )$ (assuming the within-observer variance to be the same for every observer):
We calculated these variances and covariances using the colorimetric error propagation theory [28,29]. The tristimulus values in Eq. (16–18) were calculated using the reference observer.
\begin{align}\sigma _{u^{\prime}}^2(\lambda )&= {({60Y + 12Z} )^2}/{({X + 15Y + 3Z} )^4}\mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _X^2(\lambda )d\lambda + \notag \\ &\quad {({ - 60X} )^2}/{({X + 15Y + 3Z} )^4}\mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _Y^2(\lambda )d\lambda +\notag\\ &\quad {({ - 12X} )^2}/{({X + 15Y + 3Z} )^4} \mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _Z^2(\lambda )d\lambda \end{align}\begin{align}\sigma _{v^{\prime}}^2(\lambda )& ={({ - 9Y} )^2}/{({X + 15Y + 3Z} )^4}\mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _X^2(\lambda )d\lambda +\notag\\ &\quad {({9X + 27Z} )^2}/{({X + 15Y + 3Z} )^4}\mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _Y^2(\lambda )d\lambda +\notag \\ &\quad {({ - 27Y} )^2}/{({X + 15Y + 3Z} )^4} \mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _Z^2(\lambda )d\lambda \end{align}\begin{align}{\sigma _{u^{\prime}v^{\prime}}}(\lambda ) &=({60Y + 12Z} )({ - 9Y} )/{({X + 15Y + 3Z} )^4}\mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _X^2(\lambda )d\lambda + \notag \\ &\quad ({ - 60X)(9X + 27Z} )/{({X + 15Y + 3Z} )^4}\mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _Y^2(\lambda )d\lambda +\notag \\ &\quad ({ - 12X} )({ - 27Y} )/{({X + 15Y + 3Z} )^4} \mathop \smallint \nolimits_{380\; nm}^{780\; nm} {s^2}(\lambda )\sigma _Z^2(\lambda )d\lambda \end{align}The covariance matrix ${\mathrm{\Sigma }_{u^{\prime}v^{\prime}}}$ can be used for calculating uncertainty ellipses, but we are looking for a metric consisting of a single scalar value to describe the observer metamerism. Thus, we calculated the ellipse’s area Aellipse depending on its two radii r1 and r2, similar to Long and Fairchild [31], who use the volume of CIELAB ellipsoids to define an observer metamerism index.
The two radii for a confidence value of 95% (using the chi-squared distribution) are defined with the two eigenvalues E1 and E2 of ${\mathrm{\Sigma }_{u^{\prime}v^{\prime}}}$ by
The confidence value scaling does no effect the investigated correlation but is given for the interested reader. We define the investigated metamerism index MNi (Ni denoting the Nimeroff approach) by the area resulting of the spectral difference $\Delta s(\lambda )= {s_1}(\lambda )- {s_2}(\lambda )$. Both spectra have to be normalized to $Y = 100$ before the difference is formed.
(The tabulated variances $\sigma _X^2(\lambda ),\sigma _Y^2(\lambda ),\sigma _Z^2(\lambda )$ can be found in supp. Table S2.)
- • The Allen approach: To define a standard deviate observer Allen [16] used the standard deviation of 20 individual CMFs, which then can be used with a standard observer in color difference formulas to calculate an observer metamerism index. The traditional approach of calculating the standard deviation, using the square root of the variance, causes the loss of the spectral deviations’ sign. But as the individual observer’s CMFs deviate with different signs in different spectral ranges, which are correlated (e.g. a positive deviation around 420 nm correlates with a negative deviation around 470 nm and vice versa, c.f. Figure 2(b), (e)), this means the loss of relevant information. To find a sign for the standard deviation at each wavelength, Allen investigated the interwavelength correlations as well as the interstimulus correlation coefficients. The resulting sign patterns were used to sign the standard deviations at different wavelength sections. However, this approach was not suitable when using the simulated 1000 observers, as the resulting sign patterns weren’t as unambiguous as the ones Allen found. Instead, we used the sign of an individual ${\bar{z}_i}(\lambda )$ function’s difference to the mean function ${\bar{z}_{mean}}(\lambda )$ at $\lambda = 420\; nm$. All spectral differences of an individual observer’s CMFs to the mean observer were multiplied with his previously found sign. By averaging the so found deviations, the deviation functions ${\mathrm{\Delta }_A}\bar{x}(\lambda ),{\mathrm{\Delta }_A}\bar{y}(\lambda ),{\mathrm{\Delta }_A}\bar{z}(\lambda )$ were defined (A denoting the Allen approach) (cf. Figure 2 (b),(e)). Similar to the CIE observer metamerism index [19], we define this approach’s deviate observer functions ${\bar{x}_{devA}}(\lambda ),{\bar{y}_{devA}}(\lambda ),{\bar{z}_{devA}}(\lambda )$ by adding the deviation functions to the reference observer $\bar{x}(\lambda ),\bar{y}(\lambda ),\bar{z}(\lambda )$ (Asano’s first categorical observer).
Fig. 2. Deviation and variance functions for a 2° field size (a)-(c), and 10° field size (d)-(f), calculated using 1000 individual CMFs. (a),(d): Variance functions $\sigma _X^2(\lambda ),\sigma _Y^2(\lambda ),\sigma _Z^2(\lambda )$ for the Nimeroff approach. (b),(e): Signed standard deviation functions ${\Delta _A}\bar{x}(\lambda ),{\Delta _A}\bar{y}(\lambda ),{\Delta _A}\bar{z}(\lambda )$ for the Allen approach. (c),(f): Deviation functions ${\Delta _N}\bar{x}(\lambda ),{\Delta _N}\bar{y}(\lambda ),{\Delta _N}\bar{z}(\lambda )$ built by the first component of a principal component analysis for the Nayatani approach. All functions range from 390 nm to 780 nm with a 5 nm interval.
The deviate observer’s tristimulus values ${X_{devA}}$, ${Y_{devA}},{Z_{devA}}\; $ are calculated using Eq. (1–3) for the two spectra of a metameric pair (index 1 and 2). The resulting chromaticity difference ${\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime}}}$ is calculated using Eq. (12–14). We define this observer metamerism index MA to be the ${\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime}}}$ chromaticity difference between the two metameric spectra ${s_1}(\lambda )$ and ${s_2}(\lambda )$ perceived by the deviate observer.
- • The Nayatani approach: Nayatani et al. [32] proposed a standard deviate observer by using singular value decomposition on 20 CMFs. With the first three components, they could reconstitute the standard observer. Their deviate observer was derived from the first component. We used this approach by performing a principal component analysis (Mathworks MATLAB 2020b), based on a singular value decomposition, on the 1000 individual CMFs. To take the correlation of the ${\bar{x}_i}(\lambda ),{\bar{y}_i}(\lambda )$ and ${\bar{z}_i}(\lambda )$ functions into account, which are given for 79 wavelengths (390 nm to 780 nm, 5 nm interval), each observer’s CMFs were serialized into a single vector (length of 3 × 79). The so found first deviation function explains 69.30% (2° field size) and 76.52% (10° field size) of the variations. The deserialization of the first deviation function results in the deviation functions ${\mathrm{\Delta }_N}\bar{x}(\lambda ),{\mathrm{\Delta }_N}\bar{y}(\lambda ),{\mathrm{\Delta }_N}\bar{z}(\lambda )$ (N denoting the Nayatani approach) (cf. Figure 2(c),(f)). These are used similarly to the Allen approach to define a deviate observer ${\bar{x}_{devN}}(\lambda ),{\bar{y}_{devN}}(\lambda ),{\bar{z}_{devN}}(\lambda )$, his tristimulus values ${X_{devN}}$, $ {Y_{devN}},{Z_{devN}}$ (Eq. (1–3)) and an observer metamerism index MN.
Using these three approaches for an observer metamerism index, a correlation analysis was carried out, in which the correlation of these indexes to the average and maximum chromaticity difference of the 1000 observer’s metameric spectra was investigated. This correlation analysis was based on multiple daylight spectra for reference.
2.1.3 Daylight reference spectra
Every investigated metameric pair consists of one daylight spectrum and one LED spectrum. But, as daylight comes in many different spectral shapes, a range of daylight spectra is needed, for a representative investigation. To gather as many different daylight spectra as possible, we used simulated spectra from different daylight models together with spectrometer measurements, which we carried out. While there are rather complex daylight models, which can accurately simulate radiation spectra over a wide range of wavelengths [33–35], we focused on simplified daylight models for the visible wavelength range. The utilized reference spectra were gathered from the following four sources and filtered, for every source, to have a minimum chromaticity distance of ${\Delta _{u^{\prime},v^{\prime},10}} = 1 \times {10^{ - 3}}$ (based on CIE 1964 CMFs) to each other.
Blackbody radiation [36]: With the sun being a temperature emitter, Planck’s blackbody radiation model can be considered a simple daylight model. The modeled spectra can be considered as a rough approximation of daylight and are used here as a reference due to their relevance in lighting technology. We generated 454 spectra with color temperatures from 300 K to 88000 K, as below and above these limits no significant change in chromaticity occurs.
CIE daylight model: Judd et al. proposed a model [37] which was adapted by the CIE [20]. Based on 622 daylight samples with 10 nm intervals, it defines a daylight curve in CIE 1931 chromaticity coordinates. With these coordinates, scaling factors for three vectors are calculated, which form the predicted spectrum of daylight for a given chromaticity. It is applicable from 4000 K to 25000 K and was used to generate 102 spectra.
Daylight model from Tian et al. [38]: This model predicts three different spectra: direct, diffuse and global daylight. The latter is the mixture of the previous two. The degree of cloudiness and the zenith angle of the sun are taken into account as parameters. Thus, the filtering of an extraterrestrial solar spectrum in the wavelength range from 400 to 700 nm is calculated. Various measurements and publications exist on the spectral distribution of the extraterrestrial solar spectrum [39], but since only the visual range is of interest, no further differentiation was made and Wehrli’s solar spectrum [40] was used. We used this model to generate 900 spectra.
Spectrometer measurements: The subject tests for color matching LED and daylight required spectrometer measurements of the incident daylight (cf. section 2.2), from which we found 112 spectra with significant differences.
The UCS chromaticities of these 1568 reference spectra are shown in Fig. 3.
Fig. 3. (a) CIE 1964 chromaticities of the used daylight references in the CIE 1976 UCS. (b) The chromaticities were filtered to have a minimum distance of ${\Delta _{u^{\prime},v^{\prime},10}} = 1 \times {10^{ - 3}}\; $ to each other (per reference set).
2.1.4 LED system for the generation of metameric spectra
The second part of each metameric pair consists of a LED spectrum. As we needed to create pairs with different magnitudes of observer metamerism errors while using the same daylight reference spectrum, we used the seven LED types installed in the luminaire of the experimental setup (cf. Figure 4). These allow eight different constellations for three-channel LED systems (cf. Table 1) to generate LED metamer spectra. The seven LED types were selected to fill the visible spectrum in different spectral ranges; therefore, multiple metameric spectra per daylight reference and observer could be created with various amounts of observer metamerism.
Fig. 4. The LED spectra of a seven-channel luminaire (a) were used to define eight different three-channel gamuts (b) to simulate metameric spectra. The same luminaire was used in the mobile test room (e), which is drafted for side (c) and top view (d).
Table 1. LED combinations for simulating metameric spectra.
Based on these data, a simulation was carried out in which eight different metamerism spectra were calculated for each of the 1000 individual observers for every daylight reference spectrum. With all these pairs of metamers, the correlation of the metamerism indexes to the average and maximum chromaticity difference ${\Delta _{u^{\prime},v^{\prime}}}$ was analyzed. To compare the correlation of the observer metamerism effect to the color rendering quality of the LED spectra, the CIE 2017 color fidelity index Rf [27] was examined.
2.2.1 Mobile test room for spectral tuning and analyzing LED light to natural daylight
In addition to the simulation of color matches from LED to daylight, a mobile observation room was set up to perform a color matching test between LED light and natural daylight (cf. Figure 4 (c)-(e)). The test persons sat in front of two chambers with similar dimensions. The field size was between 34° and 50° depending on the exact head position. The right chamber was illuminated with daylight and dimmed by moving one grid lying on top of another grid to prevent exceeding the illumination level of the LED site. A diffuser plate was used to make the daylight homogeneous. The left chamber was illuminated using a temperature-stabilized, multi-channel LED luminaire. Seven channels were available (cf. Figure 4 (a)), but only three channels G, CW, WW were used for the experiments.
Each box was equipped with a spectrometer (CS2000, Konica Minolta), which measured the spectra on the bottom of the respective observation box through a hole behind the wall. The spectrometers were controlled by a custom software developed for this purpose, which controlled the test procedure automatically.
2.2.2 Automated test system
This software took care of calculating and presenting different LED spectra for the test person to compare with daylight and storing the spectrometer measurements and the test person’s input. Auditory feedback was given with a speaker every time input from the subject was received and when a new LED setting was presented. The test person had to indicate whether the two current light settings match or do not match by pressing one of two buttons on the controller. In the first trial of a test, the software calculated 24 chromaticity coordinates, which were placed on three circles (radii $2;4;6 \times {10^{ - 3}}\,{\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime},10}}$) around the daylight’s current CIE 1964 chromaticity and at the center. The first trial aimed to determine the approximate chromaticity coordinates of the individual color match. After these different light settings were presented in a quick pass-through (800 ms per setting), the test started, and the subject had to decide whether the setting is “far to a match” (negative answer) or “close to a match” (positive answer). By pressing the corresponding button on the controller, the test person navigated forward in the test procedure. After all chromaticity coordinates were evaluated, the second trial began.
Therefore, the software calculated the center of gravity (COG) from the subject's positive answers, in the UCS color diagram. Around this COG, 24 new chromaticity coordinates were calculated (on circles with radii $1;2;3 \times {10^{ - 3}}\; {\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime},10}}$ around the COG) and presented, together with the center chromaticity. This time, the subject was asked to only give a positive answer when there is no chromaticity difference perceivable. After these settings were evaluated five times in random order, the test was finished.
By triggering the spectrometer measurement (which takes a few seconds) only for positive responses, the test of a single subject was referred to a constant daylight reference as far as possible. Since all settings and measurement results were automatically logged in a database, the target coordinates of the LED luminaire were still available for the negative responses. In addition, a timer function was used to ensure that daylight was measured at least once every 10 seconds to detect potential chromaticity shifts in daylight. If a change of greater than $1 \times {10^{ - 3}}$ ${\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime},10}}$ was detected, all LED target chromaticity coordinates were shifted with the same UCS vector.
2.2.3 Color matching tests
The previously described test was accomplished by 59 persons. The setup was placed outside so that no reflected light from the walls of the surrounding buildings could fall into the daylight entry of the chamber. Each subject was tested for color vision deficiency using an Ishihara test and adapted for three minutes inside the box before the matching test started.
3. Results
3.1. Metamerism index
The correlation of the three approaches of an observer metamerism index with the average and maximum chromaticity difference of 1000 individual observers’ metameric spectra is given in Fig. 5 (a) and (b). As the Nayatani approach achieves the highest correlation, with a value of 0.957 when compared to the average chromaticity error at a 2° field size and a correlation of 0.981 for a 10° field size, this is the best-suited method for an observer metamerism index compared to the other two methods (for the proposed use case). Therefore, the recommended way for calculating an observer metamerism index is to apply the Nayatani approach with the newly proposed deviate observer functions in Fig. 5 (c) and (d) (cf. supp. Table S6). Using Eq. (29) with a pair of spectra, which are metameric for the reference observer (Asano’s first categorical observer, cf. supp. Table S1), gives the observer metamerism index MN. The correlation of MN to the average chromaticity difference of 1000 individual observers’ metameric spectra can be fitted with a linear regression (cf. Figure 6 (a), (b)). Thus, the average chromaticity difference can be calculated with
For a 2° field size the coefficients are ${p_1} = 0.4416$, ${p_2} = 1.875 \times {10^{ - 3}}$ with a coefficient of determination ${R^2} = 0.9153$. When using a 10° field size, these are ${p_1} = 0.4610$, ${p_2} = 9.119 \times {10^{ - 4}}$ and ${R^2} = 0.9621$.
Fig. 5. (a), (b): Correlation of three different approaches for an observer metamerism index and the chromaticity difference of 1000 individual observer’s metamers. Asano’s first categorical observer is used as reference observer. Thus, the metamerism index values were computed using this observer’s metameric spectra, and the chromaticity differences of the 1000 individual observer’s metamers are from his perspective. (c),(d): Recommended deviate observer functions ${\bar{x}_{devN}}(\lambda ),{\bar{y}_{devN}}(\lambda ),{\bar{z}_{devN}}(\lambda )$, built with the Nayatani approach. Using the first component of a principal component analyses on 1000 individual CMFs, deviation functions were defined and added to Asano’s first categorical observer functions $\bar{x}(\lambda ),\bar{y}(\lambda ),\bar{z}(\lambda )$ (cf. supp. Table S1) to create this deviate observer functions (cf. supp. Table S6).
Fig. 6. (a), (b): Correlation of the proposed observer metamerism index MN and the average chromaticity difference ${\Delta _{u^{\prime}v^{\prime}}}$ of 1000 individual observer’s metameric spectra. The correlation is fitted (dashed red line) with a coefficient of determination ${R^2} = 0.9153$ (a) and ${R^2} = 0.9621$ (b). Every plotted circle corresponds to a pair of metameric spectra, whose CCT is color coded from 1700K to 50,000 K. (c): Correlation of the color fidelity index Rf and the average chromaticity difference of ten categorical observers’ metamer spectra, perceived by the first categorical observer.
To investigate a possible correlation of the observer metamerism effect with color rendering quality, Fig. 6 (c) compares the color fidelity index Rf of the LED metamers with the magnitude of the observer metamerism effect. This is measured by the average chromaticity distance of the ten categorical observers’ metamer spectra, for a 10° field size.
3.2 Results of color matching tests
The color matching test was performed by 59 persons. Due to the dynamic daylight behavior, all subjects compared daylight of different CCTs. After evaluation, the tests were divided into different categories. In the following, those ten tests are used where no significant daylight shift occurred (< $1 \times {10^{ - 3}}$ ${\Delta _{\textrm{u}^{\prime}\textrm{v}^{\prime},10}}$) and where a maximum of probability for a successful match could be calculated. The subject’s average age was 29.4 years with a standard deviation of 3.98, supp. Table S7 shows their individual age and gender. As each LED setting was compared five times, from the ratio of positive to negative answers, each LED chromaticity coordinate could be given a probability of matching the current daylight. Figure 7 shows the result of the subjects along with the prediction of Asano's observers and the result of the proposed metamerism index MNayatani. The center of gravity (COG) from each subject’s color match is within the range of individual observers.
Fig. 7. Color matching results of ten (out of 59) tested observers at different CCTs of natural daylight (5746 K to 11770K), with one tested observer (red cross) per subfigure (a)-(j). The chromaticity coordinates of the natural daylight, which the test person compared, is located at the center of the red and black circles. These circles’ radii are set to the simulated average chromaticity difference of the compared metameric spectra, using 1000 individual observers (black) and the proposed metamerism index (red). The simulated chromaticities of the metameric LED spectra are additionally plotted for the 1000 individual observers (blue circle) and ten categorical observers (grey filled circle). For reference, CCT lines are shown with a 1000 K interval.
4. Discussion
The Nayatani approach achieved the highest correlation and is selected for the proposed observer metamerism index for artificial daylight. This method’s deviation functions are similar to the Allen approach (cf. Figure 2), which also achieves a high correlation (cf. Figure 5 (a), (b)). The Nimeroff approach achieves a significantly lower correlation, which might be based on the loss of the sign when using the variance (cf. Figure 2 (a), (d)).
All deviation functions have their maximum sensitivity in the blue wavelength region. Thus, the short wavelength region is more critical when it comes to universally valid metamerism pairs. This can also be seen in the CCT dependence of Fig. 6 (a) and (b), as high CCTs (e.g. blue sky spectra) lead to a large metamerism error. Also, spectra optimized for non-visual effects, which emphasize the maximum of the melanopic action spectrum around 490 nm [41,42], can produce a large observer metamerism effect if the 420 nm range is not enhanced for compensation.
The high correlation of the Nayatani approach, especially for a 10° field size, allows a linear fit with a high coefficient of determination (cf. Figure 6 (a), (b)). With this approach’s usage of only one set of deviation functions, it can be assumed that the metamerism index is suited for implementations in smart multi-channel LED systems where real-time optimizations are used for adaptive lighting [42].
The compared method of calculating the observer metamerism magnitude is based on the chromaticity differences of the individual observer’s metamers and the reference spectrum. While this simulation of individual metameric spectra is a relatively effortful in computation, it allows the analysis of the individual differences in one reference color space. The other possibility would have been the simulation of only the reference observer’s metamer and the individual observer’s chromaticity differences in their individual color spaces, which leads to similar results when comparing the average differences. We chose the prior approach for being able to investigate the individual differences in one common color space, whereby a comparison to the distribution of our own color matching tests were possible.
Optimization in literature often refers to color rendering [43,44] by maximizing the CIE color rendering index Ra [45] or the CIE color fidelity index Rf [27]. But as shown in Fig. 6 (c), the color fidelity index does not provide any reliable conclusions about the resulting observer metamerism effect of a spectrum. The exceptions are LED spectra which achieve maximal color fidelity values > 95, as they have a minimal observer metamerism effect. Yet, a slight decrease of the Rf correlates with an increased variation of the metamerism effect and thus shows that an additional metric such as the here proposed metamerism index is necessary.
Apart from quantifying the metamerism effect, which is based on comparing a probe spectrum to a reference spectrum, the deviation functions allow the investigation of the variation in the visual stimulation a single spectrum causes. This variation also affects the non-visual path, leading to interobserver differences in pupil diameter [46] and the circadian effect [47]. This variation in pupil diameter was already found to be dependent on the spectral power distribution [46,48].
With the presented test setup for color matching LED and daylight, we found the predictions of Asano’s individual observers to be consistent with our measurements (cf. Figure 7). The distribution of the respective metamerism spectra’s chromaticities, for 1000 individual observers, is mostly oriented along the isothermal lines perpendicular to the Planckian locus. Due to this shape, individual observers are more likely to perceive a greenish or reddish amount than a different CCT if LED spectra are not optimized for observer metamerism. As the proposed metric results in an average chromaticity distance, circles with the respective radii are compared to the metamers’ chromaticities’ distribution and their average distance. It can be seen that the circles’ radii are a good estimate of the variation along the CCT-curve.
5. Conclusion
This work describes and compares different calculation methods for quantifying the observer metamerism effect. While individual observer’s color matching functions enable a precise calculation of individual metameric spectra and their chromaticity differences for a reference observer, their computational effort is relatively high. Categorical observers allow a simplification. For example, display or light settings can be adapted to a user’s observer category to minimize chromaticity errors. Here, we aimed to find one general light setting for all possible users to minimize perceived chromaticity differences compared to natural daylight, as LED systems with more than three channels have to choose one from many possible spectra for a certain chromaticity. For this, we showed that our proposed observer metamerism index is a suitable simplification for minimizing the computational effort even further. Thus, smart lighting systems, which adapt in real-time to natural daylight [42] and use spectral modelling techniques [49], can apply the proposed metric to find an optimum spectrum in terms of perceived chromaticity errors. We exclusively used daylight spectra for reference, which are mostly closer to an isoenergetic spectrum (in the visible spectral range) than LED spectra. Future investigations could analyze the correlation of the proposed metamerism index for different types of reference spectra. Observer metamerism is also a challenge for wide color gamut and high dynamic range displays [31,50–52], with especially for this purpose developed metamerism indexes [31,53]. The newly developed metamerism index in this work could also be used and extended for the color management of displays consisting of more than three subpixels to find a fast and for observer metamerism optimized image transformation.
Funding
Deutsche Forschungsgemeinschaft (450636577).
Acknowledgments
We thank the German Research Foundation (DFG) by partly funding the research (Grant Number 450636577). We also thank Katherine Chatziannidou for proofreading this article.
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
The deviation functions presented in this paper are available in Table S1. These functions were built using CMFs from Asano’s observer functions database [25]. An implementation of the proposed method is available online at https://github.com/AEissfeldt/metamerism-index/.
Supplemental document
See Supplement 1 for supporting content.
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