1. Introduction

A series of objects that have the same color coordinates under a specified light source are said to be metameric—that is, they have the same color coordinates with respect to the given illuminant and observer, but different spectral reflectance functions (SRFs). If the illuminant is changed, a mismatch may occur; this is known as an illuminant-induced metameric mismatch [1]. While other forms of metameric mismatch are possible, such as observer-induced [2] or geometry-induced [3], this manuscript is limited to illuminant-induced metameric mismatch, which is sometimes referred to in this article as metameric mismatch for brevity.

Illuminant-induced metameric mismatch is one method by which an illuminant alters the chromaticity and color appearance of objects. It is distinct from and supplemental to other aspects (measures) of color rendition such as color fidelity (e.g., IES R_f, IES R_f,hj, CIE R_f) [4–7], gamut area (e.g., GAI, CQS Q_g, IES R_g) [7,8–15], local chroma shift (e.g., IES R_cs,hj) [7,16], memory colors (e.g., R_m) [17,18], and color discrimination (e.g., R_d) [19–21]. Illuminant-induced metameric mismatch is of practical importance because it commonly occurs in applications such as color reproduction, color/dye matching, and diagnostic testing involving color comparisons. For example, a urinalysis test requires matching the color of a test strip to a color rubric printed on a label, but the light source under which the visual evaluation should be performed is unknown to the lighting designer, hospital owner, and healthcare provider (tester). Absent the ability to identify the intended light source for the visual evaluation, the best chance of accurate diagnosis is with a light source predicted to minimize the likelihood of metameric mismatch for unknown SRFs. An index of illuminant-induced metameric mismatch, ascribed to a light source, could therefore be informative for situations where minimizing metameric mismatches is relevant and the specific SRFs of the objects under consideration are unknown.

Though no institutional authority or standards organization has endorsed a measure of illuminant-induced metameric mismatch, there are two research proposals on offer: The Metameric Uncertainty Index (R_t) [22] and the Metamer Mismatching Color Rendering Index (MMCRI) [23]. The goal of this work was to compare the characterization of metameric mismatch of these two measures. After comparing the two computational procedures, the respective characterizations of a set of 100,000 simulated spectral power distributions (SPDs) are analyzed to identify ways in which the performance of R_t and MMCRI differ.

2. Background

This section provides a description of the computation of R_t and MMCRI. The process for computing each measure is summarized in Fig. 1. A tabular comparison of the characteristics of the two measures is provided in Table 1.

 

Fig. 1. Flow diagrams describing the computation of R_t (left) and MMCRI (right). Note that determining MMCRI does not require the computation of CCT; the computation of CCT is only necessary if using a reference illuminant at the same CCT as the test illuminant, as was done in this analysis.

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Table 1. A tabular comparison of R_t and MMCRI

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2.1 R_t

For R_t, an actual color shift—i.e., the difference in color coordinates between a test and a reference source—is conceptualized as having two constituent parts: the base color shift, which measures the smooth hue-based pattern in color space, and the metameric color shift, which captures the uncertainty (or “noise”) with respect to that smooth pattern. The base shift is a function of position in color space and the uncertainty is modeled by the specific SRFs. For a set of color samples that are metameric with respect to a given illuminant, but exhibit color difference under a second illuminant, the base color shift can be imagined as the central tendency of those color shifts, and the metameric shift the variability with respect to that central tendency.

R_t uses key aspects of the ANSI/IES TM-30-20 computational framework [7]—namely the CAM02-UCS color space and embedded chromatic adaptation transform [24,25], the 99 Color Evaluation Samples (CES) [26], and the defined reference illuminant scheme. Conceptually, R_t exists in parallel with the other measures that use the TM-30 calculation framework. Mathematically, R_t is derived from a series of matrix operations from the color difference data for the 99 CES produced by the TM-30 calculation; see David et al. [22] for details.

The base color shift in R_t is determined by fitting a 12-parameter second-order polynomial vector field (VF) model to the actual color shifts of the 99 CES between the test and reference source. The VF model consists of two equations, one for Δa’ and one for Δb’, each of which has six coefficients. The VF is computed by determining the coefficients that minimize the sum of square error of the model fit [22]. The coefficients of the VF model are unique to the test source. The sample-specific metameric shift is captured by the difference between the actual color shift and the color shift predicted by the VF model. The metameric color shift for all samples (σ_VF), is computed as the average of the differences for all 99 CES, as shown in Eq. (1):

a′_i and b′_i are the actual coordinates of the i^th sample under the test SPD;

a′_mi, b′_mi are the coordinates of the i^th sample under the test SPD according to the vector field model.

Increasing the variability of the actual color shifts with respect to the smoothly varying VF model—i.e., increasing the magnitude of the metameric color shift (σ_VF)—decreases R_t according to Eq. (2):

Equation (2) is identical in form to the equation used to compute IES TM-30-20 R_f, including the scaling factor k, which is equal to 6.73. Because R_t and R_f use the same scaling factor, and because the metameric color shift is always a fraction of the total color shift (e.g., σ_VF < average ΔE_jab for the 99 TM-30 CES), R_t will always be greater than R_f.

R_t has a theoretical range between 0 and 100. An R_t of 100 is achieved by the TM-30 reference illuminants that are defined as causing no metameric mismatch. Values decreasing from 100 indicate an increasing magnitude of the metameric shift and, therefore, increasing likelihood of the potential for metameric mismatches. Typical light sources used in architectural interiors have R_t values greater than 85, with most phosphor-converted LEDs above 92 [27].

2.2 MMCRI

The color coordinates of a set of objects that are metameric with respect to a given light source will “explode” into a cloud of points when illuminated by a different illuminant, forming a non-concave body (i.e., a line formed by any two points on the boundary of the body falls entirely within the body) [1]. For the MMCRI calculation, the boundary of the body is termed the metameric mismatch volume (MMV) [23,28]. The volume of the MMV is taken to be an estimate of the degree of metameric mismatch induced by a light source relative to a reference source; the larger the MMV’s volume, the larger the metameric mismatch.

The MMV estimates the largest possible range of object colors that can be observed under a test illuminant for objects that were metameric with the respect to the reference illuminant. To compute the MMV, n reflectance functions are generated that are metameric to a 50% neutral grey reflector—i.e., ρ = 0.5 across the visible spectrum—with respect to a given reference illuminant. The reflectance functions are imaginary with a simple rectangular shape having only values of 0 (perfect absorptance) or 1 (perfect reflectance) and transitioning between those values a specified number of times throughout the visible range, as shown in Fig. 2. Generally, optimal reflectance functions have the “maximum possible excitation purity for a given luminous reflectance and given dominant (or complimentary) wavelength” [1]. In other words, optimal reflectance functions form the boundary of the Object Color Solid (OCS) for a given illuminant (the OCS is the shape in tristimulus space formed by all possible reflectance functions under the specified illuminant). The optimal reflectance functions used in the MMV calculation do not plot to the boundary of the test illuminant’s OCS—they cannot because they are first generated to be metameric with respect to the reference illuminant—but are optimal in the sense that they will approximately plot to the boundary of the MMV when illuminated by the test illuminant.

 

Fig. 2. Spectral reflectance functions of the square type used in the computation of MMCRI. Two separate functions are shown: one with a solid black line and another with a grey dashed line. Each has five transition wavelengths, where the reflectance function transitions between 0 (0% reflectance) and 1 (100% reflectance).

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The reflectance functions used in the computation of MMV have five distinct wavelength transitions. The MMV is generated in the tristimulus space determined by the chosen set of sensitivity functions (e.g., XYZ for the CIE 1931 2° CMFs) and is independent of any linear transformation of the original tristimulus space (e.g., from XYZ to LMS) [28,29]. Overall, the MMV is a conceptual framework without a standardized reference illuminant scheme, set of color matching functions, or sample set. For example, the reference illuminant may be a standardized illuminant, such as those from the CIE or IES, or an arbitrary illuminant. The color sensitivity functions can be CIE standard colorimetric observers, computer vision functions, or anything else. The sample set may contain 100,000, 1,000, 100, 10, or some other number of samples.

Mirzaei and Funt [23] indicate that the 50% neutral gray reflector was chosen as the location of the metamer set because it “has the largest MMV of any other combination of chromaticity coordinates.” Funt et al. [29] justify this decision by demonstrating that the MMVs for all other colors are approximately proportional to that of the neutral gray reflectance and so “provide no additional information.”

The Metameric Mismatch Volume Index (MMVI) is computed as the ratio of the MMV for the test-reference illuminant pair to the volume of the OCS for the test illuminant (Eq. (3)), producing a value between 0 (no change in the metameric nature of the generated metameric samples) and 1 (a spread of the metameric color samples so large that it fills the entire volume of the OCS of the test illuminant). The larger the MMVI, the larger the percentage of the test illuminant’s OCS volume that is filled by the color samples of the metamer set, and the worse the presumed metameric mismatch.

Note that the MMVI of Mirzaei and Funt [29] is equivalent to the metameric mismatch index, i_mm (MMI) of Logvinenko and others [28]. The term MMVI will be used henceforth for consistency with the terminology used in Mirzaei and Funt [29] where MMCRI is defined. Also note that the OCS of the test illuminant is determined with a set of (m) optimal reflectance functions distinct from the metameric reflectance functions used to determine the MMV, but still generated using optimal reflectance functions with five distinct wavelength transitions.

MMCRI is computed from the MMVI according to Eq. (4):

Mirzaei and Funt [29] indicate that the cubed root was chosen because it should be more intuitive because MMVI is based on volumes. MMCRI has a theoretical range between 0 and 100. An MMCRI of 100 is achieved by the designated reference illuminant which, by definition, does not cause metameric mismatch since the metamers are generated with respect to the reference illuminant. Values decreasing from 100 indicate an increasing volume of the MMV (and a larger MMVI), and therefore a larger magnitude of metameric mismatch. When MMCRI is calculated with the CMFs and reference illuminants of IES TM-30-20, typical light sources used in architectural interiors have MMCRI values as low as 30 (fluorescent), with most phosphor-converted LEDs above 70.

3. Methodology

3.1 Spectral power distributions

To compare the characterization of SPDs by R_t and MMCRI, a set of simulated spectra with a range of narrow to broadband emission was compiled. The Narrow, Medium, and Wide simulated SPDs from Royer and Whitehead [27] were collected. These SPDs have spectral widths from 2 nm to 11 nm (Narrow), 20 nm to 51 nm (Medium), and 50 nm to 101 nm (Wide), and a varying number of primaries (three, four, five, six, or seven) within each category. They represent a wide range of spectral variability but not necessarily the full range of possible conditions. They were generated using Gaussian distributions using a random number generator to vary the peak wavelength between 400 nm and 700 nm, the spectral width (full-width at half-maximum [FWHM]), and maximum intensity. The SPDs were generated over the spectral range of 380 nm to 780 nm. The SPDs cover the ANSI/NEMA C78.377-2017 nominal CCT quadrangles from 2700 K to 6500 K and cover a distance from the blackbody locus (D_uv) from 0.006 to -0.018, the approximate boundaries for the Basic and Extended quadrangles [30]. To supplement these data sets, a fourth set of SPDs, Monochromatic, was generated in the same manner but having only monochromatic emission lines of 1 nm width. A total of 100,000 SPDs were evaluated. The generated SPDs have a five-by-four structure that is demonstrated in Fig. 3.

 

Fig. 3. Each panel shows a representative SPD of 5,000 SPDs in the indicated SPD set (column) and subset (row). The generated SPDs have a five (subsets) by four (sets) structure. Each subset contains 5,000 SPDs and each set contains 25,000 SPDs. The total SPD set contain 100,000 SPDs. The axes have been omitted for clarity. The x-axis for each plot is wavelength with unit nanometers (nm) ranging from 380 nm to 780 nm; the y-axis of each subplot is relative output from zero to one and is unitless.

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Because the veracity of these measures has not been verified with psychophysical experiments, a gauge of face validity was needed to help navigate the comparison—specifically, smoothly varying broadband SPDs should cause fewer metameric mismatches than highly discontinuous SPDs containing high peaks, deep valleys, and very narrow spectral components. This is because narrow and discontinuous SPDs are likely to emphasize small peculiarities in the SRFs of objects, leading to more erratic color shifts. As SPDs become more structured (e.g., high peaks, deep valleys, and/or narrow spectral components,) greater metameric mismatch is expected. The SPD dataset was conceptualized and generated in part to probe the manner and degree to which R_t and MMCRI quantify this phenomenological trend.

3.2 Computation of measures

R_t was computed using a MATLAB routine that is based on the Microsoft Excel calculator published as supplemental material to David et al. [22]. Measures from IES TM-30 were computed using a MATLAB implementation of IES TM-30-20 [7]. All computation of measures was performed from 380 nm to 780 nm.

MMCRI was computed using a MATLAB module provided by Funt [31]. One thousand (n = 1,000) SRFs were used to estimate the MMV, as this was the default in the provided MATLAB module. A metamer set of 1,000 SRFs can also be shown to produce more precise results than smaller metamer sets (e.g., n = 500, n = 100, or n = 10). Three hundred (m = 300) SRFs were used to estimate the OCS as this was the default in the provided MATLAB module.

Generating the 1,000 SRFs for the metamer set was computationally intensive; computing MMCRI for a single SPD took approximately 30 seconds on a high-performance desktop computer. For comparison, R_t computed in 0.05 seconds. Computing MMCRI for a set of 100,000 SPDs would require a computation time of more than 34 days. To reduce the computation time, a fixed metamer set was used, eliminating the need to estimate the metamer set for each SPD, and reducing the total computation time to just over a week, approximately.

Because MMCRI generates a set of metameric SRFs with each computation, calculating MMCRI for a single SPD multiple times results in different values—the accuracy and precision of which depends on the number of SRFs in the metamer set. Precision and accuracy decrease as the number of estimated metamers decreases and is inversely proportional to computation time. Besides reducing computational time, using a fixed metamer set has the secondary advantage of making MMCRI deterministic (i.e., for a single SPD, the calculation always produces the same value). This is convenient for direct comparison to R_t and measures from IES TM-30, all of which are deterministic, and simplified comparison when re-computation of only a subset of the full SPD set was needed.

MMCRI does not require the use of a specific reference illuminant scheme or set of color matching functions. For direct comparison to R_t and measures from IES TM-30, MMCRI was computed using the reference illuminant scheme from IES TM-30-20 (see Table 1) and the CIE 1964 10° color matching functions.

4. Results

4.1 SPD errors

Computing MMCRI with the MATLAB implementation we had access to [28,31] produced an error for some SPDs. The error occurred during the computation of the OCS for the test illuminant and in most cases the test SPDs were narrowband. Though we could not pinpoint the exact reason for the errors, error messages from MATLAB point to the convex hull calculation. The relevant error messages from MATLAB were:

For reference, a convex hull is a subset of points, from a set of points, that forms the convex boundary around the set of points—the convex boundary is the enclosure that contains the entire line formed by any two points within the boundary. For a finite set of points, the convex hull can be visualized as the shape a rubber band would make if stretched around the points.

These error messages suggest that some spectra—most likely those that have sharp peaks, deep valleys, and/or narrowband spectral features—cause such sporadic color shifts that a convex enclosure of the OCS cannot be determined. Funt provided an error handling solution [32] that increased the number of optimal reflectance spectra (m) used to estimate the OCS of the test illuminant from 300 to 400. This error handling was implemented in MATLAB in an error try/catch loop (i.e., only using the increased OCS estimation for errored SPDs), instead of computing the higher resolution OCS estimation for all SPDs, because increasing the number of points used to estimate the OCS increases computation time.

The error handling decreased the number of SPDs with MMCRI errors, but it did not eliminate them entirely. In the final computation, MMCRI could not be determined for 2,987 SPDs (≈ 3% of the 100,000 SPDs) that were subsequently removed from the analysis. Most (but not all) of these SPDs were in the Narrow and Monochromatic SPD subsets, and many had peak wavelengths near 560 nm and shorter than approximately 430 nm. These are near wavelength regions associated with lower visual sensitivity [33–39] and where the greatest difference in SRFs is expected [40]. Additional exploration of this issue is warranted.

4.2 Comparison

Figure 4 shows boxplots of R_t (top) and MMCRI (bottom) for each of the SPD subsets. Two trends are observed: (1) On average, as the number of primaries increased, illuminant-induced metameric mismatch decreased—i.e., metric values increased—as measured by both MMCRI and R_t; (2) On average, as the spectral width of primaries increased and the SPDs became more broadband, illuminant-induced metameric mismatch decreased—i.e., metric values increased—as measured by both MMCRI and R_t. These trends can also be observed in Fig. 5, which plots histograms of R_t and MMCRI as a function of the spectral width (columns) and number of components (rows) that comprise each of the SPD sets and subsets.

 

Fig. 4. (Top) Boxplot for R_t as a function of SPD set. (Bottom) Boxplot for MMCRI as a function of SPD set. For both measures, values generally decrease with fewer number of primaries and as primaries become increasingly narrowband. The primary exception is for MMCRI and the monochromatic SPDs.

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Fig. 5. Each panel shows the histogram for R_t (yellow fill) and MMCRI (grey fill) for the indicated SPD set (column) and SPD subset (row). If color cannot be distinguished, the distribution for R_t always has the rightmost peak, and MMCRI the leftmost. The axes have been omitted for clarity. The x-axis is a number line between 0 and 100 (the theoretical range for both measures) with binning every two units for computation of the histogram. The y-axis is the frequency (count) of SPDs in the bin from zero to 2000.

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These observations are consistent with our general gauge of face validity that highly discontinuous narrowband spectra should cause more metameric mismatch than smoothly varying broadband spectra. The main exception to the above observation is the characterization of metameric mismatch by MMCRI for the Monochromatic set. As the spectral width of SPDs decreased from Wide, to Medium, to Narrow, there was a corresponding decrease in the average MMCRI. Based on this trend, MMCRI should have continued to decrease moving from the Narrow set to the Monochromatic. The opposite was observed. Particularly noteworthy is that the 3-Mono set, which has three monochromatic emission lines, had a higher average MMCRI and larger variation than all the Narrow subsets and all the other Monochromatic subsets.

The increase in MMCRI for the Monochromatic set relative to the Narrow set is more easily discernable in Fig. 6 (right). This phenomenon was not observed with R_t (Fig. 6, left). The relationship between R_t and MMCRI appears to agree well for broadband SPDs but degrade as SPDs become narrower (Fig. 7, Table 2, Table 3).

 

Fig. 6. Boxplots for R_t (left) and MMCRI (right), as a function of the generated SPD sets collapsed across the number of primaries (for example, “Mono” includes SPDs with 3, 4, 5, 6, and 7 primaries). The “x” indicates the mean, the midline (inside the shaded box) is the median, the shaded box represents the interquartile range (IQR), and the unfilled data points (circles) are outliers (computed as 1.5 * IQR). When there are no outliers, the “whiskers” extend to the minimum and maximum value; when there are outliers, they extend to the local minimum and maximum, which are respectively, the smallest and largest values that are not outliers. The outlier points may represent one or more SPDs depending on the scaling of the y-axis. The plotted outlier points approximately represent the spacing of all outliers, and always includes the most extreme point. All boxplots were created in Microsoft Excel.

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Fig. 7. MMCRI versus R_t for the generated SPD sets. The change from light to dark indicates decreasing spectral width of the primaries. Generally, as spectra move from broadband to narrowband, both R_t and MMCRI decrease. In most cases, R_t is larger than MMCRI.

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Table 2. Coefficient of determination (R²) between R_t and MMCRI for each SPD set, not considering subsets. The number of SPDs in each comparison is indicated. The color gradient demonstrates the increase/decrease in R² (green is higher, red is lower). Filtering for color fidelity (i.e., R_f ≥ 70) was done to evaluate the relationship for SPDs more relevant to lighting application. An R_f of 70 is the lowest color fidelity required to achieve any specification category of IES TM-30 Annex E [7] and is taken to be a reasonable performance floor.

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Table 3. Coefficient of determination (R²) between R_t and MMCRI for each SPD set, considering subsets. The number of SPDs in each comparison is indicated. The color gradient demonstrates the increase/decrease in R² (green is higher, red is lower). Filtering for color fidelity (i.e., R_f ≥ 70) was done to evaluate the relationship for SPDs more relevant to lighting application. An R_f of 70 is the lowest color fidelity required to achieve any specification category of IES TM-30 Annex E [7] and is taken to be a reasonable performance floor.

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R_t was greater than MMCRI for most SPDs (Fig. 7); this is mainly a function of R_t’s scaling factor, and by itself does not suggest that the two measures are substantively different quantifications of metameric mismatch. There were, however, some SPDs that have MMCRI larger than R_t and appear to be anomalous because of their disproportionately high MMCRI. These spectra have emission lines concentrated near the extents of the visible spectrum, shorter than 430 nm and longer than 630 nm. We suspect that this is because as the SPDs become narrower—and especially when much of an SPD’s emission is at the spectral extremes (e.g., very long or very short wavelengths)—the distribution of the color samples will be very oblong, potentially resulting in smaller MMV’s and thus larger values of MMCRI.

With this observation, we posited that constraining or expanding the wavelength range might also influence metameric mismatch as quantified by MMCRI or R_t. To test this, four SPD sets were generated in the same manner as the Monochromatic set, but instead with peak wavelength ranges of 380 nm to780 nm, 390 nm to 740 nm, 400 nm to 700 nm (identical to the Monochromatic set), and 400 nm to 660 nm. Five thousand SPDs were generated for each wavelength range. A new SPD set was generated for each wavelength range instead of generating an SPD set with the largest range and clipping the ends to form a new set, since clipping would change the SPD’s chromaticity. The results are plotted in Fig. 8.

 

Fig. 8. Boxplot for R_t (left series, light grey fill) and MMCRI (right series, dark grey fill) for four monochromatic SPD sets generated according to the parameters in Section 3.1 but with varying spectral ranges. The average R_t decreased slightly as the spectral range was narrowed while the average MMCRI increased. The variation in both measures increase with increasing spectral range, though the variation was larger for MMCRI. The large variation in MMCRI with the widest spectral range is notable, with achievable values as high as 97.

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Before performing this analysis, we expected some difference in the characterization of metameric mismatch as a function of an expanded wavelength range, because as the wavelength range was extended, the simulated spectra could concentrate proportionally more of their radiation near the ends of the visible spectrum away from the peaks of visual sensitivity. In accord with this expectation, there was a slight average decrease in R_t as the wavelength range was extended (reading Fig. 8, from right to left, for the light gray series), where the widest wavelength range had a slightly lower average R_t—i.e., worse metameric mismatch—than the smallest wavelength range. This trend was not observed for MMCRI. Instead, MMCRI increased with increasing wavelength range (reading Fig. 8, from right to left, for the dark gray series). The variation in MMCRI also varied greatly as a function of the wavelength range. Notably, MMCRI values in the set with the largest wavelength range (380 nm to 780 nm) varied across most of MMCRI’s achievable range with a maximum value of 97. This is larger than the maximum MMCRI of 93 (Wide set) for all simulated SPDs (Table 2). It is contrary to face validity for a three-component monochromatic SPD to outperform the best SPD from the Wide set. The influence of the wavelength range is large compared to the relative changes in spectral width. This is anomalous and indicative of a mischaracterization by MMCRI.

The number of anomalous spectra varied as a function of wavelength range, where a larger range produced more anomalies. Similar to the spectra that caused errors in the MMCRI computation, these spectra had peak wavelengths most concentrated at wavelengths shorter than 420 nm, but also local concentrations near 500 nm, 580 nm, and 675 nm. Increasing the wavelength range also increased the number of errors produced by the MMCRI computation.

To compare R_t and MMCRI directly, metric values were converted to z-scores. The difference between the z-scores for R_t and MMCRI are plotted as a function of SPD subset (Fig. 9, top) and SPD set (Fig. 9, bottom). R_t and MMCRI were in relatively strong agreement for broadband SPDs (e.g., Wide and Medium sets). The difference between R_t and MMCRI was pronounced for the Narrow and Monochromatic sets, whereby MMCRI rated these light sources as inducing relatively less metameric mismatch than R_t (exemplified by the large negative Δz-score). This trend can also be seen Fig. 10 which plots the histograms of the z-scores.

 

Fig. 9. The difference between the z-scores of R_t and MMCRI as a function of SPD subset (top) and SPD set (bottom). The horizontal black dashed line shows a Δz-score value of zero. The difference between R_t and MMCRI is larger for narrower spectra, expressed by the large variation in Δz-score.

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Fig. 10. Each panel shows the histogram of z-scores for R_t (yellow fill) and MMCRI (grey fill) for the indicated SPD set (column) and SPD subset (row). If color cannot be distinguished, the distribution for MMCRI mostly has the highest peak. The axes have been omitted for clarity. The x-axis is a number line between -7 and 7 (the z-score range for all values) with binning every 0.4 units for computation of the histogram. The y-axis is the frequency (count) of SPDs in the bin width from zero to 1200.

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A two-sample Kolmogorov–Smirnov test (K-S) [41,42], a nonparametric test of the equality of continuous one-dimensional probability distributions, was performed to determine if the two “samples” (i.e., distributions of R_t and MMCRI values) were drawn from the same distribution. The test was performed using a plug-in for Microsoft Excel. The null-hypothesis states that the “two samples are drawn from the same distribution” or said another way, “the distributions of the two samples are the same.” We rephrase the null hypothesis here to state that R_t and MMCRI are equivalent assessments of illuminant-induced metameric mismatch.

All paired comparisons—i.e., Wide, Medium, Narrow, and Monochromatic SPD sets—were highly significant (p < 0.0001) indicating that the two samples did not come from the same distribution; this suggests that R_t and MMCRI were not statistically equivalent quantifications of illuminant-induced metameric mismatch for any of the SPD sets. The cumulative probably plots of Fig. 11 show that R_t and MMCRI behaved similarly for broadband light sources (e.g., the Wide subset), but diverged as spectra became increasingly narrowband. This is also shown by the decreasing coefficient of determination with narrower spectra (reading Table 2 and Table 3 from bottom to top).

 

Fig. 11. Cumulative probability functions for each SPD set. Metric values and SPD set are indicated in the legend. The distributions for R_t nearly overlap, so only the distribution for the Wide SPD set is shown to avoid clutter. The two-sample Kolmogorov-Smirnov test for all SPD set comparisons suggests that R_t and MMCRI are different quantifications of illuminant-induced metameric mismatch for all SPD sets (p < 0.0001 for all four), though R_t and MMCRI appear to behave similarly for broadband SPDs and less so for narrowband SPDs.

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5. Discussion

R_t performed in general accordance with face validity. That is, R_t predicted increasing illuminant-induced metameric mismatch (i.e., lower metric values) for light sources with increasingly narrowband spectral features. On average, MMCRI also trended as expected, but there were some situations where MMCRI produced unexpected results—e.g., some monochromatic SPDS achieved a higher score for MMCRI than even the most broadband SPDs. R_t and MMCRI agreed relatively well for broadband SPDs with an increasing difference in the characterization of metameric mismatch as spectra incorporated more narrow spectral features. Despite some similarities between the two, significant two-sample Kolmogorov–Smirnov tests for all SPD sets suggest that the measures were different quantifications of metameric mismatch.

Are these differences of practical importance for applied lighting? Table 2, Table 3, and Fig. 12 show select results for SPDs with R_f ≥ 70. Filtering SPDs for R_f ≥ 70 increased correlations between R_t and MMCRI for the Monochromatic and Narrow SPD sets—presumably related to the removal of the SPDs with disproportionately high values of MMCRI—and decreased correlation for the Medium and Wide SPDs. Considering all SPDs simultaneously, correlation was slightly increased when filtering for R_f ≥ 70 (Table 3). Figure 13 shows the cumulative probability distributions for R_t and MMCRI considering only SPDs with R_f ≥ 70. Though the difference in distributions was less pronounced (e.g., cross reference with Fig. 11, which does not have an R_f lower bound of 70), the K-S test was highly significant for all SPD set comparisons (p < 0.0001 for all four), suggesting that R_t and MMCRI were not equivalent quantifications of illuminant-induced metameric mismatch even for SPDs with modest to high average color fidelity.

 

Fig. 12. (Left) MMCRI versus R_t for light sources with R_f ≥ 70. This filtering was done to evaluate the relationship for SPDs more relevant to lighting application. An R_f of 70 is the lowest color fidelity required to achieve any specification category of IES TM-30 ANNEX E [7] and is taken to be a reasonable performance floor. When constraining to R_f ≥ 70, the relationship between MMCRI and R_t is tighter and there are no SPDs with MMCRI values disproportionately higher than other SPDs in their set. (Right) The difference between the z-scores of R_t and MMCRI as a function of SPD set. The differences between R_t and MMCRI agree more strongly for the Med and Wide sets than for the Nar and Mono sets.

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Fig. 13. Cumulative probability functions for the SPD sets as indicated, considering only SPDs with R_f ≥ 70. The differences between MMCRI and R_t are smaller (as compared to Fig. 11), but differences are still present and highly significant (p < 0.0001 for all four).

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Separate from their numerical characterizations of illuminant-induced metameric mismatch, there are practical differences between MMCRI and R_t that are relevant to lighting application. These differences are contrasted below:

 

Fig. 14. The number of times (vertical axis) the indicated wavelength (horizontal axis) was included as a transition wavelength for each of the five transitions in the 1,000 SRFs of the MMCRI computation. Transition wavelengths are disproportionately high near 457, 546, and 604 nm which are consistent with the crossover wavelengths of metamers documented by others [43–46].

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At root, R_t and MMCRI were conceptualized with different end-uses in mind. MMCRI was theorized to fully describe a set of all reflectance functions that are metameric under an arbitrary test illuminant in comparison to an arbitrary reference illuminant. It does so by computing a metameric mismatch volume that is a theoretical maximum. This approach is likely useful in machine-based colorimetry and computational color constancy—applications that include camera sensor design, image calibration and noise reduction, and mapping between image acquisition, image display, and associated image-based color correction. While the authors of MMCRI also identify lighting design as a potential application for MMCRI [28], the algorithm’s lack of concordance with physical reality is a practical shortcoming as much as it is a theoretical strength. Meanwhile, R_t was theorized to be applicable to illuminant-induced metameric mismatch in the built environment. It carries forward the concepts developed in TM-30, including use of the 99 CES that were explicitly constrained to fall within the NCS color volume. R_t does not endeavor to probe the limits of what is theoretically possible. Rather, it is intentionally limited to physically plausible reflectance functions viewed by human observers.

6. Conclusion

Using simulated sets of Gaussian spectral power distributions with multiple spectral components and spectral widths, we have demonstrated that R_t and MMCRI behave similarly, predicting less metameric mismatch for broadband spectra and more for narrower spectra. Importantly, statistical tests suggest that despite some similarities, R_t and MMCRI are potentially different characterizations of illuminant-induced metameric mismatch. This is also true for light sources more applicable to general lighting applications (i.e., R_f ≥ 70). Separate from their differences in numerical quantification, we demonstrate some characteristics of the MMCRI computation that are potentially problematic for applied lighting that are not present with R_t. These include erroring of the computational procedure, substantial variability as a function of the wavelength range used to simulate an SPD, disproportionately high MMCRI values assigned to monochromatic SPDs that should be expected to have poor performance, long computation times, lack of a deterministic solution, and a sample set that is likely to penalize light sources for color shifts that are unlikely to occur in the built environment. We recommend that future work about illuminant-induced metameric mismatch include psychophysical experimentation to determine if and how such measures can be specified to control metameric mismatch with real object samples under realistic viewing conditions.