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Abstract
The areal dimension in chromaticity diagrams is often used to evaluate the range of reproducible colors (or color gamut) of a display; however, its meaning is not fully understood. For consistent and meaningful color gamut comparisons, the metric needs a color space with a common chromatically adapted white point, which is not considered in legacy chromaticity gamut area metrics. This study demonstrates the validity of chromaticity gamut area metrics and clarifies the proper way to use the metrics by comparing the color gamuts of synthetic additive displays with a range of sampled RGB primaries with different white points. The study reports high correlations of 0.98–0.99 between the xy chromaticity gamut area and color gamut volume for additive displays with a common white point at 6500 K and above. However, using the “gamut rings” 2D visualization method, this study also demonstrates that use of different white points cause a failure of the chromaticity gamut area metric.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Almost a decade has passed since the wide-gamut system colorimetry for ultra-high-definition television was standardized in Rec. ITU-R BT.2020 (Rec. 2020) [1], while more recently, the red, green, and blue (RGB) primaries were also adopted for high dynamic range (HDR) television in Rec. ITU-R BT.2100 [2]. Since then display technologies have made rapid progress to achieve vivid color reproduction. In general, current “wide-gamut” displays use the Rec. 2020 container, which partially reproduces the Rec. 2020 gamut. Therefore, consistent metrology standards regarding the range of reproducible colors (or color gamut) are important.
According to the International Lighting Vocabulary [3] and Report 168:2005 [4] published by the CIE, a color gamut is regarded as the volume of a 3D color space utilized in reproduction and media applications. To interpret and compare color gamuts, it is assumed that the observer has adapted to the peak white luminance and chromaticity (i.e., white point) of the display to ensure that the gamuts form as 3D solids in a device-independent cylindrical color space comprising chroma, hue, and lightness. In addition, the color space for volumetry must be perceptually homogeneous with a common white point. In the IEC 62977-2-1 method [5], the measured XYZ tristimulus values of the gamut boundary are scaled linearly so that Y = 100 for the peak white level of the display. Then, the Bradford chromatic adaptation transform [6] is applied to the normalized XYZ values to obtain the device-independent D50-relative tristimulus values, enabling consistent and meaningful comparisons to be made by transforming the relative values into the CIE 1976 (L*, a*, b*) color space.
Decades before Rec. 2020 was standardized, the “display color gamut” was visualized as a triangle formed by connecting the chromaticity coordinates of the RGB primaries in the CIE 1931 (x, y) chromaticity diagram [7]. Although the genesis of this areal metric is unknown, its use remains popular. Recently, the use of the CIE 1976 (u′, v′) chromaticity diagram [8] has gained popularity based on a plausible argument for its nominal (but far from perfect) uniformity [9]. As such, these two areal units have been used concurrently to describe the larger “color gamut” of the two areas without dispute. However, a counter-argument proposed by Masaoka and Nishida [10] advocates for a return to the classic CIE 1931 (x, y) diagram based on analysis that shows impressively high correlation between the xy area and the color gamut volume (CGV) in the CIELAB color space. Following fierce debates at standards-setting organizations, the RGB triangle area is now regarded as distinct from the CIE definition of color gamut and called the “chromaticity gamut area” at IEC TC 110 [5] and ICDM [11]. The xy chromaticity gamut area is now recognized as a proxy metric for the CGV of additive RGB displays.
However, the chromaticity gamut area metrics fail to consider the chromatic adaptation state of the observer. Significantly, Masaoka and Nishida simulated the CGV in a D65 CIELAB color space for additive RGB displays using the D65 white point only. Although this assumption is reasonable for mastering monitors, consumer displays typically have cooler white points than D65, up to a correlated color temperature (CCT) of 12000 K [12]. Smith et al. [13] simulated the CGVs of synthetic displays using randomly sampled RGB primaries and white points, and reported that the xy area did not correlate adequately with the CGV with highly saturated RGB primaries or in cases of variation in the white point. This raises the question of whether the xy metric should be discarded. Consequently, this study utilizes computer simulations to examine the validity and limitations of the xy chromaticity gamut area metric for additive RGB displays with various RGB primaries and different daylight CCTs for the white points.
2. Simulation of chromaticity and color gamuts
2.1 Sampling RGB primaries and white points
Figure 1 shows the u′v′ chromaticity coordinates of the sampled RGB primaries and white points. The sampled RGB primaries consisted of 24 red, 33 green, and 15 blue primaries, which were identical to those reported in [10]. They were sampled as evenly as possible within the regions enclosed by the extended boundaries of the Rec. ITU-R BT.709 (Rec. 709) [14] RGB triangle (namely, the horseshoe-shaped spectrum locus and purple boundary, including the RGB primaries for Rec. 709), Adobe RGB [15], DCI-P3 [16], and Rec. 2020. The last of these was used as the reference gamut. In total, combinations involving 11,880 RGB primary sets were sampled. Any of the possible RGB combinations enclosed the Rec. 709 RGB triangle. White points were sampled at four daylight CCTs: 5000 K, 6500 K, 9300 K, and 12000 K. The chromaticity coordinates of the CCTs were calculated using the CIE method [17]. The RGB luminance ratios were scaled to achieve the desired white points assuming perfect additivity and infinite contrast.
Fig. 1. Diagram of the u′v′ chromaticity coordinates of sampled RGB primaries and white points. Dark and light gray triangles indicate the Rec. 709 and Rec. 2020 RGB triangles, respectively.
2.2 Chromaticity gamut area computation
The chromaticity gamut areas of the sampled RGB sets were computed using the xy and u′v′ diagrams. In addition to the triangular areas, the Rec. 2020 coverage ratios were computed for each diagram. The ratio was computed as (ADISP∩AREF)/AREF, where ADISP and AREF are the areas of a specific sampled RGB triangle and the Rec. 2020 RGB triangle, respectively, and ADISP∩AREF represents the overlap of the two triangles. The vertices and area of the intersection polygon were computed using the Sutherland–Hodgman algorithm [18] and the shoelace formula [19], respectively. Note that the sampled white points were not used in this computation.
2.3 Color gamut volume computation
The CGV was simulated in the CIELAB color space for each RGB primary set according to the IEC 62977-2-1 method [5]. Nonlinear RGB input levels (R′, G′, B′) were uniformly sampled from the faces of the cube-shaped RGB color-encoding space to determine the gamut boundary, as shown in Fig. 2. Each RGB cube face comprised an 11 × 11 lattice, yielding a total of 602 R′G′B′ sets. A display gamma of 2.2 was assumed. Additionally, the tessellation order was determined during the RGB sampling stage. The tristimulus values were calculated from the R′G′B′ values and transformed to the relative D50 tristimulus values using the Bradford chromatic adaptation transform [6]. Next, the relative (X, Y, Z) coordinates were converted to (L*, a*, b*) coordinates, and the 3D solid was sliced at L* values of 0.5, 1.5,…, 99.5, to obtain 100 constant-L* loci, with the volume approximated by summing the areas of the 100 loci. The volume coverage ratio was calculated as (VDISP∩VREF)/VREF, where VDISP is the CGV of the display for a specific sampled RGB primary set, VREF is the CGV of Rec. 2020, and VDISP∩VREF is the intersection volume of the two gamuts. The intersection area between two loci was computed for each L*. The vertices and area of the intersection polygon were computed using the Sutherland–Hodgman algorithm [18] and the shoelace formula [19], respectively, with VDISP∩VREF calculated by summing the 100 intersection areas between the two gamuts.
Fig. 2. Procedure for sampling the surface of the Rec. 2020 gamut: (a) RGB encoding color space tessellated via Delaunay triangulation; (b) tessellated gamut surface in the D50 CIELAB color space.
3. Simulation results
Figure 3 shows the u′v′ and xy chromaticity coordinates of two RGB primary sets, RGB_1 and RGB_2, which were selected as illustrative examples from the sampled RGB sets. Their Rec. 2020 area-coverage ratios in the xy diagram are identical (0.789). It is observed that the yellow–green–cyan region is larger for the xy coordinates, whereas the red–magenta–blue region is larger for the u′v′ coordinates.
Fig. 3. Diagrams of the (a) u′v′ and (b) xy chromaticity coordinates of two sets of RGB primaries: RGB_1 and RGB_2.
Figure 4 compares the size and Rec. 2020 coverage ratios of the u′v′ area with those of the CGV at CCTs of 5000 K, 6500 K, 9300 K, and 12000 K for the sampled RGB primary sets. The Pearson coefficients (r) indicate the strength of the association between the areal and volumetric dimensions. The purple and magenta circles indicate RGB_1 and RGB_2, respectively. Figure 5 compares the xy area with the CGV.
Fig. 4. Comparison between the u′v′ area and CGV at CCTs of (a) 5000 K, (b) 6500 K, (c) 9300 K, and (d) 12000 K, and the comparison between the Rec. 2020 u′v′ area-coverage ratios and Rec. 2020 CGV coverage ratios at CCTs of (e) 5000 K, (f) 6500 K, (g) 9300 K, and (h) 12000 K for the sampled RGB primary sets. The magenta and purple circles indicate the results for RGB_1 and RGB_2, respectively (the corresponding RGB triangles are shown in Fig. 3). Note that r represents the Pearson coefficient between the u′v′ area and the CGV for each CCT.
Fig. 5. Comparison between the xy area and CGV at CCTs of (a) 5000 K, (b) 6500 K, (c) 9300 K, and (d) 12000 K, and the comparison between the Rec. 2020 xy area-coverage ratio and Rec. 2020 CGV coverage ratio at CCTs of (e) 5000 K, (f) 6500 K, (g) 9300 K, and (h) 12000 K for the sampled RGB primary sets. The magenta and purple circles indicate the results for RGB_1 and RGB_2, respectively (the corresponding RGB triangles are shown in Fig. 3). Note that r represents the Pearson coefficient between the xy area and the CGV for each CCT.
From these comparisons, it is evident that the u′v′ gamut area cannot be used as a meaningful proxy metric for the CGV. High correlations of 0.98–0.99 between the xy gamut area and the CGV were found at CCTs of 6500 K and above. Moreover, at 6500 K, the volume coverage ratios for RGB_1 and RGB_2 were almost identical. However, the volume coverage ratio for RGB_1 increased as the CCT increased, whereas the opposite trend was observed for RGB_2. Notably, these differences are not distinguishable using the chromaticity gamut area metrics.
4. Gamut rings approach
The popularity and misapplication of chromaticity area metrics persist, partially because of the difficulty associated with visualizing the CGV; from any single perspective view, parts of the gamut hull will be obscured. This issue can be solved by representing the CGV in 2D using the “gamut rings” [5,11,20].
The gamut ring framework unwraps the volumetric information and maps it into rings in a polar coordinates diagram, thus providing the information needed to describe the ranges of hue, chroma, and lightness. Figure 6 shows the gamut rings of the Rec. 2020 gamut. The rings are stacked radially around the original L*-axis of the CIELAB color space while retaining the CIELAB hue angle. The radial coordinate represents the CIELAB volume per hue angle. The polar coordinates can be converted to Cartesian coordinates representing the root sum square (RSS) values of the CIELAB a* (a*RSS) and CIELAB b* (b*RSS) coordinates. The areal dimension of the ring diagram can be used to inform volumetric measurements of the color gamut. The rings are delineated according to the constant L* loci and the area of each locus extending outward from the center numerically matches the volume of the slices of the CGV summed from L* = 0 to the L* value of the specific locus. In the case of 10 rings, the area of the innermost ring (with no hole) equals the volume constructed by L* slices 0–10 of the CIELAB gamut envelope. The next outer ring is stacked cumulatively around the tenth L* locus and has an area equal to the volume of L* slices 10–20, and so on. Ultimately, the last ring equals the volume of L* slices 90–100. The outline area (i.e., the area of the hundredth L* locus) corresponds to the total color gamut. The full code for implementing the gamut ring framework is available on GitHub [21]. The code conforms with the IEC standard, as it supports input data comprising XYZ triplets accompanied by the 602 input R′G′B′ triplets used for sampling.
Fig. 6. Simulation of the Rec. 2020 gamut rings. Cartesian coordinates represent the RSS values of CIELAB a* (a*RSS) and CIELAB b* (b*RSS). In the gamut volume calculation, the relative XYZ tristimulus values of the Rec. 2020 gamut sampled with the D65 white point were converted to those of the D50-relative values.
Figure 7 shows the gamut rings of the two RGB primary sets (RGB_1 and RGB_2), assuming white points at CCTs of 5000 K and 12000 K. The dotted locus in each plot shows the outline of the Rec. 2020 gamut rings (see Fig. 6). For the RGB_1 primary set at 5000 K [Fig. 7(a)], the green portion of the gamut protrudes from the Rec. 2020 gamut but does not cover the yellow portion of the Rec. 2020 gamut sufficiently. By comparison, the RGB_1 primary set at 12000 K [Fig. 7(b)] is expanded regarding the yellow gamut while the green portion still protrudes beyond Rec. 2020, resulting in higher Rec. 2020 volume coverage. For the RGB_2 primary set [Figs. 7(c) and (d)], the green portion of the gamut is enclosed by the Rec. 2020 gamut and shrinks when the CCT of the white point increases, resulting in lower Rec. 2020 volume coverage at higher CCTs.
Fig. 7. Simulations of the gamut rings of the two RGB primary sets (RGB_1 and RGB_2). (a, b) Gamut rings of RGB_1 corresponding to white points at 5000 K and 12000 K, respectively. (c, d) Gamut rings of RGB_2 at 5000 K and 12000 K, respectively. The color gamuts were computed according to the D50 CIELAB color space. The dotted locus in each plot indicates the outline of the Rec. 2020 gamut rings shown in Fig. 6.
5. Discussion
Smith et al. [13] concluded that the correlation between the xy area and CGV is “less good for wide gamuts with highly saturated colorants (e.g., BT.2020) or where there is variation in the white point (DCI-P3).” (BT.2020 is identical to Rec. 2020.) In my analysis, however, significantly low correlations were not found for cases where the white point is fixed. The fundamental difference between the simulation of Smith et al. and mine is the size of the chromaticity regions used for sampling the RGB primaries. They sampled RGB primaries from larger regions and included many gamuts smaller than Rec. 709. According to the left plot in Fig. 7 of [13], which uses data set B of Smith et al. with 1,000 synthetic gamuts, the deviation of the Rec. 2020 gamut (BT.2020) from a linear fit seems to be very small with an expected high correlation of 0.98–0.99, although their random sampling makes quantitative verification impossible. More extensive simulations with emphasis on narrowband primaries would be useful to confirm correlation between the xy area and CGV for deeply saturated primaries. In cases where there is variation in the white point, both simulation results show significant lack of correlation between the xy area and CGV.
The gamut rings can provide more useful insights into the color gamut information than the chromaticity gamut area in terms of volume, lightness, and hue angle. Furthermore, gamut ring diagrams are applicable to any type of display or other output media, including multichromatic displays [20] and printers, provided that their color gamut hulls are represented in a 3D color space. This clearly superior method of visualizing color capability should lead to serious reconsideration of the use cases in which the xy gamut area metric is valid. First, the xy chromaticity gamut area remains valid for additive RGB displays whose white point matches the reference. This case can apply to broadcast monitors and color management displays. The xy area metric should not be used for non-additive output media, such as multichromatic displays [13,20] and printers. Comparing the xy areas of displays with different white points is illogical in terms of color perception. Second, the xy gamut area metric can be applied in the development of optical devices, such as RGB light sources, phosphors, and filters. In such cases, immediate knowledge of the (x, y) chromaticity is useful, and the assumption that the white point equals the reference must be applied before constructing the display.
6. Conclusions
This study utilized computer simulations to test the validity of the xy chromaticity gamut area metric. The conclusions are as follows:
- The xy area can be used as a proxy metric of color volume only for additive RGB displays with a common white point at a CCT of 6500 K and above with high correlations of 0.98–0.99.
- The areal dimension is invalid for comparing displays with different white points because chromatic adaptation is not considered.
- The gamut ring framework quantitatively and intuitively visualizes the adaptation-state-dependent color gamut in a 2D diagram.
Basic knowledge of the color appearance model is essential not only to prevent the misuse of chromaticity gamut area metrics but also to understand the concept of gamut rings. For decades, performance evaluation has tended to ignore the gamut volume, favoring instead areal representations that are easier to plot in two dimensions. Now, the gamut ring transform exhibits display color capability in a visually proportionate manner. It is anticipated that a user-friendly, stand-alone application that computes the color volume and visualizes its gamut rings will be available from standards-setting organizations in the near future.
Acknowledgments
The authors wish to thank the eminent experts of the International Committee for Display Metrology (ICDM) for the fruitful discussions and the anonymous reviewers for their insightful comments and suggestions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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