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Abstract
A new gamut boundary descriptor (GBD) is presented in this paper. Unlike the traditional methods to perform sampling in RGB space and to interpolate the data in a uniform color space (UCS), the proposed method is designed in a reverse manner, i.e., to sample the data in UCS. This unique characteristic makes it simple and yields a smooth and accurate boundary. The proposed method can be divided into three simple steps. Firstly, coarse sampling is performed to make a rough estimate of the boundary. In step two, dense sampling is conducted to refine the boundary. Finally, the cusp is identified to achieve the accurate boundary. An experiment was carried out to compare the performance of the proposed method with the two commonly used SMGBD and Mountain Range methods. The results showed that the proposed method can give the most accurate and smooth boundary. It can be considered for practical applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Displays are ubiquitous in daily life, acting as a point of contact between users and digital content. Different display technologies, such as cathode ray tube (CRT), liquid-crystal display (LCD), and organic light-emitting diode (OLED), are used for different types of displays. CRT technology dominated the display industry for decades and is now being replaced by LCD technology. OLED displays have recently shown a promising future in both mobile devices and larger televisions and monitors. Their color characteristics, however, differ due to their different imaging principles, resulting in a range of color rendering capabilities. Furthermore, there are many different imaging standards, such as the traditional sRGB standard [1], DCI-P3 [2] for digital cinema reference projectors, Rec. 2020 [3] for ultra-high definition (UHD) TVs, and Adobe RGB [4], which is a de-facto standard in professional color processing. Different color transformations between digital inputs and tristimulus value outputs will be defined by different standards. As a result, the reproducible colors differ between displays.
Color gamut refers to the range of colors that can be reproduced [5]. It is an important display property that is primarily determined by the display primaries. The two-dimensional (2D) gamut defined in a color diagram and the three-dimensional (3D) gamut defined in a uniform color space (UCS) are the two main types of gamut available. The shape of a 2D gamut is typically a triangle defined by its RGB primaries, as shown in Fig. 1. Three imaginary displays defined in sRGB, DCI-P3, and NTSC were included. They had distinct primaries and thus different reproducible colors. Furthermore, the gamut area of a display varies across color spaces. This is due to the uniformity of the color space used. The CIE 1976 u'v’ chromaticity diagram is traditionally thought to be more uniform than the CIE1931 xy chromaticity diagram, implying that the gamut area calculated in the u'v’ diagram should better reflect the rendering capability of a display. However, a recent study found that the gamut area in the xy diagram corresponds more closely to human perceptions [6]. As a result, it is recommended to demonstrate the 2D gamut in the CIE1931 xy diagram.
Although a color diagram is a simple and convenient way to demonstrate the 2D gamut, it is generally considered to be rarely sufficient. CIE [7] defines the color gamut as the volume of a 3D color space in reproduction and media applications. This is due in part to the fact that chromaticity cannot reflect the color appearance and in part to the absence of the lightness or brightness dimension in a chromaticity diagram. As a result, when comparing the reproducible colors of displays, a 3D gamut involving a lightness dimension is more preferred.
Adding a luminance dimension, either linear or log, to form a 3D solid, e.g., xyY or u'v'Y, is a convenient way to extend a 2D gamut into a 3D gamut. However, such a simple extension is not always uniform [7]. To better reflect the color gamut, more advanced UCSs or color appearance models (CAMs) were proposed. CIELAB, which was introduced by CIE in 1976 to predict perceptual color differences, might be the most well-known color gamut [8]. For decades, it has been the de facto standard in the imaging industry, and the International Color Consortium (ICC) uses it when performing device profiling.
Despite its simplicity, CIELAB was later discovered to not correspond well to perceptual color differences, prompting the development of some non-Euclidean color difference formulas, e.g. $\Delta {E_{CMC}}$[9], $\Delta {E_{94}}$ [10], $\Delta {E_{00}}$ [11], etc.. While these modifications performed well in some ways, they are not tied to any specific color space and cannot provide perceptual color appearance attributes. As a result, they cannot be used to demonstrate a color gamut. Later, Luo et al. proposed the CIECAM02 [12] color appearance model (along with an associated UCS, CAM02-UCS [13]), and was recommended by CIE as a new standard. It can predict not only relative color attributes such as lightness, chroma, and hue, but also absolute color attributes i.e., brightness and colorfulness. Both combinations, i.e., UCS formed by lightness, chroma, and hue or brightness, chroma, and hue, can be used to represent a color gamut. However, it should be noted that the absolute attributes have not been thoroughly evaluated. Revised versions of CIECAM02 and CAM02-UCS were recently proposed and named CAM16 and CAM16-UCS [14]. They are only intended to solve unexpected computational failures, and their prediction performance remains unchanged.
The same color will be mapped into different color coordinates in these color spaces, resulting in different color gamut shapes. Figure 2 shows the DCI-P3 gamut in CIELAB and CAM02-UCS. The hue slice in CIELAB color space, as shown, is similar to a triangle while being slightly convex in regions of low lightness. The gamut in CAM02-UCS is much rounder, indicating that the gamut is convex in most regions. As a result, correctly describing a gamut in a more complicated uniform color space (e.g., CAM02-UCS) would be more difficult than in a simple one (e.g., CIELAB) [5].
As advised by Morovic [5], the gamut estimation methods can be divided into two major categories, i.e., medium-specific methods, and generic geometric methods. Medium-specific methods are either based on directly extracting a GBD from a specific device characterization model, or they rely on assumptions about the regularities of medium gamuts that gamuts in general do not satisfy. Herzog [15] proposed gamulyt, a gamut boundary descriptor based on the fact that most media gamuts have the shape of a distorted cube, with vertices black, white, red, green, blue, cyan, magenta, and yellow. Hence, the medium gamut can be easily represented by a particularly small number of points. Similarly, Ou-Yang et al. [16] proposed a gamut boundary descriptor algorithm based on gamut shape estimation. In their method, the gamut is sliced in terms of lightness, and each slice is expressed by color gamut boundary apexes with the same brightness. Those methods, however, are predicated on prior knowledge of gamut in a specific UCS and cannot be easily extended if the convexity of a gamut is unknown.
Generic methods are, by definition, generalized approaches for defining a device's gamut boundary. Such a method has no strict requirements, and it can be easily extended to any medium gamut in any color space. The convex hull method [17] and the alpha shape method [18] are two of the most well-known methods in this category. The convex hull method considers the gamut to be the smallest convex polyhedron in 3D space and can be easily implemented using the Qhull method [19]. However, this method always overestimates the gamut, resulting in some unintended consequences. The parameterized shape description is provided by the alpha shape method. The alpha parameter was used to control the shape of the gamut. The resultant gamut is the same as the convex hull method in a special case where alpha is infinity. In general, alpha controls a gamut's convexity or concavity, and its value is empirical.
Segmentation-based methods are a more popular alternative to the convex hull and alpha shape methods, in which the color space is firstly segmented and a representative color is chosen for each segment. The segment maxima method [20] is one of the most popular in this category. In this method, a spherical segmentation is used, and the representative color is chosen as the extreme color for each segment, namely a color with the maximum distance to the center of the lightness axis. Using those extreme colors, other surface points can then be calculated using the piece-wise bilinear method [21] or the triangulation method [22]. The number of segments used in this method, as well as the interpolation methods used, are critical to the final output gamut surface. In addition, for uneven gamuts, an interpolation method to form a convex surface is preferred since concavity will cause some unwanted problems in real applications.
Mountain Range [23] is another popular method. The 3D color surface is regarded as a 2D function in this method, with lightness and hue being inputs and chroma value being output. The gamut in a UCS can be initially described using a dense sampling on the surfaces of an RGB cube, and other surface points can be acquired using a triangulation and interpolation algorithm [21]. Dense sampling is preferred for creating a precise surface gamut. However, a dense sampling in the RGB space cannot guarantee a dense sampling in the UCS due to the uneven nature of the RGB space. As a result, the choice of sampling points is critical for this method.
According to the preceding introduction, a majority of the current methods construct the gamut boundary using data collected from the surface of a color solid formed by the device channels, such as the RGB cube for a three-channel display. However, due to the discrepancy in space uniformity between the device space and the uniform color space, determining the sampling strategy on the device space is challenging, which has an impact on the precise description of the gamut boundary. The corresponding data points in the UCS will have an irregular distribution if the sampling points on the device space are not properly chosen. As a result, it is inevitable to introduce an interpolation method to construct surface gridding for subsequent applications, such as gamut mapping. This often resulted in an unsmoothed boundary in a UCS. In this paper, a new method to accurately describe the gamut boundary is proposed. Unlike the previously discussed algorithms, this new method samples points in the UCS and collects points within the reproducible gamut range. As a result, the complicated interpolation procedure is avoided, and a smooth gamut boundary with high precision can be guaranteed.
2. Precise gamut boundary description
The new gamut boundary descriptor for displays is presented in this section. A detailed introduction is provided here to build the boundary for the most common three-channel (RGB) displays. This method, however, can be extended for multi-chromatic displays with more than three channels, as will be discussed in the discussion section.
As is known, an RGB display can be regarded as a well-behaved additive color colorimetry system. This indicates that the display characterization model can be expressed using both colorimetric values and an Electro-Optical Transfer Function (EOTF) [24], known as the gamma function. The EOTF was used to define the luminance factor of ${L_{C,{d_C}}}$ for a digital input ${d_C}$ for each channel. For an 8-bit display, a maximum digital input of 255 leads to a maximum luminance factor of unity. This structure is given in 1.
First of all, the method to determine whether a color is within the display gamut should be introduced. The available range of the digital input is fixed, e.g., from 0 to 255 for an 8-bit display. With such a constraint, the reproducible color gamut cannot be expanded indefinitely. In other words, a point outside the color gamut will result in a digital input that is less than 0 or greater than 255. As a result, such a constraint can be used to determine whether a color is reproducible or not.
In our case, all of the sampled points are located within a UCS. So, they should be first converted into XYZ values using the reverse color appearance model. Afterwards, the XYZ values can then be transformed into the RGB values using the reverse display characterization model. It will be easy to determine whether a color is out of gamut or not based on the RGB values. This procedure is depicted in Fig. 3.
Based on these facts, a simple but effective gamut boundary descriptor can be completed in three steps.
Step 1. Coarse sampling in the UCS
A coarse sampling is performed in the UCS to accelerate the speed of boundary construction. Figure 4 illustrates a hue slice in the UCS, where all of the dots represent the sampled data points. Those points will be converted into their corresponding RGB values using the workflow shown in Fig. 3 to ensure if they are within the color gamut. A dot in red indicates that it is within the gamut, while a dot in black indicates that it is outside of the gamut. The true gamut boundary is defined in the dividing space between the outermost red dot and the innermost black dot.
Fig. 4. A hue slice in the uniform color space to show the coarse sampling. The lightness spacing is fixed at 5 units and the chroma spacing is fixed at 2 units.
Step 2. Dense sampling in the dividing space
The surface boundary can be roughly identified from Step 1, i.e., within the dividing space between the outermost red point and the innermost black point. In this step, dense sampling is performed to precisely locate the boundary.
It is clear from Fig. 5 that the precise boundary can be refined further using dense sampling. Similar to Step 1, red indicates a point within the gamut and black indicates a point outside the gamut. As a result, arbitrary precision can be obtained by controlling the sample step in the Chroma direction. The boundary is easily obtained by connecting all of the in-gamut points with the highest chroma value. However, the dashed line boundary from lightness 50 to lightness 60 is not smooth enough. This is where the cusp locates, which means a point with the highest chroma within a hue slice. And such an unsmoothed curve is caused by an insufficient sampling in the lightness direction. As a result, a final step is required to elaborate the boundary near the cusp.
Step 3. Cusp elaboration
Fig. 5. A hue slice in the uniform color space to show the dense sampling. The small dotted points represent the newly sampled data points, which are located in the dividing space between the outermost red point and the innermost black point. The dashed line represents the estimated gamut boundary.
Similarly, dense sampling is performed around the cusp, which is located in a region defined by the first three points with the highest chroma values, in this case from lightness values of 50 to 70. As is shown in Fig. 6, the cusp is chosen from among these sampled points and a precise location is achieved. The final constructed boundary is illustrated in Fig. 7.
Fig. 6. Further elaboration of the cusp region. A dense sampling is performed around the cusp to form a smoother gamut boundary.
2.1 For slices having a concaved boundary
The display gamut is known to be nearly convex, implying that for some hues, the gamut might be slightly concave. This is true for a display gamut in CIELAB color space. In this case, it is not recommended to use the maximum chroma to construct the boundary. Once step 1 or step 2 is completed, all of the points within the gamut are determined. The boundary points can then be identified as shown in Fig. 8, following steps as below:
- 1. Calculate the chroma difference between two adjacent points that have been identified as being within gamut for each lightness level.
- 2. Examine all of the chroma differences to see if any are larger than the sampling step specified in Step 1 or Step 2. A chroma difference greater than the sampling step indicates a gap between two adjacent points, denoting that this part of the gamut is concave. If not, the gamut in this lightness level is not concave.
- 3. Keep the first two points on either side of the gap. These two points are on the boundary of the gamut.
- 4. Iterate the above step for all the lightness levels and leave all the points on the boundary.
Fig. 8. Figure 8. Method to construct a slightly concaved boundary. The chroma difference is larger than the sampling step, indicating the boundary is concave in that region.
After collecting all of the points on the boundary, they can be connected to form the final boundary. The pitching angle, which is defined by the vector from the center of the lightness axis to a point on the boundary and the y-axis, can be used to determine the order of these points.
3. Experimental validation
The proposed method is validated using the sRGB and DCI-P3 standard system colorimetry. Both of them are imagery-perfect additive color colorimetry systems and also the de facto standards in the imaging industry. The gamut of DCI-P3 is much larger than sRGB and can be regarded as a good representative of commercial displays with a wider gamut. Two commonly used gamut boundary descriptors, namely the SMGBD method and the Mountain Range method, were also included as references to fully investigate the performance of the proposed method.
As is stated in Section 2, the standard characterization model for an sRGB system includes both colorimetric values and an Electro-Optical Transfer Function. The former component is given in Eq. (1) and the latter component is given in Eq. (2). A conversion relationship between the digital inputs and the tristimulus outputs was established by combining those two equations.
It should be noted that the number of sampling points was critical to the precision of the gamut boundary construction. A 20 by 20 grid was used for each of the six faces of the RGB cube in both the SMGBD and Mountain Range methods, resulting in a total of 2168 sampling points. This is consistent with their initial recommendations. However, the proposed method had a completely different structure than the previous two methods, indicating that defining the number of sampling points was difficult. As a result, the sampling step was used for the proposed method and it was controlled unity for lightness and 0.1 units for chroma (dense sampling).
UCS is another important factor influencing the performance of a GBD. The most commonly used UCS, CIELAB color space, and the advanced UCS, CAM02-UCS, were both included in this study.
For the testing data, a 33 by 33 grid was used for each face of the RGB cube, and the points used to construct the boundary were removed, yielding a total of 6138 colors. This is a relatively large dataset that is considered to be capable of reflecting the performance of each method.
Table 1 shows the testing results. There are two types of color difference (CD) reported. As shown in Fig. 9, Pt represents a test point from the RGB cube surfaces and its distance to the constructed boundary reflects the performance of a GBD algorithm. CD1 represents the distance between Pc1 and Pt, and CD2 represents the distance between Pc2 and Pt. CD2 is generally larger than CD1 for most of the cases. Generally, GBD is developed for gamut mapping applications. And the typical mapping directions are towards a point in the lightness axis or the center of the lightness axis (L* of 50), which corresponds to CD1 or CD2 respectively. As shown in Table 1, all methods gave a reasonable prediction error, and it is clear that the proposed method always gave the smallest mean predicted color difference, indicating its performance is the best among all these three GBDs tested.
Fig. 9. Illustration of the color differences presented in Table 1. Po represents the center of the lightness, Pt represents the test point, Pc1 represents the cross point of line Pt PO and the constructed gamut boundary, and Pc2 represents the cross point of a horizontal line and the constructed gamut boundary. The distance between Pc1 and Pt is the first type of color difference (CD1) and the distance between Pc2 and Pt is the second type of color difference (CD2). The plus mark before CD1 and CD2 means Pt is outside the boundary
Table 1. The calculated color difference between predicted and measured data points. Mean1 and Mean2 represent the mean of CD1 and CD2, respectively. Max1 and Max2 represent the maximum of CD1 and CD2, respectively.
It is worth noting that all three GBDs had a very high Max2 value in CIELAB color space. The reason is demonstrated in Fig. 10. The upper region of the boundary is shown to be near the horizontal line, making it difficult to allocate the intersection point. As a result, the calculated difference will be large, resulting in a larger error.
The advantages of this method include not only high precision but also smooth output and the preservation of the true shape of the boundary. As illustrated in Fig. 11, the proposed method produces the smoothest gamut boundary of the three methods studied. Furthermore, the proposed method was the only one that detected slight concavity. The other two GBDs, on the other hand, are poor predictors of concavity. This is not surprising given that these two ‘forward’ methods sample in RGB space and interpolate in UCS. Such minor concavity is likely to be overlooked if the sampling is not well-positioned, particularly when interpolation is performed in the UCS. It is preferable if all of the sampled points in the UCS are evenly spaced to construct a precise boundary. However, determining the sampling points in the RGB space is difficult because there is no direct connection between the RGB values and their corresponding color coordinates in a UCS. This is not an issue for the ‘reverse’ method, which performs sampling in the UCS. This again demonstrates the superiority of the proposed method.
Fig. 10. Since the boundary in the upper region is near horizontal, there is no doubt that the CD2 is relatively large.
Fig. 11. The comparison between the proposed method and the other two studied methods. Ground truth boundary is provided as a reference. It is obtained by an exhaustive sampling of the RGB cube surfaces and a limit to the hue range (±0.2° at the hue slice).
4. Discussion
The proposed method presents a new inspection into gamut boundary descriptors and is demonstrated to provide a high description precision when compared to the other two commonly used methods, namely the SMGBD method and the Mountain Range method. As distinct from traditional methods, the new GBD is unlikely to be affected by boundary concavity and can achieve arbitrary precision as long as an accurate display characterization model is provided. The derived gamut boundary is always smooth and is expressed using the outmost color samples in a constant hue slice, indicating that it is easily applicable to other gamut-related applications such as gamut mapping or gamut volume estimation. Moreover, the implementation of this method is not complicated, making it a potential candidate for real-world applications.
To fully comprehend this method, its limitations should also be discussed. To begin, the precision of this method is determined by two factors: the display characterization model and the judgment method used to determine whether a point is within the gamut or not. A precise description of the gamut necessitates dense sampling in the UCS, and all sampled data points must be judged to see if they fall within the gamut. At this moment, this is accomplished by converting back into the RGB space and judging by the corresponding RGB values. This is obviously true for an additive colorimetry system that is well-behaved.
However, the performance for some low-end displays is unsatisfactory, and for those displays, a 3D-LUT is a common solution for color control. As a result, determining whether a point is within the gamut using its RGB values will be difficult. For those cases, the following method is applied based on color difference.
- 1. Consider a color point (p1) that has been sampled in a UCS. The reverse 3D-LUT can be used to convert it back to the XYZ value and then to the RGB value.
- 2. The forward 3D-LUT can then be used to obtain the predicated color attributes (p2) with this RGB value. As a result, a color difference between p1 and p2 is obtained. P1 is within the color gamut only if the color difference is small enough, within a preset tolerance.
The computation workload is relatively large for the proposed method. Though tedious effort on interpolation is avoided, the conversion of color space is inevitable. However, this should be not an issue for modern computers.
5. Conclusion
A new simple yet more precise gamut boundary descriptor is proposed. It consists of two major components: dense sampling in the uniform color space and a way to determine whether or not the sampling points are within the gamut. Its performance was verified together with two widely used GBDs, namely the SMGBD and the mountain range methods. The method is promising and can be used in real-world applications.
Funding
Fundamental Research Funds for the Provincial Universities of Zhejiang (GK219909299001-019); National Natural Science Foundation of China (61775190).
Acknowledgement
The authors would like to acknowledge supports from TUV Rheinland (Shanghai) Co., Ltd..
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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