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Abstract
Based on the difference between multi-primary displays (MPDs) and three-primary displays, we propose a new definition for evaluating the color gamut volume (CGV) to explore the upper limit of MPDs, which could theoretically represent all colors that MPDs can display. The proposed definition corrects the defects in the L*a*b* color space that arise when calculating the CGV of MPDs. In view of the high computational complexity of this method, we propose a simplified scheme with a small margin of error. Additionally, we verify the new definition with experiments on a six-primary projector. This method is helpful in guiding the selection of light sources and the evaluation of MPDs, and also has great reference value to calculate the target gamut for gamut mapping in MPDs.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Approximately 90% of human perception information is received through the eyes, and color perception forms an important part of this [1]. At the beginning of the last century, Schrodinger et al. proposed that human color perception comes from three types of visual cells, which perceive red, green, and blue colors [2,3]. This implies that humans perceive colors in three dimensions. To describe color quantitatively, CIE proposed the CIE1931XYZ color space and simplified it into an xy chromaticity diagram [4]. Consequently, to describe the color of the human eye perception accurately, the CIE1976 L*a*b* color space was proposed, which is currently the most widely used uniform color space [4]. The gamut in the L*a*b* color space is called color gamut volume (CGV).
Pointer first measured colors in nature, and obtained a set of real surface colors, considered to be the color gamut of the natural environment [5]. To describe the colors that a display system can display accurately, MacAdam proposed a theory to calculate the most saturated color that a light source can achieve, called the optimal color. The optimal color can also be recognized as the color gamut boundary of a display system [6,7]. Masaoka later improved the computing speed of MacAdam’s algorithm [8]. Wang further improved MacAdam’s method for calculating the CGV of three-primary display systems. Based on the current three-primary display system, Wang used the characteristic that the luminance of each primary can be adjusted individually and proposed a method to calculate the CGV by controlling the luminances of the three primaries, replacing the previous method of controlling the reflectivity of different wavelengths [9]. Ouyang proposed a method for measuring the CGV boundary of a display system by directly measuring specific feature points [10]. These studies guarantee accurate calculations and CGV measurements, providing a better understanding of the theoretical and practical CGVs of display systems. However, a CGV usually exhibits a complex shape in the L*a*b* color space, which requires observers to change their views to comprehend the 3D solid. Therefore, based on the L*a*b* space, Masaoka developed a 2D gamut ring for improved visualization [11].
Currently, the most widely used display system is based on the three primaries. However, in the xy chromaticity diagram, three-primary display systems have evident limitations, such as the latest Rec. 2020 color standard, which covers approximately 64% of the total area only [12,13,14]. To increase the gamut of the display system, it is necessary to increase the number of primary colors to approach the boundary, which results in a multi-primary display (MPD) [15]. The MPDs imply that none of the primaries are obtained by mixing other primaries in the system; otherwise, they are termed multichromatic displays [16]. Early MPDs predominantly used LEDs or light bulbs as light sources, so the growth of gamut brought by the multi-primaries scheme was not noticeable, due to low monochromatic saturation [14]. And now, lasers are gradually entering a wide range of applications in display light sources. Using laser sources, properly designed MPDs can provide a nearly ultimate gamut, which can maximize the advantages of MPDs [14]. Lin proposed and fabricate a four-primary-color (MPC) quantum-dot down-converting film composed of red, yellowish green, bluish green, and blue subpixels, which could achieve four-primary display. A verification platform was built up using a laser projector, and the measured results show that the quantum-dot film can expand the chromaticity gamut area to 118.60% of Rec. 2020 and can cover the entire Pointer’s gamut [17]. Wen proposed a specific calculation method for four primaries and obtained the relative primary luminances and CGVs for four-primary displays. However, this method cannot be generalized to more primaries [18]. Zhu proposed a simplified calculation method suitable for six-primary display systems and demonstrated it using a six-primary display prototype [19]. In our previous work, we demonstrated a general method that can obtain the luminance of all primaries of MPDs with different numbers of primaries and can further obtained the CGVs [20]. This method provides all luminance solutions for primaries in MPDs and can be extended to MPDs with any number of primaries. Combined with the calculation and measurement of CGVs, the theoretical framework of MPDs has been significantly developed.
In the CIE1931 xy chromaticity diagram, it is clear that the gamut area increases gradually with an increasing number of primaries. If an MPD consists of a three-primary display and newly added primaries, the gamut coverage of the original three-primary display should be exactly 100% because the MPD is extended from it; therefore, it should contain the complete color rendering capability, as shown in Fig. 1(a). However, in our previous work on MPDs, a counterintuitive phenomenon was observed in the L*a*b* color space [20]. When new primaries were gradually added based on the three-primary display, although the CGV value increased, some parts of the gamut shrunk. In other words, increasing the number of primaries reduces some parts of the gamut. This phenomenon is common in the gamut calculation of MPDs in the L*a*b* color space. After a series of analyses, we found that when adding a new primary in the display system, in order to ensure that the total lightness in the L*a*b* color space remains at 100, the luminances of the previous primaries must be reduced to match it. Because of the reduction of the previous primaries, some colors available in the three-primary display could not be achieved in the newly built MPDs. This implies that the abilities of the original primaries are sacrificed to achieve the goal of MPDs, which is an unacceptable waste for design and optimization. This indicated a flaw in the calculation method.
To solve the above problems, we propose a new definition to explore the upper limit of CGVs in MPDs and demonstrate it using an ideal four-primary and six-primary display system. This can also be extended to N-primary (N > 3) display systems. To reduce the computational time, we propose a simplified scheme to calculate the CGVs suitable for this case. This new definition can also solve the problem in the L*a*b* color space when calculating the CGVs of the MPDs. Combined with the calculation of CGVs, it can be used to guide the selection of wavelength, spectrum width, and luminance of primaries with various light sources, such as LEDs, OLEDs, and lasers. In this study, we begin with the three-primary display system by gradually adding new primaries to form an MPD to observe the changes in CGVs. For simplification, the discussion in this paper is limited to well-behaved additive color colorimetry systems [21]. To explore the basic properties of the display system, the color adaptation transformation of color appearance brought by different illuminants is not considered.
2. Simulation
2.1 New definition of CGV for MPDs
The calculation was demonstrated with ideal three-, four-, and six-primary display systems. The chromatic coordinates used are listed in Table 1. The parameters of the original three-primary display were from Rec. 2020 [12], called the Rec. 2020 display system. The chromatic coordinates of the white point were set to (0.3127, 0.3290) in this study, corresponding to Rec. 2020.
For three-primary displays, according to the CIE 1931 standard colorimetric system, the tristimulus values were calculated as follows:
Table 1. Chromatic coordinates in the simulation.
The ratio of the Y stimuli of the three primaries for mixing white can subsequently be obtained, which is the ratio of the luminances of the three primaries. In common calculations, ${Y_{white}}$ is set to 100. Based on the method used in our previous work [20], we assumed that the Y stimulus of the newly add primaries was ${k_j}Y_{({j + 3} )}^{\prime}$ (j $\ge $ 1), and $Y_{({j + 3} )}^{\prime}$ was set to a constant. The newly add primaries could be represented with the original three primaries, as follows:
This set of inequalities is the solution space of a MPD and the dimensions of the space is (N-3). The Y stimuli of primaries corresponding to the points (${k_1}$, ${k_2}$, …, ${k_j}$, …) in the solution space are:
The Y stimuli can be proportionately scaled up or down as appropriate to match the luminances of real display systems. It is worth mentioning that the solution space includes all the possible solutions in MPDs, because a point outside the solution space would cause the Y stimuli of some primaries to be less than zero. Meanwhile, the selection of the initial three primaries does not affect the final result. Although the solution space will change, the Y stimuli of primaries corresponding to the point in solution space is the same.
According to the parameters of the three-primary display in Table 1, Eq. (17) provides the Y stimuli of the three-primary display shown in Table 1.
For the four-primary display in Table 1, set the $Y_{G2}^{\prime}$ to 100 for simplicity, and according to Eqs. (7)–(12), there is:
According to Eq. (15), the solution space is:
The Y stimuli are:
The CGV value varied with ${k_1}$, as shown in Fig. 2(a). Figure 2(b) shows the maximum CGV when ${k_1}$ = 0.38, which is 2115700. The method used to calculate the CGV was proposed by Masaoka [22]. All the simulations and calculations in this paper were proceeded in MATLAB.
Fig. 2. (a) Solution space and corresponding CGV values of the four-primary display; (b) maximum CGV when ${k_1}$ = 0.38; (c) CGVs of the three-primary (black) and four-primary display in Table 1, when ${k_1}$ = 0.38; (d) Gamut rings when L* = 100, where the angles of the straight borderlines correspond to the CIELAB hue angles.
As mentioned above, to maintain the lightness limit at 100 in the L*a*b* color space, the approach for calculating the CGV can lead to a problem in which the Y stimuli of the previous primaries must be reduced to match the newly added primaries; this implies that some colors available in the three-primary display cannot be achieved in MPDs. As shown in Fig. 2(c) and (d), it is clear that some part of the volume in the three-primary display is not within the CGV of the four-primary display.
To solve this problem, we propose that, for a certain MPD, its CGV equals the superposition of all CGVs that correspond to all points in the solution space, which implies that its whole CGV should include all possible colors the system can theoretically obtain. For simplicity, we refer to this as the superposition of all CGVs (SCGVs). It should be noted that when calculating SCGV, we need to take into account the points on the boundary of the solution space. So Eq. (15) becomes:
Although the point on the boundary means that the Y stimuli of some primaries is 0, which is not fit the definition of MPDs, these cases still belong to the situation that the system can achieve. The SCGV value of the four-primary display in Table 1 is 2196600, as shown in Fig. 3(a), which is 3.8% larger than that in Fig. 2(b). Here, the SCGV was obtained using ${k_1}$ values at intervals of 0.02. For a more precise result, we can take smaller interval values of ${k_1}$, and the value of the SCGV would simultaneously grow larger. To be more intuitive, 2D gamut rings were used, as shown in Fig. 3(b). It can be seen that in a specific set of solutions of the four-primary display, the volume of the overall gamut and volume of the green part increased significantly compared with the Rec. 2020 display system when adding a green primary; however, the gamut of some non-green parts decreased, which fits our previous description. The SCGV includes the gamut of the three primary colors and gamut of ${k_1}$ = 0.38, which is in line with our expectations and solve the problem in the L*a*b* color space when calculating the CGVs of the MPDs. It can be seen that, SCGVs were actually caused by multiple choices of primary luminances in MPDs.
Fig. 3. (a) the SCGV; (b) comparison of gamut rings when L* = 100, where the angles of the straight borderlines correspond to the CIELAB hue angles.
The six-primary display was calculated in the same manner. The three newly added primaries R1, G2, and B2 can also be represented by the original three primaries R2, G1, and B1, as follows:
Using ${k_1}$, ${k_2}$, and ${k_3}$ to control the Y stimuli of the newly added primaries, the solution space is:
The Y stimuli of all primaries are as follows:
2.2 Simplified scheme to obtain the SCGV
It is computationally expensive to obtain the SCGV of MPDs by traversing the entire solution space. Therefore, a simplified calculation scheme is proposed, which can quickly obtain the outline of the SCGV. According to the method proposed by Ouyang [10], the gamut boundaries of MPDs correspond to each channel digital information scalar, and the vertices of the CGV should be equal to the maximum luminance of each primary and their superposition. This is suitable for a specific MPD with a particular set of primary luminances, but it is not suitable in this case because the boundaries of the SCGV originate from the superposition of different CGVs with different primary luminance sets. A specific six-primary display system has been calculated above; therefore, the same system is used for further demonstration of the scheme.
Fig. 4. (a) CGV of the six-primary display when ${k_1}$ = 0.42, ${k_2}$ = 0.22, and ${k_3}$ = 0.01; (b) SCGV of the six-primary display, where the defect in the bottom is due to the reduction of the calculation accuracy to balance the calculation time; (c) solution space of the six-primary display; (d) comparison of gamut rings when L* = 100, where the angles of the straight borderlines correspond to the CIELAB hue angles.
Because the SCGV is obtained in the entire solution space, its vertices should be equal to the maximum Y stimuli that can be achieved in the solution space. According to Eqs. (22)–(24), the maximum Y stimuli of each primary and their superpositions are calculated in the solution space, instead of the channel digital information scalar. The maximum Y stimuli and corresponding chromaticity coordinates of each primary and their superpositions are listed in Table 2. The values of ${k_1}$, ${k_2}$, and ${k_3}$ corresponding to the maximum Y stimuli are also provided. The dash in Table 2 indicates that the value of the parameter did not affect the results of Y stimuli.
Table 2. Maximum Y stimuli of each primary and their superpositions in solution space
The results are shown in Fig. 5(a). The connection rule of contour lines is from the peak Y stimuli of a single primary to the peak Y stimuli of the sum of the two primaries containing the original primary, and then to the peak Y stimuli of the sum of the three primaries containing the original two primaries, and so on, finally reaching a maximum Y stimulus of 100. This follows the basic rules of color mixing. Figure 5(b) shows the contour lines transforming to the L*a*b* color space. The volume of the black outline is 2553056, which is only 1.8% larger than the theoretical value. Additionally, the outline of the boundary matches well with the SCGV, as shown in Fig. 5(c)(d)(e). Therefore, this simplified method is suitable for SCGV calculation.
Fig. 5. (a) SCGV in xy color space; (b) SCGV in the L*a*b* color space, where the black line represents the gamut drawn based on the data in Table 2, and the blue volume is the theoretical SCGV from Fig. 4(b); (c), (d), and (e) are the three views of (b).
3. Experiment
To verify the existence of SCGV, a six-primary laser projection display system was demonstrated, as shown in Fig. 6. The chromatic coordinates of all primaries are shown in Table 3. The display distance was 6.3 m, and the image size was 3.50 ${\times} $ 1.96 m2. The screen was a white wall, and the light sources were laser diodes. All the chromaticity coordinates and luminances in the experiments were measured using a spectral color luminance meter (SRC-600, EVERFINE, China). All of our measurements, including chromaticity coordinates, were based on the reflection light from the white wall, so there was no need to take into account the different reflectivity of the wall to different colors of light. The measurements were performed in a dark room to eliminate the influence of ambient light. The measured chromaticity coordinates and the maximum output luminances of primaries are listed in Table 3. In order to distinguish, we used the value of Y stimuli to calculate the CGV and SCGV in L*a*b* color space, and they were normalized based on the upper limit of L*. The values of luminances were used to refer to the primary luminance settings in the projection system, which is proportional to the Y stimuli.
Fig. 6. (a) Six-primary display system; (b) measured chromatic coordinates of the six primaries; (c) the display picture [23].
Table 3. chromaticity coordinates, and maximum output luminances of laser diodes
To verify the above methods, we calculated the solution space and the theoretical SCGV, then used the six-primary prototype to prove that the SCGV exists. R2, G2, and B2 were chosen as the initial three primaries to calculate the Y stimuli of the remaining primaries. According to Eqs. (1)–(16), the coordinates of the white point were set to D65, and there is:
The solution space is:
The Y stimuli of the primaries are:
After traversing the solution space, we obtained the theoretical SCGV, shown in Fig. 8 drawn with blue lines. The SCGV value is 2261000 with ${k_1}$ values in intervals of 0.02, ${k_2}$ values in intervals of 0.04, and ${k_3}$ values in intervals of 0.01.
Fig. 8. (a) SCGV in the L*a*b* color space, where the black line represents the gamut drawn based on the measured data in Table 5, and the blue volume is the theoretical SCGV of the six-primary display prototype; (b), (c), and (d) are the three views of (a).
We then used the simplified scheme to obtain the outline of the SCGV. The maximum Y stimuli of the six primaries in the whole solution space are shown in Table 4. Considering the working conditions of the laser diodes in Table 3, the Y stimuli were decreased by 0.5594 times to stabilize the laser diodes and make the G2 laser diodes reach their maximum luminance, as shown in Table 4. In the experiment, we adjusted the electric current of the laser diodes and measured the luminance with the SRC-600 meter until the adjusted values were close to the theoretical values. The deviation was mainly caused by the stability of the laser diodes.
Table 4. Theoretically maximum Y stimuli and adjusted set luminance of the six-primary display system
After traversing the solution space, the theoretical vertices of the SCGV were obtained, as shown in Table 5. According to the corresponding k value, we calculated the theoretical luminances of the six primaries at this time and set them in the prototype, then measured the color coordinates, as shown in Table 5. After increasing the measured luminances by (1/0.5594) times to match the Y stimuli, the contour of the SCGV in the L*a*b* color space is shown in Fig. 8 with black lines. The volume of the black outline is 2429800, which is 7.4% larger than the theoretical result 2261000. The reason for the large gap between experimental and theoretical calculations is that we reduced the calculation accuracy to improve the calculation speed, and only 1148 CGVs in the solution space were calculated to get the SCGV. For comparison, in the simulation part above, we calculated over 4500 CGVs and get the 1.8% margin of error. Additionally, the outline of the boundary matches well with the SCGV. Therefore, the accuracy of the experiment is acceptable. This proves the existence of SCGVs in MPDs and also proves the correctness of our simplification scheme to obtain SCGVs.
Table 5. Maximum luminances of each primary and their superpositions in solution space
4. Discussion
The clear difference between three-primary displays and MPDs has been widely discussed. Additionally, SCGVs and CGVs exhibit fundamental differences in MPDs. The original CGV calculation method still uses the concept of a three-primary display, ignoring the characteristics of MPDs with more degrees of freedom. Therefore, a new perspective is required for studying MPDs. The solution spaces calculated in this study were all under ideal conditions, in which the luminances of the primaries were not restricted. In practical applications, there is usually an upper limit on the luminance of each primary, which implies that the solution space has more constraints. However, this does not change the overall picture; the SCGV calculated according to the solution space still represents the total set of colors that the system can display. The reason we choose to modify the defect in L*a*b* color space rather than select other uniform color spaces is that the L*a*b* color space is still the most widely used uniform color space at present, and it is very important to keep the total lightness the same in the comparison of CGV in different display systems. Otherwise, the display system with higher luminance is almost always the one with larger CGV. Therefore, we still choose to use the L*a*b* color space. Similarly, this scheme to calculate SCGVs can be applied to other uniform color spaces. The scheme for calculating CGVs in this paper comes from Masaoka [22] and all coordinates of the volume are first calculated in CIEXYZ color space and then converted to other color spaces. Therefore, SCGVs in other color spaces can be obtained by directly converting the coordinates in CIEXYZ color space. If the SCGV is adopted, more advanced gamut mapping methods are required to achieve color gamut. Currently, there are no mature MPDs that can achieve this, and that is the future direction of our work.
5. Conclusion
Based on the difference between the three-primary displays and MPDs, we proposed a new definition of the CGV of MPDs, named SCGV, which could represent the total colors the MPDs could display. The calculation of the SCGVs is demonstrated with an ideal four-primary and six-primary display system, which is helpful in guiding the evaluation of MPDs. This definition can also solve the problem of the L*a*b* color space when calculating the CGVs of MPDs. In view of the high computational cost of this method, a simplified scheme that can quickly obtain the contour and volume values of the SCGV with a small margin of error was proposed. We further proved the existence of SCGV and the correctness of the simplified scheme by experiments. This method could also be extended to N-primary display systems (N > 3). Combined with the calculation of SCGVs, it can be used to guide the selection of wavelength, spectrum width, and luminance of primaries with various light sources, such as LEDs, OLEDs, and lasers. In future work, we will explore other applications of SCGVs in different uniform color spaces. On this basis, a new gamut mapping theory suitable for SCGVs will be further on the schedule.
Funding
National Key Research and Development Program of China (2021YFF0307804).
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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