1. Introduction

The human sensation of color is created from three types of photoreceptors, and all color-encoding systems are derived from this fundamental signal. All color systems employed for image-color encoding are three-dimensional (3D). Thus, all colors can be described accurately with each of its dimensions [1,2].

For visualization, two-dimensional (2D) CIE chromaticity diagrams have been created to map the chromaticity of light sources [3]. There are several standard colorimetric systems, described with an xy chromaticity diagram, representing the chromaticity gamut area [3] of current display devices, such as: Adobe RGB [4] as a standard in professional color processing, Rec. 709 for HDTV [5], and Rec. 2020 for UHDTV [6], as shown in Fig. 1. According to Grassmann's law, any color can be represented by three non-collinear primaries mixes in the chromaticity diagram. The closed shape connected by the chromaticity coordinates of multiple non-collinear primary colors is called the chromaticity gamut area because the colors in the region can be expressed by mixing the primary colors [3], as shown in Fig. 1. In particular, Rec. 2020 provides the largest chromaticity gamut area that covers other major standard colorimetric systems, and a chromaticity gamut area that approaches the Rec. 2020 standard is regarded as a wide-color gamut. These standards were adopted by display engineers because light sources are mixed in a linear fashion on the diagram. The possible chromaticity of a three-component system can be easily mapped if the three components are additive in nature. When using the chromaticity diagram to describe the color capability, the white luminance of the display is assumed to be the sum of its primaries; this also applies to multi-color displays. However, the chromaticity diagram does not map the full color capability, but only two components. When three-component input signals are passed through color management systems built into the display, the chromaticity diagram may not represent the true chromaticity gamut area of the display system [7]. Additionally, a modern display may have more than three primaries, such as RGBW, RGBY, or RRGGBB systems [8,9,10]. The actual color gamut of such display systems cannot be deduced from a chromaticity diagram. Therefore, a more robust method is required to measure the color range capability of display systems.

 

Fig. 1. Several standard colorimetric systems

Download Full Size | PDF

`The CIE 1976 L*a*b* color space provides proper visualization of the actual 3D color gamut volume [3]. The CIE recommends the use of the CIE 1976 L*a*b* color space for the evaluation of display color gamut volume. This color space is perceptually uniform in chromaticity and demonstrates good homogeneity across all the three axes. Thus, distances in any direction and at any length scale are perceptually uniform. This is key to understanding the range of colors existing in a particular color gamut volume. A display color gamut volume can be rendered in three dimensions as a hull in the CIE 1976 L*a*b* color space. By measuring the rendered colors produced by the code values on the primary signal gamut surface, we can create such a hull and measure its volume. This single value, the CIE 1976 L*a*b* color gamut volume, provides the true size of the color gamut relative to each other. However, determining the size of color gamut volume is not everything. To better understand color capability, it is valuable to know where the range of hue, saturation, and brightness fall. To do this, it is necessary to inspect the 3D shape of the gamut hull in CIE 1976 L*a*b* from many perspectives [7]. Masaoka developed a robust device independent color sampling and CGV computation that works with both normal and irregular gamut hulls and can efficiently produce the new 2D visualization “Gamut Rings” that surpasses the information content of all previous 2D color capability representations [10]. In the proposed 2D plot of gamut rings, the areal dimension corresponds to the volume within an arbitrary luminance and hue range, enabling an intuitive approach to grasp the color gamut volume of a complex shape in a quantitative manner. Gamut rings can also be combined with a reference color space to allow the proportional visualization of the color gamut volume.

In 1935, MacAdam proposed one of the first methods to obtain the theoretical maximum chromaticity gamut area of object colors under a particular illuminant; currently known as MacAdam limits [11,12]. However, limited by time, he did not regard the gamut as a solid to calculate its volume. The calculation of the gamut of real surface colors was carried out by Pointer [13]. Masaoka [14] improved MacAdam’s algorithm and increased its efficiency. Wang et al. [15] further improved MacAdam's method to calculate the color gamut volume for a display system. For current common display systems, the luminance of each primary color can be adjusted individually. Wang used this characteristic and proposed a method to calculate the color gamut volume by controlling the luminances of three primaries, replacing the previous way of controlling the reflectivity of different wavelengths. Ou-Yang and Huang [16] proposed methods to measure the color gamut volume of an actual display system. He proposed an approach to directly obtain the colors on the color gamut volume surface. By measuring chromaticity coordinates and luminances of these colors, he could directly draw the outline of the color gamut volume. These works enable us to calculate and measure the color gamut volume of display systems, which provides a better theoretical and practical understanding of the color gamut volume.

To improve the viewing experience and see more vivid colors, one effective way is to enlarge the color gamut volume of display systems. There are two methods for enlarging the color gamut volume to display more colors. One method is widening three RGB primary display to exhibit a larger color gamut volume by means of high-chromaticity (pure) RGB colors [17]. In particular, the color gamut volume can be enlarged using highly saturated primaries. However, using this method, the color gamut volume is still limited. According to the calculation results, the maximum coverage of triangles is only approximately 73% of a xy chromaticity diagram [18]. The other method is employing a four-or-more primary display by adding multiple primaries. Masaoka et al. [19] proposed that displays based on the mixing of RGB with additive non-primary colors should be called “multi-chromatic” displays. He examined the color gamut volume for a variety of simulated and measured multi-chromatic displays using combinations of white and/or secondary color channels, such as cyan, magenta, and yellow. However, while his method increases the mixed color of the three primary colors, it reduces the luminances of the three primary colors according to the proportion. As he emphasized, such display can only be called “multi-chromatic,” not “multi-primary”. In contrast, other studies, including this study, considered the case with new primary colors that cannot be generated by mixing the first three primary colors, implying more than three primaries are used in the display system, which can be called multi-primary displays (MPDs) [8,9,20]. Employing MPDs to extend the color gamut volume is more advantageous than narrow-spectrum trichromatic display systems. As Song et al. [18] described, six-primary display systems can reach more than 95% of the xy chromaticity diagrams. However, the peak luminance setting of the primaries in MPDs has not been properly discussed. When facing four or more primary numbers, the luminance setting is not unique owing to metamerism. This problem prevents us from fully describing MPDs.

Wen [9] proposed a method to obtain the relative primary luminances and the color gamut volume for four-primary displays. However, the method could not be generalized to more primaries. Ou-Yang and Huang [20] proposed a preliminary construction method to obtain the gamut solid of MPDs: by selecting specific brightness to construct specific cross-section of color gamut volume, the whole solid can be obtained. However, the result is limited and too imprecise to be of practical guidance; furthermore, the problem of determining the primary brightness is not addressed. Song et al. [18] proposed a calculation method based on a computer algorithm to determine the optimization of the primary wavelength and lightness. However, when selecting wavelength parameters, they used the coverage rate of the CIE xy chromaticity diagram as a basis, which was not in line with the original intention of their study of adopting the CIE L*a*b* color space. Furthermore, the exhaustive method is laborious and unsuitable for practical analysis.

Sharmaand Rodríguez-Pardo [21] and Rodríguez-Pardo and Sharma [22] provided a more complete demonstration of MPD design modeling, color management gamut, and color control representation to provide a unified comprehensive framework. The results provided not only a theoretical framework interlinking the geometry of the tristimulus gamut with the geometry of the metameric control set, but also to connect to applications in color reproduction, providing insight into alternative strategies that have been proposed for MPD color control. This work has great significance for computer image processing. However, CIE XYZ color space is not a good color space to describe the color gamut volume because it is not uniform for human eyes. The sides of the polytopes and the vectors in CIE XYZ color space will become irregularly curved in a uniform color space such as CIE L*a*b* color space; therefore, the transformation process will be too complex to draw a conclusion from. To obtain an evaluation of the performance of the display system consistent with the human eyes, such as the color gamut volume, it is necessary to use a uniform color space.

Based on previous studies, the peak luminance of all the primaries should mix to a white color, which is a necessary limitation for MPDs. After confirming the coordinates of the white point, the selection of the luminance setting of the light source according to the needs is an important research topic. In most of the above articles, the authors directly discussed the gamut property of MPDs or the color management of the system while ignoring the parameter setting of the light source, or just give a special solution of the light source parameters to illustrate. However, to build an MPD, the most fundamental problem is how to choose the wavelength spectrum width and the luminances of the light sources to meet the specific needs. The choices are usually diversified, which suggests the general solution has not been well discussed. Consider it from a simple point of view: choosing a light source with higher than the actual requirement can add unnecessary cost and difficulty to the design of the optical path. At present, the relevant literatures have not given a complete theoretical description. Further, the physical limit of the whole system can be obtained only after the light source is determined; on this basis, we can further explore the properties of MPDs.

In this study, a new and concise method for calculating the peak luminance of the primaries was introduced under the premise of ensuring additivity. Provided the chromaticity coordinates of primaries and white points, we can obtain all possible luminance combinations of primaries within a few steps; the display color gamut volume can be further calculated in the CIE L*a*b* color space. By optimizing the peak luminance of each primary, the theoretical maximum color gamut volume, or the color gamut volume satisfying specific requirements, can be obtained. As an example, we demonstrated the calculation process of luminance combinations from four- to six-primary display systems and the corresponding color gamut volumes. Furthermore, the algorithm was extended to N primaries (N > 6), and a general representation was established. Using this method, we have built a six-primary display system and use it to verify the validity of the method. Combined with the calculation of color gamut volumes, the theory can be used to guide the selection of wavelength, spectrum width, and luminance settings of primaries. This method is also suitable for display systems with common types of light sources, such as LEDs, OLEDs, and lasers.

2. Method

This is a theoretical framework for MPDs. Our method is based on the three-primary display, and the MPD is obtained by gradually adding more primaries. For the three-primary display, according to the CIE 1931 standard colorimetric system, the trichromatic value of the mixed color is calculated as follows:

In common calculations, the mixed color is white. After determining the chromaticity coordinates of the primaries and white point, we calculate the ratio of the Y stimulus of the three primaries for the next step.

When the target chromaticity coordinate is inside the triangle formed by the coordinates of the three primaries in the xy chromaticity diagram, the peak luminances of the three primaries are all positive. However, when the target color is outside the triangle, the luminances of the three primary colors may have negative values. Therefore, in mathematical form, the fourth primary color can be represented by the first three primaries. Setting its luminance as a constant Y₁, we can obtain:

Combined with Eq. (1) and Eq. (2), when the chromaticity coordinates of the first three primaries and the fourth primary is known, $Y_{red}^{\prime}$, $Y_{green}^{\prime}$ and $Y_{blue}^{\prime}$ can be calculated. Note that some of these three values may be negative. Adjusting the luminance of the fourth base color with the coefficient k₁ and combined with Eq. (3), we obtain:

The implication of Eq. (5) and Eq. (6) is that, to increase the luminance of the fourth primary, the luminances of the first three primaries will be reduced proportionately. In Eq. (6), each term in parentheses and k₁Y₁ are considered as the luminance of the four primaries. The peak luminance of each primary needs to be greater than zero if a four-primary display is to be achieved, and is:

Each value of k₁ corresponds to a set of four primary displays with the same white point setting. With each value of k₁, the peak luminance of the four primaries is:

The luminance of the fourth primaries can be scaled up or down proportionately as appropriate with k₁.

Further, for a five-primary display, its luminance Y₂ is set as a constant for calculation; similarly, with Eq. (1) and Eq. (2), the fifth primary can be represented by the first three primaries as:

Adjusting the luminance of the fifth primary with the coefficient k₂ and combined with Eq. (5), we obtain:

The peak luminance of all five primaries needs to be greater than zero in Eq. (11) if a five-primary display is to be achieved:

Obviously, the corresponding solution region is a closed two-dimensional graph. The horizontal and vertical axes are k₁ and k₂, respectively.

For a six-primary display, similarly, calculation is done based on the above five-primary display. The calculation of the parameters is consistent with the above:

According to the above experience, we can directly write the constraint conditions without derivation, and the parameters required are as follows:

The solution region is a solid in three-dimensional space, and all coordinates in the region (k₁, k₂, k₃) satisfy the condition of a six-primary display. The peak luminance of each primary is expressed as:

In the case of more primaries display (N > 6), the final constraint condition should be N, and the result should be a (N - 3)-dimensional region. For the remaining primaries, the corresponding peak luminance can be calculated in a similar manner. The solution region of N-primary display (N > 6) is determined as following:

The coordinates of the points (k₁, k₂, k₃, k₄, …) in the solution region can be directly used to obtain the peak luminance of all primaries. The luminances of the added primaries can be scaled up or down proportionately as appropriate with the corresponding k.

The luminances can be scaled up or down proportionally to adapt to different conditions. Consequently, we can further calculate all possible color gamut solids that meet the requirements, such as the need for a specific primary luminance or the case of the maximum color gamut volume. This is a theoretical framework for MPDs.

In the following section, we will use examples to illustrate the method.

3. Simulation

In this section, we will use examples to illustrate the method. Here, we use the Rec.2020 standard to demonstrate. For a display system, the peak luminance of all its primaries will be mixed to white color; we set the luminance of white to 100 to match the ceiling of L* in L*a*b* color space. The primary coordinates specified in the Rec. 2020 standard are R (0.708, 0.292), G (0.170, 0.797), and B (0.131, 0.046), which correspond to the spectral colors at 630, 532, and 467 nm, respectively. The coordinates of the white point here and in the following calculation are all set to D65 (0.3127, 0.3290) [6].

All calculations in this study were performed in MATLAB. Here, we assumed that a display system with such parameters exists and was used to demonstrate the theory. According to Eq. (1), (2), and (3), the luminance of white color is set as 100, and the peak luminance of the three primaries is:

The color gamut volume is shown in Fig. 2(b) using the algorithm proposed by Wang [15]. Figure 2(c) shows the gamut ring as Masaoka proposed when L* = 100 [10]. The total area in Fig. 2(c) is equal to the volume of the gamut in Fig. 2(b).

 

Fig. 2. (a) Chromaticity coordinates of primaries in the Rec. 2020 standard; (b) Rec. 2020 color gamut volume in CIE L*a*b* color space; (c) gamut ring when L* = 100, where the angles of the straight borderlines correspond to the CIELAB hue angles. The white point was set to D65.

Download Full Size | PDF

Considering the lack of a green part in the Rec. 2020 chromaticity gamut area, we introduced the spectral color with a wavelength of 520 nm as the fourth primary color for calculation. The corresponding chromaticity coordinates were (0.0743, 0.8336). For convenience, we set its luminance as Y₁ = 100 for simplification. Combined with Eq. (1), (2), and (4), the result is:

Substituting the calculation results into Eq. (6), the result is as follows:

Each value of k₁ corresponds to a set of four primary displays with white points at (0.3127, 0.3290).

The method used here and later to calculate the color gamut volume of MPD was proposed by Masaoka [23]. Figure 3 shows the color gamut volume trend as k₁ changes. It can be observed that when k₁ = 0.38, the color gamut volume of the four-primary display reaches its maximum value of 2115700. The maximum color gamut volume is shown in Fig. 3(b).

 

Fig. 3. (a) Range of k₁ and the corresponding color gamut volume; (b) maximum gamut when k₁ = 0.38; (c) gamut ring when L* = 100. The white point was set to D65.

Download Full Size | PDF

Further, for a five-primary display, we can assume the fifth primary as the spectral color with a wavelength of 660 nm and coordinates (0.7300, 0.2700). Its luminance is also set as Y₂ = 100 for calculation convenience; similarly, based on the Eq. (1), (2) and (9), we can calculate the parameters:

Substituting the calculation results into Eq. (11), the result is as follows:

The corresponding solution region is shown in Fig. 4(a). The luminances of all primaries can be obtained in the same manner. When (k₁, k₂) = (0.45, 0.24), the color gamut volume of this five-primary display reached its maximum value of 2257600. The corresponding color gamut volume in the solution region is shown in Fig. 4(b).

 

Fig. 4. (a) Solution region of the five-primary display; (b) corresponding color gamut volume values in the solution region.

Download Full Size | PDF

For a six-primary display, the common combination of the primaries is two sets of RGB primaries. As mentioned, we introduced the spectral color with a wavelength of 445 nm as the sixth primary for calculation, and the chromaticity coordinates were (0.1611, 0.0138). The chromaticity coordinates of the white point were still (0.3127, 0.3290). Similarly, we can obtain:

Using Eq. (14), the solution space is shown in Fig. 5.

 

Fig. 5. Solution region of the six-primary display.

Download Full Size | PDF

The solution region is a solid in three-dimensional space, and all coordinates in the region (k₁, k₂, k₃) satisfy the condition of a six-primary display, as shown in Fig. 5. By traversing all the points in the solution region, we obtained the maximum color gamut volume as shown in Fig. 6(a), where (k₁, k₂, k₃) = (0.42, 0.22, 0.01), and the maximum color gamut volume is 2352200. Rather than comparing the above calculation results relating to three, four, and six primaries, Fig. 6(b) offers a more intuitive presentation of the increase of color gamut volume.

 

Fig. 6. (a) Maximum color gamut volume of the six-primary display; (b) comparison of different gamut rings when L* = 100. The white point was set to D65.

Download Full Size | PDF

For a six-primary display composed of two groups of RGB primaries, a simple method was employed to calculate the peak luminance of each primary. That is, two groups of RGB primaries with the same point are mixed, thus resulting in a six-primary display with the same white point. A set of solutions can be obtained by adjusting the ratio between the total luminance of the two groups of the RGB primaries.

However, this algorithm will cause the degree of freedom of the solution to decrease by two; thus, the final solution set will be a line in the solution space. For calculation, we can choose the first set of RGB primaries with the chromaticity coordinates R₁(0.708, 0.292), G₁(0.17, 0.797), and B₁(0.131, 0.046), and the second set with R₂(0.7300, 0.2700), G₂(0.0743, 0.8336), and B₂(0.1611, 0.0138). The results are shown in Fig. 7. Similar conclusions can be obtained for other types of MPDs with different primary numbers, which theoretically can also indirectly prove the validity of the proposed algorithm in this paper.

 

Fig. 7. Solution space. The solid bounded by the blue lines is the solution region of the six-primary display above. The black line is the solution set obtained by using two sets of three RGB primaries mixed to white point, its two endpoints correspond to the two sets of three RGB primaries display, respectively.

Download Full Size | PDF

Note that, according to the above theoretical results, we used three-primary display as the initial for calculation. For example, for a six-primary display composed of two groups of RGB primaries, the available options of the initial RGB combination will be 2^3 = 8. If the initials are not defined as RGB, there can be 6 × 5 × 4 = 120 options. According to the above calculation, the method of selecting the three primaries does not affect the final result. That is, the corresponding peak luminance values of the six primaries in the solution space obtained by each selection method is consistent. The solution space of this computation includes all the possible solutions. This is easy to understand because a point not in the solution region must result in the luminance of one or more primaries being zero or negative.

Another example of the result when the algorithm is applied on a six-primary display is shown in Fig. 8; the spectral colors used were 660, 638, 550, 520, 465, and 445 nm. White point is still set at (0.3127, 0.3290). When (k₁, k₂, k₃) = (0.015, 0.25, 0.017), the maximum color gamut volume is 2347400. As seen in Fig. 8(c), the use of 550 nm six-primary display shows the improvement in the yellow part, but an absence in the green part.

 

Fig. 8. (a) Solution space with 660, 638, 550, 520, 465, and 445 nm spectral colors; (b) the maximum color volume; (c) comparison of gamut rings when L* = 100. The white point was set to D65.

Download Full Size | PDF

It can be seen that, according to the method proposed in this paper, combined with the calculation of color gamut volume, we can see the advantages and disadvantages of different wavelength selection.

4. Experiment

To verify the above theoretical calculations, we built a six-primary laser projection display system, as shown in Fig. 9. The display distance is 6.3 m, and the image size is 3.50 ${\times} $ 1.96 m. The screen was a white wall and the light sources were the 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, are based on the reflection light from the white wall, so there is no need to take into account the different reflectivity of the wall to different colors of light. All measurements were performed in a dark room. The chosen laser wavelength, measured chromaticity coordinates, and the maximum output luminance are listed in Table 1. The laser wavelength used is consistent with the example in Fig. 8(a)-(b). In the process of increasing the luminance of the laser diodes, the chromaticity coordinates remained the same.

 

Fig. 9. (a) Six-primary display system; (b) display status and test environment; (c) the display video [24]; (d) measured coordinates of the six primaries.

Download Full Size | PDF

Table 1. Central wavelength, chromaticity coordinates, and maximum output luminance of laser diodes

View Table | View all tables in this article

To verify the above methods, we calculated the solution space according to the measured coordinates and used the points in the solution space to calculate the theoretical luminances of the six primaries at this time. After setting the laser diodes to the expected value, we tested whether the final mixed color was the theoretical white point, and whether the color gamut volume was consistent with the theoretical prediction. First, we chose 445, 520, and 660 nm as the initial three primaries to calculate the luminances of the remaining primaries. The coordinates of the white point were set at (0.3127, 0.3290), and the luminance of the white point was set at 100. The corresponding luminance parameters of all primaries are shown in Table 2.

Table 2. Wavelength of the light source and the corresponding luminance parameters

View Table | View all tables in this article

Thereafter, setting ${Y_1}$ = ${Y_2}$ = ${Y_3}$ = 100, the calculation results can be obtained as follows:

Theoretical solution space was calculated according to the actual measured chromaticity coordinates in Fig. 10. The reason for the slight difference between solution space in Fig. 10 and the example in Fig. 8(a) is the difference in chromaticity coordinates, which is due to the spectral width. By traversing all points in the solution space, we obtained the maximum color gamut volume when (k₁, k₂, k₃) = (0.02, 0.26, 0.018) as 2232800. The luminance ratios of the six primaries are shown in Table 3. Considering the working conditions of the laser diodes and increasing the luminance, we increased the luminance by 1.2162 times to stabilize the laser diodes and make 520 nm laser diodes reach its maximum luminance, as shown in Table 3. The luminance of the six primaries was set according to the calculated results. 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 is mainly caused by the stability of the laser diodes and the regulation limit. The final coordinate of the white point measured was (0.3067, 0.3329), which is close to the theoretical value (0.3127, 0.3290). Figure 11 shows the comparison of gamut rings which have the same wavelength and different spectral width of primaries. It can be easily seen that increasing the spectrum width will reduce the color gamut volume.

 

Fig. 10. Theoretical solution space calculated according to the actual measured chromaticity coordinates.

Download Full Size | PDF

 

Fig. 11. Comparison of gamut rings when L* = 100: red one was calculated with laser diodes chromaticity coordinates, blue one is the example from Fig. 8. The white point was set to D65.

Download Full Size | PDF

Table 3. Calculated luminance and actual set luminance of the six-primary display system

View Table | View all tables in this article

In addition, we measured the vertex chromaticity coordinates of the color gamut solid according to the methods of Ou-Yang and Huang [16], and then drew the color gamut solid. The measured color gamut volume is 2277800, which is close to the theoretical value 2232800. The difference between the two results was 2%, and their shapes were the same. The main cause of this deviation is the enlargement of the boundary. From the theoretical calculation diagram, the boundary of the solid is convex or concave, as shown in Fig. 12(c)-(e). For the actual condition, because of the measurement accuracy and difficulty, we could only connect the vertices with straight lines, as shown in Fig. 12(b), which introduces a small error.

 

Fig. 12. (a) Theoretical color gamut volume when (k₁, k₂, k₃) = (0.020, 0.260, 0.018); (b) actual measured color gamut volume; (c) Theoretical gamut and actual measured gamut cross section at L* = 20; (d) Theoretical gamut and actual measured gamut cross section at L* = 50; (e) Theoretical gamut and actual measured gamut cross section at L* = 80. The white point was set to D65.

Download Full Size | PDF

To further validate the method, we added a constraint of making the luminance of 550 nm laser diodes larger than 520 nm since the maximum output luminance of 550 nm laser diodes is higher than 520 nm. By calculation, when (k₁, k₂, k₃) = (0, 0.38, 0.017), the solution with the largest gamut in the solution space is sought under the constraint conditions, as shown in Table 4. As shown in Fig. 9(d), the chromaticity coordinates of 638 nm laser are located on the line between 550 nm laser and 660 nm lasers, so the brightness increase of 550 nm laser will inevitably lead to a brightness decrease of 638 nm laser, until it goes down to zero. Furthermore, considering the working conditions of the laser diodes and increasing the luminance, we increased the luminance by 1.5268 times. After setting the luminances as in Table 4, the final coordinates of the white point measured were (0.3066, 0.3386), which are also close to the theoretical values (0.3127, 0.3290). The theoretical color gamut volume calculated was 2205000, and actual measured gamut are 2252900, which has a difference of 2.1%. The color gamut volume results are shown in Fig. 13.

 

Fig. 13. (a) Theoretical color gamut volume when (k₁, k₂, k₃) = (0, 0.38, 0.017); (b) actual measured color gamut volume; (c) Theoretical gamut and actual measured gamut cross-section at L* = 20; (d) Theoretical gamut and actual measured gamut cross-section at L* = 50; (e) Theoretical gamut and actual measured gamut cross-section at L* = 80. The white point was set to D65.

Download Full Size | PDF

Table 4. Calculated luminance and actual set luminance of the six-primary display system with the constraint of Y_550nm>Y_520nm

View Table | View all tables in this article

From Fig. 12(c)-(e) and Fig. 13(c)-(e), it can be easily seen that the theoretical calculation and actual measurement of color gamut volume are in good agreement. The main source of deviation is the convex or concave part and hue deviation in CIE L*a*b* color space when brightness is low.

5. Discussion

According to the above analysis, the proposed method is in good agreement with the experimental results. Because this method does not limit the type of light sources and the coordinates of white point, it is suitable for all common types of display systems with different light sources, such as LEDs, OLEDs, and lasers. In addition, few parameters are required in the calculation process, which is limited to the chromaticity coordinates of the light sources and white point; thus, significantly simplifying the design and test difficulty. Furthermore, because the luminance range of each primary can be determined, combined with the calculation of color gamut volume, it is more convenient to choose appropriate light sources according to the actual situation and cost. For example, as shown in Fig. 11, we can calculate the chromaticity coordinates of primary colors with different wavelengths and spectral widths, and then calculate their color gamut volume. After comparison, we can select a suitable light source according to our required conditions. Different chromaticity coordinates are used in theoretical calculations and experiment calculations, which correspond to different wavelengths and spectral widths of light sources. Therefore, the proposed method has a good reference significance in the design of MPDs, especially in the selection of the wavelength and spectral width of the light source, as well as the luminance setting.

Furthermore, the white color used as reference point here can be replaced by another color, which will turn the question into color reproduction. By calculating the luminances of all primaries for a particular mixed color in MPDs, we can convert colors from three-primary displays to MPDs without loss. This will become the next step of our focus on the direction of MPD color management.

In addition, for a certain color, the mixed ratio of the three primaries is settled in the three-primary display. For MPDs, the same color can correspond to different proportions of luminances of the primaries, rather than just a fixed proportion, which traditional three-primary display systems cannot perform. Therefore, the addition of more primaries increases the color gamut volume, as well as the redundancy of colors in the display system. During the long-term use of the display system, the intensity attenuation of the laser causes attenuation of the color gamut volume. The lifetime and attenuation degree of different light sources are also different, and the redundancy owing to multiple mixing combinations can significantly reduce this effect.

6. Conclusion

In this study, a new and concise method for calculating the peak luminance of primaries was introduced. When the coordinates of the white point are provided, we can obtain all possible luminance combinations of primaries with the coordinates of primaries only. The display color gamut volume can be further calculated in the CIE L*a*b* color space. The theoretical maximum color gamut volume can be obtained easily by optimizing the peak luminance of the primaries. The algorithm can also be extended to N (N > 6) primary display systems, and a general representation was established. Using this method in this study, we have built a six-primary display system and used it to verify the validity of the method. Combined with the calculation of color gamut volume, the theory can be used to guide the selection of the wavelength, spectrum width, and luminances of primaries. This method is suitable for common display systems with various light sources, such as LEDs, OLEDs, and lasers. To the best of our knowledge, this method provides the most complete and accurate theoretical basis for MPD construction to date, in addition to great guiding significance. As MPDs continue to become more commonplace in the near future, this method could have a wide range of applications. One study that we are going to conduct in the future is maximizing the energy efficiency of MPDs. In addition, using the method to calculate the primary color luminances required to obtain any color in the color gamut volume, which has a certain significance in color reproduction, is also of interest. In future work, we will continue to explore the color encodings of MPDs based on this method, so that the colors can be displayed more accurately as the color gamut volume is enlarged. Combined with metameric control set, we can further explore the mitigation of observer metamerism, letting the observers see more natural and accurate colors.