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Abstract
In using a laser light source, it becomes possible to realize an ultra-wide display gamut that approaches the human color vision limit. This paper introduces a method for extremely large gamut optimization for different primary numbers, and it offers a primary set that produces a nearly ultimate gamut. Considering display lightness, we calculated wavelength selection and lightness design of a display with 3-9 primaries in the CIELAB uniform color space (UCS) by optimizing the coverage of the optimal color gamut. Theoretically the maximum gamut area of a laser display with 3-12 primaries in the CIE xy and CIE u’v’ chromaticity diagrams is also calculated for comparison. We recommend 6 primaries as a reasonable choice, since the coverage reaches 97.6% of the optimal color. Taking into account the luminance efficacy of radiation (LER) and feasible laser wavelengths in practice, we get a practical design of wavelengths and power for a laser projection display with 6 primaries, which covers 96.6% of the optimal color gamut.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser display is able to provide a wide gamut due to excellent monochromaticity of laser light source, and could reproduce more realistic and abundant natural colors. As the development of laser technology, more and more visible wavelengths can be provided by laser. It becomes possible to realize an ultra-wide gamut which is far beyond current display devices’ gamut and even ultimately reaching up to human eyes’ color vision limit. In this paper, we aim to discover the possibility of an ultra-wide gamut laser display for reproducing almost any visible or natural object color, which can be called a “nearly-ultimate gamut display”. Apparently, it could create a more vivid scene by reproducing more saturated colors. Using multi-primary method, properly designed laser display is capable of providing an almost ultimate gamut.
According to currently used evaluation method, display gamut is represented by a primary triangle in CIE xy or CIE u’v’ chromaticity diagram [1]. There are several standard system colorimetries representing the gamut of current display devices, such as Adobe RGB as a de facto standard in professional color processing [2], SMPTE RP 431-2 for the reference digital cinema projector [3], Rec. 709 for HDTV [4], and Rec. 2020 for UHDTV [5]. Specially, Rec. 2020 provides the largest gamut that covers the major standard system colorimetries including three standards listed above [6], and gamut that approaches to Rec. 2020 gamut is generally regarded as a wide color gamut. However, display gamut of Rec. 2020, as shown in Fig. 1, only covers 64.74% of CIE xy chromaticity diagram. Some cyan and violet color still could not be reproduced under Rec. 2020 standard. Even theoretically largest coverage generated by 3-primary display cannot reach up to more than 85% of chromaticity diagram (73.85% for xy coordinate and 83.49% for u’v’ coordinate). From this aspect, current 3-primary display solutions are far from the demand of nearly-ultimate gamut display.
Fig. 1 Chromaticities for Adobe RGB, SMPTE RP 431-2, Rec. 709, Rec. 2020 and theoretically largest coverage RGB primary sets in (a) xy diagram and (b) u’v’ diagram.
On the other hand, although area coverage in chromaticity diagram has been widely taken as an evaluation criterion, yet it is insufficient for gamut optimization. When evaluating brightness and white balance of display devices, the color gamut is considered to form a solid inherently in a three-dimensional perceptual color space [7, 8]. Due to the lack of lightness information, area coverage in chromaticity diagram cannot represent discernable color number correctly. It is important to use an appropriate metric to measure the color space coverage when designing wide gamut display. Relatively, evaluation based on volume coverage is more comprehensive for an ultra-wide gamut display. Therefore, a volume-based evaluation method needs to be developed for gamut optimization.
Ajito [9] took the lead of volume-based wide gamut design by presenting a six-primary projection display. The design chose CIELUV uniform color space as an evaluation space, and take the coverage of Pointer’s gamut [10] as a reference for gamut evaluation. By discretely changing the cut-off wavelength of 6 filters, maximum gamut coverage can be found. Due to the light source of this design are filtered RGB lights from LCD projection, which are non-monochromatic, the design realized a gamut coverage of 99.6% Pointer’s gamut, which is excellent for an LCD projection display, but still far from the human color vision limit. Since lightness of a primary color varies by correlated cut-off wavelength, the calculating process did not take lightness into optimization, and display white point of practical device was re-adjusted after system assembling.
Wen [8] presented a method for selecting display primaries to match a target color gamut based on maximizing shape similarity between display gamut and target gamut in CIELAB uniform color space, and offered a design example for laser display. Wen’s method gives an excellent result under 3-primary circumstance. When facing 4 or more primary numbers condition, there is not only wavelength but lightness needs to be optimized due to metamerism, and Wen did not offer a solution for multi-primary condition. Both Ajito’s and Wen’s methods need to be improved for multi-primary laser display gamut optimization due to the demand of lightness optimization.
There is another significant challenge for multi-primary display gamut optimization that computation amount increases rapidly when variable evaluation step goes narrower. For a multi-primary display which requires not only wavelength but also lightness optimization, it is hardly possible to evaluate all wavelength and luminance sets due to huge amount of computation.
The studies reviewed so far have not presented a systematic method for multi-primary gamut optimization, and it has been a problem for multi-primary gamut optimization in UCS. In this paper, an optimization method is presented for primary color design of extremely large gamut display. With optimization algorithm adopted, the calculation process is computationally efficient and gives an excellent result. We analyzed gamut of 3-9 primaries display in CIELAB uniform color space, and optimized corresponding wavelength and luminance sets for each primary color number using our method. Comparison between the display gamut of our result and other researchers’ designs was made. And finally, taking feasible laser wavelength and luminance efficacy of radiation (LER) into consideration, we discussed feasibility for nearly-ultimate gamut laser display and presented a set of wavelength and power of laser source for a 6-primary laser projection display. As the result shows, this design is able to cover 96.6% of optimal color gamut while LER rising up to 189.7lm/W. By using multi-primary laser light source, display gamut can be extremely expanded up to nearly human vision limit so that almost any natural object color could be displayed.
2. Gamut evaluation and optimization method
2.1 Evaluation UCS
Our method is planned to be a volume-based evaluation. In this light, a perceptually uniform color space (UCS) has to be chosen as an evaluation space. Ajito [9] chose CIELUV uniform color space to maximize display gamut of a 6-primary projection display, while Masaoka [11] and Wen [8] preferred CIELAB uniform color space because it has a better uniformity, and moreover, CIELAB UCS system was universally applied in the colorant industry and for color specification [6].
As three-dimensional color spaces, CIELUV and CIELAB UCS only consider the normalized lightness for the third dimension by setting the peak white as 100. In fact, absolute brightness is not presented by lightness. A more comprehensive way is to use absolute brightness coordinated color model, such as CAM02-UCS or CAM16-UCS. However, both CAM02-UCS and CAM16-UCS are computationally expensive compared with CIELAB space [12, 13]. In our optimization application, it is a too large task in computation due to the complexity in these models. Besides, we regard chromatic performance more significant than brightness in primary color design. Therefore, we chose CIELAB space as an optimization UCS.
2.2 Target Gamut
Larger gamut volume in UCS means more colors are able to be reproduced by the device. However, it is not guaranteed to provide a better display performance. For example, a projection display with bright surrounding illumination may performs well for reproducing some bright colors, but it is not able to display dark scenes very well due to a bright background, though its gamut volume in UCS may be larger than an LCD display. It is hard to tell if a display performs well merely using gamut volume. As a result, in our application, evaluation of display gamut in UCS should be a process that calculates volume coverage of a target gamut, which contains all colors constrained by the limit of human color vision features.
For most traditional applications, target gamut is usually chosen as Pointer’s gamut [6–9, 11]. However, Pointer’s gamut contains only a small part of object color since it was defined using the available colorants at some considerable time ago [14], and Pointer’s gamut does not include color data of newly developed dyes and pigments [6]. In fact, Pointer’s gamut is too small to be the target gamut in our application since three-primary display with monochromatic RGB light sources is theoretically able to cover 99.9% of Pointer’s gamut [7], and four-primary laser display could easily cover all colors in Pointer’s gamut. It is far from human color vision limit when display gamut merely covers the colors in Pointer’s gamut. To make a comprehensive evaluation and optimization of display gamut, the target gamut or database has to reach the limit of human color vision. In this light, we finally choose optimal color gamut [15, 16] as target gamut, whose boundaries are also known as MacAdam limits, representing the theoretical maximum gamut of object colors under a given illuminant [17]. Masaoka [11] and Wen [8] also used optimal color gamut as target gamut in primary color design and optimization of laser display.
2.3 Calculation and optimization method
Gamut volume of an N-primary display is defined by 2N variables (N primary wavelengths and correlated N lightness). In CIEXYZ space, gamut of an N-primary display is a parallel polyhedron spanned by vectors P1, P2, …, PN, whose direction represents primary chromaticity coordinate (or particularly, wavelength), and length represents maximum lightness of that primary. When transformed into CIELAB uniform color space, this parallel polyhedron is changed into an irregular solid G, in which all colors constitute the display gamut in CIELAB space. Thus display gamut G is denoted as
where is the operator transforming CIEXYZ color space to CIELAB color space, Pi is the ith primary color vector in CIEXYZ space, αi is lightness weight of ith primary color, and c is the display color set in CIEXYZ space.As shown in Fig. 2, a 3-primary display gamut G is denoted in orange, while optimal color gamut is a fixed solid GT and denoted in blue. By changing wavelengths and lightness of N primaries, shape and volume of G correspondingly change, so is the intersection part of G and GT. We could finally find the maximum intersection volume by changing wavelengths and lightness successively. Volume of the intersection part, or equivalently, coverage over target gamut W, is selected to be the optimization objective, which is denoted as
However, excessive computations are required when number of primary colors or fineness of variables rises. For an N-primary display, it is extremely time consuming to evaluate all wavelength and luminance sets when N is larger than 3. For example, when N equals 4, there is 2N = 8 variables, and if each variable has 100 possible values, literally 1008 times of intersection volume calculation have to be done, which is hardly possible. Therefore, optimization algorithm has to be taken for computation efficiency. Interior-point algorithm was adopted into our optimizing method. This optimization algorithm is built in Optimization Toolbox of MATLAB. In Sec. 3, the comparison shows our method using this algorithm gives an excellent output.
In our method, the gamut optimization problem is divided into two subsidiary problems: lightness optimization under a given set of wavelengths and wavelength optimization. Lightness optimization is carried on because of metamerism brought by multi-primary display—if the design white point is set to, for example, D65, there are different lightness weights for a fixed wavelength set to constitute D65, and lightness needs to be optimized for a larger gamut. We treat both lightness and wavelength optimization as constrained nonlinear programming problems, and take fmincon function in MATLAB to find the optimum.
Generally, a constrained nonlinear programming problem is specified by
where f (x) is the multivariable object function and x is the variable vector, and constrains can be specified by those five equations. During lightness and wavelength optimization, constrains are configured separately.For lightness optimization of N-primary display, variable vector x is set as an N-dimensional vector whose elements represent for N lightness values. Constrains are set by
where lb is an N-dimensional zero vector; Xi, Yi, Zi in Aeq are tristimulus values of normalized ith primary, and XW,YW,ZW in beq are tristimulus values of the design white point. Equation makes lightness values are non-negative and ensures that white point remains unchanged during iteration. Other parameters in Eq. (3) are left unevaluated.For wavelength optimization, variable vector x is an N-dimensional vector whose elements represent for N wavelength values. Constrains are set by
where lb and ub are N-dimensional vectors. Equation representing wavelength range of 380–780nm; A is an N–1-by-N matrix and b is an N–1-dimensional zero vector. Equation in Eq. (3) makes sure that the gamut polygon is clockwise-oriented. Other parameters in Eq. (3) are left unevaluated.The return value of f (x) in both lightness and wavelength optimization, which is the optimization objective, is set to the negative of volume coverage W calculated by Eq. (2). Volume of GT is constant and calculated in advance. To calculate volume of the intersection part, gamut solid was sliced into 101 pieces in CIEXYZ space along the Y axis. When transformed into CIELAB uniform color space, those pieces are corresponding to equidistant L* values of 0, 1, …, 100. Gamut solid of the optimal color gamut GT, which is chosen as target gamut, is also sliced into 101 equidistant L* pieces. Intersection area in each L* piece between display gamut G and target gamut GT is calculated, and volume of intersection part is eventually accumulated by integrating intersection area of each piece. The calculation flow is shown in Fig. 3.
3. Results and analysis
Using the method presented above, we made gamut optimization under different primary numbers for laser display. According to the saturating trend between gamut size and primary number, a suitable primary number is recommended to perform a nearly-ultimate gamut without increasing too much system complexity.
With wavelength range set to 380–780nm and white point set to D65, wavelength and correlated lightness values were optimized, and we obtained the rising trend of 3-primary to 9-primary display’s optimal gamut coverage in CIELAB UCS, which is shown in Fig. 4. It suggests that 97.6% of optimal color gamut was covered when primary number reaches up to 6, and coverage tends to be saturated (at about 98.9%) if primary number is larger than 8. Figure 5 shows optimal color gamut coverage of 6-primary display in several L* slices. Blue line represents the boundary of optimal color gamut, and orange line represents the boundary of intersection part between optimal color gamut and display gamut.
Fig. 5 Optimal color coverage of 6-primary display at (a) L* = 99, (b) L* = 80, (c) L* = 70, (d) L* = 50, (e) L* = 30, (f) L* = 10. It can be seen that optimal color is almost fully covered at each slice, even at high lightness and low lightness slices.
To confirm the optimality of our result, we calculated the maximum chromaticity diagram area coverage of 3-12 primaries, and made comparison among area optimum in chromaticity diagrams, several typical wide-gamut designs and our optimization results.
Chromaticity coordinates data is prepared, which varies by wavelengths from 380nm to 780nm with step of 1nm. Maximum area is calculated using optimization algorithm proposed by Keikha et al [18], which is designed to find the maximum inscribe N-polygon of a convex polygon. To make sure the boundary of chromaticity diagram to be a convex polygon, before optimization, we extracted the convex hull of CIE xy and CIE u’v’ coordinate data of 380–780nm visible light, and got 179 vertices in xy diagram and got 163 vertices in u’v’ diagram as optimizing database. Figure 6 shows the optimized wavelength sets of 3-12 primaries display, and the growing trend between primary number and correlated maximum gamut area.
Fig. 6 Optimized wavelength sets in (a) CIE xy diagram (b) CIE u’v’ diagram and gamut area growing trend in (c) CIE xy and (d) CIE u’v’ diagram.
We firstly made 3-primary coverage comparison among display standards, best chromaticity diagram coverage, Masaoka’s [11], Wen’s [8] designs and our result, which is shown in Table 1. The largest optimal color coverage optimized by us is realized by a wavelength set of 460nm, 524nm and 766nm. It is clear that our design has an obviously larger coverage than all the other primary color sets.
Table 1. Optimal color coverage comparison of 3-primary display (White point: D65).
For multi-primary applications, there are few comparable researches since most multi-primary designs are based on non-monochromatic light sources, whose display gamut are far from approaching human color vision limit. Instead, we picked 6-primary optimized wavelength set and added ± 5nm disturbance to four primary wavelengths (except for the two primaries near to visible spectrum ends), and calculated optimal color coverage of these 81 wavelength sets. We also took wavelength sets that maximize 6-primary area coverage in xy and u’v’ diagram and calculated the optimal color coverage in UCS. Comparison is made among the optimal color coverage of all 83 wavelength sets in both CIELAB and CIELUV space, results are shown in Fig. 7. It suggests that our optimization has excellent performance in both CIELAB and CIELUV space.
Fig. 7 Comparison among all 83 6-primary wavelength sets (blue dots) and optimized result (dot with orange circle). It can be seen that our result performs excellently in both two spaces. Note that best u’v’ and xy coverage results in a relatively poor performance in UCS. This is mainly caused by two reasons, the irrelevance between area and volume color model, and the optimal color coverage criterion in our optimization. High area coverage may not result in a high optimal color coverage in UCS.
4. Gamut optimization for laser projection display
Applications that able to achieve a gamut up to human color vision limit are restricted by hardware techniques. Here, for example, we offer a solution for multi-primary laser projection display to realize nearly-ultimate color gamut using our optimization method. The design white point is D65, and total luminous flux is 20000lm.
4.1 Optical efficiency
Blue and red lights that close to the ends of visible spectrum contributes to a wider gamut. However, compared with other colors, those wavelengths have a poor performance at optical efficiency because of the decrease of luminosity function at spectrum ends. For example, as the optimization result of 3-primary display, the largest coverage corresponding to a wavelength set of 460nm, 524nm and 766nm requires a power ratio of blue, green and red lights about 2:3:20000 with a low LER of 0.11lm/W, which is obviously inefficient and impossible for physically realizing.
To balance the trade-off between color gamut and LER, we choose different wavelength optimization ranges and make optimization respectively. These ranges were achieved by symmetrically shrinking blue and red wavelengths ends, centering at 550nm. Figure 8 shows gamut and LER changing trend under different wavelength range.
From Fig. 8, we can see that with wavelength optimization range shrinking, LER shows a continuously ascending trend while volume coverage shows a flat-and-decrease trend. This means we are able to find a primary color design that performs both high LER and large gamut. When wavelength optimization range changes from 380 to 780nm to 450–650nm, gamut coverage falls less than 1% for 3 to 6-primary display, and coverage falls less than 1.5% for 7-primary display. When wavelength optimization range goes narrower than 460–640nm, coverage falls down obviously. Meanwhile, LER could reach up to larger than 150lm/W when wavelength optimization range was narrower than 450–650nm. It can be concluded that with an LER rising up to 150lm/W, gamut coverage still performs well for 3-primary and multi-primary laser displays. Taking manufacture complexity of a multi-primary system into account, 6-primary display may probably be the best choice for this laser projection display.
4.2 Suitable laser
Due to the limitation of laser technology, theoretically optimized wavelength may not practically exist. Figure 9 shows the wavelengths of semiconductor laser and solid-state laser currently available from a laser manufacturer. It can be seen that there are fewer available wavelengths around 500nm and 540nm. According to six primary color wavelengths optimized in the last section, we selected the nearest wavelength sets in the available wavelengths and calculated the corresponding gamut coverage. Finally, the wavelength of 450nm, 480nm, 505nm, 526nm, 552nm and 650nm were adopted. As an example, we assume the total flux of the designed projector is 20000lm. Corresponding power distribution (representing lightness optimization results) was shown in Table 2. To create an intuitively understanding of the display gamut, display brightness is also roughly estimated by assuming projecting on a 150 inch screen with a gain factor of 1. It covers 96.6% of optimal color gamut, and the LER reaches up to 189.7lm/W. Figure 10 shows gamut coverage of 6-primary display in several L* slices. Blue line represents the boundary of optimal color gamut, and orange line represents the boundary of intersection part between optimal color gamut and display gamut. It shows that in each slice, display gamut covers almost whole area of optimal color gamut.
Table 2. Characteristics of 6-primary laser projection display (white point: D65).
Fig. 10 Optimal color gamut coverage of 6-primary laser projection display at (a) L* = 99, (b) L* = 80, (c) L* = 70, (d) L* = 50, (e) L* = 30, (f) L* = 10.
Practical 3-primary solution is also generated using similar method. The wavelength of 460nm, 523nm and 650nm were adopted. Compared with the volume gamut of Rec. 2020 specified by the wavelength of 467nm, 532nm and 630nm, our result has a higher optimal color coverage of 78.4% (Rec. 2020 covers 75.0% optimal colors as shown in Table 1).
5. Conclusion
In this paper, we discussed feasibility for ultra-wide gamut (up to human color vision limit) display. We presented a new gamut optimization method for extremely large gamut display like laser display, and made optimization for 3-primary and multi-primary laser display using this method. We offered feasible primary color solutions for a 6-primary laser projection display, which covers 96.6% of optimal color gamut in UCS with a high LER of 189.7lm/W. From this result, we can conclude that human color vision limit could be nearly reached by using multi-primary laser display, and our studies can be a primitive scheme for primary color design of large gamut laser display.
Further study is also needed to improve the details such as metamerism brought by narrow band primaries [19], and the optimization method needs to be improved for high brightness display by adopting absolute brightness coordinated color appearance model.
Funding
National Key Research and Development Plan (2016YFB0401900).
Acknowledgments
We are indebted to Prof. M. R. Luo of Zhejiang University for helpful information and fruitful discussion.
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