1. Introduction

Laser-projection technology has been valued by people since all-solid-state lasers became available in the late 1990s [1]. Compared with conventional projection lamps, lasers exhibit superior characteristics such as higher brightness, better monochromaticity, smaller divergence angle, and longer service life [2,3]. Therefore, lasers serve as the optimal exhibition technology for modern art performances, grand opening ceremonies, and various cinemas.

However, speckle emerges when coherent or partially coherent laser beams are scattered from an optically rough surface [4]. The speckle manifests as dark and bright granules, which deteriorate the image quality. In the earlier stages, the methodology of the speckle measurement and evaluation had not been established. Many publications focused on speckle-reduction methods and devices [5–8] and researchers proposed their own methodologies for speckle measurement, which caused some difficulties in fair comparisons among different speckle-related studies on the laser projectors [9]. Therefore, the international standardization of the methodology was expected.

In addition, the speckle phenomenon is related to the colors in laser-projection systems. In a pure-white image on the screen, the colored granular noise is termed as “color speckle”. Based on several studies [10–12], the International Electrotechnical Commission (IEC) illustrated the optical measuring methods of color speckle (IEC 62906-5-4:2018) [13]. The calculations of the variances parallel and perpendicular to the red, green, and blue (RGB) directions are complicated, which is difficult for speckle comparisons between different projectors. Additionally, the results and discussions of the above studies were based on a two-dimensional (2D) chromaticity diagram, that is, the CIE 1976 u’v’ diagram or CIE 1931 xy diagram. However, color is a three-dimensional (3D) perceptual physical quantity that includes lightness, chroma, and hue. Lightness information is missing in the 2D chromaticity diagram, which is inconsistent with the human-eye perception.

In colorimetry, the CIELAB uniform 3D color space has been universally used in color specification and color-difference measurements since 1976 [14]. In the latest document (IEC 62977-2-1:2021) [15], the CIELAB color gamut volume (CGV) is defined as an optical parameter of a display system that exhibits color-rendering capacity. However, the CIELAB color space does not accurately predict perceived color differences [16] in high-dynamic-range (HDR) applications. In addition, it possesses a limited range of normalized lightness $({0 \le {L^\ast } \le 100} )$. Recently, Dolby proposed an HDR encoding space called IC_TC_P [17]. This color space followed the same structure as IPT space [18] but replaced its power function nonlinearity by the perceptual quantizer (PQ) function for better HDR encoding. Based on the structure of IC_TC_P, Luo et al. developed a more uniform color space, J_za_zb_z, which is appropriate for a wide luminance range, from 0.001 to 10,000 cd/m² [19]. The perceptual uniformity, prediction of small and large color differences, and linearity in iso-hue directions (most challenging is the blue hue) of J_za_zb_z space are proved to be superior in their following studies [20,21]. From the above discussions, experts in colorimetry were devoted to creating a more uniform and higher dynamic-range encoding color space. With the development of colorimetry, it is of great significance to develop the color-speckle theory from a 2D chromaticity diagram to a 3D color space. In 2018, Kinoshita et al. investigated the color speckle in CIELAB space but found it meaningless because the data exceeds ${L^\ast } = 100$ [22]. Recently, Kinoshita et al. proposed the color-speckle theoretical analysis in CIE xyY color space [23]. The relation between the chromaticity and illuminance can be analyzed in the 3D space. However, the CIE xyY color space is not good enough from a viewpoint of human color perception. A perceptually uniform and high-dynamic-range encoding color space is crucial for color-speckle analysis, which is consistent with human-eye perception. Therefore, J_za_zb_z space is the most appropriate for the color-speckle application.

In general, for a better visual experience, color and speckle are two essential issues for researchers in laser-projection displays. To enhance color, multi-primary displays [24] were developed to extend the color gamut. For color optimization, Song et al. analyzed the color gamut of 3-primary to 9-primary display in the CIELAB color space [25]. Zhu et al. demonstrated a six-primary laser system that is compatible with 2D and 3D displays [26]. These studies on multi-primary displays represent an important direction in color development. However, with respect to color speckle, current studies and standards are still restricted to the three-primary system, including Kinoshita’s latest study [23]. It is necessary to investigate the color speckle in multi-primary displays.

This paper presents a color-speckle assessment method based on 3D J_za_zb_z color space. It is appropriate for both three-primary and multi-primary systems. Compared with current studies based on 2D chromaticity diagrams, the proposed method based on the perceptually uniform 3D color space is more aligned with human color-speckle perception. In J_za_zb_z color space defined by non-normalized lightness, the color-speckle characteristics in different projectors with different luminance values can be compared using numerical values and spatial distribution. A simplified ellipsoid model is proposed for intuitive and visualized comparison, which is crucial for commercial color-speckle assessment applications. Additionally, the extension to multi-primary systems supplements color-speckle evaluation in multi-primary displays.

The main process of the proposed method is as follows: given the chromaticity coordinates of the laser sources and target white point, the spatial distribution of the color speckle can be simulated and measured. The results were analyzed in the J_za_zb_z color space. Five physical quantities, namely, the two color variances, ellipsoid semi-major axis, volume, and hue angle, were defined to describe the color-speckle characteristics. Experimental verifications were individually implemented using a three-primary laser projector and six-primary laser projector. The results of the simulation and measurement were consistent for the color-speckle assessment in the color space. We believe that this method will accelerate the development of standardized color-speckle assessments.

2. Theoretical analysis of color speckle

In this section, the algorithm for calculating the color speckle is deduced based on the luminance unit in the three-primary laser systems. The expression for the color speckle is derived from the monochromatic speckle contrast values of different primary colors. Additionally, the extension of these formulas is implemented in multi-primary laser systems. The color-speckle distribution was evaluated in the J_za_zb_z color space. With the definition of the color-speckle characteristics, the severity of the color speckle can be analyzed.

2.1 Calculation of the color speckle in three-primary laser systems

According to the optical measuring methods of color speckle (IEC 62906-5-4), in a three-primary laser system, the tristimulus values X, Y, and Z (XYZ) of the white point are expressed as follows:

To simplify Eq. (2), the tristimulus values of the RGB primary colors can be defined as follows:

In Eq. (4), if the values of (${L_\textrm{R}},\; {L_\textrm{G}},\; {L_\textrm{B}})$ are equal to ($1,\; 0,\; 0)$, ($0,\; 1,\; 0)$, and ($0,\; 0,\; 1)$, respectively, the left results of the equations correspond to the RGB primary colors. When RGB light sources are incoherent, (${L_\textrm{R}},\; {L_\textrm{G}},\; {L_\textrm{B}})$ is equal to ($1,\; 1,\; 1)$, which generates a uniform white image. When speckle exists, the RGB luminance of n points in space satisfies a specific distribution. Under this condition, the white image suffers from colorful granules of color speckle. The luminance spatial distributions of the three-primary colors can be expressed as follows:

By Eq. (5), XYZ of the white image with color speckle can be expressed as follows:

In the experiments, the chromaticity coordinates $({x_{\textrm{R},\textrm{G},\textrm{B}}},{y_{\textrm{R},\textrm{G},\textrm{B}}})$ and luminance ${Y_{\textrm{R},\textrm{G},\textrm{B}}}$ of the RGB sources, which are reflected by an ideal diffuse screen, can be measured with a spectroradiometer. Combined with the definition of the CIE 1931 xy chromaticity coordinates, the rest of the tristimulus values, ${X_{\textrm{R},\textrm{G},\textrm{B}}}$ and ${Z_{\textrm{R},\textrm{G},\textrm{B}}}$, are calculated. Subsequently, the Monte Carlo method can be implemented to simulate the spatial distribution of the color speckle [10,13]. The luminance spatial distribution ${L_\textrm{N}}\; ({\textrm{N} = \textrm{R},\textrm{G},\textrm{B}} )$ with color speckle in Eq. (5) can be calculated as follows:

2.2 Calculation of color speckle in multi-primary laser systems

In the previous section, the formulas and discussions were based on a three-primary laser system. Currently, in an m-primary laser system ($m \ge 3$), XYZ of the white point in Eq. (4) can be rewritten as follows:

Therefore, XYZ of a white image with color speckle is expressed as follows:

In the experiments, the chromaticity coordinates $({x_j},{y_j})$ and luminance ${Y_j}$ ($j = 1,2, \ldots ,m$), which are reflected by an ideal diffuse screen, can be measured using the spectroradiometer. Subscript j denotes a specific primary color. ${X_j}$ and ${Z_\textrm{j}}$ were calculated as described in Section 2.1. Similarly, the Monte Carlo method was implemented to simulate the spatial distribution of the color speckle. The subscript of the luminance spatial distribution ${L_\textrm{N}}$ in Eq. (7) satisfies the following:

Because of various combinations of solutions for the target white point in multi-primary laser systems, the discussions in this article are based on a given solution. The general solution and related optimization were investigated by Yao et al. [28].

2.3 Simulation results of the color speckle in J_za_zb_z space

In this section, the properties of color speckle in J_za_zb_z space are illustrated, based on the above formulation. Although the CIELAB space is the most universally used color space for color-difference measurements, as mentioned in the introduction, it cannot represent the color-speckle distribution due to the limited luminance range. Therefore, the J_za_zb_z space [27] is selected in this paper. The superiority of perceived color uniformity and wide luminance range provides possibilities for color-speckle comparison between different laser projectors.

With Eq. (3), given the arbitrary spectral distribution and average power ratio of the RGB laser sources, the CIE 1931 xy chromaticity coordinates and luminance values can be calculated. In the experiments, this calculation was achieved using a spectroradiometer. Therefore, the initial values in the simulation are the chromaticity coordinates and luminance instead of the spectral distribution and average power ratio. According to Eq. (11), the color-speckle distribution for an arbitrary multi-primary system can be simulated. For simplicity, the simulations in this paper are implemented by three- and six-primary systems. The initial values for the three- and six-primary systems are presented in Tables 1 and 2, respectively. The three-primary system is also called an RGB system. The white point was set to D65 (0.3127, 0.3290) [19,29], and the total luminance was normalized to 100 cd/m² for comparison.

Table 1. The chromaticity coordinates and luminance of a three-primary system in simulation

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Table 2. The chromaticity coordinates and luminance of a six-primary system in simulation

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First, the simulation is performed with a three-primary system in a symmetric condition, which indicates that the monochromatic speckle contrast values of the RGB colors are the same. In Fig. 1, the blue hull represents the CGV [30] of the three-primary system in J_za_zb_z color space, which manifests the color-rendering capacity of the system. The algorithm used to calculate the surface boundary of the hull is similar to those used in the studies by Masaoka [31] and Wang [32]. The hull peak denotes the white point of the system. The randomly distributed orange spots in space represent the color-speckle distribution in Eq. (6). Additional figures including three views are provided in Appendix A1.

 

Fig. 1. Color-speckle distribution for a three-primary system in a symmetric condition in J_za_zb_z color space. (a) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 10\%$. (b) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 50\%$. (c) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 100\%$.

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When no speckle exists, it can be inferred that the spots focus on the white point, and the color of the image exhibits uniform pure white. With an increase in RGB speckle contrast, the luminance of the corresponding color fluctuates randomly. In this case, the spots were distributed both inside and outside the blue hull, and the space occupied by the spots increased. When ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 10\%\; ({\mathrm{\alpha } = 100} )$, random spots accumulated in a small space around the white point, as shown in Fig. 1(a). The shape of the data spots was similar to an ellipsoid. When ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 50\%\; ({\mathrm{\alpha } = 4} )$, the spots occupy a larger space, as shown in Fig. 1(b). The data spots were still centered at the white point. However, with the hull constraint, the contours of the data spots were distorted. When ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 100\%\; ({\mathrm{\alpha } = 1} )$, the spots spread to the boundary in Fig. 1(c). The center of the data spots was maintained at the white point. The shape of the data spots was constrained by the bottom of the hull, which appeared to be enlarged. The data spots were randomly distributed, and the district near the white point had a slightly larger spot density.

In speckle evaluation for practical systems, the speckle contrast value is typically below 25% $({\mathrm{\alpha } = 16} )$. Discussions for 40%, 50%, or higher speckle contrast values were measured in experiments under laboratory conditions [33–35]. Such situations are based on a single solid-state laser with a narrow spectral width, and few speckle-reduction devices are used. However, commercial projectors have been integrated with various devices for speckle reduction. The practical measurement focuses on the speckle contrast value of approximately 10% and even lower, which is close to the threshold of the human-eye speckle perception [36]. Various studies have recommended a threshold value of 4% [6,37] or 5% [38]. In addition, according to the deduction from the central limit theorem, when parameter $\mathrm{\alpha }$ is sufficiently large, the Gamma distribution of ${L_\textrm{N}}$ approaches a Gaussian distribution. When the speckle contrast values are lower ($\mathrm{\alpha }$ is larger), the Gaussian approximation will be more accurate. In subjective speckle measurement by the detector, the finite numbers of points with Gaussian distributions could be limited in a finite space statistically. Therefore, an ellipsoid model in J_za_zb_z space is proposed to simplify the maximum color differences of all data spots in all directions from the white point, which forms the ellipsoid surface and data boundary around the white point. The specific distribution of the spots inside is ignored. In this way, the color-speckle characteristics for human-eye maximum perception can be intuitively compared though different ellipsoids instead of complex data spots. These visualized results are intuitive and understandable for customers to select laser projectors, which has great significance in practical application. In this study, the fitting ellipsoid by maximum-color-difference simplification was calculated using the nlinfit function in MATLAB. From a rigorous viewpoint, the variances along every primary-color direction similar to Kinoshita’s study [23] are more accurate to investigate the statistical distribution. However, the lines between the RGB primary colors and white point in the chromaticity diagram are slightly bent in J_za_zb_z space and similar discussions are more complicated in a multi-primary system. Relevant investigations on accurate calculation may be analyzed in further research.

Four physical quantities for color-speckle characteristics are presented to evaluate the severity of color speckle in J_za_zb_z color space in Figs. 2(a) and (b). For practical applications and Gaussian approximation, the sampling points were set when the speckle contrast value was below 25%. Similar to the variances in CIE xy or uv diagram [12,13], the color variances in J_za_zb_z space, $\sigma _{az}^2$ and $\sigma _{bz}^2$, are defined as follows:

 

Fig. 2. Relationship between color-speckle characteristics and speckle contrast in J_za_zb_z space. (a) Variances. (b) Ellipsoid semi-major axis and volume.

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Subsequently, the simulation was performed under asymmetric conditions. One of the RGB monochromatic speckle contrast values was much higher than the others. The three combinations of speckle contrast values ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}}$ are equal to (2%, 2%, 20%), (2%, 20%, 2%), and (20%, 2%, 2%), respectively. The shape of data spots under this condition is similar to a rod in Fig. 3. Additional figures including three views are provided in Appendix A2. The direction of the rod manifests as the direction of the most severe speckle. The projection of the rod onto the ${a_z} - {b_z}$ plane corresponds to two specific hue angles that match the direction of the color with the highest contrast value and its reverse extension. In this case, the aforementioned four quantities are insufficient for the asymmetric condition. If the ellipsoid vertex hue angles are defined by the hue angles of the two semi-major axis vertices, these two hue angles possess a difference value of 180°. In the fitting calculation, owing to the deviation from the ellipsoid center to point (0, 0) in the ${a_z} - {b_z}$ plane, the value is not strictly equal to 180°. For simplicity, the ellipsoid vertex hue angle ${H^z}$ is defined by one of these two hue angles, with a value between 0° and 180°, and the deviation error is ignored. The combination of the ellipsoid vertex hue angle and semi-major axis can accurately describe the asymmetric condition when one of the colors possesses much higher speckle contrast than the others. Therefore, the five physical quantities for the color-speckle characteristics in J_za_zb_z space are listed in Table 3. For an ideal uniform color space, three combinations of RGB speckle contrast values are assumed to manifest equal ellipsoid shapes. However, owing to the lack of uniformity, the ellipsoid values are different in different combinations. There are large differences between two variances in every combination of the asymmetric speckle conditions. The ellipsoid vertex hue angle indicated the direction of the most severe speckle, corresponding to the direction of the primary color.

 

Fig. 3. Color-speckle distribution for a three-primary system in an asymmetric condition in J_za_zb_z color space. (a) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 2\%,\; 2\%,\; 20\%$. (b) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 2\%,\; 20\%,\; 2\%$. (c) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 20\%,\; 2\%,\; 2\%$.

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Table 3. The color-speckle characteristics in J_za_zb_z space

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Additionally, the color-speckle distribution for a six-primary system in the color space was simulated in a symmetric condition. Compared with the three-primary system, the blue hull in the J_za_zb_z space changes slightly and maintains relatively round corners and well symmetry in Figs. 4(a)–(c). Additional figures including three views are provided in Appendix A3. The tendency of the data spots with different monochromatic speckle contrast values was similar to that of the three-primary system. With an increase in the speckle contrast, the data spots grew as an ellipsoid from the white point. When the value is higher than 25%, the ellipsoid shape is distorted by the hull constraint. When it reached 100%, the shape of the data spots appeared to be an enlarged hull of the six-primary system.

 

Fig. 4. Color-speckle distribution for a six-primary system in a symmetric condition in J_za_zb_z color space. (a) ${C_{\textrm{s} - \textrm{N}}} = 10\%$. (b) ${C_{\textrm{s} - \textrm{N}}} = 50\%$. (c) ${C_{\textrm{s} - \textrm{N}}} = 100\%$. ($\textrm{N} = {\mathrm{\lambda }_1},{\mathrm{\lambda }_2}, \ldots ,\textrm{}{\mathrm{\lambda }_6})$

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Furthermore, considering that the J_za_zb_z color space manifests a better performance with wide-range luminance data, it is possible to compare the color speckle of different projectors in the same color space. In the following simulation, the luminance normalization of 100 cd/m² was cancelled. The contrast value of each primary color was equal to 10%. The different luminance conditions of the two systems were simulated and listed in Table 4. For the same system, with an increase in luminance, all four quantities increase, except for the constant hue angle. Under equal luminance conditions, the six-primary system possesses smaller values of color-speckle characteristics than the three-primary system, which indicates a smaller spot space and milder color speckle.

Table 4. The color-speckle characteristics in different luminance conditions in J_za_zb_z space

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Moreover, it is worth noting that the color-speckle characteristics with 100 cd/m² in the three-primary system are similar to those with 500 cd/m² in the six-primary system. The simulated color-speckle distributions of the two systems under this luminance condition are shown in Fig. 5. Additional figures including three views are provided in Appendix B1. To simplify the graph, the CGV hulls were omitted. Blue spots represent the color speckle of the three-primary system and orange spots represent the color speckle of the six-primary system. Because of the higher luminance, the orange spots were higher than the blue spots in space. The fitting ellipsoids of the two collections of data spots are shown in Fig. 5(b), with the corresponding colors. Although the speckle contrast values for each primary color were equal in the simulation, the ellipsoids of the two systems were not the same. The ellipsoid for the three-primary system is flatter and longer.

 

Fig. 5. Simulated color-speckle distribution for two systems in J_za_zb_z color space. (a) Data spots without hulls. (b) Fitting ellipsoids of the data spots.

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3. Experimental setup and measurement methods

Color-speckle measurement was systematically performed according to international standards [13,41]. The apparatus was then placed in a dark room at room temperature. A schematic of this process is shown in Fig. 6. It can be divided into three parts: projector, screen, and measurement system. The details of each part are as follows.

 

Fig. 6. Schematic layout of the experimental setup.

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Two laser digital light processing projectors were used in the experiments. Both possessed a resolution of 2,048 × 1,080 pixels. One is a three-primary system which exhibits 10,000 lumens of luminous flux. The other was a six-primary system that exhibited 4,500 lumens. The color coordinates of each primary color were the same as those in the Tables. 1 and 2. They were measured using a spectroradiometer (SRC-600, EVERFINE, China). The matte white screen satisfied approximately complete diffuse scattering. Both the projection and measurement directions were vertical to the screen. The specific measurement system was similar to that used in a previous study [6]. A monochrome complementary metal oxide semiconductor (CMOS) camera (CMV2000, CMOSIS, Belgium) was used to simulate observers. The pixel size was 5.5 × 5.5 µm². The lens system before the CMOS chip is composed of two lenses. One possessed a fixed focal length of 50 mm and was named the imaging lens. The other was a focus-tunable lens (EL-16-40, Optotune, Switzerland). The clear aperture (CA) was set to 3.2 mm [42] and the integration time was 50 ms [43]. The distance between the CA and imaging chip was 28.5 mm. A neutral density filter was used to obtain the linear response of the camera.

The calculation for the color-speckle distribution includes two fundamental calculation schemes introduced in the document (IEC 62906-5-4). The simulation involved the superposition of the statistically simulated spatial distribution of the RGB monochromatic speckle with RGB speckle contrast values. The measurement involved superposition of the measured spatial distribution of the RGB monochromatic speckle. In the experiments, the projector was operated to generate single-color images for each primary laser source. Therefore, both the monochromatic speckle contrast values and measured spatial distribution for each primary color were obtained from separate pure-color images, which correspond to the two aforementioned schemes. After the measurements, each pixel on the detector exhibited unique chromaticity coordinates. The chromaticity coordinates of a sufficient number of pixels in space are the measured values in Eq. (6), which represents the measured color-speckle distribution. In our experiments, the calculated area was 100 × 100 pixels, which corresponds to the random number n in the simulation. In addition, the monochromatic speckle contrast values for each primary color can be used to calculate the simulated color-speckle distribution using Eq. (7). The effectiveness of both the schemes was verified through simulations and measurements. Therefore, both schemes can manifest the color-speckle characteristics of the projector. Furthermore, such schemes can be extended to multi-primary systems by Eq. (11).

4. Results and discussions

Based on this procedure, the color-speckle measurement protocol was conducted as follows. Initially, measurements were conducted using a three-primary projector. The projecting distance was 5 m, which generated an image size of 3.81 × 2.01 m². The measurement distance was set at 2 m. The measured luminance of the pure-white image was 492.5 cd/m². Both monochromatic and color speckle values were measured using separate pure-color images. The monochromatic speckle contrast values were calculated to be ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 8.77\%,\; 7.79\%,\; 4.12\%$. The color-speckle distributions for the three-primary system in J_za_zb_z space are shown in Fig. 7. Additional figures including three views are provided in Appendix A4. With respect to the speckle contrast value, red speckle was the most annoying. Therefore, the data spots are stretched in the red-green direction in the color space. The data spots of the two schemes are similar in J_za_zb_z space. In the district around the white point, data spots with a high density of the two schemes show good consistency. The slight differences between the two collections of data spots mainly emerge at the edges, which leads to deviation from the ellipsoid shape.

 

Fig. 7. Color-speckle distribution for a three-primary system in J_za_zb_z color space with ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 8.77{\%},\textrm{}7.79{\%},\textrm{}4.12{\%}$. (a) Simulation. (b) Measurement.

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Additionally, the color-speckle characteristics for the three-primary projector are listed in the top half of Table 5. In J_za_zb_z space, the two color variances, ellipsoid semi-major axis, volume, and hue angle of both schemes were approximately consistent in consideration of measurement errors. Horizontally, the variance $\sigma _{az}^2$ is larger than the variance $\sigma _{bz}^2$ because the red and green speckle contrast values are larger than blue. Vertically, the ellipsoid volume exhibits a slightly greater difference than the semi-major axis between the two schemes. This may result from the magnitude of the volume being proportional to the third power of the semi-major axis, and thus, the error will be larger. Qualitatively, the ellipsoid vertex hue angle is approximately along the red-green direction, which corresponds to the direction of the maximum speckle contrast.

Table 5. The color-speckle characteristics for two different systems in J_za_zb_z color space

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Subsequently, the experiment was conducted using a six-primary projector. The projection distance was 3 m, which generated an image size of 1.78 × 0.94 m². The measurement distance was set at 2 m. The measured luminance of the pure-white image was 747.9 cd/m². The monochromatic speckle values were calculated as ${C_{\textrm{s} - \textrm{N}}} = 4.75\%,\; 6.60\%,\; 12.40\%,\; 5.50\%,\; $ $3.17\%,\; 3.48\%$ ($\textrm{N} = {\mathrm{\lambda }_1},{\mathrm{\lambda }_2}, \ldots ,\textrm{}{\mathrm{\lambda }_6})$.The color-speckle distributions for the six-primary system in the J_za_zb_z space are shown in Fig. 8. Additional figures including three views are provided in Appendix A5. With respect to the contrast value, the yellowish-green speckle at ${\mathrm{\lambda }_3}$ was the most annoying. Therefore, the data spots were stretched in the yellowish-green direction in the color space. The data spots of the two schemes show good consistency.

 

Fig. 8. Color-speckle distribution for a six-primary system in J_za_zb_z color space with ${C_{\textrm{s} - \textrm{N}}} = 4.75\%,\; 6.60\%,\; 12.40\%,\; 5.50\%,\; 3.17\%,\; 3.48\%$. ($\textrm{N} = {\mathrm{\lambda }_1},{\mathrm{\lambda }_2}, \ldots ,\textrm{}{\mathrm{\lambda }_6})$ (a) Simulation. (b) Measurement.

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The color-speckle characteristics for the six-primary projector are listed in the bottom half of Table 5. Compared with the three-primary projector, the five physical quantities of both schemes for the six-primary projector exhibit better consistence. This may result from two reasons: one is that the measurement points of the six-primary projector are twice as many as those of the three-primary projector; the other is that the speckle contrast values of the six-primary projector are more asymmetric. In this condition, one speckle contrast value is much higher than the rest. The speckle distributions may be more aligned with Gaussian hypothesis and the measurement errors are relatively small.

Furthermore, the measured speckle distributions of the two systems in J_za_zb_z space are shown in Fig. 9 for comparison. Additional figures including three views are provided in Appendix B2. Blue spots represent the color speckle of the three-primary system and orange spots represent the color speckle of the six-primary system. The orange spots are higher than the blue spots because of their higher luminance. The fitting ellipsoids of the data spots are shown in Fig. 9(b), with the corresponding colors. In Fig. 9(b), two ellipsoids are distinguished because the upper ellipsoids are slimmer than the lower ones. The color-speckle characteristics of the two systems in the J_za_zb_z space are listed in Table 5. The two color variances of the two systems were different. The ellipsoid semi-major axis of the six-primary system is slightly larger than that of the three-primary system. However, the ellipsoid volume of the six-primary system was approximately half that of the three-primary system. Given the current observation conditions, the color speckle for the six-primary system was milder than that for the three-primary system due to the smaller ellipsoid. The comparison is based on the difference in the simplified ellipsoids. From a rigorous viewpoint, the accurate color-speckle distribution may be investigated by the color variances [13,23] along the RGB or other directions corresponding to the primary colors of multi-primary systems in the ${a_z} - {b_z}$ plane. However, the bent lines along these directions in J_za_zb_z space and complicated calculations by various multi-primary colors may be difficult for application. Furthermore, it is crucial to quantize the values of the color-speckle characteristics that correspond to the human speckle discrimination threshold in the color space. Relevant research may be conducted using similar methods in color-discrimination experiments to quantify the human-eye color-speckle perception.

 

Fig. 9. Measured color-speckle distribution for two systems in J_za_zb_z color space. (a) Data spots without hulls. (b) Fitting ellipsoids of the data spots.

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In summary, there was some deviation between the measurement and simulation results. This may have resulted from several potential reasons. First, the calculation method is incomplete, which is just an approximation without considering the different scale factors of the RGB speckle grains, respectively. Additionally, the ellipsoid simplification also causes some deviation from the original data spots. Furthermore, an unnoticeable error is the precision limit of the 8-bit gray level in [0, 255]. The light intensity captured by the CMOS for every pixel is an integer with limited precision. Therefore, the measured spots in Fig. 9(a) (see Appendix B2) show discontinuity in space, which differs from the high-precision random spots in Fig. 5(a). Moreover, there are other errors, such as voltage errors in different images, CMOS background noise, and intrinsic inhomogeneity of light intensity. Therefore, the measured spot distribution did not satisfy the statistical hypothesis in the simulation. These errors always exist in speckle measurements, but few studies have mentioned the solutions. On the one hand, various studies focus on the speckle-reduction devices and the results are based on the effectiveness with their utilization. Such errors did not influence speckle-reduction conclusions. On the other hand, these errors can be ignored when the speckle contrast is high. However, when a practical speckle-reduced projector exhibits a speckle contrast value close to human speckle perception (4%), these errors have a significant effect on the measurement. Both the CMOS camera and the data processing software need to support a higher bit depth for higher data precision. Accurate speckle measurement close to human speckle perception is a challenge in the current research. We will spare no effort to explore the solution in the next step.

5. Conclusions

In this paper, we propose and demonstrate a color-speckle assessment method based on 3D J_za_zb_z color space. The proposed method is suitable for both three-primary and multi-primary systems. Given the chromaticity coordinates of the laser sources and the target white point, the spatial distribution of the color speckle can be simulated and measured. The results were analyzed in the J_za_zb_z space. Five physical quantities, namely, the two color variances, ellipsoid semi-major axis, volume, and hue angle, were defined to describe the color-speckle characteristics. Experimental verifications were individually implemented using a three-primary laser projector and six-primary laser projector. The results of the simulation and measurement were consistent for color-speckle assessment in the color space. Furthermore, the color-speckle characteristics of different projectors with different luminance values can be compared in the same color space by different ellipsoids. The intuitive and visualized comparison is understandable for customers to select laser projectors. It provides possibilities for general color-speckle evaluation in different projectors, which is crucial for the development of standardized color-speckle assessments. In future research, if a more uniform color space is defined, the comparison with the values of defined quantities using this method will be more accurate. Additionally, we believe that it will also serve as a blueprint for speckle evaluation of multi-primary laser displays.

Appendix A1

The simulated color-speckle distributions in a symmetric condition for a three-primary system were plotted in J_za_zb_z space (see Fig. 10). The contrast values of each primary color were equal to 10%, 50%, and 100%, respectively. The luminance value was 100 cd/m². The black spot on the hull peak represents the white point.

 

Fig. 10. Color-speckle distribution for a three-primary system in a symmetric condition in J_za_zb_z color space with ${a_z} - {b_z}$, ${a_z} - {J_z}$, and ${b_z} - {J_z}$ planes. (a)-(c) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 10{\%}$. (d)-(f) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 50{\%}$. (g)-(i) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 100{\%}$.

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Appendix A2

The simulated color-speckle distributions in an asymmetric condition for a three-primary system were plotted in J_za_zb_z space (see Fig. 11). Three combinations of speckle contrast values ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}}$ are equal to (2%, 2%, 20%), (2%, 20%, 2%), and (20%, 2%, 2%), respectively. The luminance value was 100 cd/m². The black spot on the hull peak represents the white point.

 

Fig. 11. Color-speckle distribution for a three-primary system in an asymmetric condition in J_za_zb_z color space with ${a_z} - {b_z}$, ${a_z} - {J_z}$, and ${b_z} - {J_z}$ planes. (a)-(c) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 2{\%},\textrm{}2{\%},\textrm{}20{\%}$. (d)-(f) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 2{\%},\textrm{}20{\%},\textrm{}2{\%}$. (g)-(i) ${C_{\textrm{s} - \textrm{R},\textrm{G},\textrm{B}}} = 20{\%},\textrm{}2{\%},\textrm{}2{\%}$.

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Appendix A3

The simulated color-speckle distributions in a symmetric condition for a six-primary system were plotted in J_za_zb_z space (see Fig. 12). The contrast values of each primary color were equal to 10%, 50%, and 100%, respectively. The luminance value was 100 cd/m². The black spot on the hull peak represents the white point.

 

Fig. 12. Color-speckle distribution for a six-primary system in a symmetric condition in J_za_zb_z color space with ${a_z} - {b_z}$, ${a_z} - {J_z}$, and ${b_z} - {J_z}$ planes. (a)-(c) ${C_{\textrm{s} - \textrm{N}}} = 10\%$. (d)-(f) ${C_{\textrm{s} - \textrm{N}}} = 50\%$. (g)-(i) ${C_{\textrm{s} - \textrm{N}}} = 100\%$. ($\textrm{N} = {\mathrm{\lambda }_1},{\mathrm{\lambda }_2}, \ldots ,\textrm{}{\mathrm{\lambda }_6})$

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Appendix A4

The simulated and measured color-speckle distributions for a three-primary system were plotted in J_za_zb_z space (see Fig. 13). The luminance value was 492.5 cd/m². The black spot on the hull peak represents the white point.

 

Fig. 13. Color-speckle distribution for a three-primary system in J_za_zb_z color space with ${a_z} - {b_z}$, ${a_z} - {J_z}$, and ${b_z} - {J_z}$ planes. (a)-(c) Simulation. (d)-(f) Measurement.

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Appendix A5

The simulated and measured color-speckle distributions for a six-primary system were plotted in J_za_zb_z space (see Fig. 14). The luminance value was 747.9 cd/m². The black spot on the hull peak represents the white point.

 

Fig. 14. Color-speckle distribution for a six-primary system in J_za_zb_z color space with ${a_z} - {b_z}$, ${a_z} - {J_z}$, and ${b_z} - {J_z}$ planes. (a)-(c) Simulation. (d)-(f) Measurement.

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Appendix B1

The simulated color-speckle distributions for the two systems were plotted in J_za_zb_z space (see Fig. 15). The ellipsoids in Figs. 15(d)–(f) are plotted using the fimplicit3 function in MATLAB. Owing to the defects caused by the plotting precision, the ellipsoids appeared slightly angular.

 

Fig. 15. Simulated color-speckle distribution for two systems in J_za_zb_z color space with ${a_z} - {b_z}$, ${a_z} - {J_z}$, and ${b_z} - {J_z}$ planes. (a)-(c) Data spots without hulls. (d)-(f) Fitting ellipsoids of the data spots.

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Appendix B2

The measured color-speckle distributions for the two systems were plotted in J_za_zb_z space (see Fig. 16). Owing to the limited precision of the gray level, the measured spots show discontinuity in space. This phenomenon in blue spots is more serious than that in orange spots.

 

Fig. 16. Measured color-speckle distribution for two systems in J_za_zb_z color space with ${a_z} - {b_z}$, ${a_z} - {J_z}$, and ${b_z} - {J_z}$ planes. (a)-(c) Data spots without hulls. (d)-(f) Fitting ellipsoids of the data spots.

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Funding

National Key Research and Development Program of China (2021YFF0307804).

Acknowledgments

L. Deng thanks the National Key Research and Development Program of China for helping identify collaborators for this work. We are indebted to Prof. M. R. Luo of Zhejiang University and his teams for the open access to the MATLAB code for the J_za_zb_z color space.

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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