## Abstract

Displays with a larger color gamut to represent the images of the small color gamut are emphasized in the display development trend recently. Resulting from the vigorous development of Light Emitting Diodes (LEDs), the solutions to enlarge the color gamut which is formed a polygon area by adding multiple primary colors are possible, easier and inexpensive considerably. Therefore, how to determine the Gamut Boundary Description (GBD) plays a significant role for the applications of the multiple primary color displays, where the primaries form a convex polygon in CIE xy space. The paper proposed a method to construct the three-dimension color volume of GBD from the two-dimension polygon gamut area precisely regardless of that how many multiple primary colors the displays have. The method is examined in detail by the simulations and experiments, and proved it to fulfill from tri-primary color device to N-primary color device.

© 2007 Optical Society of America

## 1. Introduction

Many vision applications need accurate reproduction of color images, such as desktop publication and the networked office system. Besides, some systems requiring the original color of object, especially like medical treatment, military applications, or visual communication, are very important and developed currently. The color gamut shape employing three primaries on color chromaticity diagram is triangular, but there are large region of color chromaticity diagram outside the triangle. These insufficient colors are either clipped or mapped to similar colors of the reproducible color gamut by gamut mapping techniques. Generally, the reproduced colors are still not identical to the original, although gamut mapping techniques were studied to compensate for difference in gamut between various display devices [1–8].

Obviously, the primary colors of display apparatuses determine the color gamut. The implementation methods for enlarging color gamut are divided into two different manners for seeking to display more colors. One is a widened tri-primary color display that could exhibit a wider triangular gamut by means of high-chromaticity (pure) RGB colors. In other words, the color gamut of a display can be expanded by the use of high saturation primary colors. However, the color gamut is still limited within the triangle color gamut. The other method is a four or above multi-primary color display which have a polygon gamut by adding multiple primary colors, where the displays system that the number of primary colors is greater than three are called multi-primary color display (MPD). Color gamut expansions by means of MPD could be easier and cheaper than by widened tri-primary color displays [5]. Multi-primary color displays can be carried out in several ways, such as exploiting pure phosphors in addition to RGB ones; using various narrow-band filters, and making use of the spectrum of the light source. And six-primary liquid crystal display (LCD) projection displays, five-primary digital light processing (DLP) projection displays, and a four-primary or six-primary LCD have been developed enthusiastically [1–8]. Recently, the research of MPD is implemented by LED popularly as the vigorous development of LED.

MPD using more than three primary colors are performed by 5-primary DLP projection, 6-primary LCD projection, and a 4-primary LCD. Therefore, the color decomposition method is handling to decompose the conventional three primary colors into multi-primary control values, where there are various choices of control values for a set of tristimulus values is due to the 3-to-N-dimensional transformation [9]. These methods apply a chroma ratio interpolation with gamut boundary information, and the first step to implement these methods is obtaining gamut boundary information on the MPD. The gamut boundary information is using for implementing color gamut mapping, therefore, the gamut boundary information plays an important role for color gamut mapping. A color gamut is the region of colors that are physically realizable by a device or that are contained in as image. Typically the color space, such as CIELAB, is three dimensional, and the color gamut is finite subsets of it [10]. When colors are transferred from one device into another, such as from a cathode ray tube (CRT) monitor to LCD monitor, gamut mapping is a theory to transform colors from one gamut to another appropriately [11]. In another words, color gamut mapping is a theory of adapting (mapping) the color points of the source gamut to color points in the target gamut. The gamut mapping should preserve as much of the original color appearance as possible, in other words, representing original color by other device is the chief purpose. The quality of the computed gamut mapping depends among other factors on how good the accurate of the gamut boundaries approximate to the real gamut, where the gamut boundary is the shell of the gamut region that is the entire colors of a device or an image. In other words, color gamut mapping requires the capabilities of the exact color gamut boundary of a device gamut represented in a color space [12]. Therefore, the determination of gamut boundaries is a task for a number of purposes. It is easy to understand what range of colors achievable on a given color reproduction medium and is present in a given image. It can also be done to see what color gamut is predicted by a characterization model and, hence, to see how good quality of the model for the calculation of gamut boundaries. Finally, it can also be done to allow for the use of a gamut mapping algorithm in a color reproduction system, and a set of approaches used for calculating gamut boundaries has a different degree of suitability for the satisfaction of the above mentioned motives [13].

Data describing a gamut boundary in perceptual color space are usually stored in a gamut boundary descriptor (GBD). The approaches and discussions for the GBD are well developed and introduced at the following. An approach to compute color gamut boundaries fast and geometrically accurate without sacrificing geometric accuracy of the computed gamut boundary was proposed by Joachim Giesen [10]. This approach is well used to compute device gamut boundary with a polyhedral surface and proposed as a good compromised between quality and time. The segment maxima GBD (SMGBD), which is described by a matrix containing the most extreme colors for each segment of color space, was introduced by Morovic [13]. Face Triangulation Method to compute the physical gamut boundary of a color printer was proposed by Paolo Pellegri [12]. The algorithm of gamut boundary determination of CRT display in perceptual color space was proposed by Yong Wang. It requires characterization using gain-offset-gamma model at first, then the equal lightness loci on the surface of digital-to-analog converter (DAC) value space are computed and the gamut boundaries in the perceptual color space are visualized [14]. The descriptor for a LCD display gamut boundary surfaces with a form of Zernike polynomial to express the six gamut boundary surfaces was introduced by Qingmei Huang [15, 16]. An analytical method for compact GBD was introduced by Herzog, where the kernel gamut of unit cube is distorted to match to the device gamut [17]. And there are still some related discussions about the color gamut boundary [18–24].

The approaches and descriptors mentioned above for constructing the color gamut boundary have the same first step to implement. The same first step is to obtain the color points on the gamut boundary surfaces as the base to construct the gamut boundary. There are two approaches to obtain these color points on the gamut boundary surfaces. One approach is to obtain the all combination color points by the N channels for N-primary color device, and then the color points on the gamut boundary surfaces are picked out from the all combination color points [23, 24]. But a big part of color points un-on the gamut boundary surface is waste for constructing the gamut boundary, and the process of picking out the operative color points on the gamut boundary surface form these waste points un-on the gamut boundary surface is consumed extra time. Therefore another better approach is directly to obtain the color points on the gamut boundary surfaces by the three channels for tri-primary color device [10,12–22]. Base on the approach, the measurement of color points on the gamut boundary surface for constructing gamut boundary surface saves much time without measuring the these waste points un-on the gamut boundary surface, and it is efficient to pick out the operative color points on the gamut boundary surface form these waste points un-on the gamut boundary surface.

This paper proposes a theory for the approach to obtain the color points on the gamut boundary surfaces directly. The theory bases on the theory of the prior paper [25] and will be introduced first. Further the approach to obtain the color points on the gamut boundary surfaces directly by the N channels for N-primary color device, where the shape of N primaries in CIE xy space is a convex polygon, will be proposed later. Finally, the proposed theory is discussed with different gamma curve in detail by the simulations, and is examined and proved by experiments.

## 2. Theory

#### 2.1 The traditional approach for GBD of tri-primary color device

Any device is limited to the gamut determined by the primary colors. The all combinations of the all color point in the gamut are composed by N channels for N-primary color device. For example, there is a tri-primary color device with digital information (R, G, B) as its input, where R, G, and B refer to the red, green, and blue channel digital information scalar varied from 0 to 255 in the digital-to-analog converter (DAC). The number of the total color combinations is (2^{8})^{3}= 16,777,216. Further there are (2^{ζ})^{N}=2^{ζN} color combinations on the N-primary color device with ζ, bit digital information scalar varied for 0 to 2^{ζ}-1. A big part of color points are not on the gamut boundary surface. These points off the gamut boundary surface are waste for constructing gamut boundary surface, and the process of picking out the operative color points on the gamut boundary surface form these waste points is consumed extra time. Therefore the better approach to obtain directly the color points of the gamut boundary surface is introduced at the following.

The approach to obtain the color points on the gamut boundary surfaces directly by the three channels for tri-primary color device is proposed in prior papers [10,12–22]. These color points on the gamut boundary surface can be obtained by setting one of the device coordinates at 0% or 100% for each coordinate in turn, and varying the other two coordinates. For example, there is a tri-primary color device with three image planes of digital information, R, G, B, as its input, where R, G, and B refer to the red, green, and blue channel digital information scalar varied from 0 to 255 in the DAC, and there are (2^{8}-2)^{3} color points that are inside the gamut and not on the gamut boundary surface. Therefore, there are only (2^{8})^{3}-(2^{8}-2)^{3}=3×2×(2^{8})^{2}-3×2^{2}×2^{8}+2^{3} color points on the gamut boundary surface. Further there are (2^{M})^{3}-(2^{M}-2)^{3}=3×2×(2^{M})^{2}-3×2^{2}×2^{M}+2^{3} color combinations on the tri-primary color device with the channel digital information scalar of M bit varied for 0 to 2^{M}-1. But the approach mentioned above is just only appropriately using for tri-primary color device, the powerful approach appropriately using for N-primary color device, where N is large than three, is proposed in the following.

#### 2.2 The proposed approach for GBD of N-primary color device

The approach to obtain the color points on the gamut boundary surfaces directly by the N channels for N-primary color device is proposed in the following. This approach bases on the theory of the gamut boundary versus various brightness in the prior paper [25], therefore, the theory will be introduced first. Secondly the approach for tri-primary color device is discussed by the theory in detail. Further, the approach for N-primary color device is inferred from the one for four-primary color device finally.

### 2.2.1 Color gamut boundary under various brightness

Brightness of color displays, however, strongly restricts the color gamut of displays. A theory to analyze the relation between brightness and color gamut for the multi-primary color display is proposed [25]. From the color mixing theory, the area of color gamut on the color coordination would shrink smaller when brightness grows up, where the polygon region of color gamut at different brightness is determined by the apexes. These apexes can be determined by the following theory. As considering color gamut on N-primary color, we will discuss the color gamut boundary of *Y’*, where N is an integral large than two, the coordinates (x_{i}, y_{i}) of N-primary color with the maximal luminous flux Y_{i} are respected as C_{i} that i is the integral from 0 to N-1, and *Y’* is the luminous flux after reduction for mixing the appropriate color. When Y_{i} + Y_{i+1} + … + Y_{j} < *Y’* < Y_{i-1} + Y_{i} + Y_{i+1} + … + Y_{j} and Y_{i} + Y_{i+1} + … + Y_{j} < *Y’* < Y_{i} + Y_{i+1} + … + Y_{j} + Y_{j+1}, where i and j are the integrals from 0 to N-1, and as *Y’* increases, we can analogize that two apexes of the color gamut boundary approach to the two neighbor primary colors nearest the j-i+1 close primary colors, where j-i>=0 and the shape of the N primary color chromaticites is convex polygon. The above characterization is represented by the math form, B[M_{1}, M_{2}]=B[P(P(C_{i}, C_{i+1},…, C_{j}), C_{i-1}), P(P(C_{i}, C_{i+1},…, C_{j}), C_{j+1})] =B[P(G, C_{i-1}), P(G, C_{j+1})], where G represents the center-of-gravity positions of C_{i}, C_{i+1},…, C_{j-1}, and C_{j}, B[M_{1}, M_{2}] represents the color gamut boundary of the line linked by the two apexes, M_{1} and M_{2}, and P(C_{0}, C_{1}, C_{2}, C_{3}, …, C_{N}) represents the center-of-gravity positions of C_{0}, C_{1}, C_{2}, C_{3}, …, and C_{N}. In other words, the mixing color coordinate G is at the center-of-gravity position of weights Y_{i}/y_{i} at C_{i} , Y_{i+1}/y_{i+1} at C_{i+1},…, and Y_{j}/y_{j} at C_{j}. Therefore there are three relations about G and the two apexes of the color gamut boundary, M_{1} and M_{2}, as shown in following equations:

Following the conclusion, we obtain the color gamut boundary of multi-primary color display system, and the color gamut boundary are the lines linked by these available color gamut boundary apexes M with the same brightness, when total maximal luminous flux of the close primary colors is smaller than total luminous flux *Y’*, and when total luminous flux *Y’* is smaller than the total luminous flux which is the sum of total maximal luminous flux of the close primary colors and one of the two neighbor primary color luminous flux.

For tri-primary color device, the chromaticity coordinates of tri-primary colors, red(R), green(G), and blue(B), are C_{r}(x_{R}, y_{R}), C_{g}(x_{G}, y_{G}), and C_{b}(x_{B}, y_{B}) individually, and the maximal value of three primary color luminous flux, Y_{R}, Y_{G}, and Y_{B}, are Y_{R,Max}, Y_{G,max}, and Y_{B,max} individually. The gamut boundary presented in the Fig. 1 is given as an example.

### 2.2.2 The proposed approach for GBD of tri-primary color devices

When these available color gamut boundary apexes with the gradually various brightness are linked, the shell of color gamut can be separated into 3×(3-1) surfaces as Fig. 2(a). These 3× (3-1) surfaces can be divided into two loops, where each loop has three surfaces. The each channel digital information scalars of red C_{r}(α, 0, 0), green C_{g}(0, (3, 0), and blue C_{b}(0, 0, γ) are α, β, and γ respectively in DAC, where C_{r}(α, 0, 0), C_{g}(0, β, 0), and C_{b}(0, 0, γ) are representative the color coordinates of such digital information of red, green, and blue; C_{r}, C_{g}, and C_{b} are C_{r}(255, 0, 0), C_{g}(0,255, 0), and C_{b}(0, 0, 255) that are the 100% of C’_{r}, C’_{g}, and C’_{b} respectively, where the multiple representations C’ indicate colors of the weakest extreme of primaries; and C_{M}(255, 0, 255) is equal to P(C_{b}, C_{r}) that represents the center-of-gravity position of C_{r} and C_{b}. Here we set that the each channel scalar of digital information has 8 bit quantization steps in DAC. Therefore, the all possible color combinations could represent by digital information (α, β, γ), where channel digital information scalars, α, β, and γ, has 2^{8} quantization steps. The surfaces of first loop at the lower brightness can be defined by the curves linked with five apexes. For example, one of the surface in first loop is defined by the curves linked with the volume apexes, C’_{b}(0, 0, 0), C’_{r}(0, 0, 0), C_{r}(255, 0, 0), C_{M}(255, 0, 255), C_{b}(0, 0, 255), as Fig. 2(a) showing. The colors in the surface are only combined with blue and red. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalar of green at 0%, and varying the other two digital information scalars of red and blue. If ζ is the number of bits in DAC, the all possible color combinations could represent by various digital information (α, β, γ), where each scalar, α, β, and γ, has 2^{ζ} quantization steps. The total possible color combinations of this gamut boundary surface obtained by setting the digital information as (α, 0, γ), where α and γ are
various from 0 to 2^{ζ}-1, are 2^{ζ}×2^{ζ}=2^{2ζ} types. In the same way, the other surfaces in first loop are
defined by the curves linked with the volume apexes, C’_{r}(0, 0, 0), C’_{g}(0, 0, 0), C_{g}(0, 255, 0), C_{Y}(255, 255, 0), C_{r}(255, 0, 0), and the curves linked with the volume apexes, C’_{g}(0, 0, 0), C’_{b}(0, 0, 0), C_{b}(0, 0, 255), C_{C}(0, 255, 255), C_{g}(0, 255, 0) respectively, where C_{Y}(255, 255, 0) and C_{C}(0, 255, 255) are equal to P(C_{r}, C_{g}) and P(C_{g}, C_{b}) that are represents the center-of-gravity positions of C_{r} and C_{g}, C_{g} and C_{b} respectively. Therefore, the total possible color combinations of the gamut boundary surface in this two surfaces can be obtained respectively by setting the digital information scalar of blue at 0%, and varying the other two digital information scalars, and by setting the digital information scalar of red at 0%, and varying the other two digital information scalars, just like to setting digital information as (α, β, 0) and (0, β, γ) respectively, where α, β, and γ are various from 0 to 2^{ζ}-1. Therefore the total possible color combinations of the gamut boundary surface in first loop can be obtained respectively by setting one specific digital information scalar at 0, and varying the other two digital information scalars.

The surfaces of the second loop are defined conveniently at a vertical view of Fig. 2(a), as Fig. 2(b) showing. The surfaces of second loop at the higher brightness can be defined by the curves linked with four apexes. For example, one of the surface in second loop is defined by the curves linked with the volume apexes, C_{r}(255, 0, 0), C_{Y}(255, 255, 0), C_{W}(255, 255, 255), C_{M}(255, 0, 255), where CW(255, 255, 255) is equal to P(C_{r}, C_{g}, C_{b}) that represents the center-of-gravity position of C_{r}, C_{g}, and C_{b}. The colors in the surface are combined with red, green and blue, where the digital information scalar of red holds at 100%. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalar of red at 100%, and varying the other two digital information scalars. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (2^{ζ}-1, β, γ), where β and γ are various from 0 to 2^{ζ}-1, are 2^{ζ}×2^{ζ}=2^{2ζ} types. In the same way, the other surfaces in second loop are defined by the curves linked with the volume apexes, C_{g}(0, 255, 0), C_{C}(0, 255, 255), C_{W}(255, 255, 255), C_{Y}(255, 255, 0), and the curves linked with the volume apexes, C_{b}(0, 0, 255), C_{M}(255, 0, 255), C_{W}(0255 255, 255), C_{C}(0, 255, 255) respectively, where . Therefore, the total possible color combinations of the gamut boundary surface in this two surfaces of the surfaces can be obtained respectively by setting the digital information of green at 100%, and varying the other two digital information, and by setting the digital information of blue at 100%, and varying the other two digital information, just like to setting the digital information as (α, 2^{ζ}-1, γ) and (α, β, 2^{ζ}-1) respectively, where α, β, and γ are various from 0 to 2^{ζ}-1. Therefore the total possible color combinations of the gamut boundary surface in second loop can be obtained respectively by setting one specific digital information scalar at 2^{ζ}, and varying the other two digital information scalars.

Therefore the color gamut volume is constructing with six boundary surfaces defined by the curves linked mentioned above. The number of the total colors on the gamut boundary is
(2^{ζ})^{3}-(2^{ζ}-2)^{3}=6×2^{2};-12×2^{ζ}+8. In this equation, the first item, 2^{2ζ}, respects the color number on each surface, the second item, 2^{ζ}, respects the color number on the curve linked with the color gamut boundary apexes, where the curve are showing in the Fig. 2(a), and the final item, 8, respects the number of the volume apexes.

### 2.2.3 The proposed approach for GBD of four-primary and N-primary color devices

As four-primary color device, when these available color gamut boundary apexes with the gradually various brightness are linked, the color gamut boundary surface will be divided into 4×(4-1) surfaces. For four-primary colors device, the each channel digital information scalars of C_{1}(α, 0, 0, 0), C_{2}(0, β, 0, 0), C_{3}(0, 0, γ, 0) and C_{4}(0, 0, 0, κ) are α, β, γ, and κ respectively in DAC, where C_{1}(α, 0, 0, 0), C_{2}(0, β, 0, 0), C_{3}(0, 0, γ, 0)and C_{4}(0, 0, 0, κ) are representative the color coordinates of such digital information, (α, 0, 0, 0), (0, β, 0, 0), (0, 0, γ, 0), and (0, 0, 0, κ); C_{1}, C_{2}, C_{3}, and C_{4} are C_{1}(255, 0, 0, 0), C_{2}(0, 255, 0, 0), C_{3}(0, 0, 255, 0) and C_{4}(0, 0, 0, 255) that are 100% of C’_{1}, C’_{2}, C’_{3}, and C’_{4} respectively; P(C_{1}, C_{2}, C_{3}, …, C_{N}) represents the center-of-gravity positions of C_{1}, C_{2}, C_{3}, …, and C_{N}. Here we set that the each channel scalar of digital information has 8 bit quantization steps in DAC. Therefore, the all possible color combinations could represent by digital information (α, β, γ, κ), where channel digital information scalars, α, β, γ, and κ, has 2^{8} quantization steps. There are (4–1) loops of surfaces in the 4-primary color gamut, and each loop contains 4 surfaces, where the surfaces of the first loop are defined by five vertex point, and the surfaces of the other loops are defined by four vertex points. The surfaces of first loop at the lower brightness can be defined by the curves linked with five apexes. For example, one of the surface in first loop is defined by the curves linked with C’_{1}, C’_{2}, C_{2}, P(C_{1}, C_{2}), and C_{1} as Fig. 3(a) showing. The colors in the surface are only combined with C_{1} and C_{2}. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalars of C_{3} and C_{4} at 0%, and varying the other two digital information scalars of C_{1} and C_{2}. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (0, 0, γ, κ), where γ and κ are various from 0 to 2^{ζ}-1, are 2^{ζ}×2^{ζ}=2^{2ζ} types. In the same way, the other surfaces in first loop are defined by the curves linked the five vertex points listed at Table 1. The total possible color combinations of the gamut boundary surface in each surface can be obtained respectively by setting two specific digital information scalars at 0, and varying the other two digital information scalars as Table 1 showing.

The surfaces of the second and third loop are defined conveniently at a vertical view of Fig. 3(a), as Fig. 3(b) showing. The surfaces of second loop can be defined by the curves linked with four apexes. For example, one of the surface in second loop is defined by the curves linked with C_{1}, P(C_{1}, C_{2}), P(C_{4}, C_{1}, C_{2}), and P(C_{4}, C_{1}). The colors in this surface are only combined with C_{1}, C_{2}, and C_{4}. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalar of C_{1} at 100%, the digital information scalar of C_{3} at 0%, and varying the other two digital information scalars of C_{2} and C_{4}. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (2^{ζ}-1, β, 0, κ), where β and κ are various from 0 to 2^{ζ}-1, are 2^{ζ}×2^{ζ}=2^{2ζ} types. In the same way, the other surfaces in second loop are defined by the curves linked the four vertex points listed at Table 2. The total possible color combinations of the gamut boundary surface in each surface can be obtained respectively by setting one digital information scalar at 0, one digital information scalar at 2^{ζ}-1, and varying the other two digital information scalars as Table 2 showing.

In the same way, the surfaces of third loop can be defined by the curves linked with four apexes, too. For example, one of the surface in third loop is defined by the curves linked with P(C_{1}, C_{2}), P(C_{1}, C_{2}, C_{3}), P(C_{1}, C_{2}, C_{3}, C_{4}), and P(C_{4}, C_{1}, C_{2}). The colors in this surface are combined with C_{1}, C_{2}, C_{3}, and C_{4}. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalars of C_{1} and C_{2} at 100%, and varying the other two digital information scalars of C_{3} and C_{4}. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (2^{ζ}-1, 2^{ζ}-1, γ, κ) are 2^{ζ}×2^{ζ} =2^{2ζ} types. In the same way, the other surfaces in third loop are defined by the curves linked the four vertex points listed at Table 3. The total possible color combinations of the gamut boundary surface in each surface can be obtained respectively by setting two specific digital information scalars at 2^{ζ}-1, and varying the other two digital information scalars as Table 3 showing.

Therefore the color gamut volume is constructing with twelve boundary surfaces defined by the curves linked mentioned above. The number of the total colors on the gamut boundary is 12×2^{2ζ}-24×2^{ζ}+14. In this equation, the first item, 2^{2ζ}=, respects the color number on each surface, the second item, 2^{ζ}, respects the color number on the curve linked with the color gamut boundary apexes, where the curve are showing in the Fig. 3(a), and the final item, 14, respects the number of the volume apexes.

For N-primary colors device, the each channel digital information scalars of C_{1}(D_{1}, 100%, …, 0), C_{2}(0, D_{2}, 0, …, 0), …, and C_{N}(0, 0, …, 0, D_{N}) are D_{1}, D_{2}, … , and D_{N} respectively in DAC, where C_{1}(D_{1}, 0, …, 0), C_{2}(0, D_{2}, 0, …, 0), …, and C_{N}(0, 0, …, 0, D_{N}) are representative the color coordinates of such digital information, (D_{1}, 0, …, 0), (0, D_{2}, 0, …, 0), …, and (0, 0, …, 0, D_{N}); C_{1}, C_{2}, …, and C_{N} are C_{1}(255, 0, …, 0), C_{2}(0, 255, 0, …, 0), …, and C_{N}(0, 0, …, 0, 255) that are 100% of C’_{1}, C’_{2}, …, and C’_{N} respectively; P(C_{1}, C_{2}, C_{3}, …, C_{N}) represents the center-of-gravity positions of C_{1}, C_{2}, C_{3}, …, and C_{N}. Here we set that the each channel scalar of digital information has ζ bit quantization steps in DAC. Therefore, the all possible color combinations could represent by digital information (D_{1}, D_{2}, …, D_{N}), where channel digital information scalars, D_{1}, D_{2}, … , and D_{N}, has 2^{ζ} quantization steps. As N-primary color device, when these available color gamut boundary apexes with the gradually various brightness are linked, the color gamut boundary surface will be divided into N×(N-1) surfaces. There are (N-1) loops of surfaces in the N-primary color gamut, and each loop contains N surfaces, where the surfaces of the first loop are defined by five vertex point, and the surfaces of the other loops are defined by four vertex points. The surfaces of first loop at the lower brightness can be defined by the curves linked with five apexes. For example, the first surface in first loop is defined by the curves linked with C’_{1}, C’_{2}, C_{2}, P(C_{1}, C_{2}), and C_{1}. The colors in the surface are only combined with C_{1} and C_{2}. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalars of C_{3}, C_{4}, …, and C_{N} at 0%, and varying the other two digital information scalars of C_{1} and C_{2}. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (D_{1}, D_{2}, 0, 0, …, 0), where D_{1} and D_{2} are various from 0 to 2^{ζ}-1, are 2^{ζ}×2^{ζ}=2^{2ζ} types. In the same way, the n^{st} surfaces in first loop are defined by the curves linked the five vertex points listed, too. The n^{st} surface in first loop is defined by the curves linked with C’_{n}, C’_{n+1}, C_{n+1}, P(C_{n}, C_{n+1}), and C_{n}. Here we set that n+m is equal to n+m-N, if n+m is large then N. The colors in the surface are only combined with C_{n} and C_{n+1}. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalars of C_{1}, C_{2}, … , C_{n-1}, C_{n+2}, and C_{N} at 0%, and varying the other two digital information scalars of C_{n} and C_{n+1}. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (0, 0, …, D_{n}, D_{n+1}, 0, 0, …, 0), where D_{n} and D_{n+1} are various from 0 to 2^{ζ}-1, are 2^{ζ}×2^{ζ}=2^{2ζ} types. The total possible color combinations of the gamut boundary surface in each surface can be obtained by setting N-2 digital information scalars at 0%, and varying the other two neighbor digital information scalars respectively as Table 4 showing.

The surfaces of the other loops can be defined by the curves linked with four apexes. For example, n^{st} of the surface in second loop is defined by the curves linked with C_{n}, P(C_{n}, C_{n+1}), P(C_{n-1}, C_{n}, C_{n+1}), and P(C_{n-1}, C_{n}). Here we set that n+m is equal to n+m-N, if n+m is large then N. The colors in this surface are only combined with C_{n-1}, C_{n}, and C_{n+1}, where the digital information scalar of C_{n} holds at 100%. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalars of C_{n} at 100%, the digital information scalars of C_{1}, C_{2}, … , C_{n-2}, C_{n+2}, …, and C_{N} of the device coordinates at 0%, and varying the other two digital information scalars of C_{n-1} and C_{n+1}. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (0, 0, …, D_{n-1}, 2^{ζ}-1, D_{n+1}, 0, 0, …, 0) are 2^{ζ}×2^{ζ}=2^{2ζ} types as Table 4 showing. As m^{st} loop, n^{st} of the surface in m^{st} loop is defined by the curves linked with P(C_{n}, …, C_{n+m-2}), P(C_{n}, , …,C_{n+m-1}), P(C_{n-1}, C_{n}, …, C_{n+m-1}), and P(C_{n-1}, C_{n}, …, C_{n+m-2}). The colors in this surface are only combined with C_{n-1}, C_{n}, …, and C_{n+m-1}, where the digital information scalar of C_{n}, C_{n+1}, …, and C_{n+m-2} hold at 100%. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalars of C_{n}, C_{n+1},…, and C_{n+m-2} at 100%, the digital information scalars of C_{1}, C_{2}, …, C_{n-2}, C_{n+m}, …, and C_{N} at 0%, and varying the other two digital information scalars of C_{n-1} and C_{n+m1}. In other words, the total possible color combinations of this gamut boundary surface obtained by setting the digital information as (0, 0, …, D_{n-2}, 2^{ζ}-1, …, 2^{ζ}-1, D_{n+1}, 0, 0, …, 0) are 2^{ζ}×2^{ζ}=2^{2ζ} types as Table 4 showing. As N-1^{st} loop, n_{st}1 of the surface in N-1^{st} loop is
defined by the curves linked with P(C_{n}, …, C_{n+N-3}), P(C_{n}, , …,C_{n+N-2}), P(C_{1}, C_{2}, …, C_{N}), and P(C_{n-1}, C_{n}, …, C_{n+N-3}). The colors in this surface are combined with C_{1}, C_{2}, …, C_{N}, where the digital information scalar of C_{1},…,C_{n-3}, C_{n}, …, and C_{N} hold at 100%.. Therefore, the total possible color combinations of this gamut boundary surface can be obtained by setting the digital information scalars of C_{1}, …,C_{n-3}, C_{n}, …, C_{N} at 100%, and varying the other two digital information scalars of C_{n-2} and C_{n-1}. In other words, the total possible color
combinations of this gamut boundary surface obtained by setting the digital information as
(2^{ζ}-1, …, 2^{ζ}-1, D_{n-2}, D_{n-1}, 2^{ζ}-1, …, 2^{ζ}-1) are 2^{ζ}×2^{ζ}=^{2ζ} types as Table 4 showing.

Therefore the color gamut volume is constructing with N×(N-1) boundary surfaces defined by the curves linked mentioned above. The number of the total colors on the gamut boundary is N×(N-1)×2^{2ζ}-2N×(N-1)×2^{ζ}+(N×(N-1)+2). In this equation, the first item, 2^{2ζ}, respects the color number on each boundary surface, the second item, 2^{ζ}, respects the color number on the curve linked with the color gamut boundary apexes, and the final item, (N×(N-1)+2), respects the number of the volume apexes.

## 3. Simulations

The primary color coordinates, brightness, and its gamma curves of the brightness versus gray levels are obtained, then the all possible color combinations on the gamut boundary can be simulated through proposing digital information by color mixing theory. Therefore, the simulations are discussed for various gamma curves to analyze the relation between the various gamma curves and the distributions of the colors on the boundary surface.

First, the gamma curves of the brightness (B) versus gray levels (L) can be presented as

where *a* is gamma value, and B_{M} is the maximum brightness. The two set gamma values (*a*
_{R}, *a*
_{G}, *a*
_{B}, *a*
_{Y}) of primary color, (0.6, 1, 2.2, 3.4) and (1, 1, 1, 1) respectively, are discussed at the following. The curve of the two set gamma curve are showing as Fig. 4(a), where the data of red, green , blue, and yellow are marked with cross, square, point, and circle respectively.. The four-primary color coordinates, C_{R} , C_{G}, C_{B}, and C_{Y}, are (0.5882, 0.3872), (0.2933, 0.6425), (0.1449, 0.1121), and (0.4036, 0.5964) respectively, and its maximum brightness are 16.24, 88.51, 11.48, and 68.73 respectively. The all colors on twelve gamut boundary surfaces can be mixing by the proposing digital information of (α, β, γ, κ) like Table 1, 2, and 3, where α, β, γ, and κ are the level indexes. The all possible color combinations on the gamut boundary are simulated and showing as Fig. 4(b) and Fig. 4(c), where Fig. 4(b) and Fig. 4(c) are the sets of gamma value (*a*
_{R}, *a*
_{G}, *a*
_{B}, *a*
_{Y}), (0.6, 1, 2.2, 3.4) and (1, 1, 1, 1). From Fig. 4(b) and Fig. 4(c), we find that the framework constructing the boundary is invariable with different. For an example, there are 4×(4-1)=12 surfaces constructing of the gamut boundary for four-primary color device. Further the intensity of colors on the some boundary surfaces, just like the boundary surface of higher brightness in Fig. 4(b), is thick. In other words, the intensity of colors on the boundary surfaces is various with different gamma curve. Therefore, if each primary color gamma curves are much inappropriate so as to the much un-uniform intensity of color distribution on the boundary, and there are more errors to construct gamut boundary. The un-uniform intensity of color distribution on the boundary in La_{*}b^{*} space is more clearly, as Fig. 4(d) and Fig. 4(e) showing, where the Fig. 4(b) and Fig. 4(c) transferring into in La^{*}b^{*} space are Fig. 4(d) and Fig. 4(e). At lower brightness, the intensity of color distribution on the boundary is sparse.4. Typographical style

## 4. Experiments for the Tri-primary and Four-primary color displays

The all possible color combinations on the gamut boundary can be obtained by measuring the colors of the proposed digital information. The experiments are discussed on Tri-primary and Four-primary color displays respectively at the following. We perform a tri-primary color device, a LCD monitor, and a four-primary color device constituting by two projectors, a LCD projector and a DLP projector, to prove our theory.

First, the tri-primary color device, a LCD monitor (LG L1970HR-WF), and a color analyzer (CA-210) are used for color signal source and receiver respectively, and experiment framework is shown in Fig. 5(a). And the photo of the equipment is shown in Fig. 6(a). The screen of LCD monitor was fixed, and color analyzer location was set in the screen center. The all possible color combinations on the gamut boundary can be obtained by measuring the colors of the proposing digital information (α, β, γ) that one of α, β, γ is 0 or 255, and the others are various from 0 to 255 by 32 steps in this experiment. The all possible colors on the gamut boundary are measured and comparing with the all colors produced by uniform digital information in the gamut boundary, as Fig.7 showing, where the data on gamut boundary and in the gamut volume are marked with bigger dark point and smaller light point. The datum of all possible colors on the gamut boundary are marked with blue point, the datum of all colors in the gamut boundary are marked with red point. Clearly the colors produced by proposed digital information are on the gamut boundary, and are out of the other colors by uniform digital information in the gamut boundary.

For four-primary color device, two projectors (Projector I is a DLP projector, Seha compact 236, and projector II is a LCD projector, professional EX-2700.) and a color analyzer (Minolta CL-200) are used for color signal source and receiver respectively, and experiment framework is shown in Fig. 5(b). And the photo of the equipment is shown in Fig. 6(b). The screens of two projectors are fixed at the same location, and color analyzer location is set in the screen center. The LCD projector provides yellow colors produced by setting the digital information as (κ, κ, 0), and the DLP projector produces red, green, and blue colors produced by setting the digital information (α, β, γ), where α, β, γ, κ are various form 0 to 255. The all possible color combinations on the gamut boundary can be obtained by measuring the colors of the proposing digital information (α, β, γ, κ) as Table 1, 2, and 3 showing, where α, β, γ, κ are various form 0 to 255 by 16 steps in this experiment. The all possible colors on the gamut boundary are measured and comparing with the all colors produced by uniform digital information in the gamut boundary, as Fig. 8 showing, where the data on gamut boundary and in the gamut volume are marked with bigger dark point and smaller light point. The datum of all possible colors on twelve gamut boundary surfaces are marked with blue point, the datum of all colors in the gamut boundary are marked with red point. Clearly the colors produced by proposed digital information are on the gamut boundary, and are out of the other colors by uniform digital information in the gamut boundary.

## 5. Discussion and Conclusion

The advantage of the displays with a larger color gamut is representing the more real and nature image with more colors. As the vigorous development of LED, the implementation method for enlarging color gamut is implemented more easily for seeking to display more colors. The color gamut of the multi-primary color devices implemented by LEDs is more extensive and pure, it is more inexpensive then the one implemented by laser light source, and it is practicable for LCD displays of LED especially. Therefore, there are many researches and algorithms about color gamut and its related color applications from tri-primary color device to multi-primary color device. In this paper, the approach to directly obtain the colors on the gamut boundary for constructing the N-primary color gamut boundary is proposed. The theory is explained by the detailed research of gamut boundary condition. Further, the theory used for N-primary color device is proved. The color gamut is constructing of N×(N-1) surfaces on N-primary color device. What the color component of the colors on each surface is can be known. The number of the total colors on the gamut boundary is N×(N-1)×2^{2ζ}-2N×(N-1)× 2^{ζ}=+(N×(N-1)+2). From simulations, we find that the framework constructing the boundary is invariable with different, and the intensity of colors on the boundary surfaces is various with different gamma curve. Finally from experiments, clearly the colors produced by proposed digital information are on the gamut boundary, and are out of the other colors by uniform digital information in the gamut boundary, and our theory is proved.

Therefore, the colors on the N-primary color gamut boundary can be directly obtained without consuming the extra time for picking out them from the chaotic colors. The research about the surfaces constructing the gamut volume and the color component of the colors on the each surface are important for construction of the color gamut volume boundary; the color volume constructing of the surfaces is invariable under the device with different gamma curves, but the different gamma curves effect the distribution of the color datum. Finally there are some errors of color volume construction under un-continuous color datum whose intensity of distribution is too spare as Fig. 4(e) showing. In the same way, there will be some errors of color volume construction if the gamma curve of the brightness versus gray levels is not appropriate.

## Acknowledgments

This work is supported by National Science Council of Taiwan under the projects NSC 95-2221-E-008-120-MY3 and NSC 95-2221-E-008-119-MY3. Authors also want to thank Delta Electronics Ltd. of Taiwan for the kind assistance in experiments.

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