## Determination of Gamut Boundary Description for multi-primary color displays

Optics Express, Vol. 15, Issue 20, pp. 13388-13403 (2007)

http://dx.doi.org/10.1364/OE.15.013388

Acrobat PDF (1731 KB)

### 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

5. T. Ajito, T. Obi, M. Yamaguchi, and N. Ohyama, “Multiprimary color display for liquid crystal display projectors using diffraction grating,” Optical Engineering **38**, 1883–1888 (1999). [CrossRef]

23. P. Zolliker and K. Simon, “Continuity of gamut mapping algorithms,” Journal of Electronic Imaging **15**, 013004–013012 (2006). [CrossRef]

## 2. Theory

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

^{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.

^{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

#### 2.2.1 Color gamut boundary under various brightness

*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:

*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.

_{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

_{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.

_{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.

^{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

_{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.

_{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.

_{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.

^{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.

_{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.

^{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.

^{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

*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

## 5. Discussion and Conclusion

## Acknowledgments

## References and links

1. | Y. Wang and H. Xu, “Determination of CRT color gamut boundaries in perceptual color space,” in |

2. | S. Wen, “Design of relative primary luminances for four-primary displays,” Displays |

3. | S. Roth, I. Ben-David, M. Ben-Chorin, D. Eliav, and O. Ben-David, “Wide gamut, high brightness multipl primaries single panel projection displays,” Digest SID’03 (2005). |

4. | T. Ajito, T. Obi, M. Yamaguchi, and N. Ohyama, “Expanded color gamut reproduced by six-primary projection display,” in |

5. | T. Ajito, T. Obi, M. Yamaguchi, and N. Ohyama, “Multiprimary color display for liquid crystal display projectors using diffraction grating,” Optical Engineering |

6. | Y. Murakami, N. Hatano, J. Takiue, M. Yamaguchi, and N. Ohyama, “Evaluation of smooth tonal change reproduction on multiprimary display: comparison of color conversion algorithms,” in |

07. | M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, “Color image reproduction based on multispectral and multiprimary imaging: experimental evaluation,” in |

8. | M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, “Natural color reproduction in the television system for telemedicime,” in |

9. | J.-G. Chang, C. L.-D. Liao, and C.-C. Hwang, “Enhancement of the optical performances for the LED backlight systems with a novel lens cap,” in |

10. | P. C. P. Chao, L. -D. Liao, and C.-W. Chiu, “Design of a novel LED lens cap and optimization of LED placement in a large area direct backlight for LCD-TVs,” in |

11. | E. H. Ford, “Projection optical system for a scanned LED TV display,” in |

12. | W. Y. Lee, Y. C. Lee, K. Sokolov, H. J. Lee, and I. Moon, “LED projection displays,” in |

13. | B. A. Salters and M. P. C. M. Krijn, “Color reproduction for LED-based general lighting,” in |

14. | D.-W. Kang, Y.-T. Kim, Y.-H. Cho, K.-H. Park, W. Choe, and Y.-H. Ha, “Color decomposition method for multiprimary display using 3D-LUT in linearized LAB space,” in |

15. | J. Giesen, E. Schuberth, K. Simon, and P. Zolliker, “Toward image-dependent gamut mapping: fast and accurate gamut boundary determination,” in |

16. | H. Zeng, “Spring-primary mapping: a fast color mapping method for primary adjustment and gamut mapping,” in |

17. | P. Pellegri and R. Schettini, “Gamut boundary determination for a color printer using the Face Triangulation Method,” in |

18. | M. JÁn and L. Ronnier, “Calculating medium and image gamut boundaries for gamut mapping,” Color Research & Application |

19. | Q. Huang and D. Zhao, “The color gamut of LCD and its analytical expression,” in |

20. | Q. Huang and D. Zhao, “Gamut boundaries expressed with Zernike polynomials,” in |

21. | P. G. Herzog, “Further development of the analytical color gamut representation,“ in |

22. | M. D. F. Fritz Ebner, “Gamut mapping from below: Finding minimum perceptual distances for colors outside the gamut volume,” Color Research & Application |

23. | P. Zolliker and K. Simon, “Continuity of gamut mapping algorithms,” Journal of Electronic Imaging |

24. | H. K. Chen Hung-Shing, “Three-dimensional gamut mapping method based on the concept of image-dependence,” Journal of Imaging Science and Technology |

25. | M. Shaw, “Gamut estimation using 2D surface splines,” in |

26. | P. Zolliker and K. Simon, “Continuity of gamut mapping algorithms,” Journal of Electronic Imaging |

27. | R. Saito and H. Kotera, “Image-dependent three-dimensional gamut mapping using gamut boundary descriptor,” Journal of Electronic Imaging |

28. | M.-K. Cho, B. -H. Kang, and H.-K. Choh, “Black extraction method using gamut boundary descriptors,” in Color Imaging XI: Processing, Hardcopy, and Applications(SPIE, San Jose, CA, USA, 2006), pp. 60580O–60510. |

29. | O.-Y. Mang and H. Shih-Wei, “Design Considerations Between Color Gamut and Brightness for Multi-Primary Color Displays,” Display Technology, Journal of |

**OCIS Codes**

(120.2040) Instrumentation, measurement, and metrology : Displays

(330.1690) Vision, color, and visual optics : Color

(330.1710) Vision, color, and visual optics : Color, measurement

(330.1730) Vision, color, and visual optics : Colorimetry

**ToC Category:**

Vision, color, and visual optics

**History**

Original Manuscript: July 31, 2007

Revised Manuscript: September 1, 2007

Manuscript Accepted: September 25, 2007

Published: September 28, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Mang Ou-Yang and Shih-Wei Huang, "Determination of Gamut Boundary Description for multi-primary color displays," Opt. Express **15**, 13388-13403 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-20-13388

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### References

- Y. Wang, and H. Xu, "Determination of CRT color gamut boundaries in perceptual color space," in Electronic Imaging and Multimedia Technology IV(SPIE, Beijing, China, 2005), pp. 332-338.
- S. Wen, "Design of relative primary luminances for four-primary displays," Displays 26, 171-176 (2005). [CrossRef]
- S. Roth, I. Ben-David, M. Ben-Chorin, D. Eliav, and O. Ben-David, "Wide gamut, high brightness multiple primaries single panel projection displays," Digest SID'03 (2005).
- T. Ajito, T. Obi, M. Yamaguchi, and N. Ohyama, "Expanded color gamut reproduced by six-primary projection display," in Projection Displays 2000: Sixth in a Series(SPIE, San Jose, CA, USA, 2000), pp. 130-137.
- T. Ajito, T. Obi, M. Yamaguchi, and N. Ohyama, "Multiprimary color display for liquid crystal display projectors using diffraction grating," Optical Engineering 38, 1883-1888 (1999). [CrossRef]
- Y. Murakami, N. Hatano, J. Takiue, M. Yamaguchi, and N. Ohyama, "Evaluation of smooth tonal change reproduction on multiprimary display: comparison of color conversion algorithms," in Liquid Crystal Materials, Devices, and Applications X and Projection Displays X(SPIE, San Jose, CA, USA, 2004), pp. 275-283.
- M. Yamaguchi, T. Teraji, K. Ohsawa, T. Uchiyama, H. Motomura, Y. Murakami, and N. Ohyama, "Color image reproduction based on multispectral and multiprimary imaging: experimental evaluation," in Color Imaging: Device-Independent Color, Color Hardcopy, and Applications VII(SPIE, San Jose, CA, USA, 2001), pp. 15-26.
- M. Yamaguchi, R. Iwama, Y. Ohya, T. Obi, N. Ohyama, Y. Komiya, and T. Wada, "Natural color reproduction in the television system for telemedicime," in Medical Imaging 1997: Image Display(SPIE, Newport Beach, CA, USA, 1997), pp. 482-489.
- J.-G. Chang, C. L.-D. Liao, and C.-C. Hwang, "Enhancement of the optical performances for the LED backlight systems with a novel lens cap," in Novel Optical Systems Design and Optimization IX(SPIE, San Diego, CA, USA, 2006), pp. 62890X-62896.
- P. C. P. Chao, L.-D. Liao, and C.-W. Chiu, "Design of a novel LED lens cap and optimization of LED placement in a large area direct backlight for LCD-TVs," in Photonics in Multimedia(SPIE, Strasbourg, France, 2006), pp. 61960N-61969.
- E. H. Ford, "Projection optical system for a scanned LED TV display," in Optical Scanning 2002(SPIE, Seattle, WA, USA, 2002), pp. 111-122.
- W. Y. Lee, Y. C. Lee, K. Sokolov, H. J. Lee, and I. Moon, "LED projection displays," in Nonimaging Optics and Efficient Illumination Systems(SPIE, Denver, CO, USA, 2004), pp. 1-7.
- B. A. Salters, and M. P. C. M. Krijn, "Color reproduction for LED-based general lighting," in Nonimaging Optics and Efficient Illumination Systems III(SPIE, San Diego, CA, USA, 2006), pp. 63380F-63311.
- D.-W. Kang, Y.-T. Kim, Y.-H. Cho, K.-H. Park, W. Choe, and Y.-H. Ha, "Color decomposition method for multiprimary display using 3D-LUT in linearized LAB space," in Color Imaging X: Processing, Hardcopy, and Applications(SPIE, San Jose, CA, USA, 2005), pp. 354-363.
- J. Giesen, E. Schuberth, K. Simon, and P. Zolliker, "Toward image-dependent gamut mapping: fast and accurate gamut boundary determination," in Color Imaging X: Processing, Hardcopy, and Applications(SPIE, San Jose, CA, USA, 2005), pp. 201-210.
- H. Zeng, "Spring-primary mapping: a fast color mapping method for primary adjustment and gamut mapping," in Color Imaging XI: Processing, Hardcopy, and Applications(SPIE, San Jose, CA, USA, 2006), pp. 605804-605812.
- P. Pellegri, and R. Schettini, "Gamut boundary determination for a color printer using the Face Triangulation Method," in Color Imaging VIII: Processing, Hardcopy, and Applications(SPIE, Santa Clara, CA, USA, 2003), pp. 542-547
- M. Ján, and L. Ronnier, "Calculating medium and image gamut boundaries for gamut mapping," Color Research & Application 25, 394-401 (2000). [CrossRef]
- Q. Huang, and D. Zhao, "The color gamut of LCD and its analytical expression," in ICO20: Illumination, Radiation, and Color Technologies(SPIE, 2006), pp. 60330A-60337.
- Q. Huang, and D. Zhao, "Gamut boundaries expressed with Zernike polynomials," in Color Science and Imaging Technologies(SPIE, Shanghai, China, 2002), pp. 149-154.
- P. G. Herzog, "Further development of the analytical color gamut representation," in Color Imaging: Device-Independent Color, Color Hardcopy, and Graphic Arts III(SPIE, San Jose, CA, USA, 1998), pp. 118-128.
- M. D. F. Fritz Ebner, "Gamut mapping from below: Finding minimum perceptual distances for colors outside the gamut volume," Color Research & Application 22, 402-413 (1997). [CrossRef]
- P. Zolliker, and K. Simon, "Continuity of gamut mapping algorithms," Journal of Electronic Imaging 15, 013004-013012 (2006). [CrossRef]
- H. K. Chen Hung-Shing, "Three-dimensional gamut mapping method based on the concept of image-dependence," Journal of Imaging Science and Technology 46, 44-52 (2002).
- M. Shaw, "Gamut estimation using 2D surface splines," in Color Imaging XI: Processing, Hardcopy, and Applications(SPIE, San Jose, CA, USA, 2006), pp. 605807-605808.
- P. Zolliker, and K. Simon, "Continuity of gamut mapping algorithms," Journal of Electronic Imaging 15, 013004-013012 (2006). [CrossRef]
- R. Saito, and H. Kotera, "Image-dependent three-dimensional gamut mapping using gamut boundary descriptor," Journal of Electronic Imaging 13, 630-638 (2004). [CrossRef]
- M.-K. Cho, B.-H. Kang, and H.-K. Choh, "Black extraction method using gamut boundary descriptors," in Color Imaging XI: Processing, Hardcopy, and Applications(SPIE, San Jose, CA, USA, 2006), pp. 60580O-60510.
- O.-Y. Mang, and H. Shih-Wei, "Design Considerations Between Color Gamut and Brightness for Multi-Primary Color Displays," Display Technology, Journal of 3, 71-82 (2007). [CrossRef]

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