Color decomposition method for multiprimary

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s1 s1 s1 s1. −. −. −. −. = c hl. Mopp. Lightness. Chroma. 50. (a). (b). Figure 4. (a) Symmetrical characteristic of MDP gamut boundary on constant hue plane in ...
Color decomposition method for multi-primary display using 3D-LUT in linearized LAB space Dong-Woo Kang,* Yun-Tae Kim,* Yang-Ho Cho,* Kee-Hyon Park,* Won-Hee Choe,** and Yeong-Ho Ha∗ * School of Electrical Engineering and Computer Science, Kyungpook National Univ., ** Samsung Advanced Institute of Technology 1370, Sankyuk-dong, Buk-gu, Taegu 702-701, South Korea

ABSTRACT This paper proposes a color decomposition method for a multi-primary display (MPD) using a 3-dimensional look-uptable (3D-LUT) in linearized LAB space. The proposed method decomposes the conventional three primary colors into multi-primary control values for a display device under the constraints of tristimulus matching. To reproduce images on an MPD, the color signals are estimated from a device-independent color space, such as CIEXYZ and CIELAB. In this paper, linearized LAB space is used due to its linearity and additivity in color conversion. First, the proposed method constructs a 3-D LUT containing gamut boundary information to calculate the color signals for the MPD in linearized LAB space. For the image reproduction, standard RGB or CIEXYZ is transformed to linearized LAB, then the hue and chroma are computed with reference to the 3D-LUT. In linearized LAB space, the color signals for a gamut boundary point are calculated to have the same lightness and hue as the input point. Also, the color signals for a point on the gray axis are calculated to have the same lightness as the input point. Based on the gamut boundary points and input point, the color signals for the input point are then obtained using the chroma ratio divided by the chroma of the gamut boundary point. In particular, for a change of hue, the neighboring boundary points are also employed. As a result, the proposed method guarantees color signal continuity and computational efficiency, and requires less memory. Keywords: Multi-primary display, color decomposition, color conversion, linearized LAB

1. INTRODUCTION In the display field, wide gamut displays, including RGB-laser displays and multi-spectral displays, have recently been introduced to reproduce highly saturated colors outside the color gamut of conventional TV systems. RGB-laser displays use pure color light sources with a narrow spectral radiance, thereby producing colors that are far more saturated than those produced by other systems. However, mixing hues to produce less saturated colors is difficult, and such systems are complex and expensive. In contrast, multi-primary displays (MPDs) are able to minimize the color mismatch caused by observer metamerism and produce a wide gamut at a much lower cost than an RGB-laser display.1 Multi-primary displays using more than three primary colors are realized based on 5-primary DLP projection, 6-primary LCD projection, and a 4-primary liquid crystal display. Currently there are two main areas of research related to reproducing images on multi-primary displays. The first is image reproduction based on spectral information,2 which estimates the spectral reflectance of an object using a multi-spectral camera and attempts to overcome the problem of observer metamerism using the degree of freedom in the selection of the device control values. The second area is image reproduction based on colorimetric information,3,4,5 such as sRGB and CIEXYZ, where there are various choices of control values for a set of tristimulus values due to the 3-to-N-dimensional transformation, plus a color conversion or decomposition algorithm is needed to remove the degree of freedom in the signal selection. A variety of color decomposition methods have already been developed and used for image reproduction on an MPD. However, reproducing images on an MPD using a color decomposition method without considering the lightness, chroma, and hue does not ensure a smooth color signal tonal change, although the tristimulus values change smoothly.6 Research has already been conducted to analyze the relation between signal discontinuity and pseudo contours as regards the ∗

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Color Imaging X: Processing, Hardcopy, and Applications, edited by Reiner Eschbach, Gabriel G. Marcu, Proc. of SPIE-IS&T Electronic Imaging, SPIE Vol. 5667 © 2005 SPIE and IS&T · 0277-786X/05/$15

Figure 1. Gamut of sRGB and six-primary liquid crystal display in CIE-xy chromaticity diagram.

smoothness tonal change reproduction on an MPD when the observer’s color matching functions deviate from the standard one or when the device profile includes some errors. As such, for the implementation of a wide gamut HDTV system, the color decomposition method should ensure color signal continuity and be computationally efficient with a minimal memory requirement. Accordingly, this paper presents a color decomposition method that provides both color signal continuity and computational efficiency. The proposed method applies a chroma ratio interpolation with gamut boundary information stored in a 3D-LUT, plus it also considers the lightness, chroma, and hue change. First, to obtain gamut boundary information on the MPD, the 3D-LUT is created in linearzed LAB color space. The neighboring boundary points are used for hue changes, while the MPD color signals are calculated using a chroma ratio. In simulations and experiments used to evaluate the colorimetric matching performance, the proposed 3D-LUT decomposition method is compared with conventional methods based on the color difference in CIELAB. A graph is used to estimate the smoothness of the color signal tonal change, along with the lightness, chroma, and hue changes. In addition, the smoothness of the color signals is confirmed based on signal-separated images, which are also used to evaluate the smoothness on a six-primary liquid crystal display with device profile errors, including a quantization error and tone reproduction curve error, as contour artifacts due to color signal discontinuity resulting from device profile errors are often more of a problem than observer metamerism.

2. COLOR REPRODUCTION MODEL If the colorimetric characteristics of a display device have the linearity and additivity, the forward transform2 that converts the color signals of a multi-primary display into CIEXYZ can be expressed as follows: X    Y  = TN-signal to XYZ Z  

 S1     X bias   X max, S 1    S2   ⋅   +  Ybias  , TN - signal to XYZ =  Ymax, S1 M     Z bias   Z max, S1 S   N

X max, S 2 Ymax, S 2 Z max, S 2

L X max, S N   L Ymax, S N   L Z max, S N 

(1)

where N is the number of primary colors, [ X bias Ybias Z bias ]t is the tristimulus value of bias, and [ X max, S i Ymax, S i Z max, S i ]t is the maximum tristimulus value for the i-th primary S i (i = 1,2,L, N , 0 ≤ S i ≤ 1) with the exception of the bias value for Si . Although the color signals of an MPD can be calculated using the inverse matrix of TN - signal to XYZ , in the case of N > 3 there is no inverse matrix, as TN - signal to XYZ is not a square matrix. Thus, a color decomposition method is needed to obtain the color signals of an MPD.

3. CONVENTIONAL COLOR DECOMPOSITION METHOD Several kinds of color decomposition method have been developed to obtain colorimetric matching on a multi-primary

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Figure 2. Block diagram of the proposed 3D-LUT method.

display. As the color gamut of an MPD assumes the form of a polyhedron, there are color decomposition methods using the geometrical interpolation such as pyramid interpolation, triangle interpolation on equi-luminance plane or the center of gravity. As an example of pyramid interpolation, the matrix switching method3 proposed by T. Ajito et al. splits the color gamut into pyramid structures with a gamut boundary face, and each pyramid has an inverse matrix. A two-dimensional LUT is used to specify which matrix is selected for a given tristimulus value, then the color signals are computed using the 2-D LUT and 3× 3 inverse matrices. Although this method has a good computational efficiency, color signal discontinuities arise at the boundaries between pyramids, plus the tristimulus values change continuously. Meanwhile, as an example of triangle interpolation, the linear interpolation method proposed by H. Motomura4 uses two intersection points and a gray point on an equi-luminance plane in CIEXYZ to calculate the color signals. The points have N-color signals corresponding to the tristimulus value. Then the target value inside the MPD gamut is linearly interpolated with the three points. As a result, the continuity and smoothness of the color signal is ensured for lightness and chroma changes. Yet, for hue changes, discontinuities still occur at the boundaries of the triangle interpolation and complex calculations are involved identifying valid intersection points and computing the inverse matrix for each input value. More recently, the metameric black method was introduced,5,6 which calculates the center of gravity of the volume in ( N − 3) -dimensional space. Although this method generates continuous color signals, it is extremely computationally complex, and since computational efficiency is very important for wide gamut HDTV systems, this method is not mentioned further or doesn’t compared with other methods in this paper.

4. 3D-LUT COLOR DECOMPOSITION METHOD Both computational efficiency and color signal smoothness are very important for implementing wide gamut HDTV systems. The proposed method applies a chroma ratio interpolation using gamut boundary information stored in a 3DLUT, considering lightness, chroma, and hue changes. First, the proposed method converts CIEXYZ or standard RGB into N-dimensional color signals for a multi-primary display. In the case of standard RGB, to guarantee the linearity and mixture of the primary color signals, standard RGB is first transformed into CIEXYZ and then converted into linearized LAB color space. Figure 2 shows the signal flow of the proposed method. The input signals are converted to linearized LAB, then the lightness and hue are referred to a 3D-LUT for gamut boundary information. Thereafter, a color signal calculator operates the chroma ratio interpolation using the quantized points of the gamut boundary. Finally, the driving signals of the MPD are obtained after gamma correction. 4.1. Forward transform in linearized LAB color space Linearized LAB space is eliminated the power factor from CIELAB7, thereby ensuring the linearity and additivity of primary colors. As a cylindrical coordinate, it can define chroma and hue. As such, a 3D-LUT with gamut boundary information can be easily composed according to the quantized level of lightness and hue. Printer half-toning is also used.

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The 3× 3 transform matrix is: 0  L X        a  = TLab ⋅  Y  =  500 / X n b Z   0     

 X    0 ⋅ Y  − 200 / Z n   Z 

100 / Yn

0

− 500 / Yn 200 / Yn

(2)

where [ X n Yn Z n ]t is the tristimulus value for white as a reference and TLab is the 3× 3 transform matrix from CIEXYZ to linearized LAB. Non-linear standard RGB signals are converted to linear RGB. Then the 3× 3 transform matrix from linear RGB to linearized LAB can be expressed as follows:  L  RLinear   Lmax, R Lmax,G      = ⋅ = a T G = ⋅ , T T T   LinearRGB Lab RGB − XYZ  a max, R a max,G LinearRGB  Linear   b B   bmax, R bmax,G    Linear 

Lmax, B   amax, B   bmax, B 

(3)

where [ Lbias abias bbias ]t is the coordinate for the background light and TRGB to XYZ is the 3 × N transform matrix from linear RGB to CIEXYZ. The transform from the N-color signals of an MPD to linearized LAB is similar to equation 3. The 3 × N transform matrix for a multi-primary display is:  S1     Lbias   Lmax, S  L 1      S2    a  = TMPD ⋅   +  abias  , TMPD = TLab ⋅ TN - signal to XYZ =  amax, S1 M  b    bbias   bmax, S1   S   N

Lmax, S 2 amax, S 2 bmax, S 2

L Lmax, S N   L a max, S N   L bmax, S N 

(4)

where [ Lbias abias bbias ]t is the coordinate for the background light, TN - signal to XYZ is the 3 × N transform matrix from Ncolor signals to CIEXYZ, and [ LSi a Si bSi ]t is the coordinate for the i-th primary color. 4.2. 3D-LUT implementation To obtain the 3D-LUT including gamut boundary information, assuming that the bias value [ Lbias abias bbias ] = 0 and r number of primary colors N = 4 , the primary vector Pi can be described in linearized LAB color space and 4dimensional color space as follows: r Pi = [ LPi a Pi bPi ]Linearized LAB = [ s1 s2 s3 s4 ]4 − Dim.

1, i = k, k = 1, 2, 3, 4 , sk =  0, otherwise

(5)

Lightness [1 1 1 1] r

P2 [1 0 1 1] r Mi

δ

r P2

r P3

r P4

r P1

γ

[1 0 0 1]

[1 0 0 0] b a

Figure 3. Gamut of 4-primary MPD in linearized LAB space.

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Lightness

M [l ][h]

= [c s1 s 2 s3 s 4 ]

50

Lightness

M [l ][h] Chroma value

M opp [l ][h]

100

Color signal

= [c 1 − s1 1 − s 2 1 − s 3 1 − s 4 ]

180

Hue

0

M [l ][h] = [c s1 s 2 L s N ] Chroma

(a)

(b)

Figure 4. (a) Symmetrical characteristic of MDP gamut boundary on constant hue plane in linearized LAB space and (b) 3D- LUT with chroma and color signals of gamut boundary.

where si is the i-th primary color signal (i = 1,2,3,4) . As the color space conversion is linear, the linearized LAB values correspond to 4-dimensional color signals, plus arbitrary points on the gamut boundary contain lightness, hue, and chroma. Also, the color signals correspond to a scalar mixture of these values, as shown in figure 3, which represents the gamut of a 4-primary MPD in linearized LAB space. r r r r r M i = α ⋅ P1 + β ⋅ P2 + γ ⋅ P3 + δ ⋅ P4 = [ LM i aM i bM i ]Linearized LAB

(6)

= [ LM i H M i CM i ]Linearized LAB = [α β γ δ ]4 − Dim.

where M i is an arbitrary point on the gamut boundary and α , β , γ , and δ are scalar values (0 ≤ α , β , γ , and δ ≤ 1) . To construct the 3D-LUT, the chroma and color signals of the gamut boundary are stored in the 3D-LUT along with the quantized lightness and hue, as shown in figure 4 (b), where the 3D-LUT was constructed with gamut boundary information in the case of N = 6 . If the quantization level of the lightness and hue is 100 levels and 360 levels, respectively, the 3D-LUT size should be 100 × 360 × ( N + 1) . Yet, since the gamut of an MPD is 50% symmetric to lightness, the 3D-LUT size can be reduced to 100 × 180 × ( N + 1) , which involves a smaller amount of memory than the conventional matrix switching method. The symmetrical characteristic of an MPD gamut boundary on a constant hue plane in linearized LAB space is shown in figure 4 (a). For example, if the quantized lightness and hue are l and h, respectively, the information stored in the 3D-LUT is: M [l ][h] = [c s1 s 2 L s N ] , M [l ][h + 180] = [c 1-s1 1-s2 L 1-s N ]

(7)

4.3. Chroma ratio interpolation using gamut boundary information Under the constraint of colorimetric matching, the proposed 3D-LUT method uses the linearity of linearized LAB color space. The chroma ratio interpolation with the 3D-LUT is illustrated in figure 5(a). The gamut boundary points M LH , M LH , M LH , and M LH are quantized points of the gamut boundary corresponding to chroma and color signals. First, the boundary points M hue − L , M hue − H between the quantization level and gray axis point K with the same lightness of input are interpolated using the following equations:

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(

)

(8)

(

)

(9)

CM hue− L =

LK − Llow ⋅ CM HL − CM LL + C M LL Lhigh − Llow

S iM hue−L =

LK − Llow ⋅ S iM HL − S iM LL + S iM LL Lhigh − Llow

chroma ratio =

Cinput − C M θ

bound

C M bound − C M θ

bound

Lightness

chroma ratio =

C IPUT CM

Lightness axis

M HH

bound

MHH

Lhigh

K

M bound

θ

1 -φ

θ M hue −H

Lhigh

M hue-H

M θHL

K

MLH

Llow

M hue− L

MHL Mbound

φ

M hue − H

M θLH

θ M hue −L

Llow

Mhue-L

±θ

M LL

θ M bound

INPUT

chroma ratio =

INPUT

M θLL

Cinput CM θ

bound

(a)

(b)

Figure 5. Chroma ratio interpolation referring to 3D-LUT (a) between the quantized levels and (b) with neighboring points.

S iK =

LK , 100

i = 1,2, L , N

(10)

where CM is the chroma value for point M, Lhigh and Llow are the up and down quantized lightness values, SiM is the ith color signal of point M, and S iK is the i-th color signal of point K. Next, the boundary point M bound that has the same hue is calculated using the following equations: C M bound = sin(φ + (1 − φ )) ⋅

SiM bound =

C M hue − L C M hue− H

(11)

C M hue− L sin φ + C M hue − H sin(1 − φ )

(

)

dist ( M bound , M hue − L ) ⋅ SiM hue− H − SiM hue− L + SiM hue− L dist ( M hue − H , M hue − L )

(12)

C M hue − L sin φ dist ( M bound , M hue − L ) = dist ( M hue − H , M hue − L ) C M hue− L sin φ + C M hue− H sin(1 − φ )

(13)

Lightness

b*

Color signal Discontinuity

1 degree

r P2 [1 0 1 1]

C

[1 0 0 δ ]

B [1 0

[1 0 0 1]

[1 0 1 0]

a*

A

γ 0]

r P3

B r P4

C r P1

Constant [1 0 0 0] lightness plane

b*

A

[α 0 1 0]

(a)

a*

(b)

Figure 6. Color signal transition: (a) on constant lightness plane and (b) in linearized LAB space.

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where dist ( A, B ) is the Euclidian distance between A and B, and φ is the hue difference between the input and quantized hue. Finally, the color signals are computed using the chroma ratio CINPUT CMbound . SiINPUT =

(

)

C INPUT ⋅ SiM bound − SiK + SiK , i = 1,2, L , N C M bound

(14)

However, if the neighboring gamut boundary points are not considered, a color signal discontinuity can still exist. In figure 6, for a hue change, the regions with a color signal discontinuity caused by a signal transition are described as A, θ B, and C. Therefore, as shown in figure 5(b), the proposed method employs additional gamut boundary points, M hue −H Table 1. Simulated color difference in CIELAB.

∆Eab

∆L *

∆a * b *

Matrix switching

0.5165

0.1962

0.4695

Linear interpolation

0.2045

0.0572

0.1852

3D-LUT

0.1641

0.0492

0.1456

(a)

(d)

(g)

(b)

(e)

(h)

(c)

(f)

(i)

Figure 7. Color signal graph of (a) matrix switching, (b) liner interpolation, and (c) proposed 3D-LUT including lightness change, (d) matrix switching, (e) liner interpolation, and (f) proposed 3D-LUT including chroma change, and (g) matrix switching, (h) liner interpolation and (i) proposed 3D-LUT including hue change.

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θ and M hue − L , with ±θ degrees to guarantee a smooth color signal with a hue change. Similarly, these points can be θ computed using equations 8, 9, and 10, then the boundary point M bound with the same lightness and hue of the input point is calculated using equations 11, 12, and 13. Meanwhile, in the case of a discontinuous chroma change, when C INPUT > C M θ , the color signals are calculated using equation 14 with a chroma ratio based on CM θ and CM , bound bound bound or a chroma ratio based on the gray axis K and CM θ . As a result, the color signals of the MPD can maintain bound continuity and smoothness with a changing lightness, chroma, and hue.

5. SIMULATIONS AND EXPERIMENTS For the simulations and experiments, the input was assumed to be standard RGB or CIEXYZ. First, the colorimetric matching in CIELAB was evaluated and the results for the proposed method compared with those for the matrix switching method and linear interpolation method. Second, the smoothness of the color signals was assessed using a graph, along with the CIELAB value change, as shown by separated RGB images. The final smoothness evaluation was based on the images reproduced on an MPD. All the simulations and experiments used a six-primary liquid crystal display composed of red, green, blue, cyan, magenta, and yellow as the primary colors. 5.1. Colorimetric matching performance The performance of the color decomposition methods was evaluated by simulation using 24 uniform patches. The color difference( ∆Eab ) between the simulated and predicted colors in CIELAB is shown in table 1. The achromatic and chromatic components were also evaluated. For the colorimetric matching, all three methods produced a similar performance that was sufficient to reproduce images on an MPD.

(a)

(b)

(c)

(d)

Figure 8. RGB channel separated images: (a) Original image, (b) matrix switching method, (c) linear interpolation method, and (d) proposed 3D-LUT method.

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Table 2. Gradation pattern along with lightness, chroma and hue. Gradation pattern

L*

C*

H*

Pattern #1

15-90

20

20.25

Pattern #2

50

0-73

25

Pattern #3

50

20.25

1-360

5.2. Evaluation of color signal tonal change The smoothness of the color signals and CIELAB value changes were evaluated, as presented in table 2. Figure 7 shows the tonal change for each color signal. All three methods exhibited continuity for lightness, as shown in figure 7 (a), (b), and (c), yet matrix switching was unable to produce continuity for the chroma and hue, as shown in figure 7 (d) and (g). Linear interpolation and the proposed method produced a similar pattern for the chroma in figure 7 (e) and (f), yet for the hue gradation in figures 7 (h) and (i), the proposed method was more continuous than the linear interpolation method due to considering the neighboring gamut boundary color signals. In figure 8, the images reproduced based on the separated R, G, and B signals from the converted color signals exhibited signal discontinuity in the case of the matrix switching and linear interpolation methods. Although the hue changes were smooth, obvious signal discontinuity was observed in figure 8 (b) and (c). In contrast, the RGB separated image when using the proposed method, shown in figure 8 (d), was smoother than those from the other methods. In most display systems, the problem of a device profile error producing contour artifacts from a color signal discontinuity is more significant than observer metamerism. A device profile error contains a quantization error and tone reproduction curve error, which means that discontinuous regions can cause contours on reproduced images when the images are displayed on a multi-primary display system that includes a device profile error. 5.3. Image reproduction on multi-primary display To evaluate the effect of a device profile error on the production of contour artifacts as a result color signal discontinuity, the images reproduced on an MPD (six-primary liquid crystal display system) were captured using a digital camera. In

(a)

(b)

(c)

(d)

Figure 9. Image reproduction on MPD: (a) Original image, (b) matrix switching method, (c) linear interpolation method, and (d) proposed method.

(a)

(b)

(c)

(d)

Figure 10. Image reproduction on MPD: (a) Original image, (b) matrix switching method, (c) linear interpolation method, and (d) proposed method.

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terms of the colorimetric matching, the captured images were different from the original images observed with the naked eye. Yet the smoothness and contour was still evaluated in the captured images. In figures 9 and 10, (b), (c), and (d) show the images reproduced when using matrix switching, linear interpolation, and the proposed 3D-LUT method, respectively, while (a) is the original image with a hue change. The contours in the region with a continuous hue change are shown in Figure 9 (b) and (c), as well as Figure 10 (b) and (c). Yet, figures 9, 10 (d) were more continuous than the others, indicating that the proposed method was robust against contours caused by color signal discontinuity and produced a smooth image with a change of hue.

6. CONCLUSION This paper presented a 3D-LUT color decomposition method that can produce continuous N-color signals on an MPD using chroma ratio interpolation based on gamut boundary information stored in a 3D-LUT. Through simulations and experiments, the images reproduced using the proposed method were smoother than those produced using matrix switching and liner interpolation according to a graph and RGB signal separated images. In addition, the reproduced images captured using a digital camera revealed that a device profile error has an effect on the contours caused by signal discontinuity. When considering a change of hue, points neighboring the gamut boundary of the MPD are also employed, thereby improving the smoothness of the color signal. Also, as the 3D-LUT requires less memory, the computational efficiency is increased. Consequently, the proposed method appropriate for use with a wide gamut HDTV system. However, since the proposed method still has to quantize the gamut boundary, future research will attempt to minimize the quantization error when the gamut boundary information is described in linearized LAB space and enable the gamut mapping to use the full range of the MPD gamut.

ACKNOWLEDGMENT This work was supported by grant no. M10412000102-04J0000-03910 from the National Research Laboratory Program of the Korea Ministry of Science & Technology.

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