A Color Gamut Description Algorithm for Liquid Crystal Displays in CIELAB Space

Because the accuracy of gamut boundary description is significant for gamut mapping process, a gamut boundary calculating method for LCD monitors is proposed in this paper. Within most of the previous gamut boundary calculation algorithms, the gamut boundary is calculated in CIELAB space directly, and part of inside-gamut points are mistaken for the boundary points. While, in the new proposed algorithm, the points on the surface of RGB cube are selected as the boundary points, and then converted and described in CIELAB color space. Thus, in our algorithm, the true gamut boundary points are found and a more accurate gamut boundary is described. In experiment, a Toshiba LCD monitor's 3D CIELAB gamut for evaluation is firstly described which has regular-shaped outer surface, and then two 2D gamut boundaries ( CIE-a*b* boundary and CIE-C*L* boundary) are calculated which are often used in gamut mapping process. When our algorithm is compared with several famous gamut calculating algorithms, the gamut volumes are very close, which indicates that our algorithm's accuracy is precise and acceptable.


Introduction
Different color devices generally have different color gamuts because of their specific coloring parameters, such as the coloring principle (additive and subtractive), colorant (pigment or dyers), substrates, and light source (D50, D65, or others). Thus, a color image often looks different when it is output by different color devices. For example, a brightly displayed photo on a monitor commonly loses some color information when printed on the paper. For the purpose of matching color image's visual effect between different devices, gamut mapping is often performed in digital imaging processing systems. Gamut mapping can be defined as the color rendering process which rearranges the colors from source device gamut to destination device gamut [1]. Because source and destination gamuts mismatch each other in some regions, there exist some colors which are inside the source gamut but outside of the destination gamut, and they must be clipped or compressed into the destination gamut. Now there are dozens of gamut mapping algorithms (or GMAs) developed by different agencies [1]. For example, the LCLIP algorithm clips out-of-gamut colors onto the gamut boundary and leaves the chroma of in-gamut colors unchanged. The CUSP algorithm compresses colors towards a focus point, where the lightness and chroma are changed simultaneously. While for some universal GMAs, such as CARISMA or GCUSP, the mapping process is very complex but accurate. Because there are too many GMAs developed for the various color images, four specific GMAs are assigned within ICC workflow for convenience, which are absolute colorimetric, relative colorimetric, perceptual, and saturation [2].
Actually, these GMAs are sufficient to cope with all types of color images. However, the color inconsistency still occurs frequently when color images are transmitted from monitors to printers. As a matter of fact, one major reason which results in the gamut mapping errors comes from the gamut boundary calculation process, so it is significant to obtain the accurate gamut boundary before gamut mapping.
Gamut mapping is commonly performed in 2D coordinate, such as CIE- * * or CIE- * * coordinates [3]. The 2D gamut boundary is a cross-section from 3D CIELAB gamut, 2 The Scientific World Journal the errors within both the 3D gamut calculation process and the 2D gamut boundary calculation process will reduce the gamut mapping precision greatly.
Because there are some problems within the present gamut calculation algorithms (analyzed in Section 2), a 2D gamut boundary calculating algorithm is developed for gamut mapping process in this paper. In experiment an LCD monitor's gamut boundary is calculated and compared with other algorithms, and the experiment result shows that our algorithm is precise enough. In addition, it should be noted that the proposed algorithm is not only suitable for RGB monitors, but also for CMY (CMYK) printers or other color devices.

Different Types of Gamuts Employed in Gamut Mapping Process
In color reproduction systems, color gamut refers to the subset of colors which can be accurately represented in a given circumstance, such as within a given color image or by a certain color device. While the gamut boundary means the outer surface of the 3D gamut, or the outer contour line of the 2D gamut. Color gamut is often described in CIELAB space; this is mainly because CIELAB color spaces are independent of devices, and it is more perceptually uniform than CIEXYZ space. Generally, there are three types of color gamuts described in CIELAB space, such as 3D CIELAB gamut, 2D CIE- * * gamut, and 2D CIE- * * gamut [4]. The first gamut is often used for viewing the overall gamut of color devices or images but rarely used in gamut mapping process directly. The two other gamuts are described in 2D coordinates which are often applied in gamut mapping process, while the main task of this paper is to calculate the 2D gamut's boundary for gamut mapping algorithms.
The CIE- * * gamut is a cross-section of 3D gamut along the constant lightness plane, and its boundary is the 3D gamut's outer intersection line with the same lightness. The CIE- * * gamut is a cross-section of 3D gamut along the constant hue-angle plane, and the boundary is the intersection line at the specified hue angle. Because hue information is generally more important than lightness, the third gamut boundary is most widely used during gamut mapping.
It is significant to obtain accurate description of the 3D gamut, because the 2D gamuts are calculated from it. Several analytical models are used to predict the 3D gamut with relatively few sample colors, such as the Kubelka-Munk function [5], and Neugebauer equations [6]. Moreover, Herzog considered the device's 3D gamut as a distorted hexahedron [7] and used a deformation degree function to simulate the actual device gamut. Huang and Zhao used Zernike polynomials to determine the boundary [8], and Wang and Xu used a two-step workflow to describe CRT monitor gamut boundaries [9]. On the whole, all these prediction algorithms are based on a specific analytical model and can get accurate gamut with few sample data. However, these methods cannot be applied to all types of color devices, because the analytical expression is highly restricted by the device status such as coloring principle, substrate, inks, and ambient light sources.
If a sufficient number of sample colors are supplied, the 3D gamut can be described with the empirical algorithms. These algorithms do not focus on analyzing the devices' coloring principle or the relationship between device color space and the corresponding CIELAB values; they straightly simulate the 3D gamut with the measured CIELAB values. Thus these methods can be used for all types of color devices, even for color images gamut calculation. Within these empirical algorithms, Balasubramanian and Dalal used a modified convex hull method to compute gamut, in a way of adjusting the concave gamut surfaces into convex [10]. Cholewo and Love used alpha-shape to describe the gamut surface for an inkjet printer [11], and the gamut was controlled by an alpha parameter which was interactively selected with a visualization system. Because it is difficult to determine the optimal parameter values, Morovic proposed the segment maxima gamut boundary description (SMGBD) algorithm to compute media and device gamut [4]. However, sometimes there are no boundary points falling into certain segments, it must be created by interpolation method, which pulls errors into the gamut calculations.
In general, the analytical algorithms can be used neither for all types of color devices, nor for the color images, while, for the empirical algorithms, the calculated boundary points are not very precise, because some inside-gamut boundary points may be mistaken for boundary points. In the paper, a new gamut description algorithm is proposed for calculating liquid crystal display (LCD) monitors' 3D and 2D gamut. Similar to Herzog's algorithm, the 3D CIELAB gamut is deemed as a deformed hexahedron, and the relationship between device and CIELAB gamut is analyzed and established, but it should be noted that the calculation process is far less complicated. For the experimented LCD monitor, the gamut boundary points are firstly found out in RGB space, and these obtained points are the true boundary points which are on the surface of RGB cube. And then the RGB boundary points are converted into CIELAB values, while the 3D or 2D gamuts are described based on these boundary points.

Three-Dimensional CIELAB Gamut Description
During the gamut description for LCD monitors, the device color of RGB signals are controlled by users to form the sample data, and the corresponding CIELAB values are obtained by measuring the screen when the RGB colors are displayed. In general, the calculation of monitor gamut in CIELAB space can be divided into three steps, which are RGB sample data generating, CIELAB data measuring, and 3D or 2D gamut calculation.

Generating the Sample Data.
In order to obtain the monitor's CIELAB gamut boundary, the colors around the device gamut boundary should be selected to generate the sample data. Thus, for color monitors, the colorant gamut boundary is actually the outer surface of RGB cube, and the The Scientific World Journal (1) When the RGB sample signals are sent to the displaying driver, they will be displayed on the monitor, and the spectral transmittance is measured by using a spectrophotometer. Then the CIEXYZ Tristimulus can be calculated with the following: where , , and are the CIEXYZ tristimulus values of the reference white point, and the function is defined as follows:

Calculation of 3D CIELAB Gamut.
When a certain amount of sample colors are obtained, there will be many scattered points distributed in RGB or CIELAB space. The outer surface of RGB cube is the device gamut boundary with regular shape, and it will be severely deformed when converted into CIELAB space. For example, if the monitor's RGB cube is uniformly sampled by setting = = 9, in other words every colorant evenly changes in the range of [0 32 64 96 128 160 192 224 255], there will be 9 scattered points on each of the RGB cube edges.
The deformation between RGB cube and 3D CIELAB gamut can be analyzed by comparing the distribution of those 12 edges in these two color spaces. Figure 1(a) is the device RGB gamut boundary formed by its 12 edges, and its corresponding CIELAB gamut boundary points is shown in Figure 1(b). It is obvious that the colorimetric CIELAB gamut looks like a deformed hexahedron in [7], whereas the deformation degree varies in different locations or for different devices.
To get the complete 3D gamut description, the six faces of the RGB cube should be converted into CIELAB space. Among the sampling points above, as only a part of them located on the outer surface of gamut, the RGB boundary points should be firstly extracted from the six RGB cube faces, and they are depicted in Figure 2. It can be seen that if all these 3D scattered points are smoothly connected, the complete and closed 3D gamut can be obtained.
With those obtained boundary points in CIELAB space, Delaunay triangulation technique can be used to link them, and the final completed 3D gamut boundary is composed of a quantity of triangular facets. Since the gamut surface is partitioned into planar triangles, the volume can be calculated when an interior point is assigned. For example, when the center point O (50, 0, 0) in CIELAB space is selected, and all the vertices within each surface triangle are connected with the center point, then the CIELAB gamut is divided into many tetrahedrons. The overall gamut volume is the sum of the tetrahedrons' volumes, while each tetrahedron's volume can be calculated as below:

Two-Dimensional Gamut Boundary for Gamut Mapping Algorithms
Because gamut mapping is commonly carried out in 2D gamut boundary, such as CIE- * * or CIE- * * gamut boundary, the 2D gamut boundary should be accurately determined. The 2D gamut is a cross-section of 3D CIELAB gamut; thus the 2D gamut boundaries can be determined by slicing the 3D gamut along constant-lightness plane or constant-hue-angle plane. However, because the 3D boundary points do not distribute uniformly, when the scattered 2D boundary points are interpolated from the nearby 3D boundary points, there will be lots of errors caused. We propose a new method to calculate the 2D gamut boundary, in which the 2D boundary points are firstly found within the RGB cube, and then their CIELAB values are converted which are used to describe the gamut boundary in CIE- * * or CIE- * * coordinates.

CIE- * * Gamut Boundary Calculation.
The RGB cube has eight vertices, twelve edges, and six faces. For color monitors, the lightness (CIE- * values) of the vertices in the RGB cube varies greatly. Take the widely used ICC profile of sRGB, for example; the RGB cube's eight vertices' CIELAB values are listed in Table 1, and it can be seen that the vertices' lightness values are arranged in a sequence of below: > > > > > > > . Actually most of the RGB monitors accord with this discipline.
In addition, as the lightness changes continuously on the faces of RGB cube, when a specific lightness value is given, a constant-lightness plane will intersect with some of the RGB faces, while the intersection line corresponds to the CIE- * * gamut boundary line. To calculate the constantlightness gamut boundary, the scattered boundary points   should be firstly found, and the process can be described in the following steps.
(1) Compare the given lightness with twelve edges' lightness, and find out the boundary points on the edges equal to the given lightness value. For example, if ≤ ≤ , which means that the lightness value is within the range of edge CW, then it can be concluded that an intersection point with lightness must exist in the edge CW.
The intersection point can be found by comparing the edge's sample colors. If two closet sample colors are expressed as and , as their RGB and CIELAB values are known, the intersection point 's values can be interpolated as below (the same for and values): (2) For the constant-lightness lines on the RGB cube surface, there are generally three or four edges which contain the intersection points among all the twelve edges. As shown in Figure 3, when the edge intersection points are connected, the constant-lightness lines may form a triangle ( - * = 20) or quadrangle ( - * = 60). (3) Because the constant-lightness gamut boundary is a closed and smooth curve, several boundary points are calculated from the corresponding cube faces, for the purpose of obtaining the full description of the gamut boundary. The face intersection points can also be found by comparing the sample colors of the RGB cube face, which is similar to the process of calculating edge intersection points. Thus, when all the face boundary points are connected, the two constantlightness gamut boundaries of Figure 3 are described with more details in Figure 4.
(4) After the scattered gamut boundary points in RGB space are calculated, the CIE- * * gamut boundary can be obtained by converting the boundary points from RGB space to CIELAB space. There are actually several color conversion models, such as 3D interpolation [12][13][14], polynomial regression [15,16], and neural network [17,18]. In the paper, the polynomial regression method is used because of the model's precision and the quantity of sample colors, and the utilized polynomials are expressed as below: ( , , ) = 0 + 1 + 2 + 3 + 4 + 5 where ( = 0, 1, . . . , 19) are the coefficients which can be determined by the least square method [19,20]. Take one 6 The Scientific World Journal   constant-lightness boundary, for example; it is depicted in RGB and CIELAB spaces respectively, as shown in Figure 5.

CIE- * * Gamut Boundary Calculation.
The CIE- * * gamut boundary can be seen as a cross-section of the constant-hue-angle plane with 3D CIELAB gamut, and it is widely used in gamut mapping process. As most of the 3D gamut is calculated by the scattered sample data with interpolation method, it is hard to get the line boundary point directly using the continuous 3D gamut. Similar to the calculation of constant-lightness gamut, the constant-hueangle boundary points are firstly calculated in RGB cube and then converted into CIELAB color space and described in the CIE- * * coordinate, where the CIE- * is calculated from CIE- * and CIE- * values: * = √ * 2 + * 2 .
During description of the 2D CIE- * * gamut, the scattered boundary points are firstly calculated and then connected with Bezier curves or straight lines. Although the boundary looks smoother when Bezier curves are used, the computing efficiency is inferior to the straight lines, which is very important for the gamut mapping process. Thus, the CIE- * * gamut boundary is connected using straight lines in this paper.

Experiments
In experiment, the gamut boundaries of a Toshiba LCD monitor are calculated. Firstly, the RGB device color space is uniformly divided, and the red, green, and blue signals all range within [0 32 64 96 128 160 192 224 255]; thus there are 9 3 = 729 sample colors totally. And then the sample RGB colors are displayed on the LCD monitor; the corresponding CIELAB values are measured by using the spectrophotometer -Rite DTP94. At last, the monitor's 3D CIELAB gamut and two other 2D gamut boundaries are described using the algorithm proposed in the paper.
As shown in Figure 6, the 3D gamut is described in CIELAB color space, and it looks like a deformed hexahedron as Herzog proposed. Compared with the initial RGB cube, the CIELAB gamut also has eight vertices and six continual curved faces, although most of sample colors' relative positions have changed. From the described CIELAB gamut, the overall displaying capacity of the monitor can be evaluated, which will help to select the right gamut mapping algorithms.
Bakke et al. proposed a method for evaluating different gamut boundary calculating algorithms [21], and the two testing algorithms are compared by using a parameter of gamut volume mismatching rate, which is expressed in the following: where represents the gamut volume mismatching rate, ( ref ) is the volume of reference gamut, and ( / ref ) is the volume of colors which are inside of gamut but outside of gamut ref . In experiment, the gamuts determined by our algorithm are selected as reference gamut, and three famous gamut description algorithms, SMGBD, convex hull, and alpha shape, are employed as a contrast. Consequently, the mismatching rates are 3.2%, 2.4%, and 3.7%, respectively, which indicate that our algorithm's accuracy is very close to the three successful algorithms.
Additionally, two kinds of 2D gamut mapping boundaries, CIE- * * and CIE- * * boundaries, are calculated using the method described in Sections 4.1 and 4.2. For the testing monitor, several constant-lightness lines with the lightness from 10 to 95 are found in RGB cube and then described in CIELAB space, as shown in Figure 7. Similarly, two CIE- * * boundaries are listed in Figure 8, and the hue angles are 30, 210, and 330 respectively.

Conclusions
The gamut of monitor is generally bigger than printers; thus when a displayed image is printed, gamut mapping should be performed in advance. Gamut mapping is the technique of replacing nonprintable colors by printable ones, conserving the appearance of an image. There are two major reasons which influence the gamut mapping precision, selection of gamut mapping algorithms and the calculation of gamut boundaries. Because many successful GMAs are developed for various images and color devices, it is very significant to improve the accuracy of gamut boundaries. In the paper, a new gamut boundary algorithm for color devices is proposed, and an LCD monitor is tested with having the 3D and 2D gamut boundaries described. In experiment, the described 3D CIELAB gamut boundary has regular-shaped outer surface, and the 2D CIE- * * and CIE- * * boundaries both have smooth boundary lines. For the purpose of precision evaluation, our algorithm is compared with other famous gamut description algorithms, and the result shows that the gamut volume difference is very little, which indicates that this algorithm is acceptable. Besides, it should be noted that, although the color monitor's gamut  is described in experiment, the proposed gamut boundary calculating algorithm can also be applied to other color devices, such as cameras, scanners, or printers.