Preview of color modeling - 反射式電子紙之色彩特性化研究:以膽固醇液晶顯示器為例

Chapter 1 Introduction

1.2 Preview of color modeling

A color system can be separated into three levels including colorimetric characterization, color reproduction among various electronic multimedia, and evaluation of image quality by visual test. The flow chart of color modeling is illustrated as Fig. 1-3.

In consequence of the intrinsically colorimetric behavior subject to the operating mechanism or the adoptive material, the device-connection color space may exist an enormous discrepancy between two different devices even with identical user control signals. In order to eliminate the inconvenience of image communication, the colorimetric performance should be described with a common language in respect of standard color space. Therefore, the purpose of colorimetric characterization is to make a robust transformation between the device-connection space and the corresponding CIE colorimetry [4]. Since International Commission on Illumination (CIE) created CIE 1931 XYZ color space which is one of the first mathematically

defined color spaces in 1931, more and more studies of colorimetric characterization on different electronic devices have been developed.

Color reproduction is defined as the process by which color information from an original medium, under the original's viewing conditions, is transferred to a reproduction medium, under its viewing conditions, with the intention of the reproduction having a desired property [5], as shown in Fig. 1-4.

In terms of color reproduction, different dimensions can be identified dependent on the desired properties are concerned:

1. With reference to a reproduction only or with reference to the original and, therefore, to the relationship between the two;

2. By an application to which they relate to or to be independent of application;

3. Potentially descriptive or presenting an ideal state;

4. Measurable in quantitative terms by physical methods or expressible in qualitative terms and measurable psychovisually.

Fig. 1-3 Three levels of color modeling: colorimetric characterization, color reproduction, human evaluation

Fig. 1-4 The elements of color reproduction [5]

It is worth noting that the fourth dimension of the desired property is with respect to the evaluation of human visual test. When the different viewing conditions are introduced into the process, the reproduction of color information with high fidelity in CIE colorimetry is not equivalent to in human perception. In addition, even perceiving identical reproduction across different media, it is not represented the result of preference. Consequently, it would be beneficial to bring the psychophysical assessment into the concerned issue for achieving the best color reproduction.

In accordance with the flow chart of the color reproduction illustrated in Fig.

1-4, no matter what desired property are chosen in reproducing process, colorimetric characterization of each medium is always the fundamental step. If the colorimetric characterization is not finished yet or constructed without accuracy, another two levels of color modeling cannot progress anymore. This is the reason why colorimetric characterization is the most important work for information device.

Fig. 1-5 An example of color transformation procedure of display system [6]

A color transformation procedure of a traditional display as cathode-ray-tube (CRT) or LCD is shown in Fig. 1-5. The process from RGB bit values to tristimulus or appearance value is defined as colorimetric characterization. The corresponding dataset can be reproduced to another device if and only if the characterization of the device has been completed as well. Furthermore, the physical quantities (tristimulus or chromaticity) can be further transformed into perceptual quantities (LCh) by means of the appearance model.

1.3 Motivation, Objective and Organization

In section 1.2, we have clarified the significance of colorimetric characterization for each information media. Universality of the emerging display technology, electronic paper, is also subject to robust and reliable colorimetric characterization. Unlike mature product of flat panel display such as CRT and LCD,

whose color modeling have been built up completely, e-Papers are still operated by various novel techniques but singular architecture, as introduced in section 1.1.

Accordingly, in terms of various and complicated operating mechanism in e-Paper, a general rule of constructing colorimetric characterization is necessary.

In view of this, the objective of this paper is to introduce a common procedure of colorimetric characterization which can be utilized in every reflective e-Paper. We will take cholesteric liquid crystal display (ChLCD) for an examined case to illustrate how to accomplish a colorimetric characterization for a novel e-Paper following the proposed procedure. In addition, the performance of different characterizing method will be discussed for comparison as well.

In accordance with the motivation and objective, the thesis will be organized as following: in Chapter 2, a general procedure of colorimetric characterization including three different methods will be given. In Chapter 3, a full-color cholesteric liquid crystal display will be a studied case to clarify how to extend the general methods into an unkown e-Paper technology. In Chapter 4, the experimental results of different characterization manners applied in ChLCD will be discussed. Finally, the conclusion and discussions will be shown in Chapter 5.

Chapter 2 Colorimetric Characterization Methods

2.1 The framework of colorimetric characterization

Many color engineering works transports colors between imaging instruments through the medium of CIE colorimetry, and for this reason it is essential to create a model of the relationship between the device coordinates and the corresponding CIE colorimetry. There are three basic ways for generating device models:

1. Physical models (or color-mixing models) which include terms for different physical properties of the device, such as scattering, absorbance, and reflectance of colorant and substrates.

2. Numerical models in which a series of coefficients is defined, usually by regression from a suit of known samples, with no prior assumptions about physical behavior of the device or related media.

3. Look-up tables which define the conversion between a device space and a CIE color space at a series of coordinates within the color space, and interpolate the values for intermediate coordinates. The entries in a look-up table can be determined either by direct measurement or through a physical or numerical model.

No single device model gives can predict all the results from different sorts of device, thus a wide range of different models have been developed. Most device models are developed by measuring a sample of the colors on the media which is to be characterized in the beginning, and then defining a common relationship between

the two color spaces which are used to transform any color from one space to another.

Actually color transformations frequently include an element of two or even more of these methods. For example, a numerical method may be applied to define the relationship between printer and colorimetric values, with the exact dot areas to image found by a physical model, while the resulting function is applied to compute the entries in a look-up table of color. Then the resulting look-up table will be applied to show transformations of actual color image, as this will usually be more calculatingly efficient than computing the transformation on a pixel-by-pixel foundation. Alternatively a look-up table may be generated straightly from measurement data, in which case a lot of measurements will be required. The targets of device characterization can be outlined as follows:

Accuracy. The model should predict colors with smallest errors of the whole color space. The average color difference between predicted and measured colors should ideally be not larger than the noise performance within the system.

Visual acceptability. Discontinuities should not cause any artifact in the color model, and the errors distribution should not induce shifts in color attributes that are considered visually significant by observers.

Computational simplicity. This might be an important element where a device model is to run on personal computers or must be frequently recalculated to compensate for changes when media changes or device drift.

The minimum number of measurements. It is desirable to make the modeling process simplifier, particularly where a numerical model has to be redefined because of different reproduction conditions when substrate is changed.

Analytic invertibility. Device models may need to compute both back up

transforms (e.g. from electronic device digit counts into another domain such as CIE colorimetry) and forward transforms (e.g. into colorant amounts). If a model is not invertible analytically, replacements are to use numeric methods to reverse the transform, or to determine the model from both directions. In both cases the forward and back up models are probable to have different errors, so that a color which is transformed from one color space to another and then back again will not have identical coordinates like the original [4].

2.2 Physical model

Physical model is based on physical properties of the device and the mixture of themselves. The color-mixing laws can be divided into two parts, additive and subtractive mixing. The technical terms additive and subtractive mixing traditionally have been used to distinguish between the colored lights mixing and the colorants mixing. The three additive primaries are red, green, and blue, while the three subtractive primaries are cyan, magenta, and yellow. Additive mixing primaries produce white, while subtractive mixing primaries produce black.

The typical example of additive color mixing is LED color mixing. In LED displays, three different types in the red, green, and blue of LEDs, are used. And then, we determine the chromaticity coordinates of the mixture of three separated emission bands. Suppose that the three emission bands have spectral power densities P₁(λ), P2(λ) , and P3(λ) with peak wavelengths of λ1, λ2, and λ3, respectively. We suppose that each emission band is much narrower than any other of the three color-matching functions. We further suppose that the three LED light sources have the chromaticity coordinates(x₁, y₁), (x2, y₂), and (x3, y₃). Then the values of tristimulus are given by

Hence, the chromaticity coordinate of the multi-element light is a linear combination of the separate chromaticity coordinates weighted by the L_i factors.

The color mixing principle in the chromaticity diagram is shown in Fig. 2-1.

This diagram shows the two colors mixing with chromaticity coordinates (x1, y1) and (x₂, y₂). And for the case of two colors, we set L3=P₃=0. The mixed color will be located on the straight line of any point on the chromaticity coordinates of the two light sources. Thus, by mixing the two colors, any color (including white) located between the two chromaticity points can be created.

Fig. 2-1 also shows the mixing of three primary colors, located in the red, green, and blue locations of the chromaticity diagram. This three chromaticity points, connected by a dashed line, are general points of red, green, and blue LEDs. The area located inside the dash line, called the color gamut, shows all colors that can be generated by mixing the three colors red, green, and blue. For displays, the ability to create a great variety of colors is an important quality. It is a desirable that the color gamut provided by the three primary light sources is as large as possible to create displays able to show highly saturated colors.

Fig. 2-1 Color gamut with three primary colors [7]

The color gamut shows the entire color range that can be created from a set of three primary sources. Color gamuts are polygons positioned within the perimeter of the chromaticity diagram. For example, the color gamut is a triangle with three primary colors, as shown in Fig. 2-1. All colors created by additive mixtures of the vertices (primary colors) of a gamut, are necessarily located within the gamut. The example of additive color mixing of LED is based on the independency (orthogonal) of three discrete emission bands. This phenomenon is shown as Fig. 2-2. Therefore, trichromatic white multi-LED can depend on its optical powers quantities of three primary colors to decide its chromaticity coordinates.

However, the LED additive color mixing is based on the emitting spectrum of light source. E-Papers are one sort of reflective display. So, if colorimetric characterization of e-Papers can be done by physical model, the reflectance spectrum should be measure first. Then, decide whether the reflectance spectrum of three primary colors has scalability or not. Scalability means that spectral quantity of normalized reflectance can multiply any constant to present their color properties. The hypothesis is that its three primary reflectance spectrum are independent.

There are many examples based on the subtractive principles, ex: Beer-Bouguer law for the homogeneous medium. However, the subtractive mixing would not be used in this thesis, the principles of subtractive mixing would not be talked.

Fig. 2-2 Emission spectrum of trichromatic white multi-LED source with color temperature of 6500K (solid line) and gaussian fit (dashed line) [7]

2.3 Numerical model

To create a numerical model of the device space is first sampled colorimetrically so that the relationship between device values and tristimulus scalars is known for a given set of colors. The set is chosen so that the whole color space of device coordinates is depicted, and the known values that result are then used to derive a numerical model of the relationship between device space values and corresponding CIE color space scalars.

Polynomial regression is frequently used as mathematical tool in the numerical model. The polynomial regression is according to the assumption that the correlation between different color spaces can be approximated by a set of simultaneous equations. The schematic diagram of the regression method is represented in Fig. 2-3.

Sample points of the source color space are selected and their color specifications of the destination color space are measured. A training set is a set of data used in various areas of information science to discover potentially predictive relationships. A testing set is a set of data used in various areas of information science to assess the strength and utility of a predictive relationship.

Fig. 2-3 Schematic diagram of the regression method

To link the source and destination color specifications, different levels of polynomial regression are used, as shown in Table 2-1. The general three-variable cubic expression is given in Eq. 2.9.

3 9 36 deriving the coefficients of the polynomial between source and destination spaces.

The only necessity is that the number of selected points should be more than the number of polynomial terms which have to be calculated; otherwise, there will be no definite solutions to the simultaneous equations because there are more unknown variables than equations. By using derived coefficients, one can connect the source specifications to the simultaneous equations for computing the destination

specifications.

Table 2-1 The polynomials for color space conversion

1 2 3 CIEXYZ values of destination color space. What the value of P(R, G, B) is X, Y, or Z is depending on the coefficients representing a_x, a_y, a_z.The equation 1 of Table 2-1 is taken for an example to show what the value of P(R, G, B) is X, Y, or Z is as follows,

1z 2z 3z

ZP(R,G,B)a R+a G+a B (2.12)

Polynomial regression is an application of the multiple linear regressions of m variables, where m is a number greater than the number of independent variables. The general approach of the linear regression with m variables is given as follows:

1 1 2 2

i i i m im

p a u a u  a u

(2.13)

For the application to the polynomial regression with three independent variables R, G, and B taken for example:

u₁=R, u₂=G, u₃=B accuracy of numerical model. Numerical model with regression methods is ideal for transformation with a linear relationship. For nonlinear color space, this method does not guarantee uniform accuracy of the entire color space. In general, the accuracy improves as the number of terms in the equation increases [9, 10]. And another colorimetric characterization method in look-up table will be introduced below.

2.4 Look-up tables with interpolation

In color space transformation, the three-dimensional (3D) look-up table (LUT) with interpolation is a relatively new development. It includes three portions—packing (or partition), extraction (or find), and interpolation (or computation) [11]. The flow chart is shown the 3D LUT with interpolation consists of these three parts as Fig. 2-4.

Packing is a procedure that divides the source space domain and populates it with sample points to build up the lookup table. Generally, the table is built up by an equal-partition sampling along each axis of the source space as shown in Fig. 2-5(a).

This will give lattice points and cubes, where n is the number of levels.

The benefit of this arrangement is that it implicitly provides the information about which cell is next to which. Hence, one only needs to store the starting point and the spacing for each axis. In general, a matrix of color patches at the lattice points of the source should be made, and the destination color specifications of these color patches are measured. The corresponding values from the source and destination spaces are listed into a lookup table. Non-lattice points are interpolated by utilizing the nearest lattice points. This is the step where the extraction executes a search to choose the lattice points necessary for calculating the destination spaces are listed into a lookup table. A well-packed space can make the search simpler. In an 8-bit integer setting, for example, if the axis is divided into equal partitions where j is an integer smaller than 8, and then the nearest lattice points are given in the most significant j bits ( of the input color signals. That is to say, the input points is constrained in between the lattice points of p( and p( . This involves the operations of masking and shifting bits on computer, and these operations are faster than the comparison operation. For non-equally partitioned packing which is illustrated like

Fig. 2-5(b), a series of comparisons will be needed to confirm the locations of the nearest lattice points. It is required to further selection within the cubic lattice points.

Depending on the interpolation technique used to count the color values of non-lattice points, we have the geometrical method.

Fig. 2-4 Flow chart of 3D LUT

(a) (b) Fig. 2-5 (a) Equal-partition sampling (b) Unequal-partition sampling of color space (XYZ)

2.4.1 Examples of 3-D interpolation

Here, we take a five-level lookup table to make a color transformation from RGB device space to CIELab color coordinates for example. The first step is to choose lattice points in the original RGB space with known color digit counts. A usual method is to divide the original space equally

R: 0.0 0.25 0.5 0.75 1.0 G: 0.0 0.25 0.5 0.75 1.0 B: 0.0 0.25 0.5 0.75 1.0 Eight-bit depth: 0 64 128 192 255

The second step is to make total of 125 lattice points with all combinations of R, G, and B levels. The third step is to extract. The extraction is to find the cube which contains the point (or color) of interest. For example, an input point p (5, 88, 160) has the device RGB values of R=5, G=88, B=160.

The point is bounded in the following ranges:

lower range point upper range R: 0 < 5 < 64 G: 64 < 88 < 128 B: 128 < 160 < 192

When the bounding box is determined, the eight vertices and their corresponding color specifications are extracted in the destination space. The final step is computation. Here, we have to choose 3-D geometrical interpolation methods.

There are four common 3-D geometrical interpolation methods, including trilinear, prism, pyramid, and tetrahedral interpolation, respectively. Then, use one interpolation tool of these four methods to expend to full color space we want. The

examples of trilinear, prism, pyramid, and tetrahedral interpolation methods will be shown below.

2.4.2 Trilinear interpolation

Before we talk about trilinear interpolation, we have to introduce bilinear interpolation. In two dimensions, two variables p(x, y) should be derived, and we have four known lattice points p₀₀(x₀, y₀), p₀₁(x₀, y₁), p₁₀(x₁, y₀), p₁₁(x₁, y₁). The figure of bilinear interpolation is shown as Fig. 2-6. The detail p(x, y) formula is like this:

00 0 1 0 10 00 0 1 0 01 00

Fig. 2-6 Bilinear interpolation [12]

Fig. 2-7 Trilinear interpolation [12]

So, p(x, y, z) could be found by this trilinear interpolation formula. This method requires ten multiplications and seven additions, with thirteen pre-computed

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