Numerical model - Colorimetric Characterization Methods

Chapter 2 Colorimetric Characterization Methods

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 coefficients stored at each node.

2.4.3 Prism interpolation

If a cube is cut diagonally into two halves as shown in Fig. 2-8, a cube gets two prism shapes. For locating the point of interest, a search mechanism is needed. Owing to these two symmetric structures in the cube, a simple inequality comparison is sufficient to determine the location: if ∆x > ∆y, then the point is in prism 1; otherwise the point is in prism 2. For ∆x = ∆y, the point is located on the diagonal plane, then either prism can be used for interpolation. Each equation has six terms and uses six vertices of the prism to compute. This method requires eight multiplications and five additions with seven coefficients at each node.

∆x > ∆y

2.4.4 Pyramid interpolation

For pyramid interpolation, the cube is cut into three parts; each part takes a face as the pyramid base possessing its corners linked to a vertex in the opposite side as the apex (as shown in Fig. 2-9). For locating the interpolation point, the search mechanism given in Table 2-2 is used. Each equation has five terms and uses five vertices of the pyramid for computation. This method requires seven multiplications and four additions with five coefficients at each node.

Table 2-2 The inequality relationships and the coefficients for pyramid interpolation

Pyramid Test c₁ c₂ c₃ c₄

Fig. 2-9 Pyramid interpolation [12]

2.4.5 Tetrahedral interpolation

We will present the details of an efficient and accurate method for color interpolation which is frequently used in practical applications. There are many methods for splitting a cube into many tetrahedral, and no matter what methods we using, we should follow these principles below:

1. Try to use all vertices of a cube, and the using frequency is as many as well.

2. The cube may be all filled with tetrahedron.

3. The linking surfaces of tetrahedron should be complete, in other words, the area of linking surfaces should be the same.

Fig. 2-10 Tetrahedral interpolation

According to these principles and computation, the method of diagonal extraction in tetrahedral interpolation is usually used. As shown in Fig. 2-10, the tetrahedral interpolation [13] cuts a cube into six tetrahedrons; everyone has a triangle base.

The problem is described in Fig. 2-11. Given a tetrahedron in an (r, g, b) digit count color space and its corresponding tetrahedron in another color space (X, Y, Z), for any point in (X, Y, Z) space, determine whether the point is inside the tetrahedron or not, and if yes, determine its corresponding coordinates in the (r, g, b) color space.

To solve these problems, it is convenient to take one of the four vertices (see Fig.

2-12), say vertex P₀, as the origin and regard the edges from P₀to the other three vertices P1, P2, and P3, as vectors, P0P1, P0P2 and P0P3:

The above conditions can be proved as depicted below:

1. P0 and P are on the same side of the plane P1P2P3 ↔ α + β + γ ≤ 1.

2. P1 and P are on the same side of the plane P0 P2P3 ↔ α ≥ 0.

3. P2 and P are on the same side of the plane P1P0P3 ↔ β ≥ 0.

4. P3 and P are on the same side of the plane P1P2P0 ↔ γ ≥ 0.

What are the (r, g, b) coordinates of an inner point P? When P is inside the tetrahedron, its (r, g, b) coordinates can be determined from the decided four vertices by:

The general formula of eq. (2.22) is illustrated below:

This method requires six multiplications and three additions with three coefficients at each node.

2.4.6 Applications of 3D interpolation

Many applications of 3D interpolation in printer, monitor, and scanner calibration have been published in the literature. LUTs are applied where two domains are not easily related, as it will often be faster to locate the values and perform any interpolation required than to evaluate a complex function.

Chapter 3 Colorimetric Characterization on ChLCD

3.1 Introduction of cholesteric liquid crystal

The chiral nematic liquid crystal is also named cholesteric liquid crystal (ChLC). Unlike the traditional nematic liquid crystal, ChLC includes a chiral center in the nematic mesogenic molecules. If the twisting power for chirality is strong enough, ChLC material is integrated when the nematic liquid crystal and chiral dopant are mixed. The chirality generates an intermolecular force that results in a twist between each layer. In different cholesteric liquid crystal systems, the period of the helical pitch varies by a wide range. For the long pitch (low concentration chirality) P >> λ (where λ is the wavelength of light), the light propagation parallel to the helix axis may be characterized by a superposition of two eigenwaves having electric field vectors parallel and perpendicular to the director. For short pitch (high concentration chirality), the eigenwaves may become elliptical, and may be circular in the limiting case. The ChLC molecule structure is shown in Fig. 3-1.

Fig. 3-1 ChLC molecule structure, where P is the helical pitch with ChLC molecule rotating 360 perpendicular to Z-axis

3.1.1 Optical characteristics in different textures

ChLCs are simple to change its texture by adding electric or magnetic field.

Again, we add electric field instead of magnetic field in common cholesteric liquid crystal devices. When an electric field applied to the ChLCs cell, a texture transition happens to minimize the free energy system. The texture transition is strongly associated with dielectric anisotropic Δε, field amplitude, alignment layer, and the frequency of the external field. The typically operating modes which are used in common cholesteric liquid crystal display are described as Fig. 3-2.

When the ChLCs operate in the planar texture at zero field, the helical axis is perpendicular to the cell surface as shown in Fig. 3-2(a). It reflects the selective reflection light by Bragg diffraction. The texture would be changed into the focal conic texture when applied electric field V>=Vth(threshold voltage). In the focal conic texture, the helical axis is more or less parallel to the surface as shown in Fig.

3-2 (b). Incident light is diffracted or scattered in the forward direction and the material in this state has less reflective color appearance. And then, the high-voltage pulses can transform the liquid crystal from the focal conic texture to homeotropic texture (as Fig. 3-2 (c)). If the applied voltage is switched to a medium bias level, the liquid crystal relaxes to focal conic texture; if the applied voltage is switched to a low bias level, the liquid crystal relaxes to planar texture. The channels of switching between three textures [15] are illustrated in Fig. 3-3. It is noted that ChLCs can exhibit two stable textures at zero field. One of them is the planar texture, and the other is the focal conic texture. So the ChLCs material has the bistable property that is an advantage for display technology.

When the dielectric anisotropy Δε >0:

Fig. 3-2 Schematic diagram for different textures of the ChLCs

Fig. 3-3 Possible field-induced texture transition channels in ChLCs

3.1.2 Gray scale property of cholesteric liquid crystals

ChLCs exhibit gray scale property owing to their multi-domain structure when the planar texture and the focal conic texture are appeared simultaneously. From the imperfect planar texture, some domains can be switched into the focal conic texture if the threshold voltage is exceeded. The reflective color luminance will be decreased.

The figure of the gray scale states of a ChLCs display is illustrated in Fig. 3-4. From left to right, the states are achieved by applying voltage pulses with increasing amplitude.

Fig. 3-4 The gray scale states of a ChLCs display [16]

3.2 Physical mechanism of three-layers-stacked ChLCD

The three-layers-stacked ChLCD which is as implied by the name is composed of three layers. The top layer selectively reflects blue light, the center layer reflects green light while the bottom layer selectively reflects red light. The fundamental structure is shown as Fig. 3-5. And the stacking structure using additive color approach.

For example, blue color is exhibited with the top layer which operates in the reflective texture and another two layers operate in focal conic texture. Another two colors are vice versa [17].When the liquid crystal is in the focal conic texture, many domains still have the helical axes perpendicular to the surface of the cells because of the strong homogeneous surface anchoring, and the reflection is relatively high [18].

So, although another two layers operate in the focal conic texture, the diffusion light from these layers also influence the colors we want to exhibit.

When the helical structure is operated in planar texture (as illustrated in Fig.

3-6), normally incident white light is decomposed into right and left circular parts

with one part reflected and the other transmitted. So, that is why reflectivity of the panel cannot behind 50%.

A question comes out that—why the layer sequence is BGR, as shown in Fig.

3-5? The key point is depending on the reflectivity of each layer. Because the reflectivity is highest in blue-color state, three-layers-stacked ChLCD is made in this type. ‗The reflectance of the blue layer alone is 37% while the yellow reflectance is 30% [19] can prove this actuality. Here, Fig. 3-7 is the spectral reflection of three-layers-stacked ChLCD (FLEPia) under diffuse illumination. The spectral reflectivity of FLEPia would be decided whether it has scalability or not in chapter 4.

Fig. 3-5 A structure of three-layers-stacked ChLCD [20]

Fig. 3-6 Schematic illustration of the planar texture

Fig. 3-7 Reflection spectrum of three-layers-stacked ChLCD (FLEPia)

3.3 Numerical method

In principle, the polynomial order can be up to n-1 (where n is the number of discrete samples), although since each additional order adds a possible bend to the function (such bends being known as local maxima or minima, or relative extrema) it is desirable to keep the polynomial order as low as possible. However, how many numbers of polynomial terms are suitable for our device. So, here a flow chart is presented how the suitable polynomial term is chosen as Fig. 3-8.

At first, the measured training sets are a set of nine-level, unequally spaced lattice points (729 points) which is from 9x9x9 LUT talked in chapter 4 in the RGB space. Secondly, the measured training set may be substitute into regression equations.

The predicting CIEXYZ values of these training set will be computed. Why the color differences of training sets were calculated is because that the polynomial regression computed from training sets is only the approximate form. We can examine the average color differences of training sets to determine whether this polynomial equation reaches our criterion or not. If the ΔE00_avg. between predicting and measured

training sets is behind 1, the polynomial term will be increased to 20 terms at most.

Afterward, the CIEDE2000 color differences between the predicting and measured CIEXYZ values of testing samples will be calculated. If ΔE_{00_avg.} between them is not below 1, the polynomial term will be also increased to 20 terms at most. Finally, whatever the ΔE_{00_avg.} of training set and testing set in 20 terms polynomial are below 1 or not, the terms would not be increased anymore. Provided that the order of polynomials rises up to fourth order, the amounts of polynomial terms would increase to 60 terms mostly. This may cause calculating efficiency to drop. Therefore, polynomial of 20 terms is the most terms to be used.

Fig. 3-8 Flow chart to determine coefficients and polynomial terms

3.4 Look-up table with interpolation

The first step here used to characterize three-layers-stacked ChLCD is non-uniform packing because its gamma curve is nonlinear. This reason is explained as Fig. 3-9, the gamma curve in green channel of Fujitsu FLEPia is very nonlinear.

This form of LUT is very different from standard way of implementing LUT; in other words, the source color space is equally divided with respect to the sampling rate.

This may cause an irregular and uneven destination space for those color conversions which are nonlinear. Many techniques have been offered to work on the non-uniform color space [21].For this packing, each cube subcell will be replaced by rectangular one. The lattice points are selected using the gamma curve of RGB with respect to the normalized luminance.

When in the second step, the extraction is to find the cube which contains the point (or color) of interest. The method to choose the bounding box is like the way depicted in chapter 2. The final step is the interpolation, and trilinear, prism, pyramid, and tetrahedral interpolation methods are all chosen to use to interpolate full color space in ChLCD. Here, a flow chart is presented how the suitable packing number is chosen as Fig. 3-10.

At first, the gamma curves with respective R, G, B channel of Fujitsu FLEPia would be decided if its color conversion is linear. Secondly, if its color conversion is linear, its packing would be uniform; if its color conversion is nonlinear. The result

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