Applications of 3D interpolation - Look-up table with interpolation

Chapter 2 Colorimetric Characterization Methods

2.4 Look-up table with interpolation

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 showed that its packing would be non-uniform. Then, and the vertices of a cube for interpolation would be found by extraction. Finally, the value inside the polygon within a cube would be calculated by these four different geometrical interpolation methods, and the ΔE00_avg. between the predicted values by interpolation and the

measured values would be computed. If the average value is behind one, the packing number would be increased until the ΔE00_avg. is below one. Finally, if the ΔE00_avg. is below 1, it should be an adequate LUT. Fig. 3-9 is illustrated the gamma curve in only green channel of Fujitsu FLEPia for representative. The figure is proved that the gamma curve of Fujitsu FLEPia is very nonlinear. So, the chosen packing is non-uniform. Cubic of 3, 5, 7, and 9 non-uniform packing is chosen. And then, how the packing points are selected? The 0, 255 digit count is the necessary points.

Another packing points is picked by the error calculating by l²-norm which is lowest between the gamma curve of Fujitsu FLEPia and the curve of packing points depend on piecewise linear method. The l²-norm is also known as Euclidean norm. The equation form of the error calculating by l²-norm is shown as Eq. (3.1).

255 2

Yk, predicted = normalized luminance of predicted curve of packing points with green channel depend on piecewise linear method

Yk, measured = normalized luminance of Fujitsu FLEPia with green channel

Finally, follow the methods above, the ΔE00_avg. can be calculated.

Fig. 3-9 The gamma curve in green channel of Fujitsu FLEPia

Fig. 3-10 Flow chart to choose numbers of packing

Chapter 4 Experiment and Verification

4.1 Experimental platform

For verifying color differences between the measurement data and numerical model or LUT, the stable experimental testing platform must be set up. The main two conditions of this experimental platform are: (a) full color three-layers-stacked ChLCD; (b) spectrophotometer to measure any optical physical quantity. The three-layers-stacked cholesteric liquid crystal display for experiment was Fujitsu FLEPia [22], and the spectrophotometer was x-rite i1 pro [23]. The specifications of the i1 Pro hardware device is as Table 4-1.

Table 4-1 Specifications of i1 Pro

Optical resolution 10nm

Physical sampling interval 3.5nm

Spectral data Range: 380 ... 730 nm in 10nm steps

Measurement aperture 4.5mm diameter

Measurement geometry 45°/0° ring illumination optics, DIN 5033

Light source Gas filled tungsten (Type A)

According to the specification, the average color difference of i1 Pro is 0.4 of

∆E*94, which is small enough to be used to measure the color performance. The specifications of Fujitsu FLEPia are summarized in Table 4-2.

Table 4-2 Specifications of Fujitsu FLEPia

Size 158 x 240 x 12.5 mm

Display 8 inch

Resolution 768 x 1024 pixels

Number of Displayable Colors 260,000 colors (64*64*64)

The experimental condition is shown as Fig. 4-1, and Fig. 4-2 is the practical platform setup during our experiment. The measurement geometry is 45°/0° ring illumination optics with gas filled tungsten (Type A) light source. The experiments were done in the darkroom. Fig. 4-3 is the flow chart of experiment. The computer was sent digital values to simulator which is Matlab program to control any color in Fujitsu FLEPia. And x-rite i1 pro was use to measure CIEXYZ of Fujitsu FLEPia.

Fig. 4-1 Experimental condition figure

Fig. 4-2 Real construction figure of experiment

Fig. 4-3 Flow chart of experiment

4.2 Examination of performance in Fujitsu FLEPia

At first, the luminance of gray level and independent red, green, and blue color of 256 digit counts were measured as illustrated in Fig. 4-4. From this figure, these four gamma curves are not smooth enough to only utilize a simply analytical formula to represent their color performance, and each R, G, B curve does not overlap each other. Therefore, the characteristic curves cannot use only one fitting curve to represent them. Then, each X, Y, and Z value of green channel in Fujitsu FLEPia were also measured to test whether the normalized luminance values can represent three X, Y, and Z values or not.

Fig. 4-4 Nomalized luminance in 256 digit count of three independent channel and gray level

The figure of green channel each normalized X, Y, and Z value is shown as Fig.

4-5. Obviously, the normalized luminance curve of green channel cannot cover another two X, Z curve.

Fig. 4-5 Normalized X, Y, and Z values of green channel

32, 96, 192, and 255 gray level reflectance of each R, G, B channel were measured respectively to know whether three-layers stacked ChLCD has scalability or not. These three figures are shown below:

Fig. 4-6 Normalized reflectance of 4 digit counts in red channel

Fig. 4-7 Normalized reflectance of 4 digit counts in green channel

Fig. 4-8 Normalized reflectance of 4 digit counts in blue channel

These reflectance spectrum values of these three figures are normalized at dominant wavelength of each R, G, B channel which are 650, 560, and 490 nm, respectively. So, these normalized reflectance spectrum values are not scalable.

Besides, the normalized reflectance spectrum figure of only green color taken for example is shown that high gray level fits well, but low gray level would be influenced by dominant wavelength of red and blue channel. So the spectra of green channel in low gray level also rise up in another wave band. Because few amount of the disordered liquid crystal in focal conic state may be regarded as planar state for a specifically oblique incidence or viewing angle.Red and blue layer introduce a little bit of reflection accordingly even the two layers are focal conic state. In addition, while some scattering contributed from the randomized cholesteric LCs is very weak, it did affect the color performance of ChLCDs. Because effective quantity of another two dominant wavelength is constant, these effective quantities may influence the reflectance spectrum in low gray level more remarkable than in high gray level.

Another two channels are vice versa. Then, the figure of x, y values in CIE color

space with 32, 96, 192, and 255 digit counts of R, G, B channels and gray levels is illustrated as Fig. 4-9 (a), and the figure of x, y values in CIE color space with 32, 96, 192, and 255 digit counts of cyan (C), magenta (M), yellow (Y) is shown as Fig. 4-9 (b).

If the panel has scalability, no matter what the gray level of each R, G, B color channel is, x, y values should be the same or be little different. However, as Fig. 4-9 (a), they are not the same in this panel obviously. Moreover, x, y values in CIE color space with 32, 96, 192, and 255 digit counts of C, M, Y are also not at the same place respectively in Fig. 4-9 (b). So, physical model is not suitable for characterizing three-layers-stacked ChLCD. Another two methods will be given below.

Fig. 4-9 (a) x,y values in CIE color space with 32, 96, 192, and 255 digit counts of R,G,B channels and gray levels

Fig. 4-9 (b) x,y values in CIE color space with 32, 96, 192, and 255 digit counts of cyan, magenta, yellow

4.3 Regression data with numerical model

The regression is from RGB digit counts to CIEXYZ. The flow chart of Fig. 3-8 is followed to decide how many terms of polynomial should be used. Finally, eight polynomials of Table 2-1 with three, four, six, eight, nine, eleven, fourteen, and twenty terms are utilized for the regression. The training sets which are the same with 9x9x9 LUT are a set of nine-level, unequally spaced lattice points (729 points) in the RGB space presenting in chapter 4.4. The testing sets are 900 randomized test samples in uniform color space of Fujitsu FLEPia.

Fig. 4-10 (a) ~ (c) are shown the 900 randomized test color samples, the 900 randomized test samples in uniform a*b* color space of Fujitsu FLEPia, and the distributions of lightness with the 900 randomized test samples. These figures can prove that 900 test samples are uniformly distributed in La*b* color space. Afterward,

eight polynomials of Table 2-1 with three, four, six, eight, nine, eleven, fourteen, and twenty terms are utilized for the regression. Table 4-3 lists the average CIEDE2000 color differences using these polynomials.

Fig. 4-10 (a) The 900 randomized test color samples

Fig. 4-10 (b) The 900 randomized test samples in uniform a*b* color space of Fujitsu FLEPia

Fig. 4-10 (c) The distributions of lightness with the 900 randomized test samples

Table 4-3 Average CIEDE2000 color differences of various polynomial regressions

Polynomial term Training Testing Total

3 4.68 3.67 4.12

4 3.47 2.63 3.01

6 3.87 2.91 3.34

8 3.73 2.92 3.28

9 3.17 2.49 2.79

11 2.76 2.19 2.45

14 1.51 1.41 1.45

20 1.5 1.4 1.44

Although the ΔE_{00_avg.} of training sets in polynomials of low terms is too high, the color differences of testing sets in these low terms polynomials still were calculated. These results of Table 4-3 show that the average CIEDE2000 color differences decreases as the number of terms in the polynomial increases except six and eight terms. The reason should be that usually the number of terms in the polynomial increases as the average color differences decreases, but it does has exception.

The distributions of CIEDE2000 color differences with testing sets are plotted in Fig. 4-11. The error distribution becomes narrow and shifts toward small CIEDE2000 color differences as the number of terms in the polynomial increases except 6, and 8 terms. Although the ΔE00_avg. of 14, and 20 terms polynomials are below 1.5, the result is not good enough to predict the color performance of three-layers-stacked ChLCD. So, the LUT with interpolation method will be talked in the next section.

Fig. 4-11 Error distributions of regression

4.4 Interpolation from look-up tables

The first part of 3D LUT is packing. Depends on the gamma curve of FLEPia, cubic of 3, 5, 7, and 9 non-uniform packing is chosen by operating sequence of flow chart. Table 4-4 is shown the chosen packing points of RGB device space in cubic of 3, 5, 7, and 9 LUT, respectively. These points were selected is relying on the error which is lowest between the gamma curve of FLEPia and the curve of packing points

by piecewise linear method. Table 4-5 is illustrated these values of error.

Table 4-4 The chosen packing points in cubic of 3, 5, 7, and 9 LUT

Levels of LUT Packing digit counts

3 0, 44, 255

5 0, 88, 116, 208, 255

7 0, 88, 96, 108, 116, 208, 255

9 0, 44, 88, 96, 108, 116, 196, 208, 255

Table 4-5 The error between the gamma curve of Fujitsu FLEPia with green channel and the curve of packing points

Levels of LUT Error

3 0.689

5 0.267

7 0.258

9 0.097

The result of Table 4-5 can prove that the values of error decrease as levels of LUT increase. Fig. 4-12 is shown the gamma curve of G channel the predicting gamma curves of G channel with 3, 5, 7, 9 selected break points and piecewise linear interpolation. As levels of LUT increase, the fitting curve is like the gamma curve of

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