1. Introduction

The moirés appearing at the contact-type 3-D displays [1] are having color fringe patterns. Since they deteriorate the image quality of the displays, they are highly undesired interference phenomena that have to be minimized or eliminated in the displays. In this regard, many researchers tried to minimize the moirés [2–6], but no proper solution has been found to describe the color fringe patterns, so far. The typical method of approximating the patterns is one or two dimensional cosine functions of different frequencies [2, 3, 6~11]. In each of the two overlapped plates having line array patterns, each plate pattern is approximated by one or two dimensional cosine function of different frequencies. This approximation works reasonably for black and white line patterns but it doesn’t work well for the contact-type 3-D displays because of the pixel pattern on the display panel. The reason for this can be found by considering the structure and components consisting of the displays. The 3-D displays are composed of a display panel and a viewing zone forming optics (VZFO) on its top [12]. The display panel consists of a two dimensional array of pixels and each pixel can generate a desired color by controlling basically the intensities of its R (Red), G (Green), B (Blue) sub-pixels. The VZFO has a slightly different line pattern from the panel. The two commonly used VZFOs such as parallax barrier and lenticular have the same parallel line pattern but the line transparency is conjugate to each other; the parallax barrier is a transparent line grating in the dark background but the lenticular, a dark line grating on a transparent background. The line grating is formed by boundary lines between elemental lenses in the lenticular. These boundary lines are not dark but they have lower transparency than the lens parts. However, this is not a main cause of the colored moirés, but the line width of these boundary lines. Each of these boundary lines is not an ideal line but it has a finite width. In dealing the moirés problem, the cosine approximation considers only the line period of the pattern because the line width doesn’t affect the period. When two regular patterned plates are overlapped, the lines forming a pattern will be mixed with those forming the other pattern. However, this mixing will not bring any moiré fringe if 1) the top plate has no thickness, 2) the periods of the lines in two patterns are exactly the same, and 3) the lines in two patterns are completely matched to each other. But when any one or two, or all of these three conditions are not matched, there appear areas where lines are either sparsely or densely mixed. These areas appear repeatedly along the length direction of two plates. These repeatedly appearing densely and sparsely mixed areas are the moiré fringes. These moiré fringes can be predicted by the cosine approximation. But in the contact-type 3-D displays, the panel has no lines but pixels which are aligned in the form of a two-dimensional matrix and VZFO an array of lines with a finite width. When the panel is On without any picture, if the top plate is superposed on the panel, each line in the top plate blocks the pixels under it. If the line width is not bigger than the width of a sub-pixel, the intensity of a color sub-pixel is reduced somewhat depending on the transparency of the line because the part corresponding to the line width is blocked. Due to this blocking, the white balance of the panel becomes broken and as a consequence, a line pattern which is painted by colors like yellow, magenta or sky blue will appear on the panel. This is the color moiré fringes. When the line width is bigger than the width of a sub pixel, an appropriate color moiré fringes will appear on the panel. Furthermore, these color moiré fringes are shifting as viewers change their viewing position, and their periods are either reduced or increased as the viewing angle changes at a given viewing distance. The viewing distance also induces the period changes. These features can provide more versatile applications of the moiré fringes to anti-counterfeit security than those based on nano-sized combination of a lenticular and an image pattern as in a contact-type 3-D display [13].

In this paper, it is shown that the dark line width and its virtual increment in the line grating pattern of VZFO performs a major role in forming color moirés appearing at a contact-type 3-D display.

2. Role of the line width in forming color moirés

When the thickness of VZFO is negligible, the color combination of a color moiré is determined by the widths of the dark lines on the top pattern plate as shown in Fig. 1. Figure 1 shows Fig. 1(a) panel’s R(Red), G(Green) and B(Blue) sub-pixel array, Fig. 1(b) a line grid pattern, Fig. 1(c) a line grid having wider line width than that in Fig. 1(b), Fig. 1(d) a superposed pattern of Fig. 1(b) on Fig. 1(a), Fig. 1(e) a superposed pattern of Fig. 1(c) on Fig. 1(a), Fig. 1(f) the uniformly contracted Fig. 1(d) pattern by repeated adding of the same pattern several times, and Fig. 1(g) uniformly contracted Fig. 1(e) pattern by repeated adding of the same pattern several times. The superposed patterns show that the sub-pixels under the dark lines are blocked and not visible. Hence the hues of the superposed patterns in Figs. 1(c) and 1(d) are different because the sub-pixels in Fig. 1(d) are blocked more than those in Fig. 1(c). When the superposed patterns are uniformly contracted to make sub-pixel colors to be spectrally mixed together, the periodic color patterns in Figs. 1(d) and 1(e) show clearly different color combinations as shown in Figs. 1(f) and 1(g). These periodic color patterns are the color moiré fringes. The color combination difference between Figs. 1(f) and 1(g) is induced by the colors of the unblocked sub-pixels and the brightness reduction in proportional to the area of the unblocked part of a sub-pixel by the dark lines. Figures 1(f) and 1(g) clearly demonstrate that the line width of superposed line grid is the main factor of determining the color combination in the color moiré fringes. The brightness reduction can be calculated by assuming that the brightness of the sub-pixel will be reduced to 0.75, if 1/4 of its area is blocked and to 0.5 if 1/2 of its area is blocked by the lines when the brightness of a full sub-pixel is considered as 1 as in a display. This assumption is used to simulate the color moirés in this paper.

 

Fig. 1 Comparisons of color moiré fringes from layering a RGB pattern and two different line Grids: (a) RGB Pattern, (b) Line Grid Pattern, (c) Line Grid Pattern; wider line width than that in (b), (d) Superposed pattern of (b) on (a), (e) Superposed pattern of (c) on (a), (f) Uniformly contracted (d) pattern by adding the same contracted pattern several times, (g) Uniformly contracted (d) pattern by adding the same contracted pattern several times.

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When two patterns are distanced by a plate with a non-zero refractive index as in the contact-type 3-D displays, the colored moiré fringes appearing at two overlapped regular pattern plates are different from Fig. 1 because VZFO has a finite thickness and a non-zero refractive index. In the displays, the widths of boundary lines between elemental lenses in a lenticular plate or the dark lines between slit lines in a parallax barrier are the essential parameter to be considered first along with the plate thickness of the VZFO [14] for the accurate simulation of the colored moiré fringes. VZFO plate thickness includes the thickness of the panel glass. The plate thickness induces the portion of a sub-pixel or a number of sub-pixels to be blocked by the line width to more than what the actual size of the line width can block. The blocked number (or portion) of the sub-pixel(s) will be increased as the viewing angle increases for a given viewing distance. The reason is as the followings. When a viewer watches a contact-type multiview 3-D display at a distance along the normal line of the display panel of the display, the viewing geometry will be described as in Fig. 2.

 

Fig. 2 Viewing geometry of a Contact-Type 3-D Display.

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In Fig. 2, when the left and right edge points of the $n^{th}$ line from the center of the panel is specified as $a$ and $a^{\prime }$, respectively, $a$ and $a^{\prime }$will be matched to the points $b$ and $b^{\prime }$on the panel, respectively, because of light refraction induced by the VZFO thickness. The distance between points $b$ and $b^{\prime }$ determines the number of sub-pixels to be blocked by the $n^{th}$ line and it is calculated by Snell’s law [15].

In Fig. 2, the viewer distance from the panel, the plate thickness of the VZFO, the period of elemental lenses or line array in the VZFO, each line thickness (width) of the line array, a pixel size in horizontal direction, the refractive index of the VZFO material and the number order of the lines from the panel center are specified as $z_V$, $t$, $O_P$, $B_a$, $P_P$, $m$ and $n$, respectively. Furthermore, the viewing angles of viewing $a$ and $a^{\prime }$from the center of the panel are specified as $\theta (n)$ and $\theta (n^{\prime })$. In this case, $O_P-B_a=A_P$ is the slit width in VZFO, the distance to the points $a$ and $a^{\prime }$from an eye of a viewer can be represented as $n{O}_P$ and $n{O}_P-B_a$, respectively, and the distance between $b$ and $b^{\prime }$, $B_a(n)$ is expressed as,

Since the distances of $b$ and $b^{\prime }$ from the panel center are given above and the width of a sub-pixel is defined, the sub-pixels which will be under $B_a(n)$ will be identified.

3. Chirped panel pattern

Figure 3 shows $B_a/B_a(n)$ for $B_a$ values of 0.3222 mm (Fig. 3(a)) and 0.1611 mm (Fig. 3(b)) when a viewer is watching the panel at $z_V$distance along the panel’s normal line for the case of $t=0.68\text{mm}$, $O_P=0.4833\text{mm}$, $P_P=0.4833\text{mm}$, $m=1.5412$, and $n=0\text{to}\pm 960$, where $+$ represents the number order to right side and $-$ to left side. For the $z_V$ values, $500\text{mm}$ and $1,000\text{mm}$are considered. According to $P_P$ value, $B_a$ values of 0.3222 mm and 0.1611 mm correspond to two and a sub-pixel size, respectively. The m value is derived by taking the arithmetic average of two components consisting of the VZFO, i.e., display glass and the film. The refractive indexes (thicknesses) of the film (Polyethylene Terephthalate Polyester) and the glass are 1.6 (0.18 mm) and 1.52 (0.5 mm) [17], respectively. The refractive index of the polyester film is in the range 1.58 to 1.64 for the visible light spectrum but 1.6 is used for the calculation [16]. The arithmetic average (1.6·0.18 + 1.52·0.5)/0.68 gives the value 1.5412. $B_a/B_a(n)$ curves for $B_a=0.3222\text{mm}$ and $B_a=0.1611\text{mm}$ do not reveal noticeable differences between them due to the fact that $t\{\tan {\theta }_r(n)-\tan {\theta }_r(n^{\prime })\}/B_a$ in Eq. (1) is very small and $\tan {\theta }_r(n)$ is the same for all cases. Since aperture width, $O_P-B_a$ of the VZFO pattern is smaller than the line width $B_a$ for $B_a=0.3222\text{mm}$ and bigger for $B_a=0.1611\text{mm}$. Hence the former can be an example of a parallax barrier and the latter a lenticular. $B_a/B_a(n)$ curve is symmetric along the line dividing the panel into two equal parts in horizontal direction and increasing with increasing $n$ values. It indicates that $B_a(n)$ reduces continuously with increasing $n$ values and will be close to $B_a$ for large $n$ values. The value range of the ratio in Fig. 3 is 0.99915 ~0.999525 and 0.99958 ~0.99964 for $z_V$ values of $500\text{mm}$ and $1,000\text{mm}$, respectively, for both $B_a$ values. These values indicate that the difference between $B_a$ and $B_a(n)$ are less than 0.1% for all $n$ values used here and the difference is much smaller for $z_V=1,000\text{mm}$ than $z_V=500\text{mm}$. The difference between $B_a$ and $B_a(n)$ will be further reduced as the viewing distance increases. However, the accumulated difference between $B_a$ and $B_a(n)$ can be larger than the thickness $t$ for a large $n$ value as mentioned before. The accumulated difference can induce the moiré phenomenon, especially when $t$ is not too much smaller than $O_P-P_P$. Since $O_P-P_P$ determines period of moiré fringes, if $t$ is too small for $O_P-P_P$, visible changes in period or phase in moiré fringes will be hard to be noticed.

 

Fig. 3 $B_a/B_a(n)$ curves.

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$B_a<B_a(n)$ means that the panel pattern is virtually contracted and chirped because the difference between$B_a(n)$ and $B_a$ is reduced as the viewing distance increases. Due to the virtual contraction, the dark line width $B_a$ can block more sub-pixels than what its actual line width can. The thickness of VZFO causes the virtual contraction of the panel pattern.

4. Comparisons of color moirés obtained from simulation and experiment

Figures 4 and 5 show moiré fringes for two different $z_V$ values of 500 mm (Figs. 4(a) and 5(a)) and 1,000 mm (Figs. 4(b) and 5(b)) when $O_P=P_P=0.4833\text{mm}$. Figure 4 is when $B_a$ = $0.3222\text{mm}$ and Fig. 5 $B_a$ = $0.1611\text{mm}$. The white broken lines represent color boundaries. $B_a$ = 0.3222 mm can represent a parallax barrier case and $B_a$ = 0.1611 mm a lenticular case. The $B_a$ values of $0.3222\text{mm}$and $0.1611\text{mm}$ correspond to a sub-pixel and two sub-pixel sizes, respectively, in horizontal direction of the panel, since the size of each sub-pixel is given as 0.1611 mm, i.e., a pixel size is 0.4833 mm. Figures 4 and 5 are obtained with a 42 inch full HD LCD Monitor as shown in Fig. 6. It has the pixel size of 0.4833 mm and its horizontal size is 928 mm. As for the VZFO, a polyester film with thickness 0.18 mm is used as in Fig. 3. On this film, a line array having a period of 0.4833 mm and line width of 0.3222 mm is drawn for Fig. 4, and a line array having the same period but line widths of 0.1611 mm is drawn for Fig. 5. The film is attached on the panel surface by a plastic panel. Each of Figs. 4 and 5 shows two sets of moiré fringes corresponding to two $z_V$ values. In each set, the top and middle moiré fringes are simulated ones for $t=0.0\text{mm}$ and $t=0.68\text{mm}$, respectively, with use of Eq. (1) and the bottom one is experimentally obtained moiré fringe by layering the polyester film on the part of the panel, where all the pixels are on to display a white color. These moiré fringes are displayed on the panel simultaneously and photographed by a camera that is located 500 mm and 1,000 mm from the center of the panel. Since the glass thickness of the panel is 0.5 mm, the total thickness of VZFO plate is $0.68\text{mm}$ and the refractive index value is obtained as in Fig. 3. The length of the film is$750\text{mm}$. The line array is printed on the film with a photoplotter (UCAMCO Calibr8tor NaNO^II series) having more than 20,000 DPI (Dot per Inch) resolution [18] to secure the line accuracy and period uniformity. When the film is layered on the panel, if there is no gap between both patterns on the panel and the film, i.e., when $t=0.0\text{mm}$ case, there will not be any moiré fringe because $O_P=P_P$. In this case, when it is assumed that the line in the center of the film is aligned exactly with the center of the panel, blue and red sub-pixels in the panel become either invisible or partly visible by the line array in the film. This is because the blue sub-pixel of 960th pixel from left edge of and red sub-pixel of 960th pixel from right edge of the panel will be blocked completely by the line in the center of the film for $B_a=0.3222\text{mm}$ when the sub-pixel are aligned as red, green and blue orders from left to right. Hence only green sub-pixels in the panel will be visible as shown in Fig. 4 for $t=0.0\text{mm}$ case. For $B_a=0.1611\text{mm}$ case, only half of the blue and the red sub-pixels will be covered. Hence the color combination of each pixel viewed through the film will be represented by (0.5 red, Green and 0.5 blue) when the brightness level of each primary color is represented by 0 to 1.

 

Fig. 4 Comparisons of simulated and experimentally obtained color moiré fringes for $B_a$ = $0.3222\text{mm}$.

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Fig. 5 Comparisons of simulated and experimentally obtained color moiré fringes for $B_a$ = $0.1611\text{mm}$.

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Fig. 6 Experimental Set-up.

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The resulted color is shown in Fig. 5 for $t=0.0\text{mm}$. So no moiré fringe appears for $t=0.0\text{mm}$ case. However, in practice the distance between two patterns are not 0 because of the thicknesses of the panel glass and the film. This thickness induces a color pattern as the bottom pattern in each set of Figs. 4 and 5. The monochromatic color for $t=0.0\text{mm}$ appears only at the central part of the patterns. These color patterns are moiré fringes. They vary their phases and periods as the viewing angle and the viewing distance change. These patterns can be simulated by Eq. (1) with the parameter values given above. In a Full HD panel, there are 2,880 sub-pixels in each side of the panel from its center line. For the simulation, 1) the sub-pixels which will be under $B_a(n)$ are identified. And then, 2) the brightness of each sub-pixel is calculated depending on its degree of coverings by$B_a(n)$. If the sub-pixel is completely/a half/one third covered by$B_a(n)$, its brightness will be 0/0.5/(2/3) of its maximum brightness, i.e., 1. The simulated moiré fringes match well with the experiments, though there are color smearing in experimental results due to local gaps between the film and the panel, difficulties in matching the line center of the film to the sub-pixel and the refractive index of the polyester film in visible light is not defined as a single value but has a range. Figures 4 and 5 also indicate that more colors are involved in the moiré fringes for $z_V=500\text{mm}$ than $z_V=1,000\text{mm}$. This informs that the moiré fringes are dependent on the viewing distances and their periods are longer for larger viewing distances. Figures 4 and 5 inform that Eq. (1) is the governing equation of predicting moiré phenomenon in the contact-type 3-D displays. Figure 7 shows other example of moiré fringes when a film with $B_a=3.2389\text{mm}$and $O_P=3.4\text{mm}$is layered on the panel shown in Fig. 6. The moiré fringe alignment is the same as Figs. 4 and 5. In this case, since the period of the moiré fringes for $t=0.0\text{mm}$ is calculated as 0.560 mm, each fringe will be hardly identified. The moiré fringes shown in Fig. 7 reveal the repeated appearance of red, green and blue (RGB) lines combination. The white broken lines represent the boundaries of the combinations and the shorter white broken lines the combination the boundary shift in the film. This combination is expected because the film has approximately 273 (928 mm/3.4 mm = 272. 94) slits and each slit has the width of 0.1611 mm (3.4 mm–3.2389 mm = 0.1611 mm). Since $O_P$($=3.4\text{mm}$) corresponds to $7\times P_P(=0.4833\text{mm)}+0.0169\text{mm}$, each slit is aligned with the sub-pixel array as if it is shifting 0.0169 mm to the right when the slit number is counted from left to right. This means that 9.53 (0.1611/0.0169) slits are needed to scan a sub-pixel width, 0.1611 mm. This is shown in the moiré fringe for $t=0.0\text{mm}$at $z_V=1,000\text{mm}$ only because the camera used can cover entire panel at this distance. Each red, green or blue color consists of 9 or 10 lines, i.e., each RGB combination is composed of 28.6 (9.53 X 3) lines. Each of these lines has different color from other lines because the two neighboring primary colors, i.e., sub-pixels are differently mixed due to the spatial scanning action of each slit, though it is difficult to identify these color differences between lines in the moiré fringes. There are 9.53 RGB combinations in the moiré fringe. The moiré fringes appearing at the surface of the film consists of about 221 lines and consist of 8 + (21/25) RGB combinations for $z_V=500\text{mm}$(Fig. 7(a)) and 8 + (5/25) for $z_V=1,000\text{mm}$(Fig. 7(b)). Each RGB combination consists of 25 lines for $z_V=500\text{mm}$ and 27 lines for $z_V=1,000\text{mm}$. These inform that the number of RGB combinations increases as the thickness increases. But the increment is reduced as the viewing distance $z_V$ increases. The simulated moiré fringes for $t=0.68\text{mm}$ are not different from the film. The only difference between the simulated and the film is that the film is shifted one line to right compared with the simulated. The moiré fringe for $z_V=500\text{mm}$also indicates that the fringe is chirped. The widths of RGB combinations decrease as away from the center. This means that the colored line periods are decreasing as away from the center because the number of the colored lines is the same for the RGB combinations in the moiré fringe.

 

Fig. 7 Color moiré fringes for a film with $B_a=3.2389\text{mm}$ and $O_P=3.4\text{mm}$.

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However, no noticeable differences between RGB combinations appear at $z_V=1,000\text{mm}$ case. Figure 8 shows off-center viewing of moiré fringes at two different $z_V$ values of 500 mm (Fig. 8(a)) and 1,000 mm (Fig. 8(b)) when a film with $B_a=0.4028\text{mm}$ and $O_P=0.4833\text{mm}$ is layered on the panel in Fig. 6. The white broken lines represent color boundaries. Figure 8 is not different from Fig. 4, except the fringes look darker than those in Fig. 4. This is because the slit size of Fig. 8, 0.0805 mm is a half of that of Fig. 4, 0.1611 mm. Due to the difference in the silt size, the colors look more localized, i.e., the boundaries of different colors are clearer than those in Fig. 4. When the photographing camera is shifted to right side for $z_V=500\text{mm}$, the green color in center also shifts to the right side. Hence the left most blue color area increases. When the camera is shifted to left side for $z_V=1,000\text{mm}$, the green color in center shifted to left side. No red color is seen at the fringe. Figure 8 informs that the phases of moiré fringes shift according to the changes in viewing position of a viewer.

 

Fig. 8 Color moiré fringes when viewing positions are shifted.

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In Figs. 4, 5, 7 and 8, there are small mismatches between simulated and the film moiré fringes. These mismatches could be caused by the refractive index values. Since the values vary with wavelengths, the mismatches are normal. Figure 9 shows color moiré fringes superposed on two color images of a lotus field (Fig. 9(a)) and an eagle (Fig. 9(b)). The same simulated conditions and the VZFO films for Figs. 4 and 7 are used. The images are displayed on the TV in Fig. 6. The white rectangle on the simulated eagle image represents the active surface of the VZFO film. The colors of simulated and through the VZFO film images match closely for two different viewing distances of $z_V=500\text{mm}$ and $1,000\text{mm}$. Small color mismatches appearing at the transition regions of different colors are caused by the same reasons as for Figs. 4 and 7. It is clear in these images that the colors of the moiré fringes are not different from their corresponding moiré fringes in Figs. 4 and 7. The moiré fringe colors are dominating the colors of the displayed image.

 

Fig. 9 Moiré fringes on real images. The simulated and through the film images are compared.

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The results so far inform that the virtual increase in the line width of VZFO grating pattern is the determining factor of creating the color moirés and the colors of the through the film images are dominated by the colors composing the moirés. The color composition of the color moirés in the contact-type 3-D displays are defined by the line width of the VZFO grating pattern along with the thickness of VZFO.

Figure 10 shows the period of moiré fringes for the parameters used in Fig. 4, except the panel’s horizontal size that are assumed as 9280 mm. For the period calculation, Eq. (7) of reference (14) is used. Figure 10 shows that the moiré fringe periods for the central part of the fringes are 6.1 cm and 12.3 cm for $z_V=500\text{mm}$ (Fig. 10(a)) and $1,000\text{mm}$(Fig. 10(b)), respectively. Except the central fringes, the fringe periods are extending to infinity as the size of the panel increases. This is expected from Fig. 3 because $B_a(n)$ becomes close to $B_a$ as the panel size increases.

 

Fig. 10 Moiré fringe periods for the parameters applied to Fig. 4.

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5. Conclusion

Figures 4, 5, 7-9 assure that the color moiré fringes appearing at contact-type 3-D displays can be accurately simulated by finding the sub-pixels that will be blocked by the lines consisting of the VZFO pattern. The sub-pixel sizes will be virtually contracted because the light rays from them are refracted toward the viewer direction while they are passing through the VZFO to be viewed by the viewer. The contraction can be more when the plate thickness and/or the refractive index of VZFO are higher. However, the viewing angle at a given viewing distance is the most decisive factor of determining the amount of the contraction. The contraction will be more for the higher viewing angles. The viewing distance and panel size affect also the contraction because the viewing angle is determined by them. However they work oppositely, i.e., the contraction will be smaller as the viewing distance increase but larger as the panel size increases.

The color moiré fringes change their colors and periods as the viewing angle and distance changes. Added on the period and color changes, their phases also change as viewers change their viewing positions. These features provide more options in counterfeit security than the hologram which depends largely on the color changes with viewers’ viewing direction changes.