## Abstract

3D displays have been developed to provide users with a realistic 3D experience. Although various studies have endeavored to establish design principles for 3D displays, a generalized optimized model does not exist in the literature thus far. These circumstances have led to the manufacture of independently qualified 3D products, but expanding these applications remains a challenge. In this paper, we suggest a measurement model and an optimization method for optimized 3D display design. The proposed optimization can be applied to rotatable 3D displays and various pixel structures. Our experimental results based on manufactured displays and simulations confirm the proposed theory of optimization model.

© 2017 Optical Society of America

## 1. Introduction

3D displays have been studied and developed for more than decades in research and industrial fields. Consequently, various 3D materials have been developed for different applications. Moreover, 3D optics and 3D quality characteristics are well defined to utilize various 3D materials. In the consumer electronics market, consumers can enjoy a 3D experience through 3D displays.

To manufacture realistic 3D displays, light sources and 3D optics have been developed for the path control of light. A high resolution liquid crystal display (LCD) panel and multi-projection array [1] are used as light sources to realize 3D images with high spatial and angular resolution. Full parallax 3D images have also been achieved using micro-lens [2]. To generate horizontal parallax with improved 3D resolution, the design of parallax barriers [3–5] and lenticular lenses [6–8] has been studied. 3D diffraction gratings have enabled wide angle 3D displays [9], and tensor displays can realize a high resolution [10].

Light ray designs have been studied to enhance the image quality of 3D displays. Alignment designs of 3D optics and the panel have been improved the 3D quality [11, 12]. Designs that yield reduced 3D crosstalk have been introduced in [13–15]. The uniform design [16, 17] can generate realistic 3D images. To reduce optical artifacts induced by the panel and 3D optics, slanted designs for avoiding moiré artifacts have been studied [18–21]. These previous studies have laid down the principles of individual 3D displays. However, they do not provide an optimized measurement model for global design.

3D display characteristics vary according to the intended use. For example, 3D displays for 3D digital signage and personal mobile devices should be different. Moreover, 3D characteristics have trade-off relationships. Increasing the 3D resolution decreases the angular resolution and depth expression. Increasing the viewing angle also decreased depth expression. A generalized model is required for modeling the various use cases of 3D displays. In the generalized model, the 3D characteristics, 3D resolution, viewing angle, and 3D image quality should be described according to different 3D design parameters.

For rotatable 3D displays that can operate in both the portrait and landscape mode, the requirements for 3D quality must be satisfied for both the portrait and landscape mode. Rotatable 3D displays provide reasonable 3D quality with a liquid-crystal lens array [22, 23], square lens [24], crosstalk reduced pentile display [25], and moiré free design [26]. Nanoveu Pte Ltd manufactures rotatable 3D displays with good quality [27]. These studies and products have laid down the principles of rotatable 3D displays; however, they do not provide the principles for rotatable 3D display design with different 3D design parameters.

In this paper, we introduce a measurement model and an optimization method for overall 3D design, which can be applied to rotatable 3D displays and various pixel structures. We propose 3D display design optimization by using the 3D quality measurement model. Further, we define an energy model for 3D light rays according to the 3D display design parameters, which yields light luminance uniformity, red-green-blue pixel color off balance, and 3D moiré artifacts from panel pixels and 3D optical components. We also suggest an *l* − *α* map to design 3D displays that can operate in both the landscape and portrait mode, where *l* and *α* denote the lens pitch and slant angle of 3D optics, respectively.

## 2. 3D quality energy model

The reproduction of 3D light rays using a flat panel and 3D optics has several limitations. First, the number of light rays is much less than that in the real world. The light rays should be as uniformly distributed as possible in the spatial and angular domain. Second, 3D light rays are composed of primary colors: red, green, and blue. To generate natural color in 3D space, light rays of primary colors should be uniformly balanced. Finally, additional frequencies induced by pixels and 3D optics should be avoided. Figure 1 shows the artifacts caused by the lack of luminance uniformity, lack of color uniformity, and moiré artifacts. The 3D displays of Fig. 1, which contain artifacts, are experimental results of this study.

To measure the above characteristics, we define a 3D quality energy model with a luminance uniformity term *E _{L}*, color uniformity term

*E*, and moiré term

_{C}*E*. The overall 3D quality energy can be calculated by

_{M}*λ*

_{1},

*λ*

_{2}, and

*λ*

_{3}are weight parameters.

We introduce a light field representation and negentropy, which are used to measure luminance uniformity (*E _{L}*) and color uniformity (

*E*).

_{C}- Light field representation
A 4D light field representation is a map that can be defined as

By using the above light field representation, a light ray that passes through two parallel planes, Ω and Π, can be represented by four parameters (*x*,*y*,*s*,*t*) [28,29], as shown in Fig. 2(a). In the 3D display case, the*x*axis is located on the 3D display, and the*s*axis is the user position plane. For horizontal parallax 3D displays,*y*and*t*can be ignored. In Fig. 2(b),*p*(*x*) is the location of a pixel on the*x*axis, and its*s*value can be calculated using the following equation: where*f*is principal point length for the lenticular lens and effective thickness gap for the parallax barrier,*D*is the viewing distance, and*u*is the effective focal point of*x*[30]. The light ray of Fig. 2(b) is plotted by a line, since paraxial approximation is used. In the light field representation spatial and angular information are represented spontaneously since the x axis represents spatial information and the s axis represents angular information. Moreover, interpretation is simple since a lay represents as a dot in the light field representation. - Negentropy
Negentropy (

where*J*) is the reverse of entropy and is defined in [31] as*S*is entropy and*S*_{max}is the maximum value of the entropy. In the uniformity sense, a low negentropy indicates a uniform distribution.

#### 2.1. Luminance uniformity term

Sampled pixels of the panel are projected onto a certain view point through 3D optical components. When the sampled light rays are uniformly distributed, realistic 3D images are obtained. However, non-uniform sampled light rays yield structured artifacts. As shown in Fig. 1(a), light rays are not uniformly distributed; rather, they appear to be agglomerated and structured.

Light rays from one horizontal line have a uniform distribution since 3D optics such as the lenticular lens and parallax barriers are periodic. However, the uniformity of the light rays from several lines depends on the lens pitch and the slant angle of the design. Figure 3 shows light rays from a 3D display; the *y* value indicates the y*th* line on the panel and *V* indicates the viewing area on the *x* axis. Figure 3(a) and (d), and Fig. 3(b) and (e) show light rays from a horizontal line when *y* = 1 and *y* = 2, respectively. On comparing Figs. 3(a)–3(b) with Figs. 3(d)–3(e), it is difficult to distinguish whether they are uniformly distributed or not.

Figure 3(c) and (f) show light rays both of *y* = 1 and *y* = 2 in the light field representation. The uniformity of luminance can be easily measured in the light field representation domain since light rays are represented by dots. If we compare Fig. 3(c) and (f), we can see that the light rays in Fig. 3(f) are more uniformly distributed than those in Fig. 3(c).

To measure luminance uniformity (*E _{L}*) in the light field representation space, negentropy is used as a measure of distance to uniformity. The negentropy of luminance uniformity means that, if light rays are distributed with the same numbers in the blocks, then

*E*becomes 0. The large value of

_{L}*E*indicates that the distribution of light rays are far from a uniform distribution. The luminance values of (

_{L}*x*,

*s*) in the light field representation can be calculated using Eq. (3).

The negentropy of luminance uniformity can be defined using the concept of Eq. (4).

where the first term of Eq. (5) represents*S*

_{max}since the maximum entropy is attained when all the outcomes are equiprobable.

Here, *N _{b}* is the number of blocks,

*N*is the total number of light rays, and

_{L}*n*is the number of light rays in the

_{i}*ith*block. In Fig. 3, the total number of light rays

*N*is 120, and

_{L}*N*is 36 since the

_{b}*x*and

*s*axes are divided into six blocks. The value of

*E*for Fig. 3(c) is ten times that for Fig. 3(f).

_{L}#### 2.2. Color uniformity term

Light rays originating from a panel comprise red, green, and blue light rays, unlike real light. The color representation of a 3D display depends on the design of 3D optics. Without the uniform distribution of red, green, and blue light rays to certain viewpoints, unwanted red, green, and blue color artifacts are induced, as shown in Fig. 1(b).

Figure 4 illustrates red, green, and blue light rays in the space and light field representation. Similar to luminance uniformity, color uniformity is not represented in the light ray plot, but it is clear in the light field representation.

Similar to the luminance uniformity term, color uniformity can be calculated in the light field representation on the basis of negentropy as follows:

*N*is the number of blocks, and ${N}_{L}^{R}$, ${N}_{L}^{G}$, and ${N}_{L}^{B}$ are the total number of red, green, and blue light rays, respectively. ${n}_{i}^{R}$, ${n}_{i}^{G}$, and ${n}_{i}^{B}$ indicate the number of red, green, and blue light rays in the

_{b}*ith*block, respectively.

#### 2.3. Moiré term

3D moiré artifacts occurs when the frequencies of pixels and slanted lens generate additional visible frequencies. 3D design containing visible additional frequencies leads to the deterioration of 3D quality, as shown in Fig. 1(c). Moiré artifacts are modeled by a simplified spectral model based on Fourier analysis [32]. First, we define the model function of a 3D optical component (*o*(*x*, *y*)) and a pixel structure (*p*(*x*, *y*)). Then, a spectral model of the 3D optical components (*O*(*u*, *v*)) and the pixel structure (*P*(*u*, *v*)) is constructed by Fourier analysis. Finally, a moiré model (*M*(*u*, *v*)) is derived by the superposition of 3D optics and pixel structure.

To construct a spectral model, a lenticular lens, which is a 3D optical component with slant angle *θ* and lens pitch *L* can be approximated using a sinusoid function as

*f*is $\frac{L}{\text{cos}{\theta}^{\prime}}$ and

_{L}*θ′*is (90° −

*θ*). Its frequency response,

*O*(

*u*,

*v*), is given by

The slit barrier or parallax barrier, which is a 3D optical component having a barrier aperture size of *τ _{L}* cos

*θ*, can be approximated using a periodic rectangular function as

*represents a periodic function with period $\frac{L}{\text{cos}{\theta}^{\prime}}$ and pulse width $\frac{{\tau}_{L}}{\text{cos}{\theta}^{\prime}}$. Its frequency response is expressed as follows:*

_{o}The pixel structure of the display panel with horizontal pixel pitch *T _{H}*, vertical pixel pitch

*T*, horizontal pixel pitch without black matrix

_{V}*τ*, and vertical pixel pitch without black matrix

_{H}*τ*can be modeled by a periodic rectangular function as

_{V}3D moiré can be modeled by

Among the frequencies induced by 3D optics and panel pixels, viewers can only conceive visible frequencies. Therefore, we define the cost for moiré with the visible frequency criterion as

where*λ*(

*u*,

*v*) represents the optical modulation transfer function parameters of the human eye.

Figure 5 demonstrates the modeling of 3D optical components and panel pixels as well as their corresponding frequency response. The lenticular lens is modeled by a sinusoidal function and the pixel structure is modeled by a periodic rectangular function. The superposition of the optical component and pixel structure is analyzed by convolution in the frequency space. Owing to the superposition of 3D components and panel pixels, there exist additional frequencies (blue dots). The additional frequencies that can be perceived by viewers are in the visible frequency range, and are indicated by red dots.

## 3. 3D design optimization

To obtain an optimized design using a general model, the viewing angle and rays under a lens, which determine the 3D resolution [33], are important in addition to the 3D characteristics described in Sec. 2. The 3D characteristics can be modeled by two variables, lens pitch (*L*) and slanted angle (*θ*). Gap can be a constant value since it is usually fixed for a material. To design rotatable 3D displays, we convert the above parameters into *l* and *α*, where *l* is the number of pixels in the lens pitch *L* and *α* is tan *θ*.

If the slanted lens is designed for the landscape mode having lens pitch *L* and slant angle *θ*, then the lens pitch for the portrait mode can be expressed as

The viewing angle for the landscape mode *V _{L}* and portrait mode

*V*are

_{P}*g*is the gap between a panel and a 3D optical component. In the

*l*−

*α*map, as shown in Fig. 6,

*V*and

_{L}*V*are expressed as a vertical line and slant line, respectively. To design a 3D display with

_{P}If the 3D quality energy model is represented using the *l* − *α* map, Eq. (1) becomes

*λ*

_{1}and

*λ*

_{2}represent normalized parameters, such as $\frac{\text{max}({E}_{L})}{\text{max}({E}_{C})}$ and $\frac{\text{max}({E}_{L})}{\text{max}({E}_{M})}$, respectively.

In the *l* − *α* map *p̄* is a parameter that describes the pixel structure. Figure 7 illustrates an *l* − *α* map, which is calculated using Eq. (5) for luminance uniformity, Eq. (8) for color uniformity, and Eq. (19) for moiré.

## 4. Experimental Results

To validate the 3D model optimization, we manufactured four 3D displays and conducted simulations. We also included previously manufactured 3D displays in our *l* − *α* map. 3D displays of EyeFly3D [27] and Berkel [6] are indicated in Fig. 8. The figure shows that they have good luminance uniformity, color uniformity, and moiré characteristics.

Our desired viewing angles for landscape and portrait modes are

Figure 8 shows the valid region, which satisfies the criterion in Eq. (25). The colored value of Fig. 8 represents the energy value obtained using Eq. (24), which includes luminance uniformity, color uniformity, and moiré. *λ*_{1} and *λ*_{2} values of Eq. (24) are 4.47 and 2783, respectively. In the valid region, which is the trapezoid region, we selected four design parameters based on the 3D quality energy value, which were used for three displays with good 3D quality and one display with bad 3D quality. We manufactured four 3D displays with a 31.5 inch 4K (3,840×2,160) LCD panel and parallax barrier. The 3D parameters for the four displays are listed in the Table 1.

Figure 9 and 10 represent luminance uniformity and color uniformity in the light field representation, repectively. The number of light rays in a block in Fig. 9 is similar for all four displays but the number of red, green, and blue light rays in a block is not similar in the display with design 4, as shown in Fig. 10. Designs 1, 2, and 3 have color balanced characteristics; however, the color of design 4 is not balanced.

To verify moiré artifacts, we conducted image based simulation. Figure 11 demonstrates the simulation results for moiré artifacts. For this simulation, we generated an image superposing by the pixels and parallax barrier (top). Then, we performed a Fourier transform to obtain the frequency response. The circle indicates the visible frequency range and is located at a low frequency range since we cannot perceive high frequency components. The superposition of pixels and parallax barriers induces additional frequencies; however, the user is aware of additional frequencies in the visible frequency range only. As shown in Fig 11, we can verify that design 1, design 2, and design 3 are free of moiré artifacts; however, design 4 has moiré artifacts.

Figure 12 shows the images of the four manufactured displays. The images were obtained using a camera. 3D images obtained using designs 1, 2, and 3 have good 3D quality without luminance uniformity, color uniformity, and moiré artifacts, as verified by the 3D quality energy model. However, the 3D image obtained using design 4 has strong color uniformity and moiré artifacts.

Figure 13 shows the light profile of design 1. The crosstalk is 61.4% and 18.3% for the landscape and portrait mode, respectively. The full-width at half-maximum for the landscape and portrait mode is 2.21° and 3.49°, respectively.

We conducted simulations on 3D quality to verify the *l*−*α* map modeling. For the simulations, the full-width at half-maximum (*W _{H}*) of each light ray is assumed to be

*V*is the viewing angle.

Figure 14 shows simulation results for various *l* and *α* values. The simulation results confirm the luminance uniformity and color uniformity measurable model. Figure 15 shows simulation results for various *l* and *α* values.

## 5. Conclusions

We constructed a measurement model for the design optimization of rotatable 3D displays. We defined a 3D quality energy model with luminance uniformity, color uniformity, and moiré terms. Luminance uniformity and color uniformity were measured in the light field representation domain with the negentropy value. 3D moiré was measured using a spectral model based on Fourier analysis. Then, we ascertained the 3D moiré artifacts by using image based simulation. We also plotted luminance uniformity, color uniformity, and moiré energy on the *l* − *α* map. The viewing angles and 3D resolutions for landscape and portrait modes were represented on the *l* − *α* map. By using the *l* − *α* map, the optimized design for a rotatable 3D display could be determined.

In this paper, we did not consider the characteristics of 3D optics such as aberrations of the lenticular lens and aperture size of 3D optical films. Moreover, we did not analyze 3D qualities that have trade-off relationships among each other, such as crosstalk, 3D spatial resolution, and depth expression. Increasing the *l* value increases the angular resolution and crosstalk but decreases the 3D spatial resolution. In future research work, we will consider the characteristics of 3D optics and construct a model to represent these mutually dependent 3D quality characteristics on the *l*−*α* map. Moreover, we will consider optimum aperture size of barriers on the *l*−*α* map to reduce crosstalk.

## Acknowledgment

The first author wishes to thank Hyun Sung Chang and Byongmin Kang for useful discussions and comments.

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