## Abstract

In this paper, we propose a tilted elemental image array generation method for computer generated integral imaging display with reduced moiré patterns. The pixels of the tilted elemental image array are divided into border pixels and effective pixels. According to the optimal tilted angle, the effective pixels are arranged with uniform arrangement. Also, a pixel mapping method is proposed. Appropriate experiments are carried out and the experimental results show that not only the color moiré patterns are reduced remarkably, but also the resolution of the reconstructed 3D images are improved through the proposed method.

©2013 Optical Society of America

## 1. Introduction

Three-dimensional (3D) integral imaging, which is one of the most attractive techniques, is distinguished from other 3D display techniques. It can produce full-parallax, full-color, auto-stereoscopic 3D images without special viewing devices and it does not require any coherent light sources [1–6]. Basically, integral imaging is composed of two processes: pick-up and display/reconstruction. Since integral imaging has difficulty in capturing the elemental images (EIs) of 3D objects, computer-generated integral imaging (CGII) has been used in many research fields [7–9]. The conventional integral imaging display sets a micro-lens array (MLA) in front of a display device, such as liquid crystal display (LCD), plasma display panel, and other color flat-panel displays (FPDs). These display devices have fine pixel structure, flat surface, and high geometric positional accuracy, and they can offer bright and high resolution two-dimensional (2D) images for integral imaging display. Typically LCD is most widely used in integral imaging display [10, 11]. LCD usually has a high periodicity of pixels combined of red (R), green (G), and blue (B) sub-pixels, which is the cause of visible color moiré pattern that often appears in 3D displays. The color moiré pattern degrades the quality of reconstructed 3D images seriously.

Analyses about the moiré pattern have been done and many methods have been provided to reduce the moiré patterns of 3D displays by previous researches [12–15]. For integral imaging, a fundamental solution is using two types of optical low-pass filters: diffuser and defocusing [16]. However, the defocusing degrades the resolution of 3D images. Another effective and convenient method is tilting the MLA with a certain angle [17, 18]. The color moiré pattern is reduced and no extra components are added by this method. Many researches focus on optimal moiré-reduced angle by performing calculations or simulations for moiré analysis. Among them, visualizing the color moiré pattern in detail and analyzing the patterns using spatial Fourier transform, is a convenient method [19]. The obtained optimal moiré-reduced angle is an integer like 17°.

Several methods have been proposed for CGII, such as point retracing rendering [20], multiple viewpoint rendering [21], parallel group rendering [22, 23], viewpoint vector rendering [24–26]. All of these methods generate elemental image arrays (EIAs) for conventional integral imaging display. Compared with the conventional integral imaging display, a tilted MLA requires a corresponding tilted EIA displaying on the LCD. Therefore, these methods can’t be used directly for generating tilted EIA.

A method to generate a tilted EI is modifying the CGII algorithm and repeating the process to obtain a set of tilted EIs [18]. Multiple viewpoint rendering is used to render the perspective images, and the EIA for a slanted fly’s eye lens is generated after modified synthesis stage. However, this method ignores the pixel arrangement of tilted EI and the full-color information of a pixel is substituted by sub-pixel information. In the ideal case, a set of tilted EIs are calculated by an algorithm similar to the conventional CGII. However, the tilted EIs may have different pixel arrangements for different positions in the tilted EIA. In this case, the method needs a large number of calculations to generate different tilted EIs. Another simple approach to obtain tilted EIA is directly rotating the conventional EIA by image rotation. Normally, image rotation is composed of two basic steps: rotation and interpolation. The rotation operator performs a geometric transform which maps source pixels in input image onto target pixels in output image by rotating the input image with a user-specified angle. The rotation operator may produce holes or missing pixels, and the solution is using interpolation [27, 28]. However, interpolation causes some problems such as the interpolation error and computational load [29]. In addition, the resolution of reconstructed 3D images would be degraded by interpolation error.

In this paper, we propose a tilted EIA generation method in CGII by simultaneous consideration of both moiré-reduced angle and pixel arrangement. The proposed method transforms the conventional EIA into the tilted EIA according to the optimal tilted angle. The tilted EIA corresponds to a rectangular-type MLA, which is most widely used in integral imaging to reconstruct 3D images. In order to improve the resolution of reconstructed 3D images, a pixel mapping method without interpolation is proposed. Even though the proposed pixel mapping may introduce distortions in the 3D image, it doesn’t need any additional re-rending approach [30]. Experimental verification of the usefulness of the proposed method is also provided.

## 2. Principle of the proposed method

#### 2.1 Pixel classification in tilted EIA

In the conventional integral imaging display, the micro-lens’s pitch is assumed to be integer multiple of LCD’s pixel size, and pixels of the EIA have a one to one correspondence with the LCD pixels as shown in Fig. 1(a). However, when the micro-lens is rotated, the micro-lens edge intersects some LCD pixels as shown in Fig. 1(b). We call them border pixels. The border pixels are common to two or more adjacent micro-lenses. Since a RGB pixel only has full-color information of a ray, the border pixels cannot provide two EIs’ information. In this case the border pixels cannot provide correct information to the viewers. We define the LCD pixels that are totally covered by a micro-lens as effective pixels. Therefore, the LCD pixels are classified as border pixels and effective pixels. Similarly, the tilted EIA should be composed of the border pixels and the effective pixels for the one to one correspondence between the EIA pixels and the LCD pixels.

#### 2.2 Generation method of tilted EIA

As we know, the EIs of the conventional EIA are arranged periodically and have a uniform pixel arrangement. However, for tilted EIs, when the pixel size and micro-lens’s pitch are fixed, the number and arrangement of the effective pixels vary with the tilted angle. Figure 2 shows the effective pixel arrangements of four adjacent EIs with different tilted angles. Figure 2(a) shows that each EI with a tilted angle *φ* has a different effective pixel arrangement. Figure 2(b) shows that each EI with a tilted angle *θ* has a uniform effective pixel arrangement. We prefer to adopt the tilted angle *θ* because the titled EIs with the uniform arrangement can be calculated by the same algorithm. In other words, the tilted EIs can be generated more easily by choosing a more suitable tilted angle.

Taking the LCD’s pixel size *p* and the micro-lens pitch *d* into consideration, we combine the color moiré pattern reducing and the effective pixel arrangement uniformity to find out an optimal tilted angle. Assume that *n* and *m* represent the pixel numbers within two vertices of the micro-lens in the horizontal direction and vertical direction respectively as shown in Fig. 3. Both *n* and *m* are integers that range from 1 to *d*/*p*. The distance of the two LCD pixels which are under the two vertices *r _{nm}* can be expressed as:

*f*as:

_{nm}*θ*,

_{1}*θ*) is the angular range of the moiré pattern reducing. The optimal tilted angle

_{2}*θ*can be calculated by the following relationships:

*Findmin*function gives

*A*(

*n′*,

*m′*) the subscripts of the minimum non-negative entry of a matrix. When the values of

*n′*and

*m′*are given,

*θ*can be determined and then the effective pixel arrangement of a tilted EI is also confirmed.

Figure 4 briefly shows a rectangular coordinate of the proposed pixel mapping method. Red boxes *S _{i,j}* and blue boxes

*G*represent source pixels and target pixels respectively.

_{h,v}*i*,

*j*,

*h*and

*v*are equal to 1, 2, 3, ... . The centers of

*S*

_{1,1}and

*G*

_{1,1}are both on the origin of the coordinate. The input image is rotated by

*θ*around the origin

*o*. We assume the pixel size is unit length. Therefore, the coordinate value of

*G*’s center is (

_{h,v}*h*-1,

*v*-1). Since the direction of rotation is clockwise, transformation formula is given as:

According to Eq. (5), the four vertices *A*(*x*_{1},*y*_{1}), *B*(*x*_{2},*y*_{2}), *C*(*x*_{3},*y*_{3}) and *D*(*x*_{4},*y*_{4}) of *S _{i,j}* are calculated as:

*G*is inside of

_{h,v}*S*, we set up one to one mapping correspondence between

_{i,j}*G*and

_{h,v}*S*. The relationship between

_{i,j}*G*and

_{h,v}*S*is represented as

_{i,j}*G*=

_{h,v}*S*. We use the method of linear programming and substitute the coordinate value of

_{i,j}*G*’s center,

_{h,v}*A*(

*x*

_{1},

*y*

_{1}),

*B*(

*x*

_{2},

*y*

_{2}),

*C*(

*x*

_{3},

*y*

_{3}), and

*D*(

*x*

_{4},

*y*

_{4}), then the sufficient condition of

*G*=

_{h,v}*S*is given by

_{i,j}In the proposed pixel mapping method, the target pixels are effective pixels of the titled EIA. We find out a target pixel according to the effective pixel arrangement, and then look for its source pixel in the conventional EIs. That is the main difference between the general image rotation and our pixel mapping method.

## 3. Experimental results and discussion

To show the usefulness of the proposed method, we have implemented experiments to compare the reconstructed images’ qualities using the different tilted EIA generation methods.

In the experiments, we obtain the conventional EI by 3DS MAX. 3DS MAX, which is a 3D modeling software, has been usually used to form and capture a virtual 3D scene in integral imaging [31, 32]. It can provide a virtual camera array to simulate a MLA for capturing 3D objects as shown in Fig. 5(a). In our experiments, the virtual camera array is composed of 157 × 118 cameras. The camera’s focal length is 97.356 mm and the field of view is 20.95°. Each EI rendered by the corresponding camera has 13 × 13 pixels. The 3D model is two “footballs” as shown in Fig. 5(b). “T”, “L”, “F”, and “P” represent top view, left view, front view and perspective user view, respectively.

In our experiments, we use an iPad to display the EIAs. A rectangular-type MLA has been embedded in the iPad to reconstruct the 3D images. A CCD camera is placed in front of the iPad to capture the reconstructed 3D images as shown in Fig. 6. The parameters of our experimental set-up are given in Table 1. We get *d* = 1.25mm and *p* = 0.096mm from Table 1. The moiré-reduced angle range, which is measured by software testing and naked-eye observation, is (15°, 25°). According to Eqs. (1)-(4) and the value range, we obtain *n′* = 12, *m′* = 5, *θ* = 22.62°. And then the effective pixels’ arrangement of a tilted EI is also confirmed.

Table 2 shows the characteristics of different EIs. The notation of ‘-’ signifies that there is no border pixel. The tilted EI generated by general image rotation has no border pixels, because it doesn’t classify the pixels. The EIs that locate at the edge of the EIA are not included in the total number of EI.

The computer-generated EIAs and reconstructed 3D images of different generation methods are shown in Fig. 7. We can make the following conclusions by observing and comparing Figs. 7(a), (b), and (c). The reconstructed 3D image of the conventional CGII has severe color distortion as shown in Fig. 7(a). Figures 7 (b) and (c) show that the color moiré patterns are reduced remarkably with 22.62° rotation angle. In Fig. 7(b), the reconstructed blue-and-yellow “football” is discrete and the edge of the red-and-white “football” is blurred. However, in our method, the reconstructed 3D images have clear edge and have no distortion as shown in Fig. 7(c). The reconstructed view images from different directions by our method are shown in Fig. 8.

## 4. Conclusion

A tilted EIA generation method in CGII has been proposed. We divide the tilted EIA into border pixels and effective pixels. The uniform effective pixel arrangement makes the generation of the tilted EIs easier. A new pixel mapping method is proposed to improve the quality of the reconstructed 3D images. Experiment results show the reconstructed 3D images using the proposed method have a higher resolution and less distortion than the general image rotation method.

## Acknowledgments

This work is supported by the “973” Program under Grant No. 2013CB328802, the NSFC under Grant Nos. 61225022 and 61036008, and the “863” Program under Grant Nos. 2012AA011901 and 2012AA03A301.

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