Aberration analyses for improving the frontal projection three-dimensional display

Xin Gao; Xinzhu Sang; Xunbo Yu; Peng Wang; Xuemei Cao; Lei Sun; Binbin Yan; Jinhui Yuan; Kuiru Wang; Chongxiu Yu; Wenhua Dou

doi:10.1364/OE.22.023496

1. Introduction

Three-dimensional (3D) displays can provide more authentic scenes with more information to viewers than traditional two-dimensional displays. In recent years, a considerable amount of efforts have been dedicated to achieving 3D visual perception with different methods [1], such as binocular stereo displays, auto-stereoscopic displays, volumetric 3D displays and holographic displays. The binocular stereo displays with glasses or helmet are uncomfortable for viewers. The holographic displays can reproduce the real light field with complex equipment [2]. In addition, current holography is still hard to realize real time, large format, full-color 3D display for practical application. Auto-stereoscopic displays based on the liquid crystal panel [3,4] create the 3D images, which can provide the viewers with multiple different parallax images [3]. However, the resolution of each parallax image is greatly reduced. The 3D display based on the projector array can provide 3D images with high resolution [5–10].

The crosstalk severely affects the viewing experience including unclear 3D images, ghosts and double contours. Several factors causes the crosstalk in the auto-stereoscopic display, such as view geometry, optical properties of the optical element, sub-pixel layout of the display, the parallax of adjacent parallax images, the contrast and luminance of images, aberration, and so on [11–13]. Several methods were used to reduce crosstalk, including special optical elements, the adjustment of sub-pixel location, the image processing about image content, and so on. The V-shaped barriers are reported to reduce the crosstalk [14], it can reduce the amount of crosstalk almost by half. However, only the situation of two viewpoints is analyzed and this method still suffers from the low resolution and illumination loss. The positional relationship between sub-pixels and the lenticular sheet is analyzed [15]. By correcting the sub-pixel position, the crosstalk can be reduced. In addition, the methods of image processing are applied to deal with it [12,16]. However, the above methods are not always effective in all situations. In the frontal projection three-dimensional display, noticeable crosstalk still exists in marginal viewing zones(MVZs) after elaborately promoting projector array arrangement and image processing.

In order to solve the issue that 3D images with unclear experiences and ghosts are observed in MVZs, the Seidel aberration theory is used to analyze the relationship between aberration and the parameters of lenticular sheet. Theoretical and experimental results show that increasing radius of curvature (ROC) or decreasing aperture of the lenticular sheet can suppress the aberration and reduce the crosstalk. The 3D image with high quality are demonstrated in both mid-viewing zone(DVZ) and MVZs on the optimal viewing plane. However, the field of view (FOV) is decreased. To reduce aberration without decreasing FOV, a meniscus lens structure of lenticular sheet is designed from the theoretical analysis. Results of the modulation transfer function (MTF) and luminance distribution show that the crosstalk is well suppressed without decreasing the FOV in the MVZ.

2. Aberration analysis of the frontal 3D projection system

2.1 The distribution light-field of the viewing plane

In the 3D display system based on the frontal projection lenticular sheet, the arrangement of projector array is based on the light-field distribution in the viewing plane. Figure 1 illustrates the light-field distribution with a single projector in the viewing plane. The diffuse screen is placed in the focal plane of lenticular sheet.

Fig. 1 The light-field distribution with a single projector (a) The light path of single projector (b) The distribution of light-field with a single projector.

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As shown in Figs. 1(a), the light from the projector is focused on the front surface of the diffuse screen with the lenticular sheet, and then it is diffused to pass through the lenticular sheet in the second time. The emergent light is refracted to different directions and formed periodical viewpoints containing the same content. T represents the width of the viewing zones, and W₁, W₂ are widths of the same viewpoint in DVZ and the adjacent viewing zones respectively. The lens cell of the traditional lenticular sheet is cylindrical lens. Thin vertical light stripes represent the same view point in different viewing zones, which are distributed in the viewing plane periodically, as shown in the Figs. 1(b). In the same way, other projectors in the projector array also form corresponding periodical vertical stripes representing different viewpoints. Every viewing zone consists of these different vertical stripes. Different parallax images projected to the lenticular sheet are encoded to form a synthetic 3D image.

2.2. Aberration analysis of the lenticular sheet

The geometrical aberration is evident in the 3D display system based on the frontal projection lenticular sheet, which is an important factor for crosstalk. With increasing the number of viewpoints, crosstalk becomes more seriously. Decrease of aberration is a particular concern. For the ideal cylindrical lens, the parallel light from different angles is focused on different positions on focal plane as shown in Figs. 2(a),where $θ$ is the incident angle of parallel light, and F’, f’ represent the focal point and focal length respectively. H and H’ indicate a pair of principal plane. However, for the actual cylindrical lens, the parallel light of different angles cannot accurately focus on a plane surface due to the influence of aberration, as shown in Figs. 2(b). With the increasing the incident angle, the focal point is further deviated from the focal plane.

Fig. 2 The focal situation of the ideal and actual cylindrical lens (a) The ideal cylindrical lens (b) The actual cylindrical lens.

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In general, the aperture of lens cell in lenticular sheet is a few millimeters and the distance between projector array and the lenticular sheet is a few meters. For every lens cell, the incident light from the projector is originated from different angles. According to the principle of the reversible optical path, the process of the diffused light pass through the lenticular sheet the second time can be analyzed in the same way.

Based on above analysis, the width of vertical stripes in the viewing plane in different viewing zones should be equal for ideal lenticular sheet. Thus, W₁, W₂ should be equal as shown in Fig. 1. In fact, due to aberration increasing with the viewing angle, the lateral viewing zones are different in a large viewing angle and greater aberration exists in these viewing zones. The diffusion degree of vertical stripes in these viewing zones is seriously increased. As shown in the Fig. 3, W₂^’ is larger than W₁^’. W₁^’ and W₂^’represent the width of the vertical stripes which distributed in the viewing plane. T^’ is the width of viewing zones.

Fig. 3 The light path of single projector under the actual situation.

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In order to achieve the high quality 3D images, the crosstalk of system should be low enough [17]. Through the control of the distance of adjacent projector lens, the degree of crosstalk in DVZ can be decreased. However, the diffusion of the vertical stripes on viewing plane in MVZs is still seriously deteriorated and the crosstalk in MVZs becomes severe. To reduce the aberration increasing with increasing the viewing angle, Seidel aberration theory is used to analyze the cylindrical lens cell in the lenticular sheet. In general, the lens cell of traditional lenticular sheet in the auto-stereoscopic is single cylindrical lens. One surface of the lens cell is plane and the other is circular cross-sections. If the object light is assumed to locate in the meridian plane, the lens cell only focuses the light in the meridian direction. As one surface of cylindrical lens cell is a plane so that the focal length is independent on the thickness. Only the meridional vertical axis aberration is considered and the thickness of lens cell is ignored. In the case of incident light from infinity and the entrance pupil is located in the cylinder of every cylindrical lens cell, the following equations can be obtained,

\begin{array}{l} δ Y = - \frac{1}{2} \frac{1}{(n - 1) h ρ} [S_{1} + 3 S_{2} + 3 S_{3} + S_{4}] \\ S_{1} = \frac{(n - 1) [1 + (n + 1) {(n - 1)}^{3}]}{n^{2}} h^{4} ρ^{3} \\ S_{2} = \frac{n^{3} - 2 n^{2} + 1}{n} θ h^{3} ρ^{2} \\ S_{3} = (n - 1) θ^{2} h^{2} ρ \\ S_{4} = \frac{n - 1}{n} ρ h {}^{2}θ^{2} \end{array}

where

δ Y

is the meridional vertical axis aberration, which represents the diffusion degree on the Gaussian image plane. It is the distance between the point of meridional light on Gaussian image plane and the point of principal light on that plane. S₁~S₄ represent four Seidel coefficients and they are positive-negative.

θ

and n are the incident angle of parallel light and the refractive index of lens cell.

ρ

is the curvature and h represents the height of the marginal incident light from the optical axis.

δ Y

also indirectly reflects the diffusion degree of the vertical stripes on viewing plane. Cylindrical lens cell has only one active optical surface so that aberration can’t be eliminated but it can be reduced. According to Eq. (1), increasing ROC or reducing the aperture can reduce the meridional vertical axis aberration

δ Y

. The diffusion degree of vertical stripes on the viewing plane can also be reduced. The refractive index and thickness of lens are also the factors that related to the crosstalk and aberration. Due to the range of variation about refractive index is small and the refractive index of material manufacturing lenticular sheet is about 1.5. The refractive index is regard as the constant. After achieving the initial configuration, the thickness is chosen according to the aperture of lens. Thus, the two parameters are not regarded as variable parameters. Monte Carlo Non-Sequential Ray tracing arithmetic [18~20] is used to simulate the light-field distribution in the viewing plane. A projector is placed in the middle position of one meter away in front of the lenticular sheet. The parameters are shown in Table 1 and Table 2. P, r and n represent the aperture, ROC and refractive index respectively. L is the distance between the projector and the lenticular sheet.

Table 1. The change of the aperture

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Table 2. The change of ROC

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Figure 4 shows the distribution of light-field in the viewing plane with decreasing the aperture. Horizontal coordinate “0” represents the location of the projector in DVZ and the red circle mark represents the vertical stripes in MVZ, which deviates 400mm from the center. We can see that the width of vertical stripes in MVZ becomes narrower with decreasing aperture, which is close to the width of the vertical stripes in the DVZ. It means that the aberration is suppressed. For example, the width of vertical stripes in MVZ in Figs. 4(a) is greater than that in the DVZ, while the width of vertical stripes in MVZ in Figs. 4(d) is nearly the same with that in the DVZ.

Fig. 4 The light-field distribution with decreasing the aperture (a) P = 1.5mm (b) P = 1.0mm (c) P = 0.75mm (d) P = 0.55mm.

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Figure 5 shows that the width of vertical stripes in MVZ is reduced with increasing ROC. Based on the above analysis, increasing ROC or decreasing aperture can suppress aberration.

Fig. 5 The distribution light-field with increasing ROC (a) r = 4mm (b) r = 6mm (c) r = 8mm (d) r = 10mm.

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2.3. Experiments and discussions

Two lenticular sheets with different parameters are used in the experiment. A projector is placed in the middle position of three meters away in front of the lenticular sheet. The parameters are shown in Table 3. P, r, d represent the aperture, ROC and thickness of the lenticular sheet respectively。

Table 3. Parameters of the cylindrical lenticular sheet

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Figures 6(a) and 6(b) show the light-field distribution in the viewing plane of two lenticular sheets with different parameters. The distance of vertical stripes in DVZ and MVZ is 1350mm, and the viewing angle of vertical stripes in MVZ is $arc \tan \frac{1350}{3000} \approx 24^{°}$ . For Figs. 6(b), the width of vertical stripes in MVZ is close to the width of vertical stripes in DVZ and they are both narrow, which means that the influences of aberration are small. When the arrangement of projector array is based on the light-field distribution on viewing plane in DVZ, the overlapping part of adjacent viewpoints is little. It means there is low crosstalk between adjacent viewpoints in two viewing zones as shown in Fig. 7. 3D images with the high quality can be achieved in DVZ and MVZs at the same time. For Figs. 6(a), the diffusion degree of vertical stripes in MVZ is noticeable increased, and the width of vertical stripes in MVZ even reaches several times larger than that in DVZ, which results in the overlapping of adjacent viewpoints. A high level of crosstalk occurs in adjacent viewpoints in MVZ as shown in Figs. 8(b). Figure 7 and Fig. 8 are normalized intensity distribution which is measured by the illuminometer.

Fig. 6 The light-field distribution in the viewing plane (a) r = 5mm (b) r = 10mm.

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Fig. 7 The normalized luminance distribution in (a) DVZ and (b) MVZ with r = 10mm.

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Fig. 8 The normalized luminance distribution in (a) DVZ and in (b) MVZ with r = 5mm.

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According to the intensity measured from the adjacent overlapping part of the viewpoints in the viewing plane, the change of crosstalk degree can be obtained. Figure 9 shows the change of crosstalk degree along the horizontal coordinates. The range of horizontal coordinates is from 0 mm to 1350 mm. The crosstalk degree increase to 90% with r = 5mm, and the change of the crosstalk is below 10% with r = 10mm.

Fig. 9 The change of crosstalk degree along the horizontal coordinates.

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A projector array with 20 micro-projectors is used to project 20 parallax images on the two lenticular sheets. Figure 10 and Fig. 11 show the 3D images viewed in MVZ based on two lenticular sheets respectively. The 3D image in Fig. 10 includes blur and ghosts, where a high crosstalk degree results in a low resolution and a decreasing of 3D depth. The 3D image in Fig. 11 with high quality in DVZ and MVZ is achieved, which shows that the aberration is suppressed. The 3D clear depth of 1.2m can be perceived.

Fig. 10 The final 3D image with r = 5mm.

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Fig. 11 The final 3D image with r = 10mm.

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In Refs [6,9], the frontal projection three-dimensional displays were introduced. While the situation of crosstalk and the quality of 3D image are not discussed. The projection-type 3D display system equipping with lenticular-parallax barrier was used to improve the resolution and reduce the crosstalk [21], which minimized the width of pixel image projected by the projector. However, the situation of crosstalk and image quality in marginal viewing zones (MVZs) was not discussed and the system still suffered from the low brightness. In our system, the aberration theory is applied to analyze the crosstalk existing in MVZs. After improvement, the high quality 3D images in both DVZ and MVZs can be achieved with high resolution and high brightness.

3. Design of novel lenticular sheet to suppress aberration

Through the above analysis, increasing ROC or decreasing aperture can suppress the aberration and provide 3D images with high quality. However, the FOV is decreased. As we known, when our eyes move from one viewing zone to another, a jump is noticed [6], which brings discomfort. Increasing FOV can reduce the frequency of the jump. It is necessary to have an excellent stereo vision without decreasing FOV. There is only one surface of lens cell can be alternative in the traditional lenticular sheet, and aberration can’t be completely eliminated. To reduce the aberration, more variables should be added, such as dual-surface and aspheric surface. The aspheric considering the lens thickness, the formula can be obtained as follows,

\begin{array}{l} δ Y = - \frac{1}{2} [\sum S_{1} + 3 \sum S_{2} + 3 \sum S_{3} + \sum S_{4}] \\ δ T = - \frac{1}{2} \sum S_{1} \\ K_{T} = - \frac{3}{2} \sum S_{2} \\ X_{T} = - \frac{1}{2} (3 \sum S_{3} + \sum S_{4}) \end{array}

\begin{array}{l} \sum S_{1} = P_{1} + h_{2} P_{2} + K_{1} + h_{2}^{4} K_{2} \\ \sum S_{2} = h_{Z 2} P_{2} - (W_{1} + W_{2}) + h_{2}^{3} h_{Z 2} K_{2} \\ \sum S_{3} = \frac{h_{Z 2}^{2}}{h_{2}} P_{2} - 2 \frac{h_{Z 2}}{h_{2}} W_{2} + ϕ_{1} + ϕ_{2} + h_{2}^{2} h_{Z 2}^{2} K_{2} \\ \sum S_{4} = \frac{ϕ_{1} + ϕ_{2}}{n} \end{array}

\begin{array}{l} h_{Z 2} = \frac{d}{n} D, h_{2} = 1 - \frac{d D}{n} ϕ_{1}, ϕ_{1} = \frac{D_{1}}{D}, ϕ_{2} = \frac{D_{2}}{D} \\ D = (n - 1) (ρ_{1} - ρ_{2}) + \frac{{(n - 1)}^{2}}{n} d ρ_{1} ρ_{2} \\ P_{1} = \frac{ϕ_{1}^{3}}{n^{2} {(n - 1)}^{2}}, P_{2} = - \frac{ϕ_{1}^{3}}{n^{2} {(n - 1)}^{2}} + \frac{(n + 2) ϕ_{1}^{2}}{n {(n - 1)}^{2}} - \frac{(2 n + 1) ϕ_{1}}{{(n - 1)}^{2}} + \frac{n^{2}}{{(n - 1)}^{2}} \\ W_{1} = - \frac{ϕ_{1}^{2}}{n^{2} (n - 1)}, W_{2} = \frac{ϕ_{1}^{2}}{n^{2} (n - 1)} - \frac{(n + 1) ϕ_{1}}{n (n - 1)} + \frac{n}{n - 1} \\ K_{1} = \frac{k_{1} ϕ_{1}^{3}}{{(n - 1)}^{2}}, K_{2} = \frac{k_{2} ϕ_{2}^{3}}{{(n - 1)}^{2}} \end{array}

In the above equations,

δ T

,

K_{T}

and

X_{T}

represent meridian vertical axis spherical aberration, meridian coma and meridian field curvature respectively.

D_{1}

and

D_{2}

indicate the diopter of the first and second surface respectively.

K_{1}

and

K_{2}

represent the two aspherical coefficients of lens.

k_{1}

and

k_{2}

represent the cone coefficient of the first and second surface respectively.

ϕ_{1}

and

ϕ_{2}

are the normalized diopter of the first and second surface respectively.

ρ_{1}

and

ρ_{2}

represent the two surface curvature respectively and

D

is the diopter of lens.

h_{2}

and

h_{Z 2}

represent the height of incident light and principal light on the second lens surface respectively. d is the lens thickness and n is refractive index. A function of image evaluation MTF is applied. We can evaluate the optical crosstalk between adjacent viewpoints roughly. According to the actual viewing demand in our system, the viewing distance is 3000mm and the width of viewing zones T is 450mm. Corresponding aperture and the focal length of lenticular sheet are 1.5mm and 10mm respectively. The thickness of lenticular sheet is set to 1mm. In the following discussion, the viewing angle of vertical stripes in MVZ is 24 degrees. According to the Eq. (2), Eq. (3) and Eq. (4), the corresponding structure parameters of lenticular sheet can be achieved by eliminating each aberration in turn. While, From the Eq. (2),

δ Y

is the comprehensive result of the above three kinds of aberration, the diffusion degree can’t be reduced integrally by only eliminating one meridianal aberration. The overall aberration should be reduced, not just one aberration. A best balance should be achieved among aberration so that

δ Y

can be reduced. The aberration balancing algorithm [22] is applied. After the aberration is balanced, a satisfactory structure is achieved. The parameters are shown in Table 4 and the structure is shown in Fig. 12. The traditional cylindrical lens with P = 1.5mm, f = 10mm, d = 1mm is used to compared with the novel structure. With the novel meniscus lens structure, the meridional vertical axis aberration is well suppressed, but it has not been eliminated completely. For the ideal single cylindrical lens, although the aberration is zero, it cannot be realized.

Table 4. The designed parameters of lenticular sheet

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Fig. 12 The actual meniscus thick lens.

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The first and second surface of novel lens is the oblate cylinder. The meridional vertical axis aberration curve and MTF curve are shown in Fig. 13 and Fig. 14. We can see that the meridional vertical axis spherical aberration, meridian coma and meridian field curvature are all suppressed greatly. The MTF curve is higher than 0.9 in whole viewing angle, which means the aberration is suppressed in whole viewing angle and there is low crosstalk between adjacent viewpoints. The MTF curve of the novel structure is obviously superior to the traditional cylindrical lens. Designed lenticular sheet is consists of the novel meniscus thick lens. Compared with the traditional lenticular sheet, the Monte Carlo Non-Sequential Ray tracing arithmetic is used. The distribution of light-field is shown in Fig. 15. The corresponding normalized luminance distribution in MVZ and the change of crosstalk degree in whole viewing angle are shown in Fig. 16 and Fig. 17.

Fig. 13 The meridional vertical axis aberration curve (a) The traditional cylindrical lens (b) The actual meniscus thick lens.

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Fig. 14 MTF curve of novel structure.

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Fig. 15 The distribution of light-field (a) The traditional lenticular sheet (b) The novel lenticular sheet.

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Fig. 16 The normalized luminance distribution in MVZ (a) The traditional lenticular sheet (b) The novel lenticular sheet.

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Fig. 17 The change of crosstalk degree along the horizontal coordinates.

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In the calculation, the distance between lenticular sheet and projector is 1000 mm and the aperture, the focal length and thickness are 1.5mm, 10 mm and 1mm respectively. The refractive index is 1.5. Figures 15(a) and (b) show the light-field distribution for the traditional and novel lenticular sheet respectively. The width of vertical stripes in MVZ of Figs. 15(b) is almost equal to the width of vertical stripes in DVZ, and it is obviously narrower than the width of vertical stripes in MVZ of Figs. 15(a). Figure 16 and Fig. 17 show that the degree of crosstalk is serious in MVZ for the traditional lenticular sheet but slight for the novel lenticular sheet. With the novel lenticular sheet, the 3D images with high quality can be achieved without decreasing FOV.

There are two aspherical surfaces in the novel lens. In general, there are three ways to manufacture a lens with an aspherical surface including high-temperature molding of a glass, injection molding of optical plastics and low-temperature molding of a relatively thin coating. A method of making aspherical mold with high accuracy was introduced [23] and the shape accuracy of the mold is better than 0.2 μm. Besides, the method of mass fabrication could be realized [24]. It is based on a fast high-precision replication process using UV-curable coatings. The aspheric micro-lens array with sub-0.2- μm precision was reported [25]. The deviation of ROC, thickness, aperture or the cone coefficient may affect the optical performance of lenticular sheet. So, the tolerance of the novel lenticular sheet is considered in our system. Since the shape accuracy of the mold is better than 0.2 μm, the tolerance with 0.2 μm is investigated. The tolerance range is shown in Table 5.

Table 5. The tolerance range of novel lenticular sheet

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As shown in Fig. 18, for the deviation of cone coefficient, the MTF curves are more deteriorative than other parameters. However, they are higher than 0.85 in whole viewing angle. In order to further investigate the influence of tolerance, the light-field distribution in the situation of the deviation of cone coefficient is depicted in Fig. 19. Compared with the Figs. 15(b), although there is tolerance existed, the distribution of light-field is not noticeably changed. The 3D images with high quality can be achieved even if the tolerance existed.

Fig. 18 The MTF curve with tolerance (a) $Δ r_{1} = \pm 0.2 u m$ (b) $Δ r_{2} = \pm 0.2 u m$ (c) $Δ k_{1} = \pm 0.02$ (d) $Δ k_{2} = \pm 0.02$ (e) $Δ d = \pm 0.2 u m$ (f) $Δ P = \pm 0.2 u m$ .

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Fig. 19 The distribution of light-field (a) $Δ k_{1} = + 0.02$ (b) $Δ k_{1} = - 0.02$ (c) $Δ k_{2} = + 0.02$ (d) $Δ k_{2} = - 0.02$ .

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4. Conclusion

In summary, optimized aberration analysis and design is used to suppress the crosstalk in the frontal projection 3D display system. Theoretical and experimental results shows that increasing ROC or decreasing aperture of lenticular sheet can reduce the crosstalk and better 3D perception can be achieved in both DVZ and MVZs. A frontal projection 3D display system based on 20 micro-projectors and the lenticular sheet with the ROC of 10 mm and the size of 1.9 m × 1.2 m is demonstrated. The 3D image with the high quality is achieved in both the mid-viewing zone and MVZs in the optimal viewing plane. The 3D clear depth of 1.2m can be perceived. However, the FOV is usually decreased. To increase the FOV with low crosstalk, a novel structure of lenticular sheet consists of meniscus thick lens is presented, and the better 3D experiences can be expected.

Acknowledgments

This work is partly supported by the National Science Foundation of China (61177018), the “863” Program(2012AA011902),the Program for New Century Excellent Talents in University (NECT-11-0596), the Program of Beijing Science and Technology Plan (D121100004812001), and Beijing Nova Program (2011066).

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P(mm)	r(mm)	d(mm)	L(mm)	Size(m²)
1.5	5	8	3000	1.9 × 1.2
1.5	10	8	3000	1.9 × 1.2

P(mm)	r(mm)	d(mm)	L(mm)	Size(m²)
1.5	5	8	3000	1.9 × 1.2
1.5	10	8	3000	1.9 × 1.2

Aberration analyses for improving the frontal projection three-dimensional display

Abstract

1. Introduction

2. Aberration analysis of the frontal 3D projection system

2.1 The distribution light-field of the viewing plane

2.2. Aberration analysis of the lenticular sheet

2.3. Experiments and discussions

3. Design of novel lenticular sheet to suppress aberration

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (19)

Tables (5)

Equations (4)

Optics Express