Abstract

A scheme for the resolution-enhancement of a three-dimension/two-dimension convertible display based on integral imaging is proposed. The proposed method uses an additional lens array, located between the conventional lens array and a collimating lens. Using the additional lens array, the number of the point light sources is increased far beyond the number of the elemental lenses constituting the lens array, and, consequently, the resolution of the generated 3D image is enhanced. The principle of the proposed method is described and verified experimentally.

© 2005 Optical Society of America

1. Introduction

Integral imaging is a technique for displaying three-dimensional (3D) images using a lens array. The feature of the lens array that spatially samples and captures the light field from the objects is exploited in the integral imaging to capture and display 3D images [1]. Full color, full parallax real-time 3D images provided by the integral imaging make it attractive [27] and considerable efforts have been made to enhance its viewing parameters [816]. A novel integral imaging scheme with enhanced depth and 3D/two-dimension (2D) convertibility was recently reported [17]. In this approach, the lens array, which is located behind the transmission-type spatial light modulator (SLM) and a polymer-dispersed liquid crystal (PDLC) attached to the lens array, enhance the expressible depth range substantially and enable 3D/2D conversion. With these superior features, integral imaging acquires a much wider range of applications, and approaches the level of commercialization. This is because only 3D/2D convertible display techniques can infiltrate into commercial markets for 3D TV. Our previous work [17] using PDLC was the first demonstration of a 3D/2D convertible integral imaging technique (that has both horizontal and vertical parallaxes unlike the lenticular or parallax barrier technique). The remaining problem is resolution. In fact, depth enhancement and 3D/2D convertibility can be achieved, but at the expense of the resolution. Therefore it is necessary to develop a method that compensates for resolution degradation while retaining the useful features, i.e., depth enhancement and 3D/2D convertibility.

In this paper, we report on a resolution enhancement scheme for a depth-enhanced 3D/2D convertible display based on integral imaging. The proposed method utilizes an additional lens array to generate an excess of point light sources, thus permitting the resolution of the generated 3D images to be enhanced.

 

Fig. 1. Schematic diagram of 3D/2D convertible integral imaging (a) 2D mode (b) 3D mode

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2. 3D/2D convertible integral imaging and resolution limitation

Figure 1 shows the concept of the 3D/2D convertible integral imaging. The 3D/2D convertible integral imaging consists of a collimated incoherent light source, a PDLC, a lens array and an SLM. Conversion between the 3D and 2D modes is achieved by controlling the diffusing rate of the PDLC. In the 2D mode, the PDLC is set to be diffusive. The collimated light is scattered by the PDLC and this scattered field is then relayed to the SLM. Each pixel in the SLM is then illuminated effectively from all directions, and consequently, 2D images are observed on the SLM plane with full resolution and viewing angle. In the 3D mode, the PDLC is set to be transparent. In this case, the collimated light rays pass through the PDLC without scattering and are imaged into an array of point light sources at the focal plane of the lens array. The light rays from each point light source are modulated according to the direction of their propagation, and finally integrated into 3D images.

In the 3D mode, the resolution is limited by the number of point light sources. Figure 2 shows this point. As shown in Fig. 2, the light rays from each point light source are modulated by the SLM. Hence, a unit comprised of one elemental lens, a point light source, and the corresponding region on the SLM is effectively one pixel that emits light rays of different colors or intensities according to the observation directions and, as a result, acts as one pixel in the generated 3D images. Since the role of the elemental lens and the SLM is merely the formation and modulation of the point light sources, the key parameter for determining the resolution of the generated 3D images is the number of point light sources. That is, the resolution of the generated 3D images is the same as the number of point light sources. In order to enhance the resolution of generated 3D images, it is necessary to generate more point light sources with a smaller spacing between them. One straightforward method for achieving this end is to use a lens array with more elemental lenses and small elemental lens pitches. Such a lens array should reach the desired size of the display panel (e.g. 12-inch or 15-inch) with uniform elemental lenses whose pitch is comparable to the pixel pitch of conventional 2D displays (e.g. 200 um) and f-number is sufficiently small to preserve a reasonable viewing angle, which is difficult to realize at this time. This high requirement stimulates research for an alternate resolution compensation method that utilizes lens arrays that can be fabricated more readily.

 

Fig. 2. Limitations in resolution of the 3D/2D convertible integral imaging

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3. Configuration of the proposed method

The proposed method enhances the resolution of the generated 3D images using an additional lens array. A schematic diagram of the proposed method is shown in Fig. 3. In addition to the lens array between the PDLC and the SLM, an additional lens array is inserted between the collimating lens and the PDLC. In the 3D operation mode, collimated light rays from the light source are imaged by the first (additional) lens array into the first array of point light sources at the focal plane of the first lens array. Each elemental lens of the second (original) lens array then forms an image of this first point light source array again, and, consequently, an array of point light sources with far more point light sources than the number of the elemental lenses in the second lens array is generated at the image plane of the second lens array. Each one in this abundant number of second point light sources serves as one pixel in the generated 3D images, as explained above, and, hence, the resolution is dramatically enhanced, compared to the previous configuration shown in Fig. 1. One significant difference between the proposed method and our previous method [17] is that each elemental lens produces multiple point light sources in the proposed method, while only one point light source is produced in our previous method. Therefore, with the lens array, which can be readily fabricated, a high density of point light sources is easily obtained over a large display area, resulting in high resolution 3D images.

 

Fig. 3. Schematic diagram of the proposed method

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In the proposed method, the number of elemental images should be same as that of the second point light sources and each one should be generated with reference to the position of the corresponding point light source. Since much more point light sources are generated in the proposed method than in the previous method, the number of elemental images should be increased while the size of each one is decreased, which may result in a coarser modulation of the light rays from each point light source than in the previous method if the pixel pitch of the SLM is fixed.

The number of second point light sources, the spacing between them, and the diverging angle of the light rays from each point light source are determined by the specifications and locations of the two lens arrays in the proposed method. The geometry for this configuration is shown in Fig. 4. The first lens array with focal length f 1 and elemental lens pitch φ 1 forms the first point light source array at its focal plane. Each elemental lens with pitch φ 2 and focal length f 2 in the second lens array images these first point light sources at its image plane. Note that not all of the first point light sources are imaged by each elemental lens in the second lens array because the diverging angle of the light rays from each first point light source is limited. The diverging angle of each first point light source, ψ 1, is given by

ψ1=2tan1(φ12f1).

Since the position of the first point light source generated by k-th elemental lens in the first lens array, y 1,k, can be represented by y 1,k= 1, it illuminates the second lens array’s elemental lenses that satisfy

kφ1ltan(ψ12)<qφ2<kφ1+ltan(ψ12),

where l is the distance between the second lens array and the focal plane of the first lens array, and q is the index of the elemental lenses in the second lens array. Or equivalently, the q-th elemental lens in the second lens array is illuminated by the first point light sources whose index k satisfies klkkh where kl and kh are given by

klφ1+ltan(ψ12)=qφ2=khφ1ltan(ψ12).

From Eqs. (1) and (3), kl and kh can also be represented by

kl=qφ2φ1l2f1,
kh=qφ2φ1+l2f1.

Therefore the first point light sources satisfying (klkkh) are imaged into the second point light sources by the q-th elemental lens in the second lens array at

y2,k,q=qφ2(1+gl)kφ1gl,(klkkh)

where g is the location of the image plane of the second lens array which is given by

g=lf2lf2.

The diverging angle of each second point light source which mainly determines the viewing angle of the generated 3D images is written by

ψ2=2tan1(φ22g).

Since g is slightly larger than f 2 as indicated by Eq. (7), the viewing angle of the proposed method can be somewhat narrower than conventional method where g=f. Note that the diverging directions of the second point light sources are, in general, not parallel but vary depending on the relative position between the optic axis of the corresponding elemental lens in the second lens array and the second point light sources, as shown in Fig. 4 (shaded region). This non-parallel diverging direction is another factor in reducing the viewing angle of the proposed method.

 

Fig. 4. Geometry of the generation of a point light source array using two lens arrays

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The final array of the point light sources involves the collection of all of the point light sources imaged by all the elemental lenses in the second lens array. Through careful design using Eqs. (4)(8), a uniformly dense array of point light sources with parallel diverging directions can be obtained. For example, 2N×2N uniform point light sources can be generated using a second lens array consisting of N×N elemental lenses. Figure 5 shows an example of such a configuration. The parameters and locations of two lens arrays are adjusted so that each elemental lens in the second lens array images 3 point light sources at y 2,k,q=(q+0.5)φ 2, 2, (q-0.5)φ 2. As a result, at every position of (q±0.5)φ 2, the point light sources imaged by two neighboring elemental lenses in the second lens array are overlapped. Therefore a uniform point light source array can be obtained that is twice as dense. Note that at the overlapping point light sources the diverging regions (shaded region in Fig. 5) are not overlapped but are patched constructively. Therefore the reduction in viewing angle due to non-parallel diverging directions is effectively alleviated.

 

Fig. 5. Example of the generation of a uniformly dense point light source array

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4. Experimental results

An was performed to verify the feasibility of the proposed method. A schematic diagram of the experimental setup is shown in Fig. 6. An incoherent white light source and a collimating lens were used to illuminate the system. The PDLC was fabricated using NOA65 polymer and an E7 liquid crystal and was operated as a transparent plate at 30 V (1kHz rectangular wave) and as a diffusing plate at 0 V. The SLM used in the experiment had a 0.036 mm pixel pitch. In the experiment, we realized the example shown in Fig. 5. The first lens array consisted of 13×13 elemental lenses with a 10 mm elemental lens pitch and a 22 mm focal length. The second lens array consisted of 70×70 elemental lenses with a 1 mm elemental lens pitch and a 3.3 mm focal length. The second lens array was located at a distance of 66 mm from the first lens array. With these specifications, the uniform point light sources with double density were achieved.

Figure 7 shows the point light sources generated by the previous method [17] and the proposed method. It can clearly be seen that the density of the point light sources is uniformly doubled in the proposed method. The non-uniform brightness of the point light sources in the proposed method is the result of the overlapping of the point light sources generated by neighboring elemental lenses in the second lens array, as explained above. Note that this non-uniformity does not bring about any non-uniformity in the generated 3D images because the overlapping point light sources emit light in different diverging directions, as shown in Fig. 5.

 

Fig. 6. Schematic diagram of the experimental setup (a) previous method (b) proposed method

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Fig. 7. Part of generated point light source (a) previous method (b) proposed method

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Fig. 8. 3D image observed from different directions (a) previous method (b) proposed method

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Figure 8 shows the generated 3D images. Two letter images, ‘3’ and ‘D’, are displayed: at 20 mm in front of the second point light source plane for ‘3’, and 30 mm behind the second point light source plane for ‘D’. The different perspectives according to the different observing directions provide convincing support for the 3D nature of the generated images. From Figs. 8(a) and (b), the resolution enhancement of the proposed method can be readily seen. The viewing angle was determined to be about 18° (H)×19° (V) in the previous method and 18° (H)×15° (V) in the proposed method. No significant reduction in viewing angle is observed in the experiment since the gap between the second lens array and the second point light sources that determines the viewing angle as indicated by Eq. (8) is nearly the same in the previous method and the proposed method(3.3mm for the previous method and 3.5mm for the proposed method. See Fig. 6).

The resolution enhancement of the proposed method is revealed more clearly in Fig. 9. Figure 9 shows the result of displaying flat images 30 mm in front of the SLM. The much higher fidelity of the flat images of Fig. 9(c) compared with those of Fig. 9(b) provides convincing evidence for the resolution enhancement of the proposed method.

 

Fig. 9. Displayed flat images in the 3D mode (a) original images (b) previous method (c) proposed method

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5. Conclusion

A resolution enhancement method for a 3D/2D convertible display based on integral imaging is described. By inserting an additional lens array between the collimating lens and the PDLC, an abundant number of the point light sources is generated, far more than the number of elemental lenses in the lens array, and thus, the resolution of the 3D images is much enhanced. The experimental results support the feasibility of the proposed method.

Acknowledgment

This work was supported by the Information Display R&D Center, one of the 21st Century Frontier R&D Programs funded by the Ministry of Commerce, Industry and Energy of Korea.

References and links

1. G. Lippmann, “La photographie integrale,” Comptes-Rendus Acad. Sci. 146, 446–451 (1908).

2. F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. 36, 1598–1603 (1997). [CrossRef]   [PubMed]  

3. S. Manolache, A. Aggoun, M. McCormick, N. Davies, and S. Y. Kung, “Analytical model of a three-dimensional integral image recording system that uses circular and hexagonal-based spherical surface microlenses,” J. Opt. Soc. Am. A. 18, 1814–1821 (2001). [CrossRef]  

4. T. Naemura, T. Yoshida, and H. Harashima, “3-D computer graphics based on integral photography,” Opt. Express 8, 255–262 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-4-255. [CrossRef]   [PubMed]  

5. J.-H. Park, Y. Kim, J. Kim, S.-W. Min, and B. Lee, “Three-dimensional display scheme based on integral imaging with three-dimensional information processing,” Opt. Express 12, 6020–6032 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-6020. [CrossRef]   [PubMed]  

6. S.-H. Shin and B. Javidi, “Speckle reduced three-dimensional volume holographic display using integral imaging,” Appl. Opt. 41, 2644–2649 (2002). [CrossRef]   [PubMed]  

7. S.-H. Hong, J.-S. Jang, and B. Javidi, “Three-dimensional volumetric object reconstruction using computational integral imaging,” Opt. Express 12, 483–491 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-483. [CrossRef]   [PubMed]  

8. L. Erdmann and K. J. Gabriel, “High-resolution digital integral photography by use of a scanning microlens array,” Appl. Opt. 40, 5592–5599 (2001). [CrossRef]  

9. H. Liao, M. Iwahara, N. Hata, and T. Dohi, “High-quality integral videography using a multiprojector,” Opt. Express 12, 1067–1076 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1067 [CrossRef]   [PubMed]  

10. B. Lee, S. Jung, and J. -H. Park, “Viewing-angle-enhanced integral imaging using lens switching,” Opt. Lett. 27, 818–820 (2002). [CrossRef]  

11. J. S. Jang, Y.-S. Oh, and B. Javidi, “Spatiotemporally multiplexed integral imaging projector for large-scale high resolution three-dimensional display,” Opt. Express 12, 557–563 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-557 [CrossRef]   [PubMed]  

12. Y. Kim, J.-H. Park, S.-W. Min, S. Jung, H. Choi, and B. Lee, “Wide-viewing-angle integral three-dimensional imaging system by curving a screen and a lens array,” Appl. Opt. 44, 546–552 (2005). [CrossRef]   [PubMed]  

13. J. S. Jang and B. Javidi, “Three dimensional synthetic aperture integral imaging,” Opt. Lett. 27, 1144–1146 (2002). [CrossRef]  

14. J. S. Jang and B. Javidi, “Improved viewing resolution of 3-D integral imaging with nonstationary micro-optics,” Opt. Lett. 27, 324–326 (2002). [CrossRef]  

15. J. Hong, J.-H. Park, S. Jung, and B. Lee, “A depth-enhanced integral imaging by use of optical path control,” Opt. Lett. 29, 1790–1792 (2004). [CrossRef]   [PubMed]  

16. J. S. Jang and B. Javidi, “Large depth-of-focus time-multiplexed three-dimensional integral imaging by use of lenslets with nonuniform focal lengths and aperture sizes,” Opt. Lett. 28, 1924–1926 (2003). [CrossRef]   [PubMed]  

17. J.-H. Park, H.-R. Kim, Y. Kim, J. Kim, J. Hong, S.-D. Lee, and B. Lee, “Depth-enhanced three-dimensional-two-dimensional convertible display based on modified integral imaging,” Opt. Lett. 29, 2734–2736 (2004). [CrossRef]   [PubMed]  

References

  • View by:
  • |

  1. G. Lippmann, �??La photographie integrale,�?? Comptes-Rendus Acad. Sci. 146, 446-451 (1908).
  2. F. Okano, H. Hoshino, J. Arai, and I. Yuyama, �??Real-time pickup method for a three-dimensional image based on integral photography,�?? Appl. Opt. 36, 1598-1603 (1997).
    [CrossRef] [PubMed]
  3. S. Manolache, A. Aggoun, M. McCormick, N. Davies, and S. Y. Kung, �??Analytical model of a three-dimensional integral image recording system that uses circular and hexagonal-based spherical surface microlenses,�?? J. Opt. Soc. Am. A. 18, 1814-1821 (2001).
    [CrossRef]
  4. T. Naemura, T. Yoshida, and H. Harashima, �??3-D computer graphics based on integral photography,�?? Opt. Express 8, 255-262 (2001), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-4-255">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-4-255</a>.
    [CrossRef] [PubMed]
  5. J.-H. Park, Y. Kim, J. Kim, S.-W. Min, and B. Lee, "Three-dimensional display scheme based on integral imaging with three-dimensional information processing," Opt. Express 12, 6020-6032 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-6020">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-6020</a>.
    [CrossRef] [PubMed]
  6. S.-H. Shin and B. Javidi, �??Speckle reduced three-dimensional volume holographic display using integral imaging,�?? Appl. Opt. 41, 2644�??2649 (2002).
    [CrossRef] [PubMed]
  7. S.-H. Hong, J.-S. Jang, and B. Javidi, "Three-dimensional volumetric object reconstruction using computational integral imaging," Opt. Express 12, 483-491 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-483">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-483</a>.
    [CrossRef] [PubMed]
  8. L. Erdmann and K. J. Gabriel, �??High-resolution digital integral photography by use of a scanning microlens array,�?? Appl. Opt. 40, 5592-5599 (2001).
    [CrossRef]
  9. H. Liao, M. Iwahara, N. Hata, and T. Dohi, "High-quality integral videography using a multiprojector," Opt. Express 12, 1067-1076 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1067">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1067</a>
    [CrossRef] [PubMed]
  10. B. Lee, S. Jung, and J. -H. Park, �??Viewing-angle-enhanced integral imaging using lens switching,�?? Opt. Lett. 27, 818-820 (2002).
    [CrossRef]
  11. J. S. Jang, Y.-S. Oh, and B. Javidi, �??Spatiotemporally multiplexed integral imaging projector for large-scale high resolution three-dimensional display,�?? Opt. Express 12, 557-563 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-557">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-557</a>
    [CrossRef] [PubMed]
  12. Y. Kim, J.-H. Park, S.-W. Min, S. Jung, H. Choi, and B. Lee, "Wide-viewing-angle integral three-dimensional imaging system by curving a screen and a lens array," Appl. Opt. 44, 546-552 (2005).
    [CrossRef] [PubMed]
  13. J. S. Jang, and B. Javidi, �??Three dimensional synthetic aperture integral imaging,�?? Opt. Lett. 27, 1144-1146 (2002).
    [CrossRef]
  14. J. S. Jang, and B. Javidi, �??Improved viewing resolution of 3-D integral imaging with nonstationary micro-optics,�?? Opt. Lett. 27, 324-326 (2002).
    [CrossRef]
  15. J. Hong, J.-H. Park, S. Jung and B. Lee, "A depth-enhanced integral imaging by use of optical path control," Opt. Lett. 29, 1790-1792 (2004).
    [CrossRef] [PubMed]
  16. J. S. Jang, and B. Javidi, �??Large depth-of-focus time-multiplexed three-dimensional integral imaging by use of lenslets with nonuniform focal lengths and aperture sizes,�?? Opt. Lett. 28, 1924-1926 (2003).
    [CrossRef] [PubMed]
  17. J.-H. Park, H.-R. Kim, Y. Kim, J. Kim, J. Hong, S.-D. Lee, and B. Lee, "Depth-enhanced three-dimensional-two-dimensional convertible display based on modified integral imaging," Opt. Lett. 29, 2734-2736 (2004).
    [CrossRef] [PubMed]

Appl. Opt.

Comptes-Rendus Acad. Sci.

G. Lippmann, �??La photographie integrale,�?? Comptes-Rendus Acad. Sci. 146, 446-451 (1908).

J. Opt. Soc. Am. A.

S. Manolache, A. Aggoun, M. McCormick, N. Davies, and S. Y. Kung, �??Analytical model of a three-dimensional integral image recording system that uses circular and hexagonal-based spherical surface microlenses,�?? J. Opt. Soc. Am. A. 18, 1814-1821 (2001).
[CrossRef]

Opt. Express

T. Naemura, T. Yoshida, and H. Harashima, �??3-D computer graphics based on integral photography,�?? Opt. Express 8, 255-262 (2001), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-4-255">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-4-255</a>.
[CrossRef] [PubMed]

J.-H. Park, Y. Kim, J. Kim, S.-W. Min, and B. Lee, "Three-dimensional display scheme based on integral imaging with three-dimensional information processing," Opt. Express 12, 6020-6032 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-6020">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-24-6020</a>.
[CrossRef] [PubMed]

H. Liao, M. Iwahara, N. Hata, and T. Dohi, "High-quality integral videography using a multiprojector," Opt. Express 12, 1067-1076 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1067">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1067</a>
[CrossRef] [PubMed]

S.-H. Hong, J.-S. Jang, and B. Javidi, "Three-dimensional volumetric object reconstruction using computational integral imaging," Opt. Express 12, 483-491 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-483">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-483</a>.
[CrossRef] [PubMed]

J. S. Jang, Y.-S. Oh, and B. Javidi, �??Spatiotemporally multiplexed integral imaging projector for large-scale high resolution three-dimensional display,�?? Opt. Express 12, 557-563 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-557">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-557</a>
[CrossRef] [PubMed]

Opt. Lett.

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Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of 3D/2D convertible integral imaging (a) 2D mode (b) 3D mode

Fig. 2.
Fig. 2.

Limitations in resolution of the 3D/2D convertible integral imaging

Fig. 3.
Fig. 3.

Schematic diagram of the proposed method

Fig. 4.
Fig. 4.

Geometry of the generation of a point light source array using two lens arrays

Fig. 5.
Fig. 5.

Example of the generation of a uniformly dense point light source array

Fig. 6.
Fig. 6.

Schematic diagram of the experimental setup (a) previous method (b) proposed method

Fig. 7.
Fig. 7.

Part of generated point light source (a) previous method (b) proposed method

Fig. 8.
Fig. 8.

3D image observed from different directions (a) previous method (b) proposed method

Fig. 9.
Fig. 9.

Displayed flat images in the 3D mode (a) original images (b) previous method (c) proposed method

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

ψ 1 = 2 tan 1 ( φ 1 2 f 1 ) .
k φ 1 l tan ( ψ 1 2 ) < q φ 2 < k φ 1 + l tan ( ψ 1 2 ) ,
k l φ 1 + l tan ( ψ 1 2 ) = q φ 2 = k h φ 1 l tan ( ψ 1 2 ) .
k l = q φ 2 φ 1 l 2 f 1 ,
k h = q φ 2 φ 1 + l 2 f 1 .
y 2 , k , q = q φ 2 ( 1 + g l ) k φ 1 g l , ( k l k k h )
g = l f 2 l f 2 .
ψ 2 = 2 tan 1 ( φ 2 2 g ) .

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