Abstract

Hybrid multi-layer displays are proposed as the system combines additive light field (LF) displays and multiplicative LF displays. The system is implemented by integrating the multiplicative LF displays with a half mirror to expand the overall depth of field. The hybrid displays are advantageous in that the form factor is competitive with existing additive LF displays with 2 layers implemented by a half mirror and two panels, only half of brightness loss is experienced compared to multiplicative LF displays with 2 layers, and no time-division is required to provide images for multi-layer displays. The images for presentation planes are processed by light field factorization and optimized with the presented algorithm. Retinal images are reconstructed based on various accommodation states and display types to check the accommodation response and utilized to compare the proposed displays with existing displays. With ray tracing method, retinal images generated by the proposed displays can be obtained. To verify the feasibility of the system, a prototype of hybrid multi-layer displays was implemented and display photographs were captured with different accommodation states of camera. With the simulation results and experimental results, this system was confirmed to support accommodation cues in a range of 1.8 diopters.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (2)

T. Zhan, Y.-H. Lee, and S.-T. Wu, “High-resolution additive light field near-eye display by switchable pancharatnam– berry phase lenses,” Opt. Express 26, 4863–4872 (2018).
[Crossref] [PubMed]

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

2017 (4)

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

S. Moon, C.-K. Lee, D. Lee, C. Jang, and B. Lee, “Layered display with accommodation cue using scattering polarizers,” IEEE J. Sel. Top. Signal Process. 11, 1223–1231 (2017).
[Crossref]

J.-H. Park, “Recent progress in computer-generated holography for three-dimensional scenes,” J. Inf. Disp. 18, 1–12 (2017).
[Crossref]

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

2016 (2)

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35, 60 (2016).
[Crossref]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

2015 (3)

F.-C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues,” ACM Trans. Graph. 34, 60 (2015).
[Crossref]

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

H.-J. Yeom, H.-J. Kim, S.-B. Kim, H. Zhang, B. Li, Y.-M. Ji, S.-H. Kim, and J.-H. Park, “3D holographic head mounted display using holographic optical elements with astigmatism aberration compensation,” Opt. Express 23, 32025–32034 (2015).
[Crossref] [PubMed]

2014 (3)

X. Hu and H. Hua, “High-resolution optical see-through multi-focal-plane head-mounted display using freeform optics,” Opt. Express 22, 13896–13903 (2014).
[Crossref] [PubMed]

X. Hu and H. Hua, “Design and assessment of a depth-fused multi-focal-plane display prototype,” J. Disp. Technol. 10, 308–316 (2014).
[Crossref]

M. Hirsch, G. Wetzstein, and R. Raskar, “A compressive light field projection system,” ACM Trans. Graph. 33, 58 (2014).
[Crossref]

2013 (1)

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

2012 (1)

G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graph. 31, 80 (2012).
[Crossref]

2011 (5)

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3D: tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30, 95 (2011).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” ACM Trans. Graph. 30, 186 (2011).
[Crossref]

Y. Takaki, K. Tanaka, and J. Nakamura, “Super multi-view display with a lower resolution flat-panel display,” Opt. Express 19, 4129–4139 (2011).
[Crossref] [PubMed]

T. Shibata, J. Kim, D. M. Hoffman, and M. S. Banks, “The zone of comfort: Predicting visual discomfort with stereo displays,” J. Vis. 118, 11 (2011).
[Crossref] [PubMed]

S. Ravikumar, K. Akeley, and M. S. Banks, “Creating effective focus cues in multi-plane 3D displays,” Opt. Express 19, 20940–20952 (2011).
[Crossref] [PubMed]

2010 (3)

S. Liu and H. Hua, “A systematic method for designing depth-fused multi-focal plane three-dimensional displays,” Opt. Express 18, 11562–11573 (2010).
[Crossref] [PubMed]

K. J. MacKenzie, D. M. Hoffman, and S. J. Watt, “Accommodation to multiple-focal-plane displays: Implications for improving stereoscopic displays and for accommodation control,” J. Vis. 108, 22 (2010).
[Crossref] [PubMed]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29, 163 (2010).
[Crossref]

2009 (2)

M. Lambooij, M. Fortuin, I. Heynderickx, and W. IJsselsteijn, “Visual discomfort and visual fatigue of stereoscopic displays: A review,” J. Imaging Sci. Technol. 53, 30201 (2009).
[Crossref]

G. D. Love, D. M. Hoffman, P. J. Hands, J. Gao, A. K. Kirby, and M. S. Banks, “High-speed switchable lens enables the development of a volumetric stereoscopic display,” Opt. Express 17, 15716–15725 (2009).
[Crossref] [PubMed]

2008 (1)

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 83, 33 (2008).
[Crossref]

2005 (2)

M. Emoto, T. Niida, and F. Okano, “Repeated vergence adaptation causes the decline of visual functions in watching stereoscopic television,” J. Disp. Technol. 1, 328 (2005).
[Crossref]

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vis. 5, 7 (2005).
[Crossref]

2004 (1)

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23, 804–813 (2004).
[Crossref]

2000 (1)

1999 (1)

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788 (1999).
[Crossref] [PubMed]

1996 (1)

T. F. Coleman and Y. Li, “A reflective newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[Crossref]

1984 (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[Crossref] [PubMed]

1973 (1)

V. Krishnan, S. Phillips, and L. Stark, “Frequency analysis of accommodation, accommodative vergence and disparity vergence,” Vis. Res. 13, 1545–1554 (1973).
[Crossref] [PubMed]

Akeley, K.

S. Ravikumar, K. Akeley, and M. S. Banks, “Creating effective focus cues in multi-plane 3D displays,” Opt. Express 19, 20940–20952 (2011).
[Crossref] [PubMed]

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 83, 33 (2008).
[Crossref]

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vis. 5, 7 (2005).
[Crossref]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23, 804–813 (2004).
[Crossref]

Albert, R. A.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

Andersen, A. H.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[Crossref] [PubMed]

Bang, K.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

Banks, M. S.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

S. Ravikumar, K. Akeley, and M. S. Banks, “Creating effective focus cues in multi-plane 3D displays,” Opt. Express 19, 20940–20952 (2011).
[Crossref] [PubMed]

T. Shibata, J. Kim, D. M. Hoffman, and M. S. Banks, “The zone of comfort: Predicting visual discomfort with stereo displays,” J. Vis. 118, 11 (2011).
[Crossref] [PubMed]

G. D. Love, D. M. Hoffman, P. J. Hands, J. Gao, A. K. Kirby, and M. S. Banks, “High-speed switchable lens enables the development of a volumetric stereoscopic display,” Opt. Express 17, 15716–15725 (2009).
[Crossref] [PubMed]

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 83, 33 (2008).
[Crossref]

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vis. 5, 7 (2005).
[Crossref]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23, 804–813 (2004).
[Crossref]

Bulbul, A.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

Chen, K.

F.-C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues,” ACM Trans. Graph. 34, 60 (2015).
[Crossref]

Cho, J.

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35, 60 (2016).
[Crossref]

Coleman, T. F.

T. F. Coleman and Y. Li, “A reflective newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[Crossref]

Emoto, M.

M. Emoto, T. Niida, and F. Okano, “Repeated vergence adaptation causes the decline of visual functions in watching stereoscopic television,” J. Disp. Technol. 1, 328 (2005).
[Crossref]

Ernst, M. O.

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vis. 5, 7 (2005).
[Crossref]

Fortuin, M.

M. Lambooij, M. Fortuin, I. Heynderickx, and W. IJsselsteijn, “Visual discomfort and visual fatigue of stereoscopic displays: A review,” J. Imaging Sci. Technol. 53, 30201 (2009).
[Crossref]

Fuchs, H.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

Gao, J.

Girshick, A. R.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 83, 33 (2008).
[Crossref]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23, 804–813 (2004).
[Crossref]

Goon, A.

Hands, P. J.

Hanrahan, P.

M. Levoy and P. Hanrahan, “Light field rendering,” in “Proceedings of the 23rd Annual Conf. Computer Graphics and Interactive Techniques,” (ACM, 1996), pp. 31–42.

Heidrich, W.

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3D: tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30, 95 (2011).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” ACM Trans. Graph. 30, 186 (2011).
[Crossref]

Heynderickx, I.

M. Lambooij, M. Fortuin, I. Heynderickx, and W. IJsselsteijn, “Visual discomfort and visual fatigue of stereoscopic displays: A review,” J. Imaging Sci. Technol. 53, 30201 (2009).
[Crossref]

Hirsch, M.

M. Hirsch, G. Wetzstein, and R. Raskar, “A compressive light field projection system,” ACM Trans. Graph. 33, 58 (2014).
[Crossref]

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graph. 31, 80 (2012).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” ACM Trans. Graph. 30, 186 (2011).
[Crossref]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29, 163 (2010).
[Crossref]

Hoffman, D. M.

T. Shibata, J. Kim, D. M. Hoffman, and M. S. Banks, “The zone of comfort: Predicting visual discomfort with stereo displays,” J. Vis. 118, 11 (2011).
[Crossref] [PubMed]

K. J. MacKenzie, D. M. Hoffman, and S. J. Watt, “Accommodation to multiple-focal-plane displays: Implications for improving stereoscopic displays and for accommodation control,” J. Vis. 108, 22 (2010).
[Crossref] [PubMed]

G. D. Love, D. M. Hoffman, P. J. Hands, J. Gao, A. K. Kirby, and M. S. Banks, “High-speed switchable lens enables the development of a volumetric stereoscopic display,” Opt. Express 17, 15716–15725 (2009).
[Crossref] [PubMed]

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 83, 33 (2008).
[Crossref]

Hong, J.-Y.

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

Hu, X.

X. Hu and H. Hua, “High-resolution optical see-through multi-focal-plane head-mounted display using freeform optics,” Opt. Express 22, 13896–13903 (2014).
[Crossref] [PubMed]

X. Hu and H. Hua, “Design and assessment of a depth-fused multi-focal-plane display prototype,” J. Disp. Technol. 10, 308–316 (2014).
[Crossref]

Hua, H.

Huang, F.-C.

F.-C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues,” ACM Trans. Graph. 34, 60 (2015).
[Crossref]

IJsselsteijn, W.

M. Lambooij, M. Fortuin, I. Heynderickx, and W. IJsselsteijn, “Visual discomfort and visual fatigue of stereoscopic displays: A review,” J. Imaging Sci. Technol. 53, 30201 (2009).
[Crossref]

Jang, C.

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

S. Moon, C.-K. Lee, D. Lee, C. Jang, and B. Lee, “Layered display with accommodation cue using scattering polarizers,” IEEE J. Sel. Top. Signal Process. 11, 1223–1231 (2017).
[Crossref]

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35, 60 (2016).
[Crossref]

Jeong, J.

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

Ji, Y.-M.

Jo, Y.

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

Kak, A. C.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[Crossref] [PubMed]

Kim, D.

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

Kim, H.-J.

Kim, J.

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

T. Shibata, J. Kim, D. M. Hoffman, and M. S. Banks, “The zone of comfort: Predicting visual discomfort with stereo displays,” J. Vis. 118, 11 (2011).
[Crossref] [PubMed]

Kim, S.-B.

Kim, S.-H.

Kim, Y.

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29, 163 (2010).
[Crossref]

Kirby, A. K.

Krishnan, V.

V. Krishnan, S. Phillips, and L. Stark, “Frequency analysis of accommodation, accommodative vergence and disparity vergence,” Vis. Res. 13, 1545–1554 (1973).
[Crossref] [PubMed]

Krueger, M. W.

Lambooij, M.

M. Lambooij, M. Fortuin, I. Heynderickx, and W. IJsselsteijn, “Visual discomfort and visual fatigue of stereoscopic displays: A review,” J. Imaging Sci. Technol. 53, 30201 (2009).
[Crossref]

Lanman, D.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graph. 31, 80 (2012).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” ACM Trans. Graph. 30, 186 (2011).
[Crossref]

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3D: tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30, 95 (2011).
[Crossref]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29, 163 (2010).
[Crossref]

Lee, B.

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

S. Moon, C.-K. Lee, D. Lee, C. Jang, and B. Lee, “Layered display with accommodation cue using scattering polarizers,” IEEE J. Sel. Top. Signal Process. 11, 1223–1231 (2017).
[Crossref]

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35, 60 (2016).
[Crossref]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

Lee, C.-K.

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

S. Moon, C.-K. Lee, D. Lee, C. Jang, and B. Lee, “Layered display with accommodation cue using scattering polarizers,” IEEE J. Sel. Top. Signal Process. 11, 1223–1231 (2017).
[Crossref]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

Lee, D.

S. Moon, C.-K. Lee, D. Lee, C. Jang, and B. Lee, “Layered display with accommodation cue using scattering polarizers,” IEEE J. Sel. Top. Signal Process. 11, 1223–1231 (2017).
[Crossref]

Lee, D. D.

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788 (1999).
[Crossref] [PubMed]

Lee, S.

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35, 60 (2016).
[Crossref]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

Lee, Y.-H.

Levoy, M.

M. Levoy and P. Hanrahan, “Light field rendering,” in “Proceedings of the 23rd Annual Conf. Computer Graphics and Interactive Techniques,” (ACM, 1996), pp. 31–42.

Li, B.

Li, Y.

T. F. Coleman and Y. Li, “A reflective newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[Crossref]

Liu, S.

Love, G. D.

MacKenzie, K. J.

K. J. MacKenzie, D. M. Hoffman, and S. J. Watt, “Accommodation to multiple-focal-plane displays: Implications for improving stereoscopic displays and for accommodation control,” J. Vis. 108, 22 (2010).
[Crossref] [PubMed]

Maimone, A.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

Moon, S.

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

S. Moon, C.-K. Lee, D. Lee, C. Jang, and B. Lee, “Layered display with accommodation cue using scattering polarizers,” IEEE J. Sel. Top. Signal Process. 11, 1223–1231 (2017).
[Crossref]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35, 60 (2016).
[Crossref]

Nakamura, J.

Narain, R.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

Niida, T.

M. Emoto, T. Niida, and F. Okano, “Repeated vergence adaptation causes the decline of visual functions in watching stereoscopic television,” J. Disp. Technol. 1, 328 (2005).
[Crossref]

O’Brien, J. F.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

Okano, F.

M. Emoto, T. Niida, and F. Okano, “Repeated vergence adaptation causes the decline of visual functions in watching stereoscopic television,” J. Disp. Technol. 1, 328 (2005).
[Crossref]

Park, J.-H.

Phillips, S.

V. Krishnan, S. Phillips, and L. Stark, “Frequency analysis of accommodation, accommodative vergence and disparity vergence,” Vis. Res. 13, 1545–1554 (1973).
[Crossref] [PubMed]

Raskar, R.

M. Hirsch, G. Wetzstein, and R. Raskar, “A compressive light field projection system,” ACM Trans. Graph. 33, 58 (2014).
[Crossref]

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graph. 31, 80 (2012).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” ACM Trans. Graph. 30, 186 (2011).
[Crossref]

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3D: tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30, 95 (2011).
[Crossref]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29, 163 (2010).
[Crossref]

Ravikumar, S.

Rolland, J. P.

Saleh, B. E.

B. E. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics (WileyNew York, 1991).
[Crossref]

B. E. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics (WileyNew York, 1991).
[Crossref]

Seung, H. S.

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788 (1999).
[Crossref] [PubMed]

Shibata, T.

T. Shibata, J. Kim, D. M. Hoffman, and M. S. Banks, “The zone of comfort: Predicting visual discomfort with stereo displays,” J. Vis. 118, 11 (2011).
[Crossref] [PubMed]

Stark, L.

V. Krishnan, S. Phillips, and L. Stark, “Frequency analysis of accommodation, accommodative vergence and disparity vergence,” Vis. Res. 13, 1545–1554 (1973).
[Crossref] [PubMed]

Takaki, Y.

Tanaka, K.

Teich, M. C.

B. E. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics (WileyNew York, 1991).
[Crossref]

Wandell, B. A.

B. A. Wandell, Foundations of Vision (Sinauer Associates, 1995).

Ward, G. J.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

Watt, S. J.

K. J. MacKenzie, D. M. Hoffman, and S. J. Watt, “Accommodation to multiple-focal-plane displays: Implications for improving stereoscopic displays and for accommodation control,” J. Vis. 108, 22 (2010).
[Crossref] [PubMed]

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vis. 5, 7 (2005).
[Crossref]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23, 804–813 (2004).
[Crossref]

Wetzstein, G.

F.-C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues,” ACM Trans. Graph. 34, 60 (2015).
[Crossref]

M. Hirsch, G. Wetzstein, and R. Raskar, “A compressive light field projection system,” ACM Trans. Graph. 33, 58 (2014).
[Crossref]

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graph. 31, 80 (2012).
[Crossref]

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3D: tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30, 95 (2011).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” ACM Trans. Graph. 30, 186 (2011).
[Crossref]

Wu, S.-T.

Yeom, H.-J.

Yoo, D.

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

Zhan, T.

Zhang, H.

ACM Trans. Graph. (11)

C. Jang, K. Bang, S. Moon, J. Kim, S. Lee, and B. Lee, “Retinal 3D: augmented reality near-eye display via pupil-tracked light field projection on retina,” ACM Trans. Graph. 36, 190 (2017).
[Crossref]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29, 163 (2010).
[Crossref]

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3D: tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30, 95 (2011).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: dynamic light field display using multi-layer LCDs,” ACM Trans. Graph. 30, 186 (2011).
[Crossref]

G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graph. 31, 80 (2012).
[Crossref]

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32, 153 (2013).
[Crossref]

F.-C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues,” ACM Trans. Graph. 34, 60 (2015).
[Crossref]

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35, 60 (2016).
[Crossref]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23, 804–813 (2004).
[Crossref]

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34, 59 (2015).
[Crossref]

M. Hirsch, G. Wetzstein, and R. Raskar, “A compressive light field projection system,” ACM Trans. Graph. 33, 58 (2014).
[Crossref]

Appl. Opt. (1)

IEEE Access (1)

S. Lee, J. Cho, B. Lee, Y. Jo, C. Jang, D. Kim, and B. Lee, “Foveated retinal optimization for see-through near-eye multi-layer displays,” IEEE Access 6, 2170–2180 (2018).
[Crossref]

IEEE J. Sel. Top. Signal Process. (1)

S. Moon, C.-K. Lee, D. Lee, C. Jang, and B. Lee, “Layered display with accommodation cue using scattering polarizers,” IEEE J. Sel. Top. Signal Process. 11, 1223–1231 (2017).
[Crossref]

J. Disp. Technol. (2)

X. Hu and H. Hua, “Design and assessment of a depth-fused multi-focal-plane display prototype,” J. Disp. Technol. 10, 308–316 (2014).
[Crossref]

M. Emoto, T. Niida, and F. Okano, “Repeated vergence adaptation causes the decline of visual functions in watching stereoscopic television,” J. Disp. Technol. 1, 328 (2005).
[Crossref]

J. Imaging Sci. Technol. (1)

M. Lambooij, M. Fortuin, I. Heynderickx, and W. IJsselsteijn, “Visual discomfort and visual fatigue of stereoscopic displays: A review,” J. Imaging Sci. Technol. 53, 30201 (2009).
[Crossref]

J. Inf. Disp. (1)

J.-H. Park, “Recent progress in computer-generated holography for three-dimensional scenes,” J. Inf. Disp. 18, 1–12 (2017).
[Crossref]

J. Vis. (4)

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 83, 33 (2008).
[Crossref]

K. J. MacKenzie, D. M. Hoffman, and S. J. Watt, “Accommodation to multiple-focal-plane displays: Implications for improving stereoscopic displays and for accommodation control,” J. Vis. 108, 22 (2010).
[Crossref] [PubMed]

T. Shibata, J. Kim, D. M. Hoffman, and M. S. Banks, “The zone of comfort: Predicting visual discomfort with stereo displays,” J. Vis. 118, 11 (2011).
[Crossref] [PubMed]

S. J. Watt, K. Akeley, M. O. Ernst, and M. S. Banks, “Focus cues affect perceived depth,” J. Vis. 5, 7 (2005).
[Crossref]

Nature (1)

D. D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix factorization,” Nature 401, 788 (1999).
[Crossref] [PubMed]

Opt. Express (8)

H.-J. Yeom, H.-J. Kim, S.-B. Kim, H. Zhang, B. Li, Y.-M. Ji, S.-H. Kim, and J.-H. Park, “3D holographic head mounted display using holographic optical elements with astigmatism aberration compensation,” Opt. Express 23, 32025–32034 (2015).
[Crossref] [PubMed]

G. D. Love, D. M. Hoffman, P. J. Hands, J. Gao, A. K. Kirby, and M. S. Banks, “High-speed switchable lens enables the development of a volumetric stereoscopic display,” Opt. Express 17, 15716–15725 (2009).
[Crossref] [PubMed]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with Savart plate,” Opt. Express 24, 19531–19544 (2016).
[Crossref] [PubMed]

X. Hu and H. Hua, “High-resolution optical see-through multi-focal-plane head-mounted display using freeform optics,” Opt. Express 22, 13896–13903 (2014).
[Crossref] [PubMed]

Y. Takaki, K. Tanaka, and J. Nakamura, “Super multi-view display with a lower resolution flat-panel display,” Opt. Express 19, 4129–4139 (2011).
[Crossref] [PubMed]

T. Zhan, Y.-H. Lee, and S.-T. Wu, “High-resolution additive light field near-eye display by switchable pancharatnam– berry phase lenses,” Opt. Express 26, 4863–4872 (2018).
[Crossref] [PubMed]

S. Liu and H. Hua, “A systematic method for designing depth-fused multi-focal plane three-dimensional displays,” Opt. Express 18, 11562–11573 (2010).
[Crossref] [PubMed]

S. Ravikumar, K. Akeley, and M. S. Banks, “Creating effective focus cues in multi-plane 3D displays,” Opt. Express 19, 20940–20952 (2011).
[Crossref] [PubMed]

Sci. Rep. (1)

J.-Y. Hong, C.-K. Lee, S. Lee, B. Lee, D. Yoo, C. Jang, J. Kim, J. Jeong, and B. Lee, “See-through optical combiner for augmented reality head-mounted display: index-matched anisotropic crystal lens,” Sci. Rep. 7, 2753 (2017).
[Crossref] [PubMed]

SIAM J. Optim. (1)

T. F. Coleman and Y. Li, “A reflective newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[Crossref]

Ultrason. Imaging (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[Crossref] [PubMed]

Vis. Res. (1)

V. Krishnan, S. Phillips, and L. Stark, “Frequency analysis of accommodation, accommodative vergence and disparity vergence,” Vis. Res. 13, 1545–1554 (1973).
[Crossref] [PubMed]

Other (5)

“Microsoft hololens,” https://www.microsoft.com/en-us/hololens (2014). Accessed: 2018-04-23.

“Htc vive,” https://www.vive.com/us/product/vive-virtual-reality-system/ (2016). Accessed: 2018-04-23.

B. E. Saleh, M. C. Teich, and B. E. Saleh, Fundamentals of Photonics (WileyNew York, 1991).
[Crossref]

M. Levoy and P. Hanrahan, “Light field rendering,” in “Proceedings of the 23rd Annual Conf. Computer Graphics and Interactive Techniques,” (ACM, 1996), pp. 31–42.

B. A. Wandell, Foundations of Vision (Sinauer Associates, 1995).

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram and optically equivalent model of the proposed system. The physical displays are presented relatively transparent to the corresponding virtual displays. The half mirror synthesizes the images from two multiplicative displays. Each multiplicative set of layers is colored in a similar tone to show that consecutive depth planes are responsible by the displays placed in the different portions of the system. For simplicity, the gap between the panels, and the spacing between adjacent planes are described as identical respectively.
Fig. 2
Fig. 2 Flow chart of the display pattern optimization algorithm
Fig. 3
Fig. 3 Concise illustration of hybrid multi-layer displays based on dioptric unit: (a) a top view of rendered volumetric images and (b) a converted model with optimized images virtually floated at the presentation planes. For simplicity, sizes of dice are demonstrated as identical and the gap between adjacent dice as equidistant, which differs in actual case.
Fig. 4
Fig. 4 Retinal images reconstructed for three different displays (additive displays with 2 layers, multiplicative displays with 2 layers, hybrid displays with 4 layers) with two different accommodation states of eye (1D and 1.6D). For 2-layered displays, the spacing between consecutive image planes is 1.8D while 0.6D is guaranteed between adjacent layers for hybrid displays. The black border lines appear due to the sampling parameter in the retina plane. Enlargements are shown on the right side of the figure.
Fig. 5
Fig. 5 Normalized contrast ratio curves of multi-layer displays depending on the angular frequency and placement of the target object: hybrid displays (4 layers, red line), additive displays (2 layers, green line), multiplicative displays (2 layers, blue line) and estimated normalized contrast ratio curves in a natural viewing condition (black line). Green dashed line indicates the depth where the target object (light field origin plane in light field synthesis) is located. For additive and multiplicative displays, layers are optically located at 0.4D and 2.2D while four layers are placed at 0.4D, 1D, 1.6D and 2.2D for hybrid multi-layer displays.
Fig. 6
Fig. 6 Experimental setup of hybrid multi-layer displays for monocular eye.
Fig. 7
Fig. 7 Display photographs of hybrid multi-layer displays taken by modifying the accommodation depths of CCD. To confirm the focus cues, some parts of the images with different accommodation depths are enlarged. Enlarged image with a colored dot in the upper left side indicates that the accommodation distance of the camera is matched with the depth of rendered object. Retinal blur is observed from all results since the images are captured with small depth of focus lens. Sub-pixels of the displays are observed in the perceived images with the camera’s focal distance of 1.6D and 2.2D.

Tables (1)

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Table 1 Detailed specifications of hybrid multi-layer displays

Equations (6)

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d k = 1 1 f + d d k , M k = d k d k = f f + d k .
θ k = 2 tan 1 ( M k p p n p 2 ( d k + d e ) ) ,
l ˜ ( x , ν ) = m = 1 2 l ˜ m ( x , ν ) ,
l ˜ m ( x , ν ) = k = 1 2 t 2 ( k 1 + m ) ( p 2 ( k 1 ) + m ( x , ν ) ) = k = 1 2 t 2 ( k 1 ) + m ( 1 M 2 ( k 1 ) + m ( x + ( x ν ) ( d k d l f ) d l f + d e ) ) ,
l ˜ = l ˜ 1 + l ˜ 2 = P 1 t 1 P 3 t 3 + P 2 t 2 P 4 t 4 .
arg min { t 1 , t 2 , t 3 , t 4 } l ( P 1 t 1 P 3 t 3 + P 2 t 2 P 4 t 4 ) 2 2 , 0 t 1 , 2 , 3 , 4 1 ,

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