Wavefront aberration correction for integral imaging with the pre-filtering function array

Wanlu Zhang; Xinzhu Sang; Xin Gao; Xunbo Yu; Binbin Yan; Chongxiu Yu

doi:10.1364/OE.26.027064

Wavefront aberration correction for integral imaging with the pre-filtering function array

Wanlu Zhang, Xinzhu Sang, Xin Gao, Xunbo Yu, Binbin Yan, and Chongxiu Yu

Optics Express
Vol. 26,
Issue 21,
pp. 27064-27075
(2018)
•https://doi.org/10.1364/OE.26.027064

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Wanlu Zhang, Xinzhu Sang, Xin Gao, Xunbo Yu, Binbin Yan, and Chongxiu Yu, "Wavefront aberration correction for integral imaging with the pre-filtering function array," Opt. Express 26, 27064-27075 (2018)

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Related Topics
Table of Contents Category
- Imaging Systems, Microscopy, and Displays
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: August 10, 2018
- Revised Manuscript: September 22, 2018
- Manuscript Accepted: September 23, 2018
- Published: October 2, 2018

Accessible

Open Access

Abstract

In integral imaging, the quality of a reconstructed image degrades with increasing viewing angle due to the wavefront aberrations introduced by the lens-array. A wavefront aberration correction method is proposed to enhance the image quality with a pre-filtering function array (PFA). To derive the PFA for an integral imaging display, the wavefront aberration characteristic of the lens-array is analyzed and the intensity distribution of the reconstructed image is calculated based on the wave optics theory. The minimum mean square error method is applied to manipulate the elemental image array (EIA) with a PFA. The validity of the proposed method is confirmed through simulations as well as optical experiments. A 45-degree viewing angle integral imaging display with enhanced image quality is achieved.

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References

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Author
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Publication

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H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley Online Library, 2005).
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[Crossref] [PubMed]
C. Yu, J. Yuan, F. C. Fan, C. C. Jiang, S. Choi, X. Sang, C. Lin, and D. Xu, “The modulation function and realizing method of holographic functional screen,” Opt. Express 18(26), 27820–27826 (2010).
[Crossref] [PubMed]
X. Sang, X. Gao, X. Yu, S. Xing, Y. Li, and Y. Wu, “Interactive floating full-parallax digital three-dimensional light-field display based on wavefront recomposing,” Opt. Express 26(7), 8883–8889 (2018).
[Crossref] [PubMed]
R. C. Gonzales and R. E. Wood, Digital Image Processing (Prentice Hall, 2002).

C. W. Chen, M. Cho, Y. P. Huang, and B. Javidi, “Improved viewing zones for projection type integral imaging 3D display using adaptive liquid crystal prism array,” J. Disp. Technol. 10(3), 198–203 (2014).
[Crossref]

J. Y. Jang, D. Shin, and E. S. Kim, “Optical three-dimensional refocusing from elemental images based on a sifting property of the periodic δ-function array in integral-imaging,” Opt. Express 22(2), 1533–1550 (2014).
[Crossref] [PubMed]

2013 (3)

X. Xiao, B. Javidi, M. Martinez-Corral, and A. Stern, “Advances in three-dimensional integral imaging: sensing, display, and applications [Invited],” Appl. Opt. 52(4), 546–560 (2013).
[Crossref] [PubMed]

D. Fattal, Z. Peng, T. Tran, S. Vo, M. Fiorentino, J. Brug, and R. G. Beausoleil, “A multi-directional backlight for a wide-angle, glasses-free three-dimensional display,” Nature 495(7441), 348–351 (2013).
[Crossref] [PubMed]

J. Geng, “Three-dimensional display technologies,” Adv. Opt. Photonics 5(4), 456–535 (2013).
[Crossref] [PubMed]

2012 (1)

H. H. Kang, J. H. Lee, and E. S. Kim, “Enhanced compression rate of integral images by using motion-compensated residual images in three-dimensional integral-imaging,” Opt. Express 20(5), 5440–5459 (2012).
[Crossref] [PubMed]

2011 (1)

J. Y. Luo, Q. H. Wang, W. X. Zhao, and D. H. Li, “Autostereoscopic three-dimensional display based on two parallax barriers,” Appl. Opt. 50(18), 2911–2915 (2011).
[Crossref] [PubMed]

2010 (2)

W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
[Crossref] [PubMed]

C. Yu, J. Yuan, F. C. Fan, C. C. Jiang, S. Choi, X. Sang, C. Lin, and D. Xu, “The modulation function and realizing method of holographic functional screen,” Opt. Express 18(26), 27820–27826 (2010).
[Crossref] [PubMed]

2007 (1)

Y. Kim, J. Kim, J. M. Kang, J. H. Jung, H. Choi, and B. Lee, “Point light source integral imaging with improved resolution and viewing angle by the use of electrically movable pinhole array,” Opt. Express 15(26), 18253–18267 (2007).
[Crossref] [PubMed]

2006 (1)

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

2002 (1)

J. S. Jang and B. Javidi, “Improved viewing resolution of three-dimensional integral imaging by use of nonstationary micro-optics,” Opt. Lett. 27(5), 324–326 (2002).
[Crossref] [PubMed]

1978 (1)

Y. Igarashi, H. Murata, and M. Ueda, “3D display system using a computer generated integral photography,” Jpn. J. Appl. Phys. 17(9), 1683–1684 (1978).
[Crossref]

1931 (1)

H. E. Ives, “Optical properties of a Lippmann lenticulated sheet,” J. Opt. Soc. Am. 21(3), 171–176 (1931).
[Crossref]

1908 (1)

G. Lippmann, “La photographie integrale,” C. R. Acad. Sci. 146, 446–451 (1908).

Ai, L. Y.

Beausoleil, R. G.

Brug, J.

Chen, C. W.

Chen, Y.

J. Zhang, X. Wang, X. Wu, C. Yang, and Y. Chen, “Wide-viewing integral imaging using fiber-coupled monocentric lens array,” Opt. Express 23(18), 23339–23347 (2015).
[Crossref] [PubMed]

Cho, M.

Choi, H.

Choi, S.

Dong, X. B.

Duo, C.

Fan, F. C.

Fang, F.

L. Jiang, X. Zhang, F. Fang, X. Liu, and L. Zhu, “Wavefront aberration metrology based on transmitted fringe deflectometry,” Appl. Opt. 56(26), 7396–7403 (2017).
[Crossref] [PubMed]

Fattal, D.

Fiorentino, M.

Gao, X.

Geng, J.

J. Geng, “Three-dimensional display technologies,” Adv. Opt. Photonics 5(4), 456–535 (2013).
[Crossref] [PubMed]

Guan, Y.

Huang, Y. P.

Igarashi, Y.

Y. Igarashi, H. Murata, and M. Ueda, “3D display system using a computer generated integral photography,” Jpn. J. Appl. Phys. 17(9), 1683–1684 (1978).
[Crossref]

Ives, H. E.

H. E. Ives, “Optical properties of a Lippmann lenticulated sheet,” J. Opt. Soc. Am. 21(3), 171–176 (1931).
[Crossref]

Jang, J. S.

J. S. Jang and B. Javidi, “Improved viewing resolution of three-dimensional integral imaging by use of nonstationary micro-optics,” Opt. Lett. 27(5), 324–326 (2002).
[Crossref] [PubMed]

Jang, J. Y.

Javidi, B.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

J. S. Jang and B. Javidi, “Improved viewing resolution of three-dimensional integral imaging by use of nonstationary micro-optics,” Opt. Lett. 27(5), 324–326 (2002).
[Crossref] [PubMed]

Jiang, C. C.

Jiang, L.

L. Jiang, X. Zhang, F. Fang, X. Liu, and L. Zhu, “Wavefront aberration metrology based on transmitted fringe deflectometry,” Appl. Opt. 56(26), 7396–7403 (2017).
[Crossref] [PubMed]

Jung, J. H.

Kang, H. H.

Kang, J. M.

Karimzadeh, A.

A. Karimzadeh, “Integral imaging system optical design with aberration consideration,” Appl. Opt. 54(7), 1765–1769 (2015).
[Crossref]

Kim, E. S.

Kim, J.

Kim, Y.

Lee, B.

Lee, J. H.

Li, D. H.

J. Y. Luo, Q. H. Wang, W. X. Zhao, and D. H. Li, “Autostereoscopic three-dimensional display based on two parallax barriers,” Appl. Opt. 50(18), 2911–2915 (2011).
[Crossref] [PubMed]

W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
[Crossref] [PubMed]

Li, Y.

Lin, C.

Lippmann, G.

G. Lippmann, “La photographie integrale,” C. R. Acad. Sci. 146, 446–451 (1908).

Liu, X.

L. Jiang, X. Zhang, F. Fang, X. Liu, and L. Zhu, “Wavefront aberration metrology based on transmitted fringe deflectometry,” Appl. Opt. 56(26), 7396–7403 (2017).
[Crossref] [PubMed]

Luo, J. Y.

J. Y. Luo, Q. H. Wang, W. X. Zhao, and D. H. Li, “Autostereoscopic three-dimensional display based on two parallax barriers,” Appl. Opt. 50(18), 2911–2915 (2011).
[Crossref] [PubMed]

Martinez-Corral, M.

Martínez-Corral, M.

Martinez-Cuenca, R.

Murata, H.

Y. Igarashi, H. Murata, and M. Ueda, “3D display system using a computer generated integral photography,” Jpn. J. Appl. Phys. 17(9), 1683–1684 (1978).
[Crossref]

Navarro, H.

Pang, B.

Peng, Z.

Pons, A.

Saavedra, G.

Sang, X.

Shin, D.

Stern, A.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

Tolosa, Á.

Tran, T.

Ueda, M.

Y. Igarashi, H. Murata, and M. Ueda, “3D display system using a computer generated integral photography,” Jpn. J. Appl. Phys. 17(9), 1683–1684 (1978).
[Crossref]

Vo, S.

Wang, A. H.

W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
[Crossref] [PubMed]

Wang, K.

Wang, Q. H.

J. Y. Luo, Q. H. Wang, W. X. Zhao, and D. H. Li, “Autostereoscopic three-dimensional display based on two parallax barriers,” Appl. Opt. 50(18), 2911–2915 (2011).
[Crossref] [PubMed]

W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
[Crossref] [PubMed]

Wang, X.

J. Zhang, X. Wang, X. Wu, C. Yang, and Y. Chen, “Wide-viewing integral imaging using fiber-coupled monocentric lens array,” Opt. Express 23(18), 23339–23347 (2015).
[Crossref] [PubMed]

Wu, X.

J. Zhang, X. Wang, X. Wu, C. Yang, and Y. Chen, “Wide-viewing integral imaging using fiber-coupled monocentric lens array,” Opt. Express 23(18), 23339–23347 (2015).
[Crossref] [PubMed]

Wu, Y.

Xiao, X.

Xing, S.

Xu, D.

Yan, B.

Yang, C.

J. Zhang, X. Wang, X. Wu, C. Yang, and Y. Chen, “Wide-viewing integral imaging using fiber-coupled monocentric lens array,” Opt. Express 23(18), 23339–23347 (2015).
[Crossref] [PubMed]

Yang, S.

Yu, C.

Yu, X.

Yuan, J.

Zhang, J.

J. Zhang, X. Wang, X. Wu, C. Yang, and Y. Chen, “Wide-viewing integral imaging using fiber-coupled monocentric lens array,” Opt. Express 23(18), 23339–23347 (2015).
[Crossref] [PubMed]

Zhang, X.

L. Jiang, X. Zhang, F. Fang, X. Liu, and L. Zhu, “Wavefront aberration metrology based on transmitted fringe deflectometry,” Appl. Opt. 56(26), 7396–7403 (2017).
[Crossref] [PubMed]

Zhao, W. X.

J. Y. Luo, Q. H. Wang, W. X. Zhao, and D. H. Li, “Autostereoscopic three-dimensional display based on two parallax barriers,” Appl. Opt. 50(18), 2911–2915 (2011).
[Crossref] [PubMed]

W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
[Crossref] [PubMed]

Zhu, L.

L. Jiang, X. Zhang, F. Fang, X. Liu, and L. Zhu, “Wavefront aberration metrology based on transmitted fringe deflectometry,” Appl. Opt. 56(26), 7396–7403 (2017).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

J. Geng, “Three-dimensional display technologies,” Adv. Opt. Photonics 5(4), 456–535 (2013).
[Crossref] [PubMed]

Appl. Opt. (4)

J. Y. Luo, Q. H. Wang, W. X. Zhao, and D. H. Li, “Autostereoscopic three-dimensional display based on two parallax barriers,” Appl. Opt. 50(18), 2911–2915 (2011).
[Crossref] [PubMed]

A. Karimzadeh, “Integral imaging system optical design with aberration consideration,” Appl. Opt. 54(7), 1765–1769 (2015).
[Crossref]

L. Jiang, X. Zhang, F. Fang, X. Liu, and L. Zhu, “Wavefront aberration metrology based on transmitted fringe deflectometry,” Appl. Opt. 56(26), 7396–7403 (2017).
[Crossref] [PubMed]

C. R. Acad. Sci. (1)

G. Lippmann, “La photographie integrale,” C. R. Acad. Sci. 146, 446–451 (1908).

J. Disp. Technol. (1)

J. Opt. Soc. Am. (1)

H. E. Ives, “Optical properties of a Lippmann lenticulated sheet,” J. Opt. Soc. Am. 21(3), 171–176 (1931).
[Crossref]

Jpn. J. Appl. Phys. (1)

Y. Igarashi, H. Murata, and M. Ueda, “3D display system using a computer generated integral photography,” Jpn. J. Appl. Phys. 17(9), 1683–1684 (1978).
[Crossref]

Nature (1)

Opt. Express (9)

J. Zhang, X. Wang, X. Wu, C. Yang, and Y. Chen, “Wide-viewing integral imaging using fiber-coupled monocentric lens array,” Opt. Express 23(18), 23339–23347 (2015).
[Crossref] [PubMed]

Opt. Lett. (2)

J. S. Jang and B. Javidi, “Improved viewing resolution of three-dimensional integral imaging by use of nonstationary micro-optics,” Opt. Lett. 27(5), 324–326 (2002).
[Crossref] [PubMed]

W. X. Zhao, Q. H. Wang, A. H. Wang, and D. H. Li, “Autostereoscopic display based on two-layer lenticular lenses,” Opt. Lett. 35(24), 4127–4129 (2010).
[Crossref] [PubMed]

Proc. IEEE (1)

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

Other (2)

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley Online Library, 2005).

R. C. Gonzales and R. E. Wood, Digital Image Processing (Prentice Hall, 2002).

Supplementary Material (2)

Name	Description
» Visualization 1	the displayed 3D image with the captured EIA (without correction)
» Visualization 2	the displayed 3D image with the PEIA (with correction)

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Fig. 1 Block diagram of the proposed system: (a) capturing stage, (b) pre-filtering stage, (c) optical reconstruction stage.

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Fig. 2 (a) Reconstruction process of the volume pixel

A^{'}

. (b) Ideal wavefront map; (c) distorted wavefront map. (d) RMS wavefront aberration of the imperfect lens.

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Fig. 3 (a) 81 representative field sections of the

E I_{00}

. (b) 25 field sections in the upper-right corner of Fig. 3(a) and the corresponding field positions (these field sections are plotted, for the lens is rotationally symmetric).

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Fig. 4 (a) Imaging process of a pixel in the integral imaging display. (b) Intensity distribution of spot

A^{'}

vs. intensity distribution of an ideal spot. (c) RMS spot of

A^{'}

(central, RMS radius = 184.8um) vs. RMS spot of an ideal spot (right top corner, RMS radius = 0).

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Fig. 5 Spot diagrams of the lens unit in 25 field sections (RMS spot size: “S₅”: 3.18mm, “S₆”: 3.42mm, “S₇”: 3.68mm, “S₈”: 3.88mm, “S₉”: 0.97mm, “S₁₄”: 2.34mm, “S₁₅”: 2.59mm, “S₁₆”: 3.04mm, “S₁₇”: 3.68mm, “S₁₈”: 4.55mm, “S₂₃”: 1.70mm, “S₂₄”: 1.94mm, “S₂₅”: 2.41mm, “S₂₆”: 3.04mm, “S₂₇”: 3.88mm, “S₃₂”: 1.22mm, “S₃₃”: 1.47mm, “S₃₄”: 1.94mm, “S₃₅”: 2.59mm, “S₃₆”: 3.42mm, “S₄₁”: 0.97mm, “S₄₂”: 1.22mm, “S₄₃”: 1.69mm, “S₄₄”: 2.34mm, “S₄₅”: 3.18mm).

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Fig. 6 RMS wavefront aberrations without correction and with correction.

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Fig. 7 The simulation with the proposed method: (a) original images, (b) reconstructed images without correction, (c) reconstructed images with correction.

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Fig. 8 Optical configuration of the integral imaging display.

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Fig. 9 Two kinds of EIAs: (a) Captured EIA, (b) PEIA.

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Fig. 10 Image quality of the reconstructed 3D object is improved. The top row shows different views of the displayed 3D image with the captured EIA (see Visualization 1). The bottom row shows different views of the displayed 3D image with the PEIA (see Visualization 2).

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Tables (2)

Table 1 Surface Specifications of the lens unit.

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Table 2 Zernike coefficients of field sections S₁~S₈₁.

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Equations (11)

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\begin{array}{l} W_{00} = W_{00} (ξ^{2} + η^{2}, η y_{0}, y_{0}^{2}, ξ x_{0}, x_{0}^{2}) \\ = a_{1} (ξ^{2} + η^{2}) + a_{2} η y_{0} + a_{2} ξ x_{0} + b_{1} {(ξ^{2} + η^{2})}^{2} + b_{2} η y_{0} (ξ^{2} + η^{2}) + b_{2} ξ x_{0} (ξ^{2} + η^{2}) \\ + b_{3} η^{2} y_{0}^{2} + b_{3} ξ^{2} x_{0}^{2} + b_{4} y_{0}^{2} (ξ^{2} + η^{2}) + b_{4} x_{0}^{2} (ξ^{2} + η^{2}) + b_{5} η y_{0}^{3} + b_{5} ξ x_{0}^{3} \end{array}

\begin{array}{l} (x_{0}, y_{0}) = {\begin{cases} (- 0.9 H_{m}, 0.9 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{1}} \\ (- 0.7 H_{m}, 0.9 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{2}} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \dots \dots \\ (0.7 H_{m}, 0.9 H_{m}), \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{8}} \\ (0.9 H_{m}, 0.9 H_{m}), \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{9}} \end{cases}, \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} (x_{0}, y_{0}) = {\begin{cases} (- 0.9 H_{m}, 0.7 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{10}} \\ (- 0.7 H_{m}, 0.7 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{11}} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \dots \dots \\ (0.7 H_{m}, 0.7 H_{m}), \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{17}} \\ (0.9 H_{m}, 0.7 H_{m}), \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{18}} \end{cases} \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \dots \dots \end{matrix} \end{matrix} \\ (x_{0}, y_{0}) = {\begin{cases} (- 0.9 H_{m}, - 0.7 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{64}} \\ (- 0.7 H_{m}, - 0.7 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{65}} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \dots \dots \\ (0.7 H_{m}, - 0.7 H_{m}), \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{71}} \\ (0.9 H_{m}, - 0.7 H_{m}), \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{72}} \end{cases}, \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) = {\begin{cases} (- 0.9 H_{m}, - 0.9 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{73}} \\ (- 0.7 H_{m}, - 0.9 H_{m}), \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{74}} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \dots \dots \\ (0.7 H_{m}, - 0.9 H_{m}), \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{80}} \\ (0.9 H_{m}, - 0.9 H_{m}), \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} (x_{0}, y_{0}) \in Ω_{S_{81}} \end{cases} \end{array}

\begin{array}{l} W_{00} (ξ, η) = W_{[x]} (ξ, η) \\ = Z_{0 [x]} + Z_{1 [x]} η + Z_{2 [x]} ξ + Z_{3 [x]} [2 (ξ^{2} + η^{2}) - 1] + Z_{4 [x]} (η^{2} - ξ^{2}) + Z_{5 [x]} 2 ξ η \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} + Z_{6 [x]} [- 2 η + 3 η (ξ^{2} + η^{2})] + Z_{7 [x]} [- 2 ξ + 3 ξ (ξ^{2} + η^{2})] \\ \begin{matrix}  \end{matrix} + Z_{8 [x]} [1 - 6 (ξ^{2} + η^{2}) + 6 {(ξ^{2} + η^{2})}^{2}] \\ s u b j e c t t o (x_{0}^{A_{00}}, y_{0}^{A_{00}}) \in Ω_{x}, f o r x = S_{1}, S_{2} \dots S_{80}, S_{81} \end{array}

W_{m n} (ξ, η) = W_{00} (ξ - p m, η - p n)

\begin{array}{l} h (x, y; z) = | \frac{1}{λ^{2} l g} \iint_{Ω_{E I}} \iint_{Ω_{L}} δ (x_{0} - x_{0}^{A_{m n}}, y_{0} - y_{0}^{A_{m n}}) \times \exp {\frac{i k}{2 l} [{(ξ - x_{0}^{A_{m n}})}^{2} + {(η - y_{0}^{A_{m n}})}^{2}]} \times P_{m n} (ξ, η) \\ \begin{matrix} {\times \exp {- \frac{i k}{2 f} [{(ξ - p m)}^{2} + {(η - p n)}^{2}]} \times \exp {\frac{i k}{2 g} [{(x - ξ)}^{2} + {(y - η)}^{2}]} d x_{0} d y_{0} d ξ d η |}^{2} \end{matrix} \end{array}

P_{m n} (ξ, η) = {\begin{cases} \begin{matrix} \exp [- i k W_{m n} (ξ, η)] & {\begin{matrix} (ξ - p m) \end{matrix}}^{2} + {(η - p n)}^{2} \leq {(p / 2)}^{2} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} o t h e r w i s e \end{matrix} \end{matrix} \end{matrix} \end{cases}

h (x, y; z) = {| M \int \int_{- \infty}^{+ \infty} \exp [- i k W_{m n} (λ g u, λ g v)] \exp {- i 2 π [(x - M x_{0}^{A_{m n}}) u + (y - M y_{0}^{A_{m n}}) v]} d u d v |}^{2}

R I_{m n} (x, y; z) = I_{m n} (x_{0}, y_{0}; z) * h (x, y; z)

R I (x, y; z) = \sum_{m = - M, n = - N}^{m = M, n = N} R I_{m n} (x, y; z) = \sum_{m = - M, n = - N}^{m = M, n = N} I_{m n} (x_{0}, y_{0}; z) * h (x, y; z)

\begin{array}{l} R I (x, y; z) \begin{matrix}  \end{matrix} = \sum_{m = - M, n = - N}^{m = M, n = N} P I_{m n} (x_{0}, y_{0}; z) * h (x, y; z) \\ \begin{matrix}  \end{matrix} = \sum_{m = - M, n = - N}^{m = M, n = N} I_{m n} (x_{0}, y_{0}; z) * h^{- 1} (x, y; z) * h (x, y; z) \end{array}

\sum_{m = - M, n = - N}^{m = M, n = N} P I_{m n} (x, y; z) = \sum_{m = - M, n = - N}^{m = M, n = N} \frac{1}{F T [h (x, y; z)]} \frac{{| F T [h (x, y; z)] |}^{2}}{{| F T [h (x, y; z)] |}^{2} + K} F T [I_{m n} (x, y; z)]

Surface	Radius(mm)	Thickness(mm)	Glass	Aperture(mm)
1	20.243	5.020	K9	10
2	−20.243	0.520		10
3	16.704	4.097	K9	10
4	19.440			10

Field section	Z0	Z1	Z2	Z3	Z4	Z5	Z6	Z7	Z8
Field “S₁”	66.02	27.62	−27.62	72.47	0	−29.54	14.28	−14.28	6.58
Field “S₂”	56.66	22.42	−28.59	63.07	−5.89	−24.03	11.52	−14.69	6.53
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
Field “S₄₁”	12.91	0	0	19.46	0	0	0	0	6.61
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
Field “S₈₀”	55.66	−22.42	28.59	63.07	−5.89	−24.03	−11.52	14.69	6.53
Field “S₈₁”	66.02	−27.62	−27.62	72.47	0	−29.54	−14.28	14.28	6.58

Surface	Radius(mm)	Thickness(mm)	Glass	Aperture(mm)
1	20.243	5.020	K9	10
2	−20.243	0.520		10
3	16.704	4.097	K9	10
4	19.440			10

Field section	Z0	Z1	Z2	Z3	Z4	Z5	Z6	Z7	Z8
Field “S₁”	66.02	27.62	−27.62	72.47	0	−29.54	14.28	−14.28	6.58
Field “S₂”	56.66	22.42	−28.59	63.07	−5.89	−24.03	11.52	−14.69	6.53
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
Field “S₄₁”	12.91	0	0	19.46	0	0	0	0	6.61
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
Field “S₈₀”	55.66	−22.42	28.59	63.07	−5.89	−24.03	−11.52	14.69	6.53
Field “S₈₁”	66.02	−27.62	−27.62	72.47	0	−29.54	−14.28	14.28	6.58

Abstract

References

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