Abstract

Like classical optical design, joint digital-optical design of complex lenses requires a skilled optical designer helped by powerful optical design software. Consequently, if optimization criteria have to be modified to take into account digital post-processing, the convenient optimization environment provided by commercial optical design software needs to be preserved. For that purpose, we define a joint-design criterion based on a merit function that contains terms classically implemented in optical design software but used in a non-standard way. After validation on a simple design problem, the proposed method is applied to the design of a very fast (f/0.75) complex lens. The obtained joint-designed lens is shown to be superior to a classically designed one in terms of weight and image quality in the field.

© 2018 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  5. F. Diaz, F. Goudail, B. Loiseaux, and J.-P. Huignard, “Comparison of depth-of-focus-enhancing pupil masks based on a signal-to-noise-ratio criterion after deconvolution,” J. Opt. Soc. Am. A 27, 2123–2131 (2010).
    [Crossref]
  6. R. Falcón, F. Goudail, C. Kulcsár, and H. Sauer, “Performance limits of binary annular phase masks codesigned for depth-of-field extension,” Opt. Eng. 56, 065104 (2017).
    [Crossref]
  7. F. Diaz, M.-S. L. Lee, X. Rejeaunier, G. Lehoucq, F. Goudail, B. Loiseaux, S. Bansropun, J. Rollin, E. Debes, and P. Mils, “Real-time increase in depth of field of an uncooled thermal camera using several phase-mask technologies,” Opt. Lett. 36, 418–420 (2011).
    [Crossref]
  8. M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
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2018 (2)

2017 (1)

R. Falcón, F. Goudail, C. Kulcsár, and H. Sauer, “Performance limits of binary annular phase masks codesigned for depth-of-field extension,” Opt. Eng. 56, 065104 (2017).
[Crossref]

2015 (2)

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

H. Haim, A. Bronstein, and E. Marom, “Computational multi-focus imaging combining sparse model with color dependent phase mask,” Opt. Express 23, 24547–24556 (2015).
[Crossref]

2014 (1)

2013 (1)

2011 (2)

2010 (2)

2009 (1)

2008 (1)

2004 (1)

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. Van Der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Ashok, A.

Bansropun, S.

Besnerais, G. L.

P. Trouvé, F. Champagnat, G. L. Besnerais, G. Druart, and J. Idier, “Design of a chromatic 3D camera with an end-to-end performance model approach,” in Conference on Computer Vision and Pattern Recognition Workshops (2013), pp. 953–960.

Bronstein, A.

Burcklen, M.-A.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Carles, G.

Champagnat, F.

P. Trouvé, F. Champagnat, G. L. Besnerais, G. Druart, and J. Idier, “Design of a chromatic 3D camera with an end-to-end performance model approach,” in Conference on Computer Vision and Pattern Recognition Workshops (2013), pp. 953–960.

Chen, X.

Christensen, M. P.

T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, “Computational imaging systems: joint design and end-to-end optimality,” Appl. Opt. 47, B86–B103 (2008).
[Crossref]

T. Mirani, M. P. Christensen, S. C. Douglas, D. Rajan, and S. L. Wood, “Optimal co-design of computational imaging system,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’05) (2005), Vol. 2, pp. 597–600.

Debes, E.

Delboulbé, A.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Demenikov, M.

Denel, S.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Diaz, F.

Dong, L.

Douglas, S. C.

T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, “Computational imaging systems: joint design and end-to-end optimality,” Appl. Opt. 47, B86–B103 (2008).
[Crossref]

T. Mirani, M. P. Christensen, S. C. Douglas, D. Rajan, and S. L. Wood, “Optimal co-design of computational imaging system,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’05) (2005), Vol. 2, pp. 597–600.

Dowski, E. R.

E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems,” Proc. SPIE3779 (1999).
[Crossref]

Druart, G.

P. Trouvé, F. Champagnat, G. L. Besnerais, G. Druart, and J. Idier, “Design of a chromatic 3D camera with an end-to-end performance model approach,” in Conference on Computer Vision and Pattern Recognition Workshops (2013), pp. 953–960.

Du, H.

Duhem, F.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Falcón, R.

R. Falcón, F. Goudail, C. Kulcsár, and H. Sauer, “Performance limits of binary annular phase masks codesigned for depth-of-field extension,” Opt. Eng. 56, 065104 (2017).
[Crossref]

Fisher, R.

R. Fisher, B. Tadic-Galeb, and P. Yoder, Optical System Design, 2nd ed. (McGraw-Hill, 2008).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Goudail, F.

R. Falcón, F. Goudail, C. Kulcsár, and H. Sauer, “Performance limits of binary annular phase masks codesigned for depth-of-field extension,” Opt. Eng. 56, 065104 (2017).
[Crossref]

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

F. Diaz, M.-S. L. Lee, X. Rejeaunier, G. Lehoucq, F. Goudail, B. Loiseaux, S. Bansropun, J. Rollin, E. Debes, and P. Mils, “Real-time increase in depth of field of an uncooled thermal camera using several phase-mask technologies,” Opt. Lett. 36, 418–420 (2011).
[Crossref]

F. Diaz, F. Goudail, B. Loiseaux, and J.-P. Huignard, “Comparison of depth-of-focus-enhancing pupil masks based on a signal-to-noise-ratio criterion after deconvolution,” J. Opt. Soc. Am. A 27, 2123–2131 (2010).
[Crossref]

F. Diaz, F. Goudail, B. Loiseaux, and J.-P. Huignard, “Increase in depth of field taking into account deconvolution by optimization of pupil mask,” Opt. Lett. 34, 2970–2972 (2009).
[Crossref]

Haim, H.

Harvey, A. R.

Hui, M.

Huignard, J.-P.

Idier, J.

P. Trouvé, F. Champagnat, G. L. Besnerais, G. Druart, and J. Idier, “Design of a chromatic 3D camera with an end-to-end performance model approach,” in Conference on Computer Vision and Pattern Recognition Workshops (2013), pp. 953–960.

Jia, W.

Johnson, G. E.

E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems,” Proc. SPIE3779 (1999).
[Crossref]

Kong, L.

Koudoli, A.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Kulcsár, C.

R. Falcón, F. Goudail, C. Kulcsár, and H. Sauer, “Performance limits of binary annular phase masks codesigned for depth-of-field extension,” Opt. Eng. 56, 065104 (2017).
[Crossref]

Lee, M.-S. L.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

F. Diaz, M.-S. L. Lee, X. Rejeaunier, G. Lehoucq, F. Goudail, B. Loiseaux, S. Bansropun, J. Rollin, E. Debes, and P. Mils, “Real-time increase in depth of field of an uncooled thermal camera using several phase-mask technologies,” Opt. Lett. 36, 418–420 (2011).
[Crossref]

Lehoucq, G.

Lemonnier, F.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Leprêtre, F.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Liu, M.

Liu, X.

Loiseaux, B.

Marom, E.

Millet, P.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Mils, P.

Mirani, T.

T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, “Computational imaging systems: joint design and end-to-end optimality,” Appl. Opt. 47, B86–B103 (2008).
[Crossref]

T. Mirani, M. P. Christensen, S. C. Douglas, D. Rajan, and S. L. Wood, “Optimal co-design of computational imaging system,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’05) (2005), Vol. 2, pp. 597–600.

Neifeld, M. A.

Pauca, V. P.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. Van Der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Plemmons, R. J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. Van Der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Prasad, S.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. Van Der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Rajan, D.

T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, “Computational imaging systems: joint design and end-to-end optimality,” Appl. Opt. 47, B86–B103 (2008).
[Crossref]

T. Mirani, M. P. Christensen, S. C. Douglas, D. Rajan, and S. L. Wood, “Optimal co-design of computational imaging system,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’05) (2005), Vol. 2, pp. 597–600.

Rejeaunier, X.

Robinson, D.

D. Robinson and D. G. Stork, “Joint design of lens systems and digital image processing,” in International Optical Design (Optical Society of America, 2006), p. WB4.

Rollin, J.

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

F. Diaz, M.-S. L. Lee, X. Rejeaunier, G. Lehoucq, F. Goudail, B. Loiseaux, S. Bansropun, J. Rollin, E. Debes, and P. Mils, “Real-time increase in depth of field of an uncooled thermal camera using several phase-mask technologies,” Opt. Lett. 36, 418–420 (2011).
[Crossref]

Sauer, H.

R. Falcón, F. Goudail, C. Kulcsár, and H. Sauer, “Performance limits of binary annular phase masks codesigned for depth-of-field extension,” Opt. Eng. 56, 065104 (2017).
[Crossref]

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

Stork, D. G.

D. Robinson and D. G. Stork, “Joint design of lens systems and digital image processing,” in International Optical Design (Optical Society of America, 2006), p. WB4.

Tadic-Galeb, B.

R. Fisher, B. Tadic-Galeb, and P. Yoder, Optical System Design, 2nd ed. (McGraw-Hill, 2008).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Thales, S. A.

S. A. Thales, “HELIE night vision goggle for airborne operations,” (2014).

Torgersen, T. C.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. Van Der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Trouvé, P.

P. Trouvé, F. Champagnat, G. L. Besnerais, G. Druart, and J. Idier, “Design of a chromatic 3D camera with an end-to-end performance model approach,” in Conference on Computer Vision and Pattern Recognition Workshops (2013), pp. 953–960.

Van Der Gracht, J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. Van Der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Vettenburg, T.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Wood, S. L.

T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L. Wood, “Computational imaging systems: joint design and end-to-end optimality,” Appl. Opt. 47, B86–B103 (2008).
[Crossref]

T. Mirani, M. P. Christensen, S. C. Douglas, D. Rajan, and S. L. Wood, “Optimal co-design of computational imaging system,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’05) (2005), Vol. 2, pp. 597–600.

Yi, R.

Yoder, P.

R. Fisher, B. Tadic-Galeb, and P. Yoder, Optical System Design, 2nd ed. (McGraw-Hill, 2008).

Zammit, P.

Zhao, Y.

Zhou, Y.

Appl. Opt. (4)

J. Eur. Opt. Soc. (1)

M.-A. Burcklen, F. Diaz, F. Leprêtre, J. Rollin, A. Delboulbé, M.-S. L. Lee, B. Loiseaux, A. Koudoli, S. Denel, P. Millet, F. Duhem, F. Lemonnier, H. Sauer, and F. Goudail, “Experimental demonstration of extended depth-of-field f/1.2 visible high definition camera with jointly optimized phase mask and real-time digital processing,” J. Eur. Opt. Soc. 10, 15046 (2015).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

R. Falcón, F. Goudail, C. Kulcsár, and H. Sauer, “Performance limits of binary annular phase masks codesigned for depth-of-field extension,” Opt. Eng. 56, 065104 (2017).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optica (1)

Proc. SPIE (1)

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. Van Der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Other (7)

P. Trouvé, F. Champagnat, G. L. Besnerais, G. Druart, and J. Idier, “Design of a chromatic 3D camera with an end-to-end performance model approach,” in Conference on Computer Vision and Pattern Recognition Workshops (2013), pp. 953–960.

R. Fisher, B. Tadic-Galeb, and P. Yoder, Optical System Design, 2nd ed. (McGraw-Hill, 2008).

E. R. Dowski and G. E. Johnson, “Wavefront coding: a modern method of achieving high-performance and/or low-cost imaging systems,” Proc. SPIE3779 (1999).
[Crossref]

T. Mirani, M. P. Christensen, S. C. Douglas, D. Rajan, and S. L. Wood, “Optimal co-design of computational imaging system,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’05) (2005), Vol. 2, pp. 597–600.

D. Robinson and D. G. Stork, “Joint design of lens systems and digital image processing,” in International Optical Design (Optical Society of America, 2006), p. WB4.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

S. A. Thales, “HELIE night vision goggle for airborne operations,” (2014).

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Figures (7)

Fig. 1.
Fig. 1. Logarithmic scale (dB) display of hybrid system PSFs before deconvolution when ψ is (a) 10 km, (b) 9.6 m, and (c) 4.8 m. (d) Linear scale PSF cross sections along x and for y=0 including the PSFs’ maxima.
Fig. 2.
Fig. 2. MTF of the optical system including the binary phase mask that was optimized following (a) JIQ and (b) JSOD. Global MTF after pixel filtering and deconvolution of the system with phase mask optimized following (c) JIQ and (d) JSOD.
Fig. 3.
Fig. 3. Simulated images with respect to the object distance provided before deconvolution by the system with the JIQ-optimized phase mask (column 1) and with the JSOD-optimized phase mask (column 2), and after deconvolution by the system with the JIQ-optimized phase mask (column 3) and with the JSOD-optimized phase mask (column 4).
Fig. 4.
Fig. 4. Conventional f/0.75 lens.
Fig. 5.
Fig. 5. Logarithmic scale (dB) display of polychromatic PSF with associated RMS spot size ϕRMS. Conventional system: (a) on-axis, (c) for intermediate HFoV of 10°, and (e) for maximum HFoV of 20°. Co-designed system: (b) on-axis, (d) for HFoV of 10°, and (f) for HFoV of 20°.
Fig. 6.
Fig. 6. MTF in tangential plane (plain curve) and sagittal plane (dots) for (a) the conventional optical system and (b) the co-designed system. MTFs after deconvolution for (c) conventional system and (d) co-designed system.
Fig. 7.
Fig. 7. Simulated images with respect to HFoV Fk, provided before deconvolution by the conventional system (column 1) and by the co-designed system (column 2), and provided after deconvolution by the conventional system (column 3) and by the co-designed system (column 4).

Tables (3)

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Table 1. Binary Phase Mask Optimal Parameters and Associated Optimal Value of Defocus Setting Parameter for JIQ (Column 1) and JSOD (Column 2)

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Table 2. Relative Illumination and Distortion

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Table 3. Estimated Weight of Each Individual Lens Composing the Conventional System (Column 1) and the Co-Designed System (Column 2) Using the Code V WEI Command

Equations (9)

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hψθ(r)=hp(r)*PSFψθ(r),
Yψθ(r)=hψθ(r)*O(r)+n(r),
MSE(d,θ,ψ)=|d(r)*Yψθ(r)O(r)|2,
doptθ=argmind[1Kk=1KMSE(d,θ,ψk)].
JIQ=maxk[MSE(doptθ,θ,ψk)].
IQ(θ,ψ)=10log10SOO(ν)d2νMSE(doptθ,θ,ψ),
ζ(ν)=Snn(ν)SOO(ν).
JSOD(θ)=k=1K(wψkϕϕψkθ)2+m=1M(wmΔcΔcmθ)2+k=2Kνξ(wψk,ν,ξΔMTF)2[MTFψkθ(ξ,ν)MTFψ1θ(ξ,ν)]2+k=2K(wψkΔϕ)2(ϕψkθϕψ1θ)2,
ϕψkθ=(p=1Pq=1Q{wλp2[(δxψk,λp,qθ)2+(δyψk,λp,qθ)2]})1/2.

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