Abstract

We propose a new type of lensless camera enabling light-field imaging for focusing after image capture and show its feasibilities with some prototyping. The camera basically consists only of an image sensor and Fresnel zone aperture (FZA). Point sources making up the subjects to be captured cast overlapping shadows of the FZA on the sensor, which result in overlapping straight moiré fringes due to multiplication of another virtual FZA in the computer. The fringes generate a captured image by two-dimensional fast Fourier transform. Refocusing is possible by adjusting the size of the virtual FZA. We found this imaging principle is quite analogous to a coherent hologram. Not only the functions of still cameras but also of video cameras are confirmed experimentally by using the prototyped cameras.

© 2018 Optical Society of America

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References

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  1. R. Raskar and J. Tumblin, Computational Photography (CRC Press, 2016).
  2. S. K. Nayar, “Computational cameras: redefining the image,” Computer 39, 30–38 (2006).
    [Crossref]
  3. E. R. Dowski and W. T. Cathey, “Expanded depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [Crossref]
  4. H. Nagahara, S. Kuthirummal, C. Zhou, and S. Nayar, “Flexible depth of field photography,” in European Conference on Computer Vision (ECCV) (2008), p. 73.
  5. O. Cossairt, C. Zhou, and S. Nayar, “Diffusion coded photography for expanded depth of field,” ACM Trans. Graph. 29, 31 (2010).
    [Crossref]
  6. M. Ohta, K. Sakita, T. Shimano, and A. Sakemoto, “Rotationally symmetric wavefront coding for expanded depth of focus with annular phase mask,” Jpn. J. Appl. Phys. 54, 09ME03 (2015).
    [Crossref]
  7. E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
    [Crossref]
  8. R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).
  9. D. G. Stork and P. R. Gill, “Lensless ultra-miniature CMOS computational imagers and sensors,” SensorComm, Barcelona, Spain, 2013, https://www.iaria.org/conferences2013/filesSENSORCOMM13/Keynote_SensorComm2013.pdf .
  10. P. R. Gill and D. G. Stork, “Hardware verification of an ultra-miature computational diffractive imager,” in Proceedings of Computational Optical Sensing and Imaging (2014).
  11. M. S. Asif, A. Ayremlou, A. Sankaranarayanan, A. Veeraraghavan, and R. Baraniuk, “FlatCam: thin, bare-sensor cameras using coded aperture and computation,” arXiv: 1509.00116v2 (2016).
  12. U. D. Desai, J. P. Norris, and R. J. Nemiroff, “Soft gamma-ray telescope for space flight use,” Proc. SPIE 1948, 75–81 (1993).
    [Crossref]
  13. U. D. Desai, L. Orwig, L. Piquet, and C. C. Gaither, “X-ray telescope for small satellites,” Proc. SPIE 3442, 94–104 (1998).
    [Crossref]
  14. S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
    [Crossref]
  15. J. H. Bruning, Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), p. 409.
  16. N. Ohyama, T. Shimano, J. Tsujiuchi, and T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
    [Crossref]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 2.

2015 (1)

M. Ohta, K. Sakita, T. Shimano, and A. Sakemoto, “Rotationally symmetric wavefront coding for expanded depth of focus with annular phase mask,” Jpn. J. Appl. Phys. 54, 09ME03 (2015).
[Crossref]

2010 (1)

O. Cossairt, C. Zhou, and S. Nayar, “Diffusion coded photography for expanded depth of field,” ACM Trans. Graph. 29, 31 (2010).
[Crossref]

2009 (1)

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

2006 (1)

S. K. Nayar, “Computational cameras: redefining the image,” Computer 39, 30–38 (2006).
[Crossref]

1998 (1)

U. D. Desai, L. Orwig, L. Piquet, and C. C. Gaither, “X-ray telescope for small satellites,” Proc. SPIE 3442, 94–104 (1998).
[Crossref]

1995 (1)

1993 (1)

U. D. Desai, J. P. Norris, and R. J. Nemiroff, “Soft gamma-ray telescope for space flight use,” Proc. SPIE 1948, 75–81 (1993).
[Crossref]

1986 (1)

N. Ohyama, T. Shimano, J. Tsujiuchi, and T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[Crossref]

1978 (1)

Asif, M. S.

M. S. Asif, A. Ayremlou, A. Sankaranarayanan, A. Veeraraghavan, and R. Baraniuk, “FlatCam: thin, bare-sensor cameras using coded aperture and computation,” arXiv: 1509.00116v2 (2016).

Ayremlou, A.

M. S. Asif, A. Ayremlou, A. Sankaranarayanan, A. Veeraraghavan, and R. Baraniuk, “FlatCam: thin, bare-sensor cameras using coded aperture and computation,” arXiv: 1509.00116v2 (2016).

Baraniuk, R.

M. S. Asif, A. Ayremlou, A. Sankaranarayanan, A. Veeraraghavan, and R. Baraniuk, “FlatCam: thin, bare-sensor cameras using coded aperture and computation,” arXiv: 1509.00116v2 (2016).

Bredif, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).

Bruning, J. H.

J. H. Bruning, Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), p. 409.

Cannon, T. M.

Cathey, W. T.

Chakrabarti, S. K.

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

Cossairt, O.

O. Cossairt, C. Zhou, and S. Nayar, “Diffusion coded photography for expanded depth of field,” ACM Trans. Graph. 29, 31 (2010).
[Crossref]

Debnath, D.

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

Desai, U. D.

U. D. Desai, L. Orwig, L. Piquet, and C. C. Gaither, “X-ray telescope for small satellites,” Proc. SPIE 3442, 94–104 (1998).
[Crossref]

U. D. Desai, J. P. Norris, and R. J. Nemiroff, “Soft gamma-ray telescope for space flight use,” Proc. SPIE 1948, 75–81 (1993).
[Crossref]

Dowski, E. R.

Duval, G.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).

Fenimore, E. E.

Gaither, C. C.

U. D. Desai, L. Orwig, L. Piquet, and C. C. Gaither, “X-ray telescope for small satellites,” Proc. SPIE 3442, 94–104 (1998).
[Crossref]

Gill, P. R.

P. R. Gill and D. G. Stork, “Hardware verification of an ultra-miature computational diffractive imager,” in Proceedings of Computational Optical Sensing and Imaging (2014).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 2.

Hanrahan, P.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).

Honda, T.

N. Ohyama, T. Shimano, J. Tsujiuchi, and T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[Crossref]

Horowitz, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).

Kuthirummal, S.

H. Nagahara, S. Kuthirummal, C. Zhou, and S. Nayar, “Flexible depth of field photography,” in European Conference on Computer Vision (ECCV) (2008), p. 73.

Levoy, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).

Nagahara, H.

H. Nagahara, S. Kuthirummal, C. Zhou, and S. Nayar, “Flexible depth of field photography,” in European Conference on Computer Vision (ECCV) (2008), p. 73.

Nandi, A.

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

Nayar, S.

O. Cossairt, C. Zhou, and S. Nayar, “Diffusion coded photography for expanded depth of field,” ACM Trans. Graph. 29, 31 (2010).
[Crossref]

H. Nagahara, S. Kuthirummal, C. Zhou, and S. Nayar, “Flexible depth of field photography,” in European Conference on Computer Vision (ECCV) (2008), p. 73.

Nayar, S. K.

S. K. Nayar, “Computational cameras: redefining the image,” Computer 39, 30–38 (2006).
[Crossref]

Nemiroff, R. J.

U. D. Desai, J. P. Norris, and R. J. Nemiroff, “Soft gamma-ray telescope for space flight use,” Proc. SPIE 1948, 75–81 (1993).
[Crossref]

Ng, R.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).

Norris, J. P.

U. D. Desai, J. P. Norris, and R. J. Nemiroff, “Soft gamma-ray telescope for space flight use,” Proc. SPIE 1948, 75–81 (1993).
[Crossref]

Ohta, M.

M. Ohta, K. Sakita, T. Shimano, and A. Sakemoto, “Rotationally symmetric wavefront coding for expanded depth of focus with annular phase mask,” Jpn. J. Appl. Phys. 54, 09ME03 (2015).
[Crossref]

Ohyama, N.

N. Ohyama, T. Shimano, J. Tsujiuchi, and T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[Crossref]

Orwig, L.

U. D. Desai, L. Orwig, L. Piquet, and C. C. Gaither, “X-ray telescope for small satellites,” Proc. SPIE 3442, 94–104 (1998).
[Crossref]

Palit, S.

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

Piquet, L.

U. D. Desai, L. Orwig, L. Piquet, and C. C. Gaither, “X-ray telescope for small satellites,” Proc. SPIE 3442, 94–104 (1998).
[Crossref]

Raskar, R.

R. Raskar and J. Tumblin, Computational Photography (CRC Press, 2016).

Sakemoto, A.

M. Ohta, K. Sakita, T. Shimano, and A. Sakemoto, “Rotationally symmetric wavefront coding for expanded depth of focus with annular phase mask,” Jpn. J. Appl. Phys. 54, 09ME03 (2015).
[Crossref]

Sakita, K.

M. Ohta, K. Sakita, T. Shimano, and A. Sakemoto, “Rotationally symmetric wavefront coding for expanded depth of focus with annular phase mask,” Jpn. J. Appl. Phys. 54, 09ME03 (2015).
[Crossref]

Sankaranarayanan, A.

M. S. Asif, A. Ayremlou, A. Sankaranarayanan, A. Veeraraghavan, and R. Baraniuk, “FlatCam: thin, bare-sensor cameras using coded aperture and computation,” arXiv: 1509.00116v2 (2016).

Sarkar, R.

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

Shimano, T.

M. Ohta, K. Sakita, T. Shimano, and A. Sakemoto, “Rotationally symmetric wavefront coding for expanded depth of focus with annular phase mask,” Jpn. J. Appl. Phys. 54, 09ME03 (2015).
[Crossref]

N. Ohyama, T. Shimano, J. Tsujiuchi, and T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[Crossref]

Stork, D. G.

P. R. Gill and D. G. Stork, “Hardware verification of an ultra-miature computational diffractive imager,” in Proceedings of Computational Optical Sensing and Imaging (2014).

Tsujiuchi, J.

N. Ohyama, T. Shimano, J. Tsujiuchi, and T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[Crossref]

Tumblin, J.

R. Raskar and J. Tumblin, Computational Photography (CRC Press, 2016).

Veeraraghavan, A.

M. S. Asif, A. Ayremlou, A. Sankaranarayanan, A. Veeraraghavan, and R. Baraniuk, “FlatCam: thin, bare-sensor cameras using coded aperture and computation,” arXiv: 1509.00116v2 (2016).

Yadav, V.

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

Zhou, C.

O. Cossairt, C. Zhou, and S. Nayar, “Diffusion coded photography for expanded depth of field,” ACM Trans. Graph. 29, 31 (2010).
[Crossref]

H. Nagahara, S. Kuthirummal, C. Zhou, and S. Nayar, “Flexible depth of field photography,” in European Conference on Computer Vision (ECCV) (2008), p. 73.

ACM Trans. Graph. (1)

O. Cossairt, C. Zhou, and S. Nayar, “Diffusion coded photography for expanded depth of field,” ACM Trans. Graph. 29, 31 (2010).
[Crossref]

Appl. Opt. (2)

Computer (1)

S. K. Nayar, “Computational cameras: redefining the image,” Computer 39, 30–38 (2006).
[Crossref]

Exp. Astron. (1)

S. K. Chakrabarti, S. Palit, D. Debnath, A. Nandi, V. Yadav, and R. Sarkar, “Fresnel zone plate telescopes for X-ray imaging I: experiments with a quasi-parallel beam,” Exp. Astron. 24, 109–126 (2009).
[Crossref]

Jpn. J. Appl. Phys. (1)

M. Ohta, K. Sakita, T. Shimano, and A. Sakemoto, “Rotationally symmetric wavefront coding for expanded depth of focus with annular phase mask,” Jpn. J. Appl. Phys. 54, 09ME03 (2015).
[Crossref]

Opt. Commun. (1)

N. Ohyama, T. Shimano, J. Tsujiuchi, and T. Honda, “An analysis of systematic phase errors due to nonlinearity in fringe scanning systems,” Opt. Commun. 58, 223–225 (1986).
[Crossref]

Proc. SPIE (2)

U. D. Desai, J. P. Norris, and R. J. Nemiroff, “Soft gamma-ray telescope for space flight use,” Proc. SPIE 1948, 75–81 (1993).
[Crossref]

U. D. Desai, L. Orwig, L. Piquet, and C. C. Gaither, “X-ray telescope for small satellites,” Proc. SPIE 3442, 94–104 (1998).
[Crossref]

Other (8)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 2.

J. H. Bruning, Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), p. 409.

R. Raskar and J. Tumblin, Computational Photography (CRC Press, 2016).

H. Nagahara, S. Kuthirummal, C. Zhou, and S. Nayar, “Flexible depth of field photography,” in European Conference on Computer Vision (ECCV) (2008), p. 73.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” (Stanford University, 2005).

D. G. Stork and P. R. Gill, “Lensless ultra-miniature CMOS computational imagers and sensors,” SensorComm, Barcelona, Spain, 2013, https://www.iaria.org/conferences2013/filesSENSORCOMM13/Keynote_SensorComm2013.pdf .

P. R. Gill and D. G. Stork, “Hardware verification of an ultra-miature computational diffractive imager,” in Proceedings of Computational Optical Sensing and Imaging (2014).

M. S. Asif, A. Ayremlou, A. Sankaranarayanan, A. Veeraraghavan, and R. Baraniuk, “FlatCam: thin, bare-sensor cameras using coded aperture and computation,” arXiv: 1509.00116v2 (2016).

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Figures (15)

Fig. 1.
Fig. 1. Basic configuration of the proposed lensless camera with FZA.
Fig. 2.
Fig. 2. Formulation model of proposed lensless camera.
Fig. 3.
Fig. 3. Sample of FZA pattern for four step fringe scanning in the spatial division method (β=5.026  rad/mm2,ϕF=0,π/2,π,3π/2).
Fig. 4.
Fig. 4. Moire fringes by single parallel beam without and with fringe scanning, where sensor size is S=20  mm, pixel size M=1024, d1 is infinity, d2=1  mm, incident angle θx=θy=30°, FZA constant β=5.026  rad/mm2, and division number m=2. (a) Without fringe scanning. (b) With fringe scanning (real part).
Fig. 5.
Fig. 5. Reconstructed images from the moiré images of Fig. 4. (a) Without fringe scanning. (b) With fringe scanning.
Fig. 6.
Fig. 6. Image reconstruction simulation for a defocused point source with and without refocus adjustment of virtual FZA. The abbreviation “FS” stands for fringe scanning in each caption. (a) Moiré image without FS and refocus adjustment. (b) Reconstructed image without FS and refocus adjustment. (c) Moiré image with FS and without refocus adjustment. (d) Reconstructed image with FS and without refocus adjustment. (e) Moiré image with FS and with refocus adjustment. (f) Reconstructed image with FS and with refocus adjustment.
Fig. 7.
Fig. 7. Image reconstruction simulation with and without FS for test image. (a) Original image (256×256). (b) Moiré image with FS (512×512). (c) Reconstructed image with FS (1024×1024). (d) Expanded image of (c). (e) Moiré image without FS (512×512). (f) Reconstructed image without FS.
Fig. 8.
Fig. 8. Photograph of fabricated chromium patterned glass FZA with size of 28  mm×26  mm×1.5  mm, 2×2 FZA area size of 11.25  mm×11.25  mm, and FZA constant β=50.71  rad/mm2.
Fig. 9.
Fig. 9. Prototype lensless camera.
Fig. 10.
Fig. 10. Real-time image capturing and reconstruction demonstration of a prototyped lensless camera.
Fig. 11.
Fig. 11. Experimentally captured images: (a) raw sensor image and (b) reconstructed image, by spatial division FS lensless camera.
Fig. 12.
Fig. 12. Experimentally captured display image of 400  mm×400  mm size in the distance of 280 mm by time division FS lensless camera.
Fig. 13.
Fig. 13. Refocus experiment results using time division fringe scanning with liquid crystal device. By changing virtual FZA constant β at fringe scanning calculation, focus position could be selectively changed. (a) β=9.9  (rad/mm2). (b) β=11.4  (rad/mm2). (c) β=12.2  rad/mm2. (d) Object distances.
Fig. 14.
Fig. 14. Experimental evaluation of the contrast of FZA shadow by a point source for consideration of image resolution of the prototype time division FS lensless camera. The contrast is inverted around the radius of 800 pixels from the center. (a) An experimental sensor image by a point source (2048×2048). (b) Pixel brightness along the brightness inspection axis inspected in (a).
Fig. 15.
Fig. 15. Simulation results of overlapping two defocused point spread functions of parallel lights with several mutual incident angle shifts in diagonal direction [(a)–(f)], where S=20  mm, β=5.026  rad/mm2 (W20=80λ), β=3.770  rad/mm2 (W20=60λ). Two defocused beams generate interference fringes when overlapping. (a) (0°, 0°, 0°, 0°). (b) (2°,2°,2°,2°). (c) (4°,4°,4°,4°). (d) (6°,6°,6°,4°). (e) (8°,8°,8°,8°). (f) (18°,18°,18°,18°).

Equations (30)

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

T(r)=12{1+cos(βr2+ϕ)},
β=1{1+(d2/d1)}2β.
I(x,y)=[12kNIk{1+cos(β{(x+xk)2+(y+yk)2}+ϕF)}]cos(βr2+ϕB),
I(x,y)=12kNIk{cos(βr2+ϕB)+12cos(β{(x+xk)2+(y+yk)2}+ϕF+βr2+ϕB)+12cos(β{(x+xk)2+(y+yk)2}+ϕFβr2ϕB)}12(kNIk)cos(βr2+ϕB)+14kNIkcos{2β(r2+xkx+yky)+ϕF+ϕB}+14kNIkcos{2β(xkx+yky)+ϕFϕB},
I(x,y,ϕF,ϕB)=Ccos(θ0+ϕB)+14kNIk{cos(θ1k+ϕF+ϕB)+cos(θ2k+ϕFϕB)},
02πI(x,y,ϕF,ϕB)cosϕBdϕB=πCcosθ0+π4kNIk{cos(θ1k+ϕF)+cos(θ2k+ϕF)},
02πI(x,y,ϕF,ϕB)sinϕBdϕB=πCsinθ0+π4kNIk{sin(θ1k+ϕF)+sin(θ2k+ϕF)},
02π02πI(x,y,ϕF,ϕB)cosϕBcosϕFdϕBdϕF=π24kNIk(cosθ1k+cosθ2k),
02π02πI(x,y,ϕF,ϕB)sinϕBsinϕFdϕBdϕF=π24kNIk(cosθ1k+cosθ2k),
02π02πI(x,y,ϕF,ϕB)cos(ϕBϕF)dϕBdϕF=π24kNIkcosθ2k,
kNIkcos{2β(xkx+yky)}=4π202π02πI(x,y,ϕF,ϕB)cos(ϕBϕF)dϕBdϕF.
F[kNIkcos{2β(xkx+yky)}]=12kNIk{δ(uβxkπ,vβykπ)+δ(u+βxkπ,v+βykπ)},
kNIksin{2β(xkx+yky)}=4π202π02πI(x,y,ϕF,ϕB)sin(ϕBϕF)dϕBdϕF,
kNIkexp{i2β(xkx+yky)}=4π202π02πI(x,y,ϕF,ϕB)exp{i(ϕBϕF)}dϕBdϕF,
4π2F[02π02πI(x,y,ϕF,ϕB)exp{i(ϕBϕF)}dϕFdϕB]=kNIkδ(uβxkπ,vβykπ).
xk=d2tanθ,
u=βπxk=βπd2tanθ.
umax=βπd2tanθmax=M2S,
tanθmax=Mπ2d2βS.
f=S2tanθmax=d2βS2Mπ,
pmin=2SM.
p(r)=πβr,
p(S2m)=2mπβS,
βmax=MmπS2.
fmax=md2.
Δu=βd2πΔθ=1Sm,
Δθ=πβd2Sm,
Δh=SM·S·Δu=S2MSm.
Δh=mSM,
λ=πβd2cosθ.

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