Abstract

The effects of focusing on the resolution characteristics of integral photography (IP) are analyzed. First, there is an attempt to obtain the resolution characteristics of capture and display systems as the product of their modulation transfer functions (MTFs). Next, the relationship between this overall MTF and focusing during the capture is studied. The results show that, with focusing set at infinity, IP can provide three-dimensional images without remarkable resolution degradation over a wide range of depth.

© 2003 Optical Society of America

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References

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  1. M. G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. (Paris)(4th series) 7, 821–825 (1908).
  2. T. Okoshi, Three-Dimensional Imaging Techniques (Academic, New York, 1971).
  3. J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
    [CrossRef]
  4. F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Real-time three-dimensional pickup and display system based on integral photography,” in Novel Optical Systems and Large-Aperture Imaging, K. D. Bell, M. K. Powers, J. M. Sasian, eds., Proc. SPIE3430, 70–79 (1998).
    [CrossRef]
  5. H. Hoshino, F. Okano, H. Isono, I. Yuyama, “Analysis of resolution limitation of integral photography,” J. Opt. Soc. Am. A 15, 2059–2065 (1998).
    [CrossRef]
  6. C. B. Burckhardt, “Optimum parameters and resolution limitation of integral photography,” J. Opt. Soc. Am. 58, 71–76 (1968).
    [CrossRef]
  7. C. B. Burckhardt, R. J. Collier, E. T. Doherty, “Formation and inversion of pseudoscopic images,” Appl. Opt. 7, 627–631 (1968).
    [CrossRef] [PubMed]
  8. N. Davies, M. McCormick, M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624–3633 (1994).
    [CrossRef]
  9. J.-H. Park, S.-W. Min, B. Javidi, “Analysis of viewing parameters for two display methods based on integral photography,” Appl. Opt. 40, 5217–5232 (2001).
    [CrossRef]
  10. B. Lee, S.-W. Min, B. Javidi, “Theoretical analysis for three-dimensional integral imaging systems with double devices,” Appl. Opt. 41, 4856–4865 (2002).
    [CrossRef] [PubMed]
  11. Y. A. Dudnikov, “On the design of a scheme for producing integral photographs by a combination method,” Sov. J. Opt. Technol. 41, 426–429 (1974).
  12. F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, “Integral three-dimensional video system,” in Three-Dimensional Video and Display: Devices and Systems, B. Javidi, F. Okano, eds., Vol. 76 of Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 2000), pp. 90–116.
  13. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91–103 (1955).
    [CrossRef]
  14. P. Mertz, F. Gray, “A theory of scanning and its relation to the characteristic of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (1934).
    [CrossRef]
  15. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1922).
  16. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 2000).

2002 (1)

2001 (1)

1998 (2)

1994 (1)

N. Davies, M. McCormick, M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624–3633 (1994).
[CrossRef]

1974 (1)

Y. A. Dudnikov, “On the design of a scheme for producing integral photographs by a combination method,” Sov. J. Opt. Technol. 41, 426–429 (1974).

1968 (2)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91–103 (1955).
[CrossRef]

1934 (1)

P. Mertz, F. Gray, “A theory of scanning and its relation to the characteristic of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (1934).
[CrossRef]

1908 (1)

M. G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. (Paris)(4th series) 7, 821–825 (1908).

Arai, J.

J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Real-time three-dimensional pickup and display system based on integral photography,” in Novel Optical Systems and Large-Aperture Imaging, K. D. Bell, M. K. Powers, J. M. Sasian, eds., Proc. SPIE3430, 70–79 (1998).
[CrossRef]

F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, “Integral three-dimensional video system,” in Three-Dimensional Video and Display: Devices and Systems, B. Javidi, F. Okano, eds., Vol. 76 of Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 2000), pp. 90–116.

Brewin, M.

N. Davies, M. McCormick, M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624–3633 (1994).
[CrossRef]

Burckhardt, C. B.

Collier, R. J.

Davies, N.

N. Davies, M. McCormick, M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624–3633 (1994).
[CrossRef]

Doherty, E. T.

Dudnikov, Y. A.

Y. A. Dudnikov, “On the design of a scheme for producing integral photographs by a combination method,” Sov. J. Opt. Technol. 41, 426–429 (1974).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 2000).

Gray, F.

P. Mertz, F. Gray, “A theory of scanning and its relation to the characteristic of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (1934).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91–103 (1955).
[CrossRef]

Hoshino, H.

H. Hoshino, F. Okano, H. Isono, I. Yuyama, “Analysis of resolution limitation of integral photography,” J. Opt. Soc. Am. A 15, 2059–2065 (1998).
[CrossRef]

J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Real-time three-dimensional pickup and display system based on integral photography,” in Novel Optical Systems and Large-Aperture Imaging, K. D. Bell, M. K. Powers, J. M. Sasian, eds., Proc. SPIE3430, 70–79 (1998).
[CrossRef]

F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, “Integral three-dimensional video system,” in Three-Dimensional Video and Display: Devices and Systems, B. Javidi, F. Okano, eds., Vol. 76 of Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 2000), pp. 90–116.

Isono, H.

Javidi, B.

Lee, B.

Lippmann, M. G.

M. G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. (Paris)(4th series) 7, 821–825 (1908).

McCormick, M.

N. Davies, M. McCormick, M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624–3633 (1994).
[CrossRef]

Mertz, P.

P. Mertz, F. Gray, “A theory of scanning and its relation to the characteristic of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (1934).
[CrossRef]

Min, S.-W.

Okano, F.

H. Hoshino, F. Okano, H. Isono, I. Yuyama, “Analysis of resolution limitation of integral photography,” J. Opt. Soc. Am. A 15, 2059–2065 (1998).
[CrossRef]

J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Real-time three-dimensional pickup and display system based on integral photography,” in Novel Optical Systems and Large-Aperture Imaging, K. D. Bell, M. K. Powers, J. M. Sasian, eds., Proc. SPIE3430, 70–79 (1998).
[CrossRef]

F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, “Integral three-dimensional video system,” in Three-Dimensional Video and Display: Devices and Systems, B. Javidi, F. Okano, eds., Vol. 76 of Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 2000), pp. 90–116.

Okoshi, T.

T. Okoshi, Three-Dimensional Imaging Techniques (Academic, New York, 1971).

Park, J.-H.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 2000).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1922).

Yamada, M.

F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, “Integral three-dimensional video system,” in Three-Dimensional Video and Display: Devices and Systems, B. Javidi, F. Okano, eds., Vol. 76 of Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 2000), pp. 90–116.

Yuyama, I.

J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradient-index lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
[CrossRef]

H. Hoshino, F. Okano, H. Isono, I. Yuyama, “Analysis of resolution limitation of integral photography,” J. Opt. Soc. Am. A 15, 2059–2065 (1998).
[CrossRef]

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Real-time three-dimensional pickup and display system based on integral photography,” in Novel Optical Systems and Large-Aperture Imaging, K. D. Bell, M. K. Powers, J. M. Sasian, eds., Proc. SPIE3430, 70–79 (1998).
[CrossRef]

F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, “Integral three-dimensional video system,” in Three-Dimensional Video and Display: Devices and Systems, B. Javidi, F. Okano, eds., Vol. 76 of Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 2000), pp. 90–116.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

P. Mertz, F. Gray, “A theory of scanning and its relation to the characteristic of the transmitted signal in telephotography and television,” Bell Syst. Tech. J. 13, 464–515 (1934).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. (Paris)(4th series) (1)

M. G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. (Paris)(4th series) 7, 821–825 (1908).

Opt. Eng. (1)

N. Davies, M. McCormick, M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624–3633 (1994).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London, Ser. A 231, 91–103 (1955).
[CrossRef]

Sov. J. Opt. Technol. (1)

Y. A. Dudnikov, “On the design of a scheme for producing integral photographs by a combination method,” Sov. J. Opt. Technol. 41, 426–429 (1974).

Other (5)

F. Okano, H. Hoshino, J. Arai, M. Yamada, I. Yuyama, “Integral three-dimensional video system,” in Three-Dimensional Video and Display: Devices and Systems, B. Javidi, F. Okano, eds., Vol. 76 of Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 2000), pp. 90–116.

T. Okoshi, Three-Dimensional Imaging Techniques (Academic, New York, 1971).

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Real-time three-dimensional pickup and display system based on integral photography,” in Novel Optical Systems and Large-Aperture Imaging, K. D. Bell, M. K. Powers, J. M. Sasian, eds., Proc. SPIE3430, 70–79 (1998).
[CrossRef]

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, London, 1922).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 2000).

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Figures (13)

Fig. 1
Fig. 1

Schematic of IP for capture.

Fig. 2
Fig. 2

Schematic of IP for display.

Fig. 3
Fig. 3

Structure used to form the real image of the object by the convex lens.

Fig. 4
Fig. 4

Overall MTF (example 1) of IP.

Fig. 5
Fig. 5

Sampling of the reconstructed image.

Fig. 6
Fig. 6

Overall MTF (example 2) of IP at a spatial frequency of 1440 cpr.

Fig. 7
Fig. 7

Schematic of conventional 2-D photography for capture.

Fig. 8
Fig. 8

Change in MTF of conventional 2-D photography depending on the position of the object. The shooting lens is 10 mm in diameter, has a focal length of 50 mm, and is focused at infinity.

Fig. 9
Fig. 9

Example of overall MTF of conventional 2-D photography at a spatial frequency of 1440 cpr.

Fig. 10
Fig. 10

Reconstructed image positioned on the lens array.

Fig. 11
Fig. 11

Overall MTF of the reconstructed image on the lens array.

Fig. 12
Fig. 12

Integration range.

Fig. 13
Fig. 13

Overall MTF when z^1cz^2d.

Tables (2)

Tables Icon

Table 1 Parameters of Pickup and Display Systems for Example 1

Tables Icon

Table 2 Parameters of Pickup and Display Systems for Example 2

Equations (38)

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αc=1tan-1(νc|z1c|)νc|z1c|.
MTFc(αc)=MTFLc(αc)MTFdc(αc).
MTFd(αd)=MTFLd(αd)MTFdd(αd).
αd=(-gd/gc)αc,
z2d=(-gd/gc)z1c.
MTFT(α)=[MTFLc(α)MTFdc(α)][MTFLd(α)MTFdd(α)]=MTFc(α)MTFd(α).
MTFT(α)=MTFLc(α)MTFLd(αd).
Pc=exp[iπ(xc2+yc2)c(z)/λ],
c(z)=|1/g-1/z-1/fc|,
Pd=exp[iπ(xd2+yd2)d(z)/λ],
d(z)=|1/z+1/g-1/fd|,
MTFLc(αxc, αyc)=1ScpupilPcxc+αxcλ2, yc+αycλ2×Pc*xc-αxcλ2, yc-αycλ2dxcdyc=1Scpupilexp[i2παxcxcc(z)]×exp[i2παycycc(z)]dxcdyc,
MTFLd(αxd, αyd)=1SdpupilPdxd+αxdλ2, yd+αydλ2×Pd*xd-αxdλ2, yd-αydλ2dxddyd=1Sdpupilexp[i2παxdxdd(z)]×exp[i2παydydd(z)]dxddyd,
MTFLc(α)=1Scpupilexp[i2παxc(z)]dxdy,
MTFLd(α)=1Sdpupilexp[i2παxd(z)]dxdy,
1/g-1/z^1c-1/fc=0,
1/z^2d+1/g-1/fd=0.
c(z)=d(z)(z).
MTFLc(α)=MTFLd(α)MTFL(α).
MTFT(α)=[MTFL(α)]2,
MTFL(α)=1Spupilexp[i2παx(z)]dxdy=4rSb (D1cos φ-D2sin φ),
D1=2J1(rb)θ2+sin 2θ4-J3(rb)sin 2θ4+sin 4θ8+J5(rb)sin 4θ8+sin 6θ12-,
D2=J0(rb)sin θ-2J2(rb)sin θ2+sin 3θ6-J4(rb)sin 3θ6+sin 5θ10+J6(rb)sin 5θ10+sin 7θ14-,
β=α(L-z)/|z|,
βnyq=L/2pd.
z2=z1 f/(f+z1).
limz0MTFT[β(z)]=2J1([β(z)pπ/L](w/p))[(β(z)pπ/L)](w/p)2.
MTFL(α)=1Spupilexp[i2παx(z)]dxdy.
x+αλ22+y2=r2,
x-αλ22+y2=r2.
MTFL(α)=1S-[r2-(αλ/2)2]1/2[r2-(αλ/2)2]1/2dy-[(r2-y2)1/2-|αλ/2|](r2-y2)1/2-|αλ/2|×exp(ibx)dx=4Sb0[r2-(αλ/2)2]1/2×sinb(r2-y2)1/2-b|αλ|2dy,
MTFL(α)=4rSbcos φ0θsin(rb cos ϕ)cos ϕ dϕ-sin φ0θcos(rb cos ϕ)cos ϕ dϕ,
sin(rb cos ϕ)=2[J1(rb)cos ϕ-J3(rb)cos 3ϕ+J5(rb)cos 5ϕ-],
cos(rb cos ϕ)=J0(rb)-2[J2(rb)cos 2ϕ-J4(rb)cos 4ϕ+J6(rb)cos 6ϕ-],
0θcos mx cos x dx=sin(m-1)θ2(m-1)+sin(m+1)θ2(m+1)(m±1).
MTFL(α)=4rSbcos φ0θsin(rb cos ϕ)cos ϕ dϕ-sin φ0θcos(rb cos ϕ)cos ϕ dϕ=4rSb (D1cos φ-D2sin φ),
D1=2J1(rb)θ2+sin 2θ4-J3(rb)sin 2θ4+sin 4θ8+J5(rb)sin 4θ8+sin 6θ12-,
D2=J0(rb)sin θ-2J2(rb)sin θ2+sin 3θ6-J4(rb)sin 3θ6+sin 5θ10+J6(rb)sin 5θ10+sin 7θ14-.

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