Abstract

Integral imaging systems performance has been previously investigated with regard to different parameters such as lateral resolution, field of view, and depth of view. Those parameters are linked to one another, and, since the information capacity of an integral imaging system is finite, there are always trade-offs among them. We use the Shannon number and information capacity limit as figures of merit of integral imaging systems. The Shannon number and information capacity provide compact assessments of the system and are useful for analysis and design. The limitations on the Shannon number and the information capacity of an integral imaging system are determined by the recording and display media.

© 2004 Optical Society of America

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References

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2004

2003

2002

2001

2000

1999

O. Hadar, G. D. Boreman, “Oversampling requirements for pixilated-imager systems,” Opt. Eng. 38, 782–785 (1999).
[CrossRef]

1998

1997

1996

1992

1984

1979

1971

1969

1967

1966

1955

1949

F. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 447–457 (1949).
[CrossRef]

1908

G. Lippmann, “La photographic intergrale,” C. R. Hebd. Seances Acad. Sci. 146, 446–451 (1908).

Arai, J.

Arimoto, H.

Arrizon, V.

Bastiaans, M. J.

Bertero, M.

M. Bertero, E. R. Pike, “Signal processing for linear instrumental systems with noise: a general theory with illustrations from optical imaging and light scattering problems,” in Handbook of Statistics, N. K. Bose, C. R. Rao, eds. (Elsevier Science, Amsterdam, 1993), Vol. 10, pp. 1–45.

Boreman, G. D.

O. Hadar, G. D. Boreman, “Oversampling requirements for pixilated-imager systems,” Opt. Eng. 38, 782–785 (1999).
[CrossRef]

Buckhardt, C. B.

Caballero, M. T.

Catrysse, P. B.

Cover, T. M.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 8.

Dorsch, R. G.

Dragoman, D.

D. Dragoman, “The Wigner distribution in optics and optoelectronics” in Progress in Optics (Elsevier, Amsterdam, 1997), E. Wolf, ed., Vol. 37, pp. 1–56.

Ferreira, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Hadar, O.

O. Hadar, G. D. Boreman, “Oversampling requirements for pixilated-imager systems,” Opt. Eng. 38, 782–785 (1999).
[CrossRef]

Hoshino, H.

Ibáñez-López, C.

Isono, H.

Jang, J. S.

Javidi, B.

Jin, F.

Kaczynski, M. D.

Lippmann, G.

G. Lippmann, “La photographic intergrale,” C. R. Hebd. Seances Acad. Sci. 146, 446–451 (1908).

Lohmann, A. W.

Luckosz, W.

Marti´nez-Corral, M.

Mendlovic, D.

Ojeda-Castaneda, J.

Okano, F.

Okoshi, T.

Okui, M.

Park, S. K.

Pike, E. R.

M. Bertero, E. R. Pike, “Signal processing for linear instrumental systems with noise: a general theory with illustrations from optical imaging and light scattering problems,” in Handbook of Statistics, N. K. Bose, C. R. Rao, eds. (Elsevier Science, Amsterdam, 1993), Vol. 10, pp. 1–45.

Saavedra, G.

Schowegerdt, R.

Shannon, F. C. E.

F. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 447–457 (1949).
[CrossRef]

Stern, A.

Thomas, J. A.

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 8.

Toraldo di Francia, G.

Wandell, B. A.

Yuyama, I.

Zalevsky, Z.

Appl. Opt.

C. R. Hebd. Seances Acad. Sci.

G. Lippmann, “La photographic intergrale,” C. R. Hebd. Seances Acad. Sci. 146, 446–451 (1908).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

O. Hadar, G. D. Boreman, “Oversampling requirements for pixilated-imager systems,” Opt. Eng. 38, 782–785 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. IRE

F. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 447–457 (1949).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

T. M. Cover, J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991), Chap. 8.

D. Dragoman, “The Wigner distribution in optics and optoelectronics” in Progress in Optics (Elsevier, Amsterdam, 1997), E. Wolf, ed., Vol. 37, pp. 1–56.

M. Bertero, E. R. Pike, “Signal processing for linear instrumental systems with noise: a general theory with illustrations from optical imaging and light scattering problems,” in Handbook of Statistics, N. K. Bose, C. R. Rao, eds. (Elsevier Science, Amsterdam, 1993), Vol. 10, pp. 1–45.

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Figures (10)

Fig. 1
Fig. 1

Typical II system. (a) Pickup, (b) reconstruction.

Fig. 2
Fig. 2

One-dimensional optical signal and its two-dimensional phase space. (a) The optical signal arriving at an angle θ and collected by a lens. (b) The phase space obtained from the Wigner transform is a two-dimensional representation as a function of θ (in radians) and its reciprocal {spatial angle frequency [in cycles per radian (cpr)] α. If the FOV of the lens is Δθ and the angular resolution of each ray is Δα/2 cpr, then the acceptance range is a rectangle in the phase space. The area of the rectangle is ΔθΔα and gives the SWP.

Fig. 3
Fig. 3

FOV Δθc is determined by the condition of nonoverlapping elemental images.

Fig. 4
Fig. 4

Aperture-plate II pickup system with a pixelated sensor. The optical field passing through an array of pinholes (aperture plate) is collected by pickup optics (here illustrated by a lens) and captured by a digital sensor with pixels of size pcpxl and fill factor FFc.

Fig. 5
Fig. 5

Dependence of the Shannon number on the optimal gap g [expression (13)] for the setup in Table 1 and M=1/2, λ=0.55 μm.

Fig. 6
Fig. 6

Lenslet-array II capturing system with a pixelated sensor.

Fig. 7
Fig. 7

AMTF of an in-focus pickup system with parameters shown in Table 2. The dotted curve is the lenslet MTF [Eq. (14)], the dashed curve denotes the approximated pure sampling AMTF in expression (10), and the solid curve is the total system AMTF [Eq. (16)]. The solid and the dashed curves almost coincide. The cutoff frequencies due to diffraction, sampling, and out-of-focus are αcdiff=1818 cpr, αcs=126 cpr, and αcerrf=, respectively. It can be seen that the effect of sampling is dominant.

Fig. 8
Fig. 8

AMTF of a severely out-of-focus pickup system. The system is focusing at 0.5 m from the lenslet array while the object is located at 0.05 m from the lenslet array. The dotted curve is the MTF according to Eq. (14), the dashed curve denotes the approximated pure sampling AMTF in expression (10), and the solid curve is the total system AMTF [Eq. (16)]. It can be seen that the out-of-focus effect is dominant.

Fig. 9
Fig. 9

Optical display subsystem.

Fig. 10
Fig. 10

II imaging system with a pixelated sensor. The instantaneous angular FOV of each pixel is FFcpcpxl/Mgc.

Tables (2)

Tables Icon

Table 1 Parameters of the Capturing System

Tables Icon

Table 2 Parameters of the Capturing System

Equations (51)

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W(θ, α)=-uθ+θ2u*θ-θ2×exp(-j2πθα)dθ,
Sc=SWPc=NceSWPce,
SWPce=ΔθcΔαc,
Δθc=2tg-1wcd2gc2 tan-1pc2gc,
MTFc(α)=|OTFc(α)|=|OTFe(α)OTFcd(α)|,
OTFca(α)=1-αλwcasincwca1-αλwcaα/gc,0αwca/λ0,α>wca/λ,
αca=2gcopt/λ=wcaoptλ.
wcaopt=2gcaoptλ.
AOTFcsa(α)=n(-1)nsincpcpxlgcM α-n×OTFcaα-n gcMpcpxlsincα FFcpcpxlgcM,
|α|gcM2pcpxl,
AMTFcsa(α)cosπ pcpxlgcM α,|α|gcM2pcpxl
αcs=gcM2pcpxl.
αc=min(αca, αcs).
Sca=2Ncetan-1wed2ge2 min(αea, αcs)4Ncetan-1pc2gcmin(αea, αcs),
OTFcl(α)=1-αλwclsincwcl1-αλwclαecerr(z),0αwcl/λ0,α>wcl/λ,
ecerr(z)=1z+1gc-1fcl.
AOTFcsl(α)=n(-1)nsincpcpxlgcM α-n×OTFclα-n gcMpcpxlsincα FFcpcpxlgcM,
|α|gcM2pcpxl.
αc=min(αcdiff, αcerrf, αcd),
αcdiff=wcl/λ.
αcerrf1wclecerr(z).
Scl=4Ncetg-1wd2gcmin(αcdiff, αcerrf, αcd)4Ncetg-1pc2gcmin(αcdiff, αcerrf, αcd),
Sc4Ncetan-1wcd2gcαcd4Ncetan-1pc2gcαcd.
Nce=min(Na, Nc).
Sc4 tan-1wcd2gcgcM2pcpxlminNa, WsMpc4 tan-1pcpxl2gcgcM2pcpxlminNa, WsMpc,
ScNs,NaWsMpc.
ST=NeSWPe,
SWPe=Δα min(Δθc, Δθd),
MTFT(α)=[MTFc(α)MTFcd(α)][MTFd(α)MTFdd(α)],
AOTFdsd(α)=n(-1)nsincpdpxlgd α-nAOTFcsd×α-n gdpdpxlHintα-n gpdpxl,
|α|gd2pdpxl.
Δα=2 min(αca, αcs, αda, αsd),
Δα=2 min(αcdif, αcerrf, αcs; αddif, αderrf, αds),
αddiff=wdlλ,
αderrf1wdlederr(z).
ST=2 min(Nce, Nde)min(Δθc, Δθd)×min(αca, αcs, αda, αds),
ST=2 min(Nce, Nde)min(Δθc, Δθd×)min(αcdif, αcerrf, αcs; αddif, αderrf, αds),
CS log2(1+P/N)bits,
N=Ncd+Ndd+Nct,
Δα=2 0α2|MTF(α)|2dα0|MTF(α)|2dα.
Δθ=2 0θ2|f(θ)|2dθ0|f(θ)|2dθ,
SWPT=n=1NeSWPe(n)=n=1NeΔθ(n)Δα,
hcpxl(θ)=rectθFFcpcpxl/Mgc=1|θ|FFcpcpxl2Mgc0|θ|>FFcpcpxl2Mgc.
OTFcpxl(α)=F[h(θ)]F[h(θ)]|α=0=sincFFpcpxlαgcM.
g(x)=[f(x) * h(x)]nδ(x-nΔ),
hsys(x, ϕ)=h(x-ϕ)nδ(x-nΔ).
AOTFsys(ν)=n(-1)nsincΔν-nΔ×Hν-nΔ,|ν|12Δ,
AOTFcsd(α)=n(-1)nsincpcpxlgcM α-n×Hα-n gcMpcpxl,|α|gcM2pcpxl,
AOTFsys(ν)=exp(jπνΔ)×n[rect(x-Δ/2) * h(x)]n×exp(-j2πnνΔ)|ν|12Δ,
AOTFsys(ν)cos(πΔν),|ν|12Δ.
AOTFsys(α)cosπ pcpxlgcM α|α|gcM2pcpxl.

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