Abstract

A mathematical model for a three-dimensional omnidirectional integral recording camera system that uses either circular- or hexagonal-based spherical surface microlens arrays is derived. The geometry of the image formation and recording process is fully described. Matlab is then used to establish the number of recorded micro-intensity distributions representing a single object point and their dependence on spatial position. The point-spread function for the entire optical process for both close and remote imaging is obtained, and the influence of depth on the point-spread dimensions for each type of microlens and imaging condition is discussed. Comparisons of the two arrangements are made, based on the illustrative numerical results presented.

© 2001 Optical Society of America

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References

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  1. G. Lippmann, “La photographie intégrale,” C. R. Hebd. Seances Acad. Sci. 146, 446–451 (1908).
  2. H. E. Ives, “Optical properties of a Lippmann lenticulated sheet,” J. Opt. Soc. Am. 21, 171–176 (1931).
    [CrossRef]
  3. C. Burckhardt, “Optimum parameters and resolution limitation of integral photography,” J. Opt. Soc. Am. 58, 71–76 (1968).
    [CrossRef]
  4. C. Burckhardt, R. J. Collier, E. T. Doherty, “Formation and inversion of pseudoscopic images,” Appl. Opt. 7, 627–631 (1968).
    [CrossRef] [PubMed]
  5. T. Okoshi, Three Dimensional Imaging Techniques (Academic, London, 1976).
  6. N. Davies, M. McCormick, L. Yang, “Three-dimensional imaging systems: a new development,” Appl. Opt. 27, 4520–4528 (1988).
    [CrossRef] [PubMed]
  7. F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
    [CrossRef]
  8. M. Brewin, M. Forman, N. Davies, “Electronic capture and display of full parallax 3D images,” in Stereoscopic Displays and Virtual Reality Systems II, S. S. Fisher, J. O. Merritt, M. T. Bolas, eds., Proc. SPIE2409, 118–124 (1995).
    [CrossRef]
  9. N. Davies, M. McCormick, “Holoscopic imaging with true 3D-content in full natural colour,” J. Phot. Sci. 40, 46–49 (1992).
  10. N. Davies, M. McCormick, M. Brewin, “Design and analysis of an image transfer system using microlens arrays,” Opt. Eng. 33, 3624–3633 (1994).
    [CrossRef]
  11. M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

1999 (1)

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

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]

1992 (1)

N. Davies, M. McCormick, “Holoscopic imaging with true 3D-content in full natural colour,” J. Phot. Sci. 40, 46–49 (1992).

1988 (1)

1968 (2)

1931 (1)

1908 (1)

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

Arai, J.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

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]

M. Brewin, M. Forman, N. Davies, “Electronic capture and display of full parallax 3D images,” in Stereoscopic Displays and Virtual Reality Systems II, S. S. Fisher, J. O. Merritt, M. T. Bolas, eds., Proc. SPIE2409, 118–124 (1995).
[CrossRef]

Burckhardt, C.

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]

N. Davies, M. McCormick, “Holoscopic imaging with true 3D-content in full natural colour,” J. Phot. Sci. 40, 46–49 (1992).

N. Davies, M. McCormick, L. Yang, “Three-dimensional imaging systems: a new development,” Appl. Opt. 27, 4520–4528 (1988).
[CrossRef] [PubMed]

M. Brewin, M. Forman, N. Davies, “Electronic capture and display of full parallax 3D images,” in Stereoscopic Displays and Virtual Reality Systems II, S. S. Fisher, J. O. Merritt, M. T. Bolas, eds., Proc. SPIE2409, 118–124 (1995).
[CrossRef]

Doherty, E. T.

Forman, M.

M. Brewin, M. Forman, N. Davies, “Electronic capture and display of full parallax 3D images,” in Stereoscopic Displays and Virtual Reality Systems II, S. S. Fisher, J. O. Merritt, M. T. Bolas, eds., Proc. SPIE2409, 118–124 (1995).
[CrossRef]

Furtak, T. E.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

Hoshino, H.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Ives, H. E.

Klein, M. V.

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

Lippmann, G.

G. Lippmann, “La photographie intégrale,” C. R. Hebd. Seances Acad. Sci. 146, 446–451 (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]

N. Davies, M. McCormick, “Holoscopic imaging with true 3D-content in full natural colour,” J. Phot. Sci. 40, 46–49 (1992).

N. Davies, M. McCormick, L. Yang, “Three-dimensional imaging systems: a new development,” Appl. Opt. 27, 4520–4528 (1988).
[CrossRef] [PubMed]

Okano, F.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Okoshi, T.

T. Okoshi, Three Dimensional Imaging Techniques (Academic, London, 1976).

Yang, L.

Yuyama, I.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Appl. Opt. (2)

C. R. Hebd. Seances Acad. Sci. (1)

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

J. Opt. Soc. Am. (2)

J. Phot. Sci. (1)

N. Davies, M. McCormick, “Holoscopic imaging with true 3D-content in full natural colour,” J. Phot. Sci. 40, 46–49 (1992).

Opt. Eng. (2)

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

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Other (3)

M. Brewin, M. Forman, N. Davies, “Electronic capture and display of full parallax 3D images,” in Stereoscopic Displays and Virtual Reality Systems II, S. S. Fisher, J. O. Merritt, M. T. Bolas, eds., Proc. SPIE2409, 118–124 (1995).
[CrossRef]

M. V. Klein, T. E. Furtak, Optics (Wiley, New York, 1986).

T. Okoshi, Three Dimensional Imaging Techniques (Academic, London, 1976).

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Figures (5)

Fig. 1
Fig. 1

Two-tier optical system.

Fig. 2
Fig. 2

Transverse sections of the camera arrangement with a plane x=xP: (a) schematic representation of the intermediary, integral, and recorded optical models of a physical point P; (b) microlenses of the recording array that record a certain optical reconstruction P.

Fig. 3
Fig. 3

Microlens packaging: (a) circular-based microlenses in rectangular net packaging, (b) honeycomb-packed circular-based microlenses, (c) honeycomb-packed hexagonal-based microlenses.

Fig. 4
Fig. 4

Number of recorded microintensity distributions for a hemispherical microlens system: (a) x=const., (b) y=const., (c) z=const.

Fig. 5
Fig. 5

Close and remote imaging.

Tables (2)

Tables Icon

Table 1 Spread versus Depth at Close Imaging for Circular- and Hexagonal-Based Microlenses a

Tables Icon

Table 2 Spread versus Depth at Remote Imaging for Circular- and Hexagonal-Based Microlenses a

Equations (63)

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xk1=ak(zP+S)+xPf1f1+S+zP,
yk1=bk(zP+S)+yPf1f1+S+zP,
zk1=zPf1-zPS-S2f1+S+zP,
xk2=ak(zP+S)+xPf1f1+S+zP,
yk2=bk(zP+S)+yPf1f1+S+zP,
zk2=-zPf1+zPS+S2f1+S+zP.
xk=αk-F(xP-αk)D-|zP|,
yk=βk-F(xP-βk)D-|zP|,
zk=D+F.
dyk=D+F-|zP|D-|zP|δjk,
xP=αkdjh-xkδjhdjh-δjk,
yP=βkdjk-xkδjkdjk-δjk,
zP=Fδjkdjk-δjk-D.
Nmax=ap D-|zP|ϕr(|zP-S)+12,
Nmax=ap D-|zP|32 ϕr(|zP|-S)+12,
A(X, Y)=CDexpik2c (x2+y2)exp(-ikR)dxdy,
A(X, Y)=Cexp-ik2 f (X2+Y2)×Dexp-ik2a (x2+y2)×expikf (xX+yY)dxdy,
xˆ=-xλf,y=-yλf,rˆ=rλf,
A(X, Y)=Cexp-ik2 f (X2+Y2)×D˜exp-ikλ2f22a (x^2+y^2)×exp[-i2π(xˆX+yˆY)]dxˆdyˆ,
A(X, Y)=Cexp-ik2 f (X2+Y2)F(ψ1) * F(ψ2),
F(ψ1)=aλf2exp-iπaλf2 (X2+Y2),
F(ψ2)=CJ1(2πρrˆ)2πρrˆ,
A(ρ)=Ca(6.0736r^2+iπa/λf2)1/2exp-ik2 f ρ2×exp-βρ2πaλf2+i6.0736r^2,
β=6.0736r^2πaλf236.89λ2f4r^4+π2a2.
PSFcirc,cl(X, Y)=A(X, Y)A*(X, Y)=Ca2(36.89r^4+π2a2/λ2f4)1/2×exp-2β πaλf2 (X2+Y2).
PSFcirc,cl(X, Y)=Cαcirc,clexp-X2+Y2ucirc,cl2,
αcirc,cl=a2λ2f2(2.305d4+π2a2λ2)1/2
ucirc,cl2=0.076 d2f2a2+0.328 λ2f2d2.
PSFcirc,clATS = PSFcirc,clATS*PSFcirc,clATS.
PSFcirc,clATS(X, Y)=C(αcirc,clATS)2ucirc,clATSexp-X2+Y22(ucirc,clATS)2.
PSFcirc,clrec(X, Y)=Cαcirc,clrecexp-X2+Y2(ucirc,clrec)2,
PSFcirc,cltotal = PSFcirc,clATS*PSFcirc,clrec.
PSFcirc,cltotal(X, Y)
=C(αcirc,clATS)2αcirc,clrec(ucirc,clATS)2ucirc,clrec[2(ucirc,clATS)2+(ucirc,clrec)2]1/2
×exp-X2+Y22(ucirc,clATS)2+(ucirc,clrec)2.
PSFcirc,cltotal,k(x, y)
=C(αcirc,clATS)2αpsh,clATS(ucirc,clATS)2ucirc,clrec[2(ucirc,clATS)2+(ucirc,clrec)2]1/2
×exp-(x-xk)2+(y-yk)22(ucirc,clATS)2+(ucirc,clrec)2,
PSFcirc,clglobal(x, y)=kPSFcirc,cltotal,k(x, y),
c=f+f2a2 b,
Acirc,rm(X, Y)=CDexp-ik2a2 (x2+y2)×expikf (xX+yY)dxdy.
PSFcirc,rm(X, Y)=Cαcirc,rmexp-X2+Y2ucirc,rm2,
αcirc,rm=λ2α4f2b(2.305d4b2+π2λ6f4a4)1/2,
ucirc,rm2=0.076 d2f2b2a4+0.328 λ2f2d2.
PSFcirc,rmtotal,k(X, Y)
=C(αcirc,rmATS)2αcirc,rmrec(ucirc,rmATS)2ucirc,rmrec[2(ucirc,rmATS)2+(ucirc,rmrec)2]1/2
×exp-X2+Y22(ucirc,rmATS)2+(ucirc,rmrec)2,
PSFcirc,rmglobal(x, y)=kPSFcirc,rmtotal,k(x, y),
F(ψ2)=12π2Y(X+Y/3)×{cos[πlˆ(Y-X3)]-cos(2πlˆY)}+12π2Y(X-Y/3) {cos(2πlˆY)-cos[πlˆ(Y+X3)]}.
F(ψ2)332 l^2exp[(-4.8l^2X2-5.3l^2Y2)].
PSFhex,cl(X, Y)=Cαhex,clexp-X2uhex,cl2-Y2vhex,cl2,
αhex,cl=a2l2f2[(23.04l4+π2a2λ2)(28.09l4+π2a2λ2)]1/2,
uhex,cl=0.243 l2f2a2+0.104 λ2f2l2,
vhex,cl=0.270 l2f2a2+0.094 λ2f2l2.
PSFhex,cltotal,k(X, Y)=C(αhex,clATS)2αhex,clrecΓhex,cl×exp-X2Uhex,cl2-Y2Vhex,cl2,
Γhex,cl=(uhex,clATS)2(vhex,clATS)2uhex,clrecvhex,clrec{[2(uhex,clATS)2+(uhex,clrec)2][2(vhex,clATS)2+(vhex,clrec)2]}1/2,
Uhex,cl2=2(uhex,clATS)2+(uhex,clrec)2,
Vhex,cl2=2(vhex,clATS)2+(vhex,clrec)2,
PSFhex,rmtotal,k(X, Y)=C(αhex,rmATS)2αhex,rmrecΓhex,rm×exp-X2Uhex,rm2-Y2Vhex,rm2,
αhex,rm=a4l4λ223.04l4+π2λ6a4f428.09l4+π2λ6a4f4,
uhex,rm=0.243 l2f2b2a4+0.104 λ2f2l2,
vhex,rm=0.270 l2f2b2a4+0.094 λ2f2l2.
PSFhexglobal(x, y)=kPSFhextotal,k(x, y).

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