Abstract

We describe and compare two methods of displaying autostereoscopic three-dimensional images by integral photography. One method is to display the image in front of the lens array, and the other method is to display the image behind the lens array. We compare and discuss these two methods from the viewpoints of lateral resolution, depth resolution, and viewing angle. We also discuss the effect of the optical parameter difference in the pickup and display.

© 2001 Optical Society of America

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References

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  1. G. Lippmann, “La photographie integrale,” C. R. Acad. Sci. 146, 446–451 (1908).
  2. F. Okano, H. Hoshino, J. Arai, I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. 36, 1598–1603 (1997).
    [CrossRef] [PubMed]
  3. B. Lee, S.-W. Min, S. Jung, J.-H. Park, “A three-dimensional display system based on computer-generated integral photography,” J. Soc. 3D Broadcast. Imag. 1, 78–82 (2000).
  4. H. E. Ives, “Optical properties of a Lippmann lenticulated sheet,” J. Opt. Soc. Am. 21, 171–176 (1931).
    [CrossRef]
  5. N. Davies, M. McCormick, L. Yang, “3D imaging systems: a new development,” Appl. Opt. 27, 4520–4528 (1988).
    [CrossRef] [PubMed]
  6. J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradientindex lens-array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
    [CrossRef]
  7. C. B. Burckhardt, “Optimum parameters and resolution limitation of integral photography,” J. Opt. Soc. Am. 58, 71–76 (1968).
    [CrossRef]
  8. T. Okoshi, “Optimum design and depth resolution of lens-sheet and projection-type three-dimensional displays,” Appl. Opt. 10, 2284–2291 (1971).
    [CrossRef] [PubMed]
  9. 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]
  10. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

2000 (1)

B. Lee, S.-W. Min, S. Jung, J.-H. Park, “A three-dimensional display system based on computer-generated integral photography,” J. Soc. 3D Broadcast. Imag. 1, 78–82 (2000).

1998 (2)

1997 (1)

1988 (1)

1971 (1)

1968 (1)

1931 (1)

1908 (1)

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

Arai, J.

Burckhardt, C. B.

Davies, N.

Goodman, J. W.

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

Hoshino, H.

Isono, H.

Ives, H. E.

Jung, S.

B. Lee, S.-W. Min, S. Jung, J.-H. Park, “A three-dimensional display system based on computer-generated integral photography,” J. Soc. 3D Broadcast. Imag. 1, 78–82 (2000).

Lee, B.

B. Lee, S.-W. Min, S. Jung, J.-H. Park, “A three-dimensional display system based on computer-generated integral photography,” J. Soc. 3D Broadcast. Imag. 1, 78–82 (2000).

Lippmann, G.

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

McCormick, M.

Min, S.-W.

B. Lee, S.-W. Min, S. Jung, J.-H. Park, “A three-dimensional display system based on computer-generated integral photography,” J. Soc. 3D Broadcast. Imag. 1, 78–82 (2000).

Okano, F.

Okoshi, T.

Park, J.-H.

B. Lee, S.-W. Min, S. Jung, J.-H. Park, “A three-dimensional display system based on computer-generated integral photography,” J. Soc. 3D Broadcast. Imag. 1, 78–82 (2000).

Yang, L.

Yuyama, I.

Appl. Opt. (4)

C. R. Acad. Sci. (1)

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

J. Opt. Soc. Am. (2)

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

J. Soc. 3D Broadcast. Imag. (1)

B. Lee, S.-W. Min, S. Jung, J.-H. Park, “A three-dimensional display system based on computer-generated integral photography,” J. Soc. 3D Broadcast. Imag. 1, 78–82 (2000).

Other (1)

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

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Figures (21)

Fig. 1
Fig. 1

Conventional integral photography. (a) Pickup; (b) reconstruction.

Fig. 2
Fig. 2

Concepts of the real IP and the virtual IP.

Fig. 3
Fig. 3

Examples of elemental images and integrated images. (a) Elemental images of the real IP; (b) elemental images of the virtual IP; (c) integrated image of the real IP; (d) integrated image of the virtual IP.

Fig. 4
Fig. 4

Shape of pixels of the display panel.

Fig. 5
Fig. 5

Geometry of the real IP and the virtual IP. (a) Real IP; (b) virtual IP.

Fig. 6
Fig. 6

Geometry to calculate the lateral resolution.

Fig. 7
Fig. 7

Intensity distribution of the images of pixels.

Fig. 8
Fig. 8

Lateral resolutions of the real IP and the virtual IP.

Fig. 9
Fig. 9

Amplitude transfer function and the intensity distribution of the image of the pixel around the origin of the marginal depth plane. (a) h(u); (b) I i,r and I i,v .

Fig. 10
Fig. 10

Cross talk when the pixel fill factor is varying (ϕ = 4 cm, l c = 20 cm, b = 5 cm, and p w = 0.2 mm).

Fig. 11
Fig. 11

Cross talk when the lens pitch ϕ is varying (b = 5 cm). (a) Real IP; (b) virtual IP.

Fig. 12
Fig. 12

Cross talk when l c is varying (b = 5 cm). (a) Real IP; (b) virtual IP.

Fig. 13
Fig. 13

Cross talk when b is varying. (a) Real IP with several values of ϕ(l c = 40 cm); (b) virtual IP with several values of ϕ(l c = 40 cm); (c) real IP with several values of l c (ϕ = 2 mm); (d) virtual IP with several values of l c (ϕ = 2 mm).

Fig. 14
Fig. 14

Depth resolution of the IP method.

Fig. 15
Fig. 15

Geometry to calculate the depth resolution. (a) Real IP; (b) virtual IP.

Fig. 16
Fig. 16

Intensity distributions along the depth direction, I i,d,r (z), I i,d,v (z).

Fig. 17
Fig. 17

Spreads in depth directions Δ d,r , Δ d,v when l c is varying (b = 3 cm, ϕ = 2 mm).

Fig. 18
Fig. 18

Spreads in depth directions Δ d,r , Δ d,v when b is varying at several l c (ϕ = 2 mm). (a) Real IP; (b) virtual IP.

Fig. 19
Fig. 19

Spreads in depth directions Δ d,r , Δ d,v when b is varying at several ϕ(l c = 19 cm). (a) Real IP; (b) virtual IP.

Fig. 20
Fig. 20

Geometry to calculate the viewing angles of two IP methods. (a) Viewing angle of real IP; (b) viewing angle of virtual IP.

Fig. 21
Fig. 21

Geometry to obtain the relation between the object distance and the image distance. (a) Pickup; (b) display by the real IP; (c) display by the virtual IP.

Tables (1)

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Table 1 Specifications for the Image Integration Setup

Equations (72)

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1/gr+1/lc=1/f,
1/gv-1/lc=1/f,
gr=lcf/lc-f,
Rdisp=1/ps.
ps,r=|Mr|ps=lc+bgr ps.
ps,r=lc+blc-flcf ps.
Rr=1ps,r=lcflc+blc-fps=lcfRdisplc+blc-f.
Ulx, y=1jλgrexpj k2grx-ξ2+y-η2,
Ulx, y=Ulx, yPx, yexp-j k2f×x2+y2,
hru, v; ξ, η=1jλlc+b- Ulx, y×expj k2lc+bu-x2+v-y2dxdy,
hru, v; ξ, η=expj k2grξ2+η2×expj k2lc+bu2+v2×- Px, yexpj k21gr+1lc+b-1fx2+y2exp-j klc+b×u-ξx+v-ηydxdy,
hru, v=- Px, yexpj k21gr+1lc+b-1f×x2+y2exp-j klc+bux+vydxdy.
Px, y=rectx/ϕrecty/ϕ.
hru, v=-ϕ/2ϕ/2expj k21gr+1lc+b-1fx2+y2×exp-j klc+bux+vydxdy.
hru=-ϕ/2ϕ/2expj k21gr+1lc+b-1fx2×exp-j klc+b uxdx,
hru, v=hruhrv,
sr=1gr+1lc+b-1f,
hru=exp-j k2srulc+b2×-ϕ/2ϕ/2expj ksr2x-ulc+bsr2dx.
FCx=0xcosπ2 t2dt,
FSx=0xsinπ2 t2dt,
hru=FCk|sr|π1/2ϕ2-ulc+bsr+FCk|sr|π1/2ϕ2+ulc+bsr+jFSk|sr|π1/2ϕ2-ulc+bsr+FSk|sr|π1/2ϕ2+ulc+bsr,
Uoξ=rectξ/pw.
Ii,ru=- |hru-ξ|2|Ugξ|2dξ,
Ugξ=Uoξ/Mr,
Ii,ru=-|Mr|pw/2|Mr|pw/2 |hru-ξ|2dξ.
cross talk=ps,r/2 Ii,rudu0 Ii,rudu×100  %,
gv=lcflc+f.
Rv=1ps,v=lcflc+blc+fps=lcfRdisplc+blc+f.
sv=1gv-1lc+b-1f,
hvu=FCk|sv|π1/2ϕ2+ulc+bsv+FCk|sv|π1/2ϕ2-ulc+bsv+jFSk|sv|π1/2ϕ2+ulc+bsv+FSk|sv|π1/2ϕ2-ulc+bsv.
sv=-sr,
hvu=hru.
Mv=lc+b/gv.
Ii,vu=-|Mv|pw/2|Mv|pw/2 |hvu-ξ|2dξ.
cross talk=ps,v/2 Ii,vudu0 Ii,vudu×100  %,
hq,ru, z=FCk|srz|π1/2×ϕ2-u-qϕ1-Mrzzsrz+FCk|srz|π1/2×ϕ2+u-qϕ1-Mrzzsrz+jFSk|srz|π1/2×ϕ2-u-qϕ1-Mrzzsrz+FSk|srz|π1/2×ϕ2+u-qϕ1-Mrzzsrz,
srz=1/gr+1/z-1/f,
Mrz=-gr/z,
Uq,o,rξ=rectξ-qϕ-dq,rpw,
dq,r=qϕ grlc+b.
Ii,q,ru, z=- |hq,ru-ξ, z|2|Uq,g,rξ|2dξ,
Uq,g,rξ, z=Uq,o,rξMrz=Uq,o,r-grξz.
Ii,d,rz=q=-qmax,rqmax,r Ii,q,r0, z,
qmax,r=floorlc+b2gr,
hq,vu, z=FCk|svz|π1/2×ϕ2+u-qϕ1-Mvzzsvz+FCk|svz|π1/2×ϕ2-u-qϕ1-Mvzzsvz+jFSk|svz|π1/2×ϕ2+u-qϕ1-Mvzzsvz+FSk|svz|π1/2×ϕ2-u-qϕ1-Mvzzsvz,
svz=1/gv-1/z-1/f,
Mvz=z/gv.
Uq,o,vξ=rectξ-qϕ+dq,vpw,
dq,v=qϕ gvlc+b.
Ii,q,vu, z=- |hq,vu-ξ, z|2|Uq,g,vξ|2dξ,
Uq,g,vξ, z=Uq,o,vξMvz=Uq,o,vgvξz.
Ii,d,vz=q=-qmax,vqmax,v Ii,q,v0, z,
qmax,v=floorlc+b2gv,
ψ=2 arctanϕ/2grfor real IP,2 arctanϕ/2gvfor virtual IP,gr>f>gv.
d=2L-lc,rtanψr/2for real IP,2L+lc,vtanψv/2for virtual IP,ψv>ψr,
gp=lofplo-fp,
dqx=qϕp-xgp/lo,
1/gr+1/lr=1/fr,
lr+gr:qϕr+ϕrϕp dqx-mx=lr:qϕr-mx,
qϕrlrfplo-fp-ϕrlrfrlr-fr=xlrϕrfpϕplo-fp-m lrfrlr-fr.
lr=frlo-fpfp+fr=frfp lo,
m=ϕrϕp.
dq,bx=qϕp-xgplo+b.
lr,b+gr:qϕr+ϕrϕp dq,bx-mx=lr,b:qϕr-mx.
qϕrlr,blofplo-fplo+b-ϕrlrfrlr-fr=xlr,bϕrlofpϕplo-fplo+b-m lrfrlr-fr.
lr,b=lr+lrlo b.
1/gv-1/lv=1/fv,
lv-gv:qϕv-ϕvϕp dqx-mx=lv:qϕv-mx.
lv=lo-fpfvfp-fv=fvfp lo-2fv,
m=ϕvϕp.
lv,b-gv:qϕv-ϕvϕp dq,bx-mx=lv,b:qϕv-mx,
lv,b=lv+lvlo b.

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