Abstract

Traditional methods for superresolution have sacrificed field of view for resolution. These methods multiplexed different parts of the signals’ spectrum on different carriers, and thus managed to transfer a wider range of frequency, in a manner that is similar to frequency division multiplexing in classical communication. We propose code division multiplexing for such an application, which has been shown to have superior capabilities. To enable such mutiplexing we propose a unique setup that creates an incoherent cosine transform of the image. A theoretical analysis of the setup is obtained and later compared with the empirical results.

© 2003 Optical Society of America

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References

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  1. E. Abbe, “Beitrage zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microsk. Anat. Entwicklungsmech. 9, 413 (1873).
    [CrossRef]
  2. M. Bertero, C. De Mol, “Super resolution by data inversion,” in Progress in Optics Vol. XXXVI, E. Wolf, ed. (North-Eastland, Amsterdam, 1997), pp. 130–178.
  3. G. Toraldo Di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
  4. G. Toraldo Di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  5. H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
    [CrossRef]
  6. W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18–21 (1963).
  7. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  8. A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectry. 9, 204–206 (1960).
  9. D. Mendelovic, A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
    [CrossRef]
  10. D. Mendelovic, A. W. Lohmann, Z. Zalevsky, “Space-bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
    [CrossRef]
  11. A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  12. D. Mendelovic, A. W. Lohmann, Z. Zalevsky, “Understanding superresolution in Wigner space,” J. Opt. Soc. Am. A 17, 2422–2430 (2000).
    [CrossRef]
  13. Z. Zalevsky, D. Mendelovic, A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” J. Opt. Commun. 163, 79–85 (1999).
    [CrossRef]
  14. A. J. Viterbi, CDMA, Principles of Spread Spectrum Communication (Addison-Wesley, Reading, Mass., 1995).
  15. D. Mendlovic, Z. Zalevsky, N. Konforti, “Joint transform correlator with incoherent output,” J. Am. Opt. Soc. A 11, 3201–3205 (1994).
    [CrossRef]
  16. Shen-ge. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 6, 842–850 (1985).
    [CrossRef]

2000 (1)

1999 (1)

Z. Zalevsky, D. Mendelovic, A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” J. Opt. Commun. 163, 79–85 (1999).
[CrossRef]

1997 (2)

1994 (1)

D. Mendlovic, Z. Zalevsky, N. Konforti, “Joint transform correlator with incoherent output,” J. Am. Opt. Soc. A 11, 3201–3205 (1994).
[CrossRef]

1993 (1)

1985 (1)

Shen-ge. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 6, 842–850 (1985).
[CrossRef]

1982 (1)

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

1969 (1)

1966 (1)

1963 (1)

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18–21 (1963).

1960 (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectry. 9, 204–206 (1960).

1955 (1)

1873 (1)

E. Abbe, “Beitrage zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microsk. Anat. Entwicklungsmech. 9, 413 (1873).
[CrossRef]

Abbe, E.

E. Abbe, “Beitrage zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microsk. Anat. Entwicklungsmech. 9, 413 (1873).
[CrossRef]

Bartelt, H.

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

Bertero, M.

M. Bertero, C. De Mol, “Super resolution by data inversion,” in Progress in Optics Vol. XXXVI, E. Wolf, ed. (North-Eastland, Amsterdam, 1997), pp. 130–178.

De Mol, C.

M. Bertero, C. De Mol, “Super resolution by data inversion,” in Progress in Optics Vol. XXXVI, E. Wolf, ed. (North-Eastland, Amsterdam, 1997), pp. 130–178.

Gartner, W.

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18–21 (1963).

George, N.

Shen-ge. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 6, 842–850 (1985).
[CrossRef]

Kartashev, A. I.

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectry. 9, 204–206 (1960).

Konforti, N.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Joint transform correlator with incoherent output,” J. Am. Opt. Soc. A 11, 3201–3205 (1994).
[CrossRef]

Lohmann, A. W.

Lukosz, W.

Mendelovic, D.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, N. Konforti, “Joint transform correlator with incoherent output,” J. Am. Opt. Soc. A 11, 3201–3205 (1994).
[CrossRef]

Toraldo Di Francia, G.

Viterbi, A. J.

A. J. Viterbi, CDMA, Principles of Spread Spectrum Communication (Addison-Wesley, Reading, Mass., 1995).

Wang, Shen-ge.

Shen-ge. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 6, 842–850 (1985).
[CrossRef]

Zalevsky, Z.

D. Mendelovic, A. W. Lohmann, Z. Zalevsky, “Understanding superresolution in Wigner space,” J. Opt. Soc. Am. A 17, 2422–2430 (2000).
[CrossRef]

Z. Zalevsky, D. Mendelovic, A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” J. Opt. Commun. 163, 79–85 (1999).
[CrossRef]

D. Mendelovic, A. W. Lohmann, Z. Zalevsky, “Space-bandwidth product adaptation and its application to superresolution: examples,” J. Opt. Soc. Am. A 14, 563–567 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, N. Konforti, “Joint transform correlator with incoherent output,” J. Am. Opt. Soc. A 11, 3201–3205 (1994).
[CrossRef]

Appl. Opt. (1)

Shen-ge. Wang, N. George, “Fresnel zone transforms in spatially incoherent illumination,” Appl. Opt. 24, 6, 842–850 (1985).
[CrossRef]

Archiv. Microsk. Anat. Entwicklungsmech. (1)

E. Abbe, “Beitrage zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv. Microsk. Anat. Entwicklungsmech. 9, 413 (1873).
[CrossRef]

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

D. Mendlovic, Z. Zalevsky, N. Konforti, “Joint transform correlator with incoherent output,” J. Am. Opt. Soc. A 11, 3201–3205 (1994).
[CrossRef]

J. Opt. Commun. (1)

Z. Zalevsky, D. Mendelovic, A. W. Lohmann, “Super resolution optical systems for objects with finite sizes,” J. Opt. Commun. 163, 79–85 (1999).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

H. Bartelt, A. W. Lohmann, “Optical processing of 1-D signals,” Opt. Commun. 42, 87–91 (1982).
[CrossRef]

Opt. Spectry. (1)

A. I. Kartashev, “Optical systems with enhanced resolving power,” Opt. Spectry. 9, 204–206 (1960).

Z. Physik (1)

W. Gartner, A. W. Lohmann, “An experiment going beyond Abbe’s limit of diffraction,” Z. Physik 174, 18–21 (1963).

Other (2)

A. J. Viterbi, CDMA, Principles of Spread Spectrum Communication (Addison-Wesley, Reading, Mass., 1995).

M. Bertero, C. De Mol, “Super resolution by data inversion,” in Progress in Optics Vol. XXXVI, E. Wolf, ed. (North-Eastland, Amsterdam, 1997), pp. 130–178.

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Figures (13)

Fig. 1
Fig. 1

Wigner properties.

Fig. 2
Fig. 2

(a) Classic superresolution setup, (b) a flowchart illustrating the stages in the mathematical analysis of the optical setup.

Fig. 3
Fig. 3

Classic superresolution shown in the Wigner plane.

Fig. 4
Fig. 4

CDMA superresolution shown in the Wigner plane.

Fig. 5
Fig. 5

(a) Full optical setup, (b) a flowchart illustrating the stages in the mathematical analysis of the optical setup and the computerized image retrieval.

Fig. 6
Fig. 6

CDMA mask function broken into three subfunctions for each of the three ranges.

Fig. 7
Fig. 7

Cosine transform setup.

Fig. 8
Fig. 8

Simulation input and output: (top) original input image used, (middle) output of the system with no resolution enhancement, (bottom) output of CDMA superresolving.

Fig. 9
Fig. 9

Input image spectrum and system’s bandwidth (outlined by the rectangle).

Fig. 10
Fig. 10

CDMA mask.

Fig. 11
Fig. 11

Output of cosine setup with coherent illumination.

Fig. 12
Fig. 12

Output of the cosine setup for incoherent illumination but without a mirror.

Fig. 13
Fig. 13

Output of the cosine setup with incoherent illumination. Arrows mark where the fringes appeared.

Equations (24)

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SW=ΔxΔv.
Wx, v=- ux+x2 u* x-x2×exp-2πivxdx,
SWBx, v=1Wx, v>Wthresh0otherwise.
N=ΔxΔδ=Δvδv=ΔxΔv.
SWIvx, v  SWYvx, v.
NsignalNsystem.
CDMAv=n=-11 gnv-nv0.
givgjv=0 ij giv=0 i=-1, 0, 1  v  -v02, v02.
gridv=δv+δv-v0+δv+v0.
Ix=u0x * CDMAxgridx.
Iv=u0vn=-11 gnv-nv0 * δv+δv-1+δv+1 =u0vn=-11 gnv-nv+u0v-v0×n=-11 gnv-n+1v0+u0v+v0n=-11 gnv-n-1v0.
Ov=Iv · rectv/Δv =u0vg0v+u0v-v0g-1v+u0v+v0g1v.
R1v=Ov * gridv=u0vg0v+u0v-2v0g0v-v0+u0v+2v0g0v+v0+u0v-v0g-1v +u0v-2v0g-1v-v0 +u0vg-1v-v0 +u0v+v0g1v+u0vg1v-v0 +u0v+2v0g1v+v0.
R2v=R2vCDMAtv=u0vg0v+u0vg-1v+v0+u0vg1×v-v0=u0vCDMAtvdownsampling u0v.
Cx=expjkzjλzexpj k2z x2  Ixexpj k2z x2exp-j 2πλz xxdx =expjkzjλzexpj k2z x2  Ixexpj k2z x2exp-j 2πλz xx+Ixexpj k2z x2expj 2πλz xxdx =2 expjkzjλzexpj k2z x2  Ixexpj k2z x2cos2πλz xxdx.
Cx, t=2 expjkzjλzexpj k2z x2  Ix×expj k2z x2cos2πλz xxdx.
|Cx|2=4λ2z2  Ix1, tĪx2, texpj k2z x12exp-j k2z x22cos2πλz x1xcos2πλz x2xdx1dx2dt =4λ2z2  |Ix1|2δx1-x2expj k2z x12exp-j k2z x22cos2πλz x1xcos2πλz x2xdx1dx2 =4λ2z2  |Ix|2cos22πλz xxdx=2λ2z2  |Ix|2dx+2λ2z2  |Ix|2cos4πλz xxdx =A0+2λ2z2  |Ix|2cos4πλz xxdx.
Fx, t=Cx, thx, tgx, t+C-x, th-x, tg-x, t.
Oxt, t=expjkzjλzexpj k2z x2  Fx, t×expj k2z x2exp-j 2πλz xxdx.
|Ox¯|2=4λ2z2  Fx, tF¯x, texpj k2z x2×exp-j 2πλz xxexp-j k2z x2×expj 2πλz xxdxdxdt.
 Fx, tFx, t¯dt=hxhxgx, tgx, tcx, tcx, t +hxh-xgx, tg-x, tcx, tc-x, t +h-xhxg-x, tgx, tc-x, tcx, t +h-xh-xg-x, tg-x, tc-x, tc-x, t.
 Fx, tFx, t¯dt=hxhxδx-xcx, tcx, t +hxh-xδx+xcx, tc-x, t +h-xhxδx+xc-x, tcx, t +h-xh-xδx-xc-x, tc-x, t.
|Ox¯|2=4λ2z2  |hx|2|cx|2+|h-x|2|c-x|2+|h-x|2|cx|2+|hx|2|c-x|2exp2-j 2πλz xxdx =B0+4λ2z2  |hx|2|cx|2+|h-x|2|c-x|2exp-j 4πλz xxdx =B0+4λ2z2  |hx|2|cx|2×cos4πλz xxdx.
Entrance pupil areaHalf sphere from diffuser=2h22πz2.

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