Abstract

Objects that temporally vary slowly can be superresolved by the use of two synchronized moving masks such as pinholes or gratings. This approach to superresolution allows one to exceed Abbe’s limit of resolution. Moreover, under coherent illumination, superresolution requires a certain approximation based on the time averaging of intensity rather than of field distribution. When extensive digital postprocessing can be incorporated into the optical system, a detector array and some postprocessing algorithms can replace the grating that is responsible for information decoding. In this way, no approximation is needed and the synchronization that is necessary when two gratings are used is simplified. Furthermore, we present two novel approaches for overcoming distortions when extensive digital postprocessing cannot be incorporated into the optical system. In the first approach, one of the gratings, in the input or at the output plane, is shifted at half the velocity of the other. In the second approach, various spectral regions are transmitted through the system’s aperture to facilitate postprocessing. Experimental results are provided to demonstrate the properties of the proposed methods.

© 2001 Optical Society of America

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References

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  1. M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).
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    [CrossRef]
  3. D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. 36, 2353–2359 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
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  6. D. Mendlovic, A. W. Lohmann, “Space–bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
    [CrossRef]
  7. D. Mendlovic, 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]
  8. D. Mendlovic, D. Farkas, Z. Zalevsky, A. W. Lohmann, “High-frequency enhancement by an optical system for superresolution of temporally restricted objects,” Opt. Lett. 23, 801–803 (1998).
    [CrossRef]
  9. A. Shemer, D. Mendlovic, Z. Zalavsky, J. Garcia, P. G. Martinez, “Superresolving optical system with time multiplexing and computer decoding,” Appl. Opt. 38, 7245–7251 (1999).
    [CrossRef]

1999 (1)

1998 (1)

1997 (3)

1967 (1)

1964 (1)

1960 (1)

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

1952 (1)

M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).

Farkas, D.

Francon, M.

M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).

Garcia, J.

Kartashev, A. I.

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

Kiryuschev, I.

Konforti, N.

Lohmann, A. W.

Lukosz, W.

Martinez, P. G.

Mendlovic, D.

Paris, D. P.

Shemer, A.

Zalavsky, Z.

Zalevsky, Z.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nouvo Climento Suppl. (1)

M. Francon, “Amélioration de résolution d’optique,” Nouvo Climento Suppl. 9, 283–290 (1952).

Opt. Lett. (1)

Opt. Spectra (1)

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

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Figures (10)

Fig. 1
Fig. 1

Optical system for obtaining an increased effective aperture with a physical moving grating at the input plane and a computer-generated virtual grating at the output.

Fig. 2
Fig. 2

Main principles of coherent superresolution in the spectral domain: (a) FT of the input, (b) FT of the field distribution after it passes the first grating, (c) FT after it passes the system’s limited aperture, (d) FT of the field after it passes the second grating and before time averaging.

Fig. 3
Fig. 3

Spectral intensity after superresolution: (a) Desired output (after time averaging of the field distribution). (b) Superresolved distorted output (after time averaging of the intensity).

Fig. 4
Fig. 4

Half-velocity movement technique: (a) Spectrum of the field distribution after the spectrum passes through the second grating. (It has five orders, and V 2 = V 1/2.) (b) Intensity spectrum of (a) after time averaging by the CCD.

Fig. 5
Fig. 5

Slower-moving virtual grating FT as produced in the computer.

Fig. 6
Fig. 6

Multiple-grating approach to superresolution, which yields spectral band separation and nonoverlapping cross correlation terms (dashed lines): (a) even-order grating, (b) odd-order grating.

Fig. 7
Fig. 7

(a) Input image; (b) horizontal cross section of the input image’s spectrum.

Fig. 8
Fig. 8

(a) Distorted input image after the image passes through the system’s finite aperture; (b) horizontal cross section of the distorted input image’s spectrum.

Fig. 9
Fig. 9

Experimental results of the virtual grating method: (a) the reconstructed image without correction; (b) horizontal cross section FT of the reconstructed image without correction.

Fig. 10
Fig. 10

Experimental results of the multiple-grating approach to obtaining improved superresolution: (a) reconstructed image with correction; (b) horizontal cross section of the spectrum of (a).

Equations (21)

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aν=ŨInνrectν/Δν,bν=ŨInν+ν0rectν/Δν,cν=ŨInν-ν0rectν/Δν,
FTUInx=ŨInν=bν  δν-ν0+aν  δν+cν  δν+ν0,
ŨInν * ŨInν=cν * bν  δν-2ν0+aν * bν+cν * aν δν-ν0+aν * aν+bν * bν+cν * cν δν+bν * aν+aν * cν  δν+ν0+bν * cν  δν+2ν0,
UInxG1x-V1t=UInxn=-11 An×exp2πinv0x-V1t=U1x, t.
Ũ1v, t= U1x, texp-2πivxdx=n=-11 AnŨInv-nv0exp-2πinv0V1t=n=-11 AnŨInv-nv0expinϕ.
P˜v=rectv/Δv,
Ũ2ν, t=P˜vŨ1v, t=A-1bνexp-iϕ+A0aν+A1cνexpiϕ.
U2x, t= P˜vŨ1v, texp2πivxdv.
UOutx, t=U2x, tG2x-V2t=m Bm exp2πimν0x-V2tn An× P˜νŨInν-nν0×exp-2πinν0V1texp2πiνxdν=m,n BmAn  P˜νŨInν-nν0×exp2πixνmν0-nV1+mV2ν0tdν.
ŨOutν * ŨOutν=cν * bν  δν-2ν0+2aν * bν+cν * aν δν-ν0+3aν * aν+bν * bν+cν * cν δν+2bν * aν+aν * cν  δν+ν0+bν * cν  δ(ν+2ν0.
IX, t=|UOutX, t|2=m,nm,n AnBmAn*Bm*× P˜0ν1P˜0*ν2ŨInν1-nν0 ŨIn*ν2-nν0exp2πi[Xν1+mν0-ν2-mν0-[m-mV2+n-nV1ν0t}dν1dν2.
1τ-τ/2τ/2exp-2πiν0V1tn-n+m/2-m/2dt=1n-n+1/2m-m=00n-n+1/2m-m0,
τ=1/ν0V1=d/V1
μ1=ν1-nν0,μ2=ν2-nν0,m=m+2n-2n,
IX=|UOutX|2=mm BmBm*  n AnP˜0μ1+nν0ŨInμ1×exp2πiXμ1exp-2πiXν0ndμ1×n An*P˜0*μ2+nν0ŨIn*μ2×exp-2πiXμ2exp2πiXν0ndμ2.
IX=Mean|UOutX, t|2=mm BmBm*  n AnP˜0μ1+nν0ŨInμ1exp2πiXμ1dμ1×n An*P˜0*μ2+nν0ŨIn*μ2×exp-2πiXμ2dμ2,
μ1=ν1-nν0,μ2=ν2-nν0,m=m+n-n.
AnEven=1 |n|N-1/2n even0 |n|>N-1/2otherwise.
AnAll-AnEven=AnOdd=1 |n|N-1/2n odd0 |n|>N-1/2otherwise.
BmEven=1 |m|N-1/2m even0 |m|>N-1/2otherwise,
BmAll-BmEven=BmOdd=1 |m|N-1/2m odd0 |m|>N-1/2otherwise.

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