Abstract

Sparse aperture imaging systems are capable of producing high resolution images while maintaining an overall light collection area that is small compared to a fully filled aperture yielding the same resolution. This is advantageous for applications where size, volume, weight and/or cost are important considerations. However, conventional sparse aperture systems pay the penalty of reduced contrast at midband spatial frequencies. This paper will focus on increasing the midband contrast of sparse aperture imaging systems based on the Golay-9 array. This is one of a family of two-dimensional arrays we have previously examined due to their compact, non-redundant autocorrelations. The modulation transfer function, or normalized autocorrelation, provides a quantitative measure of both the resolution and contrast of an optical imaging system and, along with an average relative midband contrast metric, will be used to compare perturbations to the standard Golay-9 array. Numerical calculations have been performed to investigate the behavior of a Golay-9 array into which autocorrelation redundancy has been introduced and our results have been experimentally verified. In particular we have demonstrated that by proper choice of sub-aperture diameters the average relative midband contrast can be improved by over 55%.

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References

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  1. J. S. Fender, “Synthetic apertures: an overview,” Proc. SPIE 440, 2–7 (1983).
  2. S.-J. Chung, D. W. Miller, and O. L. Weck, “Design and implementation of sparse aperture imaging systems,” Proc. SPIE 4849, 181–192 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002).
    [CrossRef]
  5. M. J. Golay, “Point arrays having compact, nonredundant autocorrelations,” J. Opt. Soc. Am. 61(2), 272–273 (1971).
    [CrossRef]
  6. J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” Proc. SPIE 4091, 43–47 (2000).
    [CrossRef]
  7. J. R. Fienup, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1–8 (2002).
    [CrossRef]
  8. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts and Company, Englewood, CO, 2005), Chap. 6.
  10. Q. Wu, L. Qian, and W. Shen, “Configuration optimization of a kind of sparse-aperture system,” Proc. SPIE 6024, 602420 (2005).
    [CrossRef]

2007

2005

Q. Wu, L. Qian, and W. Shen, “Configuration optimization of a kind of sparse-aperture system,” Proc. SPIE 6024, 602420 (2005).
[CrossRef]

2002

S.-J. Chung, D. W. Miller, and O. L. Weck, “Design and implementation of sparse aperture imaging systems,” Proc. SPIE 4849, 181–192 (2002).
[CrossRef]

R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002).
[CrossRef]

J. R. Fienup, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1–8 (2002).
[CrossRef]

2000

J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” Proc. SPIE 4091, 43–47 (2000).
[CrossRef]

1995

1983

J. S. Fender, “Synthetic apertures: an overview,” Proc. SPIE 440, 2–7 (1983).

1971

Calus, J. R.

R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002).
[CrossRef]

Chung, S.-J.

S.-J. Chung, D. W. Miller, and O. L. Weck, “Design and implementation of sparse aperture imaging systems,” Proc. SPIE 4849, 181–192 (2002).
[CrossRef]

Dierking, M. P.

Duncan, B. D.

Fender, J. S.

J. S. Fender, “Synthetic apertures: an overview,” Proc. SPIE 440, 2–7 (1983).

Fienup, J. R.

J. R. Fienup, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1–8 (2002).
[CrossRef]

J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” Proc. SPIE 4091, 43–47 (2000).
[CrossRef]

Fiete, R. D.

R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002).
[CrossRef]

Golay, M. J.

Harvey, J. E.

Kotha, A.

Miller, D. W.

S.-J. Chung, D. W. Miller, and O. L. Weck, “Design and implementation of sparse aperture imaging systems,” Proc. SPIE 4849, 181–192 (2002).
[CrossRef]

Miller, N. J.

Mooney, J. A.

R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002).
[CrossRef]

Phillips, R. L.

Qian, L.

Q. Wu, L. Qian, and W. Shen, “Configuration optimization of a kind of sparse-aperture system,” Proc. SPIE 6024, 602420 (2005).
[CrossRef]

Shen, W.

Q. Wu, L. Qian, and W. Shen, “Configuration optimization of a kind of sparse-aperture system,” Proc. SPIE 6024, 602420 (2005).
[CrossRef]

Tantalo, T. A.

R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002).
[CrossRef]

Weck, O. L.

S.-J. Chung, D. W. Miller, and O. L. Weck, “Design and implementation of sparse aperture imaging systems,” Proc. SPIE 4849, 181–192 (2002).
[CrossRef]

Wu, Q.

Q. Wu, L. Qian, and W. Shen, “Configuration optimization of a kind of sparse-aperture system,” Proc. SPIE 6024, 602420 (2005).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

R. D. Fiete, T. A. Tantalo, J. R. Calus, and J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41(8), 1957–1969 (2002).
[CrossRef]

Proc. SPIE

J. S. Fender, “Synthetic apertures: an overview,” Proc. SPIE 440, 2–7 (1983).

S.-J. Chung, D. W. Miller, and O. L. Weck, “Design and implementation of sparse aperture imaging systems,” Proc. SPIE 4849, 181–192 (2002).
[CrossRef]

J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” Proc. SPIE 4091, 43–47 (2000).
[CrossRef]

J. R. Fienup, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1–8 (2002).
[CrossRef]

Q. Wu, L. Qian, and W. Shen, “Configuration optimization of a kind of sparse-aperture system,” Proc. SPIE 6024, 602420 (2005).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts and Company, Englewood, CO, 2005), Chap. 6.

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Figures (8)

Fig. 1
Fig. 1

A uniform sub-aperture diameter Golay-9 array with an expansion factor of S = 1.4.

Fig. 2
Fig. 2

MTF of a Golay-9 array with D = 1mm and S = 1.4. The spatial frequency cutoff ρmin, determined by the conservative maximum inscribed circle approach, is shown by the white circle.

Fig. 3
Fig. 3

Two-dimensional MTF for a uniform sub-aperture diameter (D = 1mm) Golay-9 array with an expansion factor of 1.4 for: (a) the R1 set of sub-apertures attenuated by 75%; (b) the R3 set of sub-apertures attenuated by 75%; and, (c) the R2 set of sub-apertures attenuated by 75%.

Fig. 4
Fig. 4

(a) The MTF of a Golay-9 pupil plane array with S = 1.4, DR1 = DR2 = 1mm and DR2 = 2.6mm; and, (b) the binary version of (a) demonstrating the nominal frequency cutoffs as determined by the conventional conservative approach (ρmin-old) and by use of the new spatial frequency availability algorithm (ρmin-new).

Fig. 5
Fig. 5

Numerical simulations of spatial frequency cutoff ρmin , fill factor α and midband average relative contrast (ARC) for varying DR2 and initial thresholds of (a) T1 = 0; and, (b) T1 = 0.01. The dots in (a) denote the values associated with the MTF shown in Fig. 4.

Fig. 6
Fig. 6

Diagram of our experimental setup.

Fig. 7
Fig. 7

An example of one set of distinct lobes for which the center-to-center pixel spacing was recorded in (a) an experimental, magnified PSF; and (b) the theoretical, unmagnified PSF.

Fig. 8
Fig. 8

Theoretical (solid) and experimental (dashed) calculations of spatial frequency cutoff ρmin, fill factor α and midband average relative contrast (ARC) using an initial threshold of T1 = 0.01 and a spatial frequency availability threshold of T2 = 0.98.

Equations (6)

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O T F ( f x , f y ) = { | h ( u , v ) | 2 } | h ( u , v ) | 2 d u d v ,
O T F ( f x , f y ) = P ( x + λ z i f x 2 , y + λ z i f y 2 ) P ( x λ z i f x 2 , y λ z i f y 2 ) d x d y P ( x , y ) d x d y ,
D eff = ρ min λ z i .
α = 3 D R 1 2 + 3 D R 2 2 + 3 D R 3 2 D eff 2 ,
A R C = 0 2 π D λ z i ρ min M T F s p a r s e ( ρ , ϕ ) ρ d ρ d ϕ 2 π D λ z i ρ min M T F f u l l y f i l l e d ( ρ ) ρ d ρ ,
M T F f u l l y f i l l e d = { 2 π [ arccos ( ρ ρ min ) ρ ρ min 1 ( ρ ρ min ) 2 ]         ρ ρ min 0                                                                        otherwise,

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