Abstract

Optical sparse aperture (OSA) can greatly improve the spatial resolution of optical system. However, because of its aperture dispersion and sparse, its mid-frequency modulation transfer function (MTF) are significantly lower than that of a single aperture system. The main focus of this paper is on the mid-frequency MTF compensation of the optical sparse aperture system. Firstly, the principle of the mid-frequency MTF decreasing and missing of optical sparse aperture are analyzed. This paper takes the filling factor as a clue. The method of processing the mid-frequency MTF decreasing with large filling factor and method of compensation mid-frequency MTF with small filling factor are given respectively. For the MTF mid-frequency decreasing, the image spatial-variant restoration method is proposed to restore the mid-frequency information in the image; for the mid-frequency MTF missing, two images obtained by two system respectively are fused to compensate the mid-frequency information in optical sparse aperture image. The feasibility of the two method are analyzed in this paper. The numerical simulation of the system and algorithm of the two cases are presented using Zemax and Matlab. The results demonstrate that by these two methods the mid-frequency MTF of OSA system can be compensated effectively.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. B. Meinel, “Aperture Synthesis Using Independent Telescopes,” Appl. Opt. 9(11), 2501 (1970).
    [Crossref] [PubMed]
  2. S.-J. Chung, D. W. Miller, and O. L. de Weck, “Design and implementation of sparse aperture imaging systems,” Proceedings of SPIE - The International Society for Optical Engineering 4849, 181–192 2002 (2002).
    [Crossref]
  3. J. S. Fender, “Synthetic apertures: An Overview,” Proceedings of SPIE - The International Society for Optical Engineering 440, 2–7 (1984).
    [Crossref]
  4. J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” Proc. SPIE 4091, 43–47 (2000).
    [Crossref]
  5. A. J. Stokes, B. D. Duncan, and M. P. Dierking, “Improving mid-frequency contrast in sparse aperture optical imaging systems based upon the Golay-9 array,” Opt. Express 18(5), 4417–4427 (2010).
    [Crossref] [PubMed]
  6. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007).
    [Crossref] [PubMed]
  7. L. M. Mugnier, G. Rousset, and F. Cassaing, “Aperture configuration optimality criterion for phased arrays of optical telescopes,” J. Opt. Soc. Am. A 13(12), 2367–2374 (1996).
    [Crossref]
  8. I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 (2005).
    [Crossref]
  9. P. S. Salvaggio, J. R. Schott, and D. M. McKeown, “Genetic apertures: an improved sparse aperture design framework,” Appl. Opt. 55(12), 3182–3191 (2016).
    [Crossref] [PubMed]
  10. K. D. Bell, R. H. Boucher, R. Vacek, and M. Hopkins, “Assessment of large aperture lightweight imaging concepts,” in Aerospace Applications Conference, 1996. Proceedings (1996), pp. 187–203.
    [Crossref]
  11. J. R. Fienup and J. J. Miller, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proceedings of SPIE - The International Society for Optical Engineering 4792, 1–8 (2002).
    [Crossref]
  12. D. Wang and S. Tao, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46(10), 13053–13057 (2007).
    [Crossref]
  13. Z. Zhou, D. Wang, and Y. Wang, “Effect of noise on the performance of image restoration in an optical sparse aperture system,” J. Opt. 13(7), 075502 (2011).
    [Crossref]
  14. E. Huggins, Introduction to Fourier Optics (McGraw-Hill, 1968).
  15. P. S. Salvaggio, J. R. Schott, and D. M. McKeown, “Validation of modeled sparse aperture post-processing artifacts,” Appl. Opt. 56(4), 761–770 (2017).
    [Crossref] [PubMed]
  16. Y. Jiang, Z. Wang, M. Zhao, and L. Zhang, “MTF-based Research on the Minimum Fill Factor in Optical Sparse Aperture System,” in Fourth International Conference on Intelligent Computation Technology and Automation(2011), pp. 612–615.
  17. H. Hotelling, “Analysis of a complex of statistical variables into principal components,” Br. J. Educ. Psychol. 24(6), 417–520 (1933).
    [Crossref]
  18. K. Elsayed and C. Lacor, “Comparison of Kriging, RBFNN, RBF and polynomial regression surrogates in design optimization,” in Eleventh International Conference of Fluid Dynamics(2013).
  19. B. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. 19(4), 285–288 (1971).
    [Crossref]
  20. B. Hunt, “A Theorem on the Difficulty of Numerical Deconvolution,” AC-20, 94–95 (1972).
    [Crossref]
  21. B. R. Hunt, “The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer,” IEEE Trans. Comput. 22(9), 805–812(1973).
  22. C. R. Vogel, “Computational Methods for Inverse Problems,” Frontiers in Applied Mathematics (2002).

2017 (1)

2016 (1)

2011 (1)

Z. Zhou, D. Wang, and Y. Wang, “Effect of noise on the performance of image restoration in an optical sparse aperture system,” J. Opt. 13(7), 075502 (2011).
[Crossref]

2010 (1)

2007 (2)

N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007).
[Crossref] [PubMed]

D. Wang and S. Tao, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46(10), 13053–13057 (2007).
[Crossref]

2005 (1)

I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 (2005).
[Crossref]

2000 (1)

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

1996 (1)

1973 (1)

B. R. Hunt, “The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer,” IEEE Trans. Comput. 22(9), 805–812(1973).

1971 (1)

B. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. 19(4), 285–288 (1971).
[Crossref]

1970 (1)

1933 (1)

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” Br. J. Educ. Psychol. 24(6), 417–520 (1933).
[Crossref]

Bell, K. D.

K. D. Bell, R. H. Boucher, R. Vacek, and M. Hopkins, “Assessment of large aperture lightweight imaging concepts,” in Aerospace Applications Conference, 1996. Proceedings (1996), pp. 187–203.
[Crossref]

Boucher, R. H.

K. D. Bell, R. H. Boucher, R. Vacek, and M. Hopkins, “Assessment of large aperture lightweight imaging concepts,” in Aerospace Applications Conference, 1996. Proceedings (1996), pp. 187–203.
[Crossref]

Cassaing, F.

Dierking, M. P.

Duncan, B. D.

Elsayed, K.

K. Elsayed and C. Lacor, “Comparison of Kriging, RBFNN, RBF and polynomial regression surrogates in design optimization,” in Eleventh International Conference of Fluid Dynamics(2013).

Fienup, J. R.

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

Hopkins, M.

K. D. Bell, R. H. Boucher, R. Vacek, and M. Hopkins, “Assessment of large aperture lightweight imaging concepts,” in Aerospace Applications Conference, 1996. Proceedings (1996), pp. 187–203.
[Crossref]

Hotelling, H.

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” Br. J. Educ. Psychol. 24(6), 417–520 (1933).
[Crossref]

Hunt, B.

B. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. 19(4), 285–288 (1971).
[Crossref]

Hunt, B. R.

B. R. Hunt, “The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer,” IEEE Trans. Comput. 22(9), 805–812(1973).

Jiang, Y.

Y. Jiang, Z. Wang, M. Zhao, and L. Zhang, “MTF-based Research on the Minimum Fill Factor in Optical Sparse Aperture System,” in Fourth International Conference on Intelligent Computation Technology and Automation(2011), pp. 612–615.

Kahrizi, M.

I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 (2005).
[Crossref]

Lacor, C.

K. Elsayed and C. Lacor, “Comparison of Kriging, RBFNN, RBF and polynomial regression surrogates in design optimization,” in Eleventh International Conference of Fluid Dynamics(2013).

McKeown, D. M.

Meinel, A. B.

Miller, N. J.

Mugnier, L. M.

Rousset, G.

Salvaggio, P. S.

Schott, J. R.

Stokes, A. J.

Tao, S.

D. Wang and S. Tao, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46(10), 13053–13057 (2007).
[Crossref]

Tcherniavski, I.

I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 (2005).
[Crossref]

Vacek, R.

K. D. Bell, R. H. Boucher, R. Vacek, and M. Hopkins, “Assessment of large aperture lightweight imaging concepts,” in Aerospace Applications Conference, 1996. Proceedings (1996), pp. 187–203.
[Crossref]

Wang, D.

Z. Zhou, D. Wang, and Y. Wang, “Effect of noise on the performance of image restoration in an optical sparse aperture system,” J. Opt. 13(7), 075502 (2011).
[Crossref]

D. Wang and S. Tao, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46(10), 13053–13057 (2007).
[Crossref]

Wang, Y.

Z. Zhou, D. Wang, and Y. Wang, “Effect of noise on the performance of image restoration in an optical sparse aperture system,” J. Opt. 13(7), 075502 (2011).
[Crossref]

Wang, Z.

Y. Jiang, Z. Wang, M. Zhao, and L. Zhang, “MTF-based Research on the Minimum Fill Factor in Optical Sparse Aperture System,” in Fourth International Conference on Intelligent Computation Technology and Automation(2011), pp. 612–615.

Zhang, L.

Y. Jiang, Z. Wang, M. Zhao, and L. Zhang, “MTF-based Research on the Minimum Fill Factor in Optical Sparse Aperture System,” in Fourth International Conference on Intelligent Computation Technology and Automation(2011), pp. 612–615.

Zhao, M.

Y. Jiang, Z. Wang, M. Zhao, and L. Zhang, “MTF-based Research on the Minimum Fill Factor in Optical Sparse Aperture System,” in Fourth International Conference on Intelligent Computation Technology and Automation(2011), pp. 612–615.

Zhou, Z.

Z. Zhou, D. Wang, and Y. Wang, “Effect of noise on the performance of image restoration in an optical sparse aperture system,” J. Opt. 13(7), 075502 (2011).
[Crossref]

Appl. Opt. (4)

Br. J. Educ. Psychol. (1)

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” Br. J. Educ. Psychol. 24(6), 417–520 (1933).
[Crossref]

IEEE Trans. Audio Electroacoust. (1)

B. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. 19(4), 285–288 (1971).
[Crossref]

IEEE Trans. Comput. (1)

B. R. Hunt, “The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer,” IEEE Trans. Comput. 22(9), 805–812(1973).

J. Opt. (1)

Z. Zhou, D. Wang, and Y. Wang, “Effect of noise on the performance of image restoration in an optical sparse aperture system,” J. Opt. 13(7), 075502 (2011).
[Crossref]

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

Opt. Eng. (2)

I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 (2005).
[Crossref]

D. Wang and S. Tao, “Experimental study on imaging and image restoration of optical sparse aperture systems,” Opt. Eng. 46(10), 13053–13057 (2007).
[Crossref]

Opt. Express (1)

Proc. SPIE (1)

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

Other (9)

S.-J. Chung, D. W. Miller, and O. L. de Weck, “Design and implementation of sparse aperture imaging systems,” Proceedings of SPIE - The International Society for Optical Engineering 4849, 181–192 2002 (2002).
[Crossref]

J. S. Fender, “Synthetic apertures: An Overview,” Proceedings of SPIE - The International Society for Optical Engineering 440, 2–7 (1984).
[Crossref]

K. D. Bell, R. H. Boucher, R. Vacek, and M. Hopkins, “Assessment of large aperture lightweight imaging concepts,” in Aerospace Applications Conference, 1996. Proceedings (1996), pp. 187–203.
[Crossref]

J. R. Fienup and J. J. Miller, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proceedings of SPIE - The International Society for Optical Engineering 4792, 1–8 (2002).
[Crossref]

E. Huggins, Introduction to Fourier Optics (McGraw-Hill, 1968).

Y. Jiang, Z. Wang, M. Zhao, and L. Zhang, “MTF-based Research on the Minimum Fill Factor in Optical Sparse Aperture System,” in Fourth International Conference on Intelligent Computation Technology and Automation(2011), pp. 612–615.

B. Hunt, “A Theorem on the Difficulty of Numerical Deconvolution,” AC-20, 94–95 (1972).
[Crossref]

K. Elsayed and C. Lacor, “Comparison of Kriging, RBFNN, RBF and polynomial regression surrogates in design optimization,” in Eleventh International Conference of Fluid Dynamics(2013).

C. R. Vogel, “Computational Methods for Inverse Problems,” Frontiers in Applied Mathematics (2002).

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Figures (15)

Fig. 1
Fig. 1 Abbe imaging principle: (a) Abbe imaging principle of single aperture system; (b) Abbe imaging principle of OSA system.
Fig. 2
Fig. 2 Relationship between filling factor and MTF aligned the baseline of the two-aperture system: (a) two sub-aperture system; (b) filling factor versus MTF.
Fig. 3
Fig. 3 MTF curves before and after compensation [10]
Fig. 4
Fig. 4 Restored images: (a) restored image without wave front-error correction; (b) restored image with wave-front error correction [15]. The author has been authorized to use of this figure.
Fig. 5
Fig. 5 Framework of SV image restoration.
Fig. 6
Fig. 6 Position map of PSF (a): position of measured PSF on image plane; (b): target plane put on object plane.
Fig. 7
Fig. 7 Phased array OSA system: (a) shaded model of OSA system; (b) 3D schematic layout.
Fig. 8
Fig. 8 Simulated space variant images: (a) SNR = 20; (b) SNR = 30; (c) SNR = 50.
Fig. 9
Fig. 9 Accuracy map of PSF acquired by interpolation (a): polynomial regression; (b): inverse distance to a power; (c): modified Shepard’s method; (d): radial basis function.
Fig. 10
Fig. 10 Restored SIV images: (a) part of original image; (b) part of blurred image; (c) part of SIV restored image; (d) Part of SV restored image.
Fig. 11
Fig. 11 Compensating missing mid-frequency information of OSA system.
Fig. 12
Fig. 12 Aperture structure of the OSA system.
Fig. 13
Fig. 13 Optical structure of the two systems in Zemax: (a) OSA system; (b) single aperture system.
Fig. 14
Fig. 14 MTF curves of the two systems and MTF curve of the equivalent aperture system.
Fig. 15
Fig. 15 Results of simulation experiment: (a) Image of OSA system; (b) Image of single aperture system; (c) Image of equivalent aperture system; (d) fused image.

Tables (2)

Tables Icon

Table 1 Average MSE of each interpolation method.

Tables Icon

Table 2 ISNR results of each group of restored image.

Equations (42)

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OTF(ξ,η)= CTF(ξ,η)CTF(ξ,η) | CTF(α,β) | 2 dαdβ .
MTF=MT F d ( f x , f y )+ 1 2 [ MT F d ( f x +(1 F 2 ) ρ Dc , f y )+MT F d ( f x (1 F 2 ) ρ Dc , f y ) ],
MTF(f)= 2 π [ arccos f f c f f c 1 ( f f c ) 2 ],
PSF i = PSF i sum( PSF i ) ,i=1,2,3,...N,
PSF i * =PSF i mean(PSF),i=1,2,3,...,N,
C ij =cov(PSF i , PSF j ),1i,jN,
ρ i = λ i j=1 n λ j ,i=1,2,3,...,N,
( r 1 r 2 r 3 ... r N )=( ei g 1 ei g 2 ei g 3 ... ei g N )( PSF 1 PSF 2 PSF 3 ... PSF N ).
( PSF 1 PSF 2 PSF 3 ... PSF N )= ( ei g 1 ei g 2 ei g 3 ... ei g N ) T ( r 1 r 2 r 3 ... r N )+mean(PSF),
PSF=( a 1 a 2 a 3 ... a k )( r 1 r 2 r 3 ... r k )+mean(PSF),
y= B c x,
B c =WΛW -1 ,
MNIDFT[Λ],
W -1 Λ 1 MN DFT[Λ],
PSF(x,y)= k=1 PCA num a k (x,y) r k (x,y)+mean(PSF),
y= B c x= i=1 MN x i * PS F i = i=1 MN x i * ( k=1 PCA num a k (x,y) r k (x,y)+mean(PSF)) = i=1 MN x i * k=1 PCA num a k (x,y) r k (x,y)+ i=1 MN x i * mean(PSF). = k=1 PCA num r k *( A k x) +x*mean(PSF)
k=1 PCA num [ A k x* r k ]+x*mean(PSF) = k=1 PCA num [ A k x*W Λ k W 1 ]+x*W Λ mean W 1 .
B c = k=1 PCA num [ A k W Λ k W 1 ]+W Λ mean W 1 .
( B c T B c +α C T C)x= B c T y,
c(m,n)= 1 8 [ 0 1 0 1 4 1 0 1 0 ].
ISNR=10 log 10 gf 2 f f α 2
( k=1 PCA num [ A k W Λ k W 1 ]+W Λ mean W 1 ) T ( k=1 PCA num [ A k W Λ k W 1 ]+W Λ mean W 1 )x+α C T Cx , =( k=1 PCA num [ A k W Λ k W 1 ]+W Λ mean W 1 ) T y
SNR=10 log 10 ( var( g ) σ n 2 ),
MSE( i, j )= k=1 m l=1 n ( PS F zemax ( k, l )PS F ip ( k, l ) ) 2 ,
MSE ave =( i=1 M j=1 N MSE( i, j ) )/( MN )
ISNR=10 log 10 gf 2 f'f 2
g(x)= 0 f cu t MTF(f) A(f)cos(fx+ θ f ),
g OSA (x)= 0 f low MT F low (f) A(f)cos(fx+ θ f )+ 0 f high MT F high (f) A(f)cos(fx+ θ f ),
g single (x)= 0 f MT F single (f) A(f)cos(fx+ θ f ),
F= 4π d 2 π D 2 ,
f Dc = D λf ,
MTF( f x , f y )= 1 4 i=1 4 j=1 4 MT F d ( f x x i x j λf , f y y i y j λf ) =MT F d ( f x , f y )+ 1 4 [MT F d ( f x ± Dd λf , f y ), +2MT F d ( f x ± Dd 2λf , f y ± Dd 2λf ) +MT F d ( f x , f y ± Dd λf )]
MTF( f x , f y )= 1 4 i=1 4 j=1 4 MT F d ( f x x i x j λf , f y y i y j λf ) =MT F d ( f x , f y )+ 1 4 [MT F d ( f x ±(1 F 2 ) f Dc , f y ) +2MT F d ( f x ± (2 F ) f Dc 4 , f y ± (2 F ) f Dc 4 ), +MT F d ( f x , f y ±(1 F 2 ) f Dc )]
f x1 = d λf = F D 2λf = F 2 f Dc ,
f x2 = (D2d) λf =(1- F ) D λf =(1- F ) f Dc .
MTF( f x , f y )= 2 π [ arccos( f x 2 + f y 2 f 0 )( f x 2 + f y 2 f 0 ) 1 ( f x 2 + f y 2 f 0 ) 2 ],
MTF( f x ,0)= 2 π [ arccos( f x2 f 0 )( f x2 f 0 ) 1 ( f x2 f 0 ) 2 ]=0.3.
F(x,y)= ω 1 A(x,y)+ ω 2 B(x,y),
a= f x2 f Dc A(f) df,
b= f x1 f x2 B(f) df,
ω 1 = a a+b ,
ω 2 = b a+b ,

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