Abstract

We explore opportunities afforded by an extremely large telescope design comprised of ill-figured randomly varying subapertures. The veracity of this approach is demonstrated with a laboratory scaled system whereby we reconstruct a white light binary point source separated by 2.5 times the diffraction limit. With an inherently unknown varying random point spread function, the measured speckle images require a restoration framework that combine support vector machine based lucky imaging and non-negative matrix factorization based multiframe blind deconvolution. To further validate the approach, we model the experimental system to explore sub-diffraction-limited performance, and an object comprised of multiple point sources.

© 2017 Optical Society of America

Full Article  |  PDF Article

Corrections

Xiaopeng Peng, Garreth J. Ruane, Marco B. Quadrelli, and Grover A. Swartzlander, "Randomized apertures: high resolution imaging in far field: erratum," Opt. Express 25, 20952-20952 (2017)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-25-17-20952

24 July 2017: Typographical corrections were made to the abstract; paragraph 1 and 2 of Section 1, paragraph 1 of Section 2.1, paragraphs 1–3 of Section 2.2, paragraph 1 of Section 3, paragraphs 1 and 2 of Section 3.1, paragraphs 1–3 of Section 3.2.1, paragraph 1 of Section 3.2.2, paragraph 1 of Section 3.2.3, paragraph 1 of Section 4, paragraph 1 of Section 4.1, paragraph 1 of Section 4.2, paragraphs 1–3 of Section 4.3, paragraph 1 of Section 4.4, paragraph 1 of Section 5, paragraph 1–4 of Section 5.2, paragraph 1 of Section 6, paragraph 1 of Section 7; Eqs. (1)–(18); the figure captions of Figs. 3, 4, and 7–10; Tables 1 and 2; the acknowledgments section; and Refs. 3–8, 10, 11, and 20–64.


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References

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2017 (1)

2016 (1)

2015 (3)

S. A. Basinger, D. Palacios, M. B. Quadrelli, and G. A. Swartzlander, “Optics of a granular imaging system (i.e. “orbiting rainbows”),” Proc. SPIE 9602, 96020E(2015).
[Crossref]

B. Judkewitz, R. Horstmeyer, I. M. Vellekoop, I. N. Papadopoulos, and C. Yang, “Translation correlations in anisotropically scattering media,” Nat. Phys. 11(8), 684–691 (2015).
[Crossref]

V. Duval and G. Peyré, “Exact support recovery for sparse spikes deconvolution,” Found. Comput. Math. 15 (5), 1315–1355 (2015).
[Crossref]

2014 (3)

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photon. 8(10), 784–790 (2014).
[Crossref]

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Comm. Pure Appl. Math. 67(6), 906–956 (2014).
[Crossref]

K. S. Ford, B. McKernan, A. Sivaramakrishnan, A. Martel, K. Koekemoer, D. Lafrenière, and S. Parmentier, “Active galactic nucleus and quasar science with aperture masking interferometry on the James Webb Space Telescope,” Astrophys. J. 783, 73–75 (2014).
[Crossref]

2013 (2)

E. J. Candès and C. Fernandez-Granda, “Super-resolution from noisy data,” J. Fourier Anal Appl. 19 (6), 1229–1254 (2013).
[Crossref]

X. Zhu and P. Milanfar, “Removing atmospheric turbulence via space-invariant deconvolution,” IEEE Trans. Pattern Anal Mach Intell. 35(1), 157–170 (2013).
[Crossref]

2012 (2)

J. Malcolm, P. Yalamanchili, C. McClanahanm, V. Venugopalakrishnan, K. Patel, and J. Melonakos, “ArrayFire: a GPU acceleration platform,” Proc. SPIE 8403, 84030A (2012).
[Crossref]

J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14, 083001 (2012).
[Crossref]

2011 (3)

M. Mazilu, J. Baumgartl, S. Kosmeier, and K. Dholakia, “Optical eigenmodes; exploiting the quadratic nature of the energy flux and of scattering interactions,” Opt. Express 19(2), 933–945 (2011).
[Crossref] [PubMed]

M. Hirsch, S. Harmeling, S. Sra, and B. Schölkopf, “Online multi-frame blind deconvolution with super-resolution and saturation correction,” Astron. Astrophys. 531, A9(2011).
[Crossref]

D. Kim, P. Protopapas, Y. Byun, C. Alcock, R. Khardon, and M. Trichas, “Quasi-stellar object selection algorithm using time variability and machine learning: Selection of 1620 quasi-stellar object candidates from MACHO Large Magellanic Cloud database,” Astrophys. J. 735(2), 68–70 (2011).
[Crossref]

2009 (2)

A. M. Bruckstein, D. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM review 51 (1), 34–81 (2009).
[Crossref]

M. Aubailly, M. A. Vorontsov, G. W. Carhart, and M.T. Valley, “Automated video enhancement from a stream of atmospherically-distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630C (2009).
[Crossref]

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]

G. Andersen and D. Tullson, “Broadband antihole photon sieve telescope,” Appl. Phys. 46, 3706–3708 (2007).

2006 (1)

D. Dunbar and G. Humphreys, “A spatial data structure for fast poisson-disk sample generation,” ACM Trans. Graph. 25 (3), 503–508 (2006).
[Crossref]

2005 (1)

K. Hirakawa and T. W. Parks, “Adaptive homogeneity-directed demosaicing algorithm,” IEEE Trans. Image. Proc. 14(3), 360–369 (2005).
[Crossref]

2004 (1)

Y. Zhang and Y. Zhao, “Automated clustering algorithms for classification of astronomical objects,” Astron. Astrophys. 422 (3), 1113–1121 (2004).
[Crossref]

1996 (1)

D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 13(3), 43–64 (1996).
[Crossref]

1995 (3)

C. Cortes and V. Vapnik, “Support-vector networks,” Mach. Learn 20(3), 273–297 (1995).
[Crossref]

J. E. Harvey, A. Kotha, and R. L. Phillips, “Image characteristics in applications utilizing dilute subaperture arrays,” Appl. Opt. 34(16), 2983–2992 (1995).
[Crossref] [PubMed]

D. Geman and C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4(7), 932–946 (1995).
[Crossref] [PubMed]

1993 (1)

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862(1993).
[Crossref]

1989 (1)

W. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2(2), 207–228, (1989).

1982 (1)

1980 (1)

J. R. Fienup and G. B. Feldkamp, “Astronomical imaging by processing stellar speckle interferometry data,” Proc. SPIE 0243, 2723–2727 (1980).

1979 (3)

A. B. Meinel, “Cost-scaling laws applicable to very large optical telescopes,” Opt. Eng 18(6), 645–647 (1979).
[Crossref]

G. C. Loos and C. B. Hogge, “Turbulence of the upper atmosphere and isoplanatism,” Appl. Opt. 18 (15), 2654–2661 (1979).
[Crossref] [PubMed]

F. Roddier, J. M. Gilli, and J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Optics 13(2), 63–65 (1979).
[Crossref]

1978 (1)

1974 (1)

K. T. Knox and B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, 45–48 (1974).
[Crossref]

1970 (3)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6(1), 85–87 (1970).

A. W. Lohmann, G. Weigelt, and B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22(24), 4028–4037 (1970).
[Crossref]

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” J. Mod. Opt. 17 (10), 761–772 (1970).

1967 (1)

P. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15 (2), 70–73, (1967)
[Crossref]

Alcock, C.

D. Kim, P. Protopapas, Y. Byun, C. Alcock, R. Khardon, and M. Trichas, “Quasi-stellar object selection algorithm using time variability and machine learning: Selection of 1620 quasi-stellar object candidates from MACHO Large Magellanic Cloud database,” Astrophys. J. 735(2), 68–70 (2011).
[Crossref]

Alloin, D. M.

D. M. Alloin and J. M. Mariotti, Diffraction-limited imaging with very large telescopes, (Springer, 2012).

Andersen, G.

G. Andersen and D. Tullson, “Broadband antihole photon sieve telescope,” Appl. Phys. 46, 3706–3708 (2007).

Andersen, T.

T. Andersen and A. Enmark, Integrated modeling of Telescopes, (Springer, 2011).
[Crossref]

Artusio-Glimpse, A. B.

Arumugam, D.

M. B. Quadrelli, S. A. Basinger, G. A. Swartzlander, and D. Arumugam, “Dynamics and control of granular imaging systems,” in Proceedings of AIAA Space Conference and Exposition(AIAA, 2015), pp. 4484.

Aubailly, M.

M. Aubailly, M. A. Vorontsov, G. W. Carhart, and M.T. Valley, “Automated video enhancement from a stream of atmospherically-distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630C (2009).
[Crossref]

Baiocchi, D.

D. Baiocchi and H. P. Stahl, “Enabling future space telescopes: mirror technology review and development roadmap,” Astro2010: The Astronomy and Astrophysics Decadal Survey, 23 (2009).

Basinger, S. A.

S. A. Basinger, D. Palacios, M. B. Quadrelli, and G. A. Swartzlander, “Optics of a granular imaging system (i.e. “orbiting rainbows”),” Proc. SPIE 9602, 96020E(2015).
[Crossref]

M. B. Quadrelli, S. A. Basinger, G. A. Swartzlander, and D. Arumugam, “Dynamics and control of granular imaging systems,” in Proceedings of AIAA Space Conference and Exposition(AIAA, 2015), pp. 4484.

Baumgartl, J.

Beavers, W.

W. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2(2), 207–228, (1989).

Beletic, J. W.

W. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2(2), 207–228, (1989).

Bruckstein, A. M.

A. M. Bruckstein, D. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM review 51 (1), 34–81 (2009).
[Crossref]

Byun, Y.

D. Kim, P. Protopapas, Y. Byun, C. Alcock, R. Khardon, and M. Trichas, “Quasi-stellar object selection algorithm using time variability and machine learning: Selection of 1620 quasi-stellar object candidates from MACHO Large Magellanic Cloud database,” Astrophys. J. 735(2), 68–70 (2011).
[Crossref]

Candès, E. J.

E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” Comm. Pure Appl. Math. 67(6), 906–956 (2014).
[Crossref]

E. J. Candès and C. Fernandez-Granda, “Super-resolution from noisy data,” J. Fourier Anal Appl. 19 (6), 1229–1254 (2013).
[Crossref]

Carhart, G. W.

M. Aubailly, M. A. Vorontsov, G. W. Carhart, and M.T. Valley, “Automated video enhancement from a stream of atmospherically-distorted images: the lucky-region fusion approach,” Proc. SPIE 7463, 74630C (2009).
[Crossref]

Carron, I.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep.4 (2014).
[PubMed]

Chardon, G.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep.4 (2014).
[PubMed]

Christou, J. C.

S. M. Jefferies and J. C. Christou, “Restoration of astronomical images by iterative blind deconvolution,” Astrophys. J. 415, 862(1993).
[Crossref]

Chung, S. J.

S. J. Chung and F. Y. Hadaegh, “Swarms of Femtosats for Synthetic Aperture Applications,” in Proceedings of the Fourth International Conference on Spacecraft Formation Flying Missions and Technologies, (2011).

Cortes, C.

C. Cortes and V. Vapnik, “Support-vector networks,” Mach. Learn 20(3), 273–297 (1995).
[Crossref]

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” J. Mod. Opt. 17 (10), 761–772 (1970).

J. C. Dainty, “The statistics of speckle patterns,” Progress in Optics14, (Elsevier, 1977), Chap. 1.
[Crossref]

Daudet, L.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: Compressive imaging using a multiply scattering medium,” Sci. Rep.4 (2014).
[PubMed]

Dholakia, K.

Dierking, M. P.

Donoho, D.

A. M. Bruckstein, D. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM review 51 (1), 34–81 (2009).
[Crossref]

Dudgeon, D. E.

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Appl. Phys. (1)

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Figures (10)

Fig. 1
Fig. 1 Schematic of a Granular Space Telescope: free-flying aperture elements distributed across a baseline D and roughly aligned to a focal point f. Inset: comparison of a diffraction-limited point source and speckles from a modeled random aperture array.
Fig. 2
Fig. 2 (a) Experimental setup of a random aperture array system. Measured broadband images of a binary source using (b) RAA with random wavefront error and (c) filled aperture without amplitude and phase mask.
Fig. 3
Fig. 3 (a) Numerical model of the RAA system witha broadband binary point source. (b) Typical RAA image with respective rms tip-tilt and piston errors 5λc/D and one wave. (c) Diffraction-limited image.
Fig. 4
Fig. 4 Comparisons of the polychromatic PSF with respective rms tip-tilt and piston error 5λc/D and one wave. (a) Filled aperture. (b) Well aligned RAA. (c) Typical RAA. (d) Lucky RAA. Line plots indicate the FWHM.
Fig. 5
Fig. 5 (a) Illustration of the image reconstruction framework; (b) histograms and fitted generalized extreme value (GEV) distributions of the broadband binary source in (left to right): the diffraction-limited image, a lucky RAA frame, a typical RAA frame of type I, and a typical RAA frame of type II.
Fig. 6
Fig. 6 (a–c) Experimental recordings of a broadband binary source, and (d) a well-resolved reconstructed image.
Fig. 7
Fig. 7 Reconstruction of a broadband binary point source with a separation of 1.22λc/D from numerical images. (a) Diffraction-limited image. (b–d) Example frames and reconstructed image of with rms tip-tilt and piston of 5λc/D and 1 wave, respectively. (e–g) Comparison of well aligned RAA with inferior reconstruction.
Fig. 8
Fig. 8 Reconstruction of a broadband binary source with a separation of 0.85λc/D from numerical images. (a) Unresolved diffraction-limited image. (b–d) Example frames and reconstructed image of with rms tip-tilt and piston of 5λc/D and 1 wave, respectively. (e–g) Comparison of well aligned RAA with inferior reconstruction.
Fig. 9
Fig. 9 Reconstruction of multiple broadband point sources with a separation of 1.22λc/D between neighboring sources from numerical images. (a) A diffraction-limited image. (b–d) Example frames and reconstructed image of with rms tip-tilt and piston of 5λc/D and 1 wave, respectively. (e–g) Comparison of well aligned RAA with inferior reconstruction.
Fig. 10
Fig. 10 Strehl ratio as a function of (a) fill factor FF and (b) wavefront error for various values of the subaperture diameter ratio Γ. (c) An illustration of RAA with a same FF (∼ 12%) but different Γ.

Tables (2)

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Table 1 Evaluation of the Experimental Results

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Table 2 Evaluation of the Numerical Results

Equations (18)

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U ( x ) = circ ( x R ) m = 1 M circ ( x c m r ) exp ( i ϕ m ( x ) )
ϕ m ( x ) = k ( Δ z m + x Δ α m )
E ( x , λ , θ ) = E 0 i λ f exp ( i 2 π λ f x x ) U ( x ) exp ( i k x θ ) d x = i E 0 A 0 m = 1 M [ Somb ( x ( Δ α m + θ ) f λ f / d ) exp ( i k m x ) exp ( i ψ m ) exp ( i k c m θ ) ]
P ( x , λ , θ ) = ( E 0 A 0 ) 2 m = 1 M n = 1 M [ Somb ( x ( Δ α m + θ ) f λ f / d ) Somb ( x ( Δ α n + θ ) f λ f / d ) cos ( ( k m k n ) x + ( ψ m ψ n ) + k ( c m c n ) θ ) ]
h ( x ) = ρ s ( λ ) ρ c ( λ ) P ( x , λ , 0 ) d λ
I ( x ) = j h ( x θ j f )
I noise ( x ) = 1 ν τ Poisson ( σ p 2 ( x ) ) + Gaussian ( 0 , σ r 2 ) ( x )
SNR = 10 log 10 μ σ n
gev ( I ( x ) ) = 1 σ e ( 1 + χ ) 1 ζ 1 exp ( ( 1 + χ ) 1 ζ )
minimize w , b , s 1 2 w 2 + C j = 1 N t s j subject to s j 0 , l j f ( q j ) 1 s j , j = 1 , , N t
minimize η 1 2 j = 1 N t k = 1 N t l j l k K ( q j , q k ) η j η k j N t η j subject to j = 1 N t η j l j = 0 , 0 η j C , j = 1 , , N t
w ^ = j = 1 N t η j l j q j , b ^ = w ^ T q j l j
minimize Q , B j = 1 N t t j log ( p j ) + ( 1 t j ) log ( 1 p j )
g n ( x ) = a * h n
G n ( ξ ) = A H n
{ a ^ , h n } = 1 { min h n 0 , a 0 1 N 2 n = 0 N | G n A H n | F 2 }
H n = H n 1 A n T G n A n T ( A n H n 1 )
A n = A n 1 H n T G n H n T ( H n A n 1 )

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