Abstract

The resolution of imaging by space and earth-based telescopes is often limited by the finite aperture of the optical systems. We propose a novel synthetic aperture-based imaging system with two physical subapertures distributed only along the perimeter of the synthetic aperture. The minimum demonstrated two-subaperture area is only 0.43% of a total full synthetic aperture area. The proposed optical configuration is inspired by a setup in which two synchronized satellites move only along the boundary of the synthetic aperture and capture a few light patterns from the observed scene. The light reflected from the two satellites interferes with an image sensor located in a third satellite. The sum of the entire interfering patterns is processed to yield the image of the scene with a quality comparable to an image obtained from the complete synthetic aperture. The proposed system is based on the incoherent coded aperture holography technique in which the light diffracted from an object is modulated by a pseudorandom coded phase mask. The modulated light is recorded and digitally processed to yield the 3D image of the object. A laboratory model of imaging with two synchronized subapertures distributed only along the border of the aperture is demonstrated. Experimental results validate that sampling along the boundary of the synthetic aperture is enough to yield an image with the resolving power obtained from the complete synthetic aperture. Unlike other schemes of synthetic aperture, there is no need to sample any other part of the aperture beside its border. Hence, a significant saving of time and/or devices are expected in the process of data acquisition.

© 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. K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
    [Crossref]
  2. M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (Wiley, 1999).
  3. A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.
  4. A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
    [Crossref]
  5. P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
    [Crossref]
  6. P. R. Lawson, Principles of Long Baseline Stellar Interferometry (NASA Jet Propulsion Laboratory, 2000).
  7. B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19, 4924–4936 (2011).
    [Crossref]
  8. Y. Kashter, Y. Rivenson, A. Stern, and J. Rosen, “Sparse synthetic aperture with Fresnel elements (S-SAFE) using digital incoherent holograms,” Opt. Express 23, 20941–20960 (2015).
    [Crossref]
  9. R. Zhu, J. T. Richard, D. J. Brady, D. L. Marks, and H. O. Everitt, “Compressive sensing and adaptive sampling applied to millimeter wave inverse synthetic aperture imaging,” Opt. Express 25, 2270–2284 (2017).
    [Crossref]
  10. J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
    [Crossref]
  11. V. Mico, Z. Zalevsky, P. García-Martínez, and J. García, “Synthetic aperture superresolution with multiple off-axis holograms,” J. Opt. Soc. Am. A 23, 3162–3170 (2006).
    [Crossref]
  12. L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16, 161–169 (2008).
    [Crossref]
  13. L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. 49, 845–857 (2010).
    [Crossref]
  14. B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18, 962–972 (2010).
    [Crossref]
  15. Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22, 20551–20565 (2014).
    [Crossref]
  16. A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25, 13883–13896 (2017).
    [Crossref]
  17. A. Bulbul, A. Vijayakumar, and J. Rosen, “Partial aperture imaging by systems with annular phase coded masks,” Opt. Express 25, 33315–33329 (2017).
    [Crossref]
  18. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  19. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography system with improved performance [Invited],” Appl. Opt. 56, F67–F77 (2017).
    [Crossref]
  20. J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

2017 (4)

2015 (1)

2014 (1)

2011 (3)

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19, 4924–4936 (2011).
[Crossref]

2010 (2)

2009 (1)

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

2008 (1)

2006 (1)

1978 (1)

K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Andelson, P.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Brady, D. J.

Brooker, G.

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

Bulbul, A.

Covey, K. R.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Czeszumska, A.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Edelstein, J.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Erskine, D. J.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Everitt, H. O.

Fienup, J. R.

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.

García, J.

García-Martínez, P.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

Granero, L.

Halverson, S. P.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Hamren, K.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Indebetouw, G.

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

Javidi, B.

Kashter, Y.

Katz, B.

Kelner, R.

Kimber, D.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Kumar, A.

Lawson, P. R.

P. R. Lawson, Principles of Long Baseline Stellar Interferometry (NASA Jet Propulsion Laboratory, 2000).

Lloyd, J. P.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Marks, D. L.

Martínez-León, L.

Mercer, T.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Mico, V.

Micó, V.

Mondo, D.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Muirhead, P. S.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Muterspaugh, M. W.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Richard, J. T.

Rivenson, Y.

Rosen, J.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shaked, N. T.

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

Soumekh, M.

M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (Wiley, 1999).

Stern, A.

Tippie, A. E.

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.

Tomiyasu, K.

K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
[Crossref]

Vanderburg, A.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Vijayakumar, A.

Wishnow, E. H.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Wright, J. T.

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Zalevsky, Z.

Zhu, R.

Appl. Opt. (2)

J. Hologr. Speckle (1)

J. Rosen, G. Brooker, G. Indebetouw, and N. T. Shaked, “A review of incoherent digital Fresnel holography,” J. Hologr. Speckle 5, 124–140 (2009).
[Crossref]

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

Opt. Express (9)

L. Martínez-León and B. Javidi, “Synthetic aperture single-exposure on-axis digital holography,” Opt. Express 16, 161–169 (2008).
[Crossref]

B. Katz and J. Rosen, “Could SAFE concept be applied for designing a new synthetic aperture telescope?” Opt. Express 19, 4924–4936 (2011).
[Crossref]

Y. Kashter, Y. Rivenson, A. Stern, and J. Rosen, “Sparse synthetic aperture with Fresnel elements (S-SAFE) using digital incoherent holograms,” Opt. Express 23, 20941–20960 (2015).
[Crossref]

R. Zhu, J. T. Richard, D. J. Brady, D. L. Marks, and H. O. Everitt, “Compressive sensing and adaptive sampling applied to millimeter wave inverse synthetic aperture imaging,” Opt. Express 25, 2270–2284 (2017).
[Crossref]

A. E. Tippie, A. Kumar, and J. R. Fienup, “High-resolution synthetic-aperture digital holography with digital phase and pupil correction,” Opt. Express 19, 12027–12038 (2011).
[Crossref]

B. Katz and J. Rosen, “Super-resolution in incoherent optical imaging using synthetic aperture with Fresnel elements,” Opt. Express 18, 962–972 (2010).
[Crossref]

Y. Kashter and J. Rosen, “Enhanced-resolution using modified configuration of Fresnel incoherent holographic recorder with synthetic aperture,” Opt. Express 22, 20551–20565 (2014).
[Crossref]

A. Vijayakumar and J. Rosen, “Interferenceless coded aperture correlation holography-a new technique for recording incoherent digital holograms without two-wave interference,” Opt. Express 25, 13883–13896 (2017).
[Crossref]

A. Bulbul, A. Vijayakumar, and J. Rosen, “Partial aperture imaging by systems with annular phase coded masks,” Opt. Express 25, 33315–33329 (2017).
[Crossref]

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. IEEE (1)

K. Tomiyasu, “Tutorial review of synthetic-aperture radar (SAR) with applications to imaging of the ocean surface,” Proc. IEEE 66, 563–583 (1978).
[Crossref]

Publ. Astron. Soc. Pac. (1)

P. S. Muirhead, J. Edelstein, D. J. Erskine, J. T. Wright, M. W. Muterspaugh, K. R. Covey, E. H. Wishnow, K. Hamren, P. Andelson, D. Kimber, T. Mercer, S. P. Halverson, A. Vanderburg, D. Mondo, A. Czeszumska, and J. P. Lloyd, “Precise stellar radial velocities of an M dwarf with a Michelson interferometer and a medium-resolution near-infrared spectrograph,” Publ. Astron. Soc. Pac. 123, 709–724 (2011).
[Crossref]

Other (4)

P. R. Lawson, Principles of Long Baseline Stellar Interferometry (NASA Jet Propulsion Laboratory, 2000).

M. Soumekh, Synthetic Aperture Radar Signal Processing with MATLAB Algorithms (Wiley, 1999).

A. E. Tippie and J. R. Fienup, “Gigapixel synthetic-aperture digital holography: sampling and resolution considerations,” in Imaging and Applied Optics, OSA Technical Digest (Optical Society of America, 2011), paper CWB1.

J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, 2017).

Supplementary Material (1)

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Figures (11)

Fig. 1.
Fig. 1. Schematic of the space-based telescope for the implementation of SMART.
Fig. 2.
Fig. 2. Laboratory model of SMART for image acquisition. CPM, coded phase mask; L0 and L1, refractive lens; LED, light-emitting diode; and DOE, diffractive optical element.
Fig. 3.
Fig. 3. Modified GSA for designing the CPM pairs with all possible permutations of eight equally separated circles along the perimeter of the aperture such that every pair is constrained to produce a uniform magnitude on the sensor plane.
Fig. 4.
Fig. 4. Experimental setup for demonstration of SMART. BS1 and BS2, beam splitters; SLM, spatial light modulator; NBS, National Bureau of Standards; L0A, L0B and B1, refractive lenses; LED1 and LED2, identical light-emitting diodes; and P, polarizer.
Fig. 5.
Fig. 5. (a)–(c), (g)–(i) Intensity patterns recorded for a point object and a resolution target for eight subapertures, respectively; (d) magnitude and (e) phase of hIR; (j) magnitude and (k) phase of hOR; (f) reconstructed image of PAIS; (l) direct imaging result using eight subapertures with diffractive lens; (m)–(o), (s)–(u) intensity patterns recorded for a point object and a resolution target for full aperture, respectively; (p) magnitude and (q) phase of hIR; (v) magnitude and (w) phase of hOR; (r) reconstructed image of full aperture imaging system; (x) direct imaging result using a full aperture with a diffractive lens.
Fig. 6.
Fig. 6. Reconstruction results for r=0.2, 0.28, 0.4, and 0.8 mm of (a) PAIS with a pair of subapertures; (b) direct imaging results through a pair of subapertures with a diffractive lens; (c) reconstruction results of PAIS with four subapertures; (d) direct imaging results through four subapertures with a diffractive lens; (e) reconstruction results of SMART with all possible permutations of subaperture pair over the four locations; (f) reconstruction results of PAIS with eight subapertures; (g) direct imaging results through eight subapertures with a diffractive lens; and (h) reconstruction results of SMART with all possible permutations of subaperture pair over the eight locations; (i) direct imaging with a single aperture at the center.
Fig. 7.
Fig. 7. Reconstruction results for r=0.2, 0.28, 0.4, and 0.8 mm of PAIS with eight subapertures at (a) z=0  cm; (b) z=1  cm, direct imaging results through eight subapertures with a diffractive lens at (c) z=0  cm; (d) z=1  cm; reconstruction results of SMART at (e) z=0  cm; (f) z=1  cm.
Fig. 8.
Fig. 8. MTF profile for PAIS and direct imaging for ring thickness of (a1) 0.2 mm and (a2) 0.4 mm; (b) MTF plots for direct imaging with various annular widths; (c1) 2D and (c2) mesh profile of MTF of direct imaging; and (d1) and (d2) PAIS with eight subapertures.
Fig. 9.
Fig. 9. Plot of the normalized SNR and visibility versus scattering rank determined by the effective pixel size of the SLM. Inset figures are the reconstruction results using object–element 1 group 3 of USFA 1951 1X negative resolution chart.
Fig. 10.
Fig. 10. Plots of MSE of SMART versus the number of permutations of subaperture pairs for subaperture radii r=0.2, 0.28, 0.4, and 0.8 mm.
Fig. 11.
Fig. 11. MSE of PAIS and direct imaging for eight subapertures and SMART with 28 permutations of subaperture pair.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

MP(r¯)=p=1Pexp[iϕp(r¯)]Circ(|r¯|R)*δ(r¯r¯p),
IIR(r¯0;r¯s,zs)=C0|ν[1λzh]F{MP(r¯)}|2*δ(r¯0zhzsr¯s),
hIR(r¯0;0,zs)=j=13IIR(r¯0;0,zs)exp(iθj).
hIR,PAIS(r¯0;0,zs)=C0j=13exp(iθj)|ν[1λzh]F{MP=N(r¯)}|2=C0j=13exp(iθj)|ν[1λzh]F{p=1Nexp[iϕj,p(r¯)]Circ(|r¯|R)*δ(r¯r¯p)}|2=C0j=13exp(iθj)|p=1NF{exp[iϕj,p(λzhr¯)]}*Jinc(R|ro¯|λzh)exp[i2πλzh(r¯p·r¯o)]|2,
hIR,PAIS(r¯0;0,zs)C0j=13exp(iθj)k=1N1l=k+1N[(F{exp[iϕj,k(λzhr¯)]}*Jinc(R|r¯o|λzh))×(F{exp[iϕj,l(λzhr¯)]}*Jinc(R|r¯o|λzh))exp[i2πλzh[(r¯kr¯l)·r¯o]]+C.C.],
hIR,SMART(r¯o;0,zs)=C0k=1N1l=k+1Nj=13exp(iθj)|ν[1λzh]F{M2(r¯)}|2=C0k=1N1l=k+1Nj=13exp(iθj)×|ν[1λzh]F{exp[iϕk,j(r¯)]Circ(|r¯r¯k|R)+exp[iϕl,j(r¯)]Circ(|r¯r¯l|R)}|2=C0k=1N1l=k+1Nj=13exp(iθj)|F{exp[iϕk,j(λzhr¯)]}*Jinc(R|r¯o|λzh)exp(i2πr¯k·r¯oλzh)+F{exp[iϕl,j(λzhr¯)]}*Jinc(R|r¯o|λzh)exp(i2πr¯l·r¯oλzh)|2.
hIR,SMART(r¯o;0,zs)C0j=13exp(iθj)×k=1N1l=k+1N[(F{exp[iϕk,j(λzhr¯)]}*Jinc(R|r¯o|λzh))×(F{exp[iϕl,j(λzhr¯)]}*Jinc(R|r¯o|λzh))exp(i2π(r¯kr¯l)·r¯oλzh)+C.C.].
o(r¯s)=mMamδ(r¯sr¯m),
IOR(r¯0;zs)=mamIIR(r¯0zhzsr¯m;0,zs).
hOR,PAIS(r¯0;zs)=j=13IOR(r¯0;zs)exp(iθj)=j=13mamIIR(r¯0zhzsr¯m;0,zs)exp(iθj)=mamhIR,PAIS(r¯0zhzsr¯m;zs).
hOR,SMART(r¯0;zs)=k=1N1l=k+1Nj=13IOR(r¯0;zs)exp(iθj)=k=1N1l=k+1Nj=13mamIIR(r¯0zhzsr¯m;0,zs)exp(iθj)mamhIR,SMART(r¯0zhzsr¯m;zs).
IIM(r¯R)=hOR(r¯0;zs)h˜IR*(r¯0r¯R;zs)dr¯0=mamhIR(r¯0zhzsr¯m;zs)h˜IR*(r¯0r¯R;zs)dr¯0=mamΛ(r¯Rzhzsr¯m)o(r¯sMT),
MSE=1MNm=1Mn=1N|OD(m,n)γOM(m,n)|2,
γ=m=1Mn=1NOD(m,n)OM(m,n)m=1Mn=1N|OM(m,n)|2.

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