Abstract

Computational imaging modalities support a simplification of the active architectures required in an imaging system and these approaches have been validated across the electromagnetic spectrum. Recent implementations have utilized pseudo-orthogonal radiation patterns to illuminate an object of interest—notably, frequency-diverse metasurfaces have been exploited as fast and low-cost alternative to conventional coherent imaging systems. However, accurately measuring the complex-valued signals in the frequency domain can be burdensome, particularly for sub-centimeter wavelengths. Here, computational imaging is studied under the relaxed constraint of intensity-only measurements. A novel 3D imaging system is conceived based on ‘phaseless’ and compressed measurements, with benefits from recent advances in the field of phase retrieval. In this paper, the methodology associated with this novel principle is described, studied, and experimentally demonstrated in the microwave range. A comparison of the estimated images from both complex valued and phaseless measurements are presented, verifying the fidelity of phaseless computational imaging.

© 2016 Optical Society of America

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References

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  1. X. Zhuge, A. G. Yarovoy, T. Savelyev, and L. Ligthart, “Modified kirchhoff migration for uwb mimo array-based radar imaging,” Geoscience and Remote Sensing, IEEE Transactions on 48, 2692–2703 (2010).
    [Crossref]
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    [Crossref]
  3. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
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  4. R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
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    [Crossref] [PubMed]
  7. A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems 27, 015005 (2010).
    [Crossref]
  8. E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Review 57, 225–251 (2015).
    [Crossref]
  9. J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
    [Crossref] [PubMed]
  10. B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
    [Crossref] [PubMed]
  11. T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” Antennas and Propagation, IEEE Transactions on 63, 593–600 (2015).
    [Crossref]
  12. Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Advances in Neural Information Processing Systems pp. 739–747 (2015).
  13. M. Grant, S. Boyd, and Y. Ye, “Cvx: Matlab software for disciplined convex programming,” (2008).
  14. B. Fuchs and L. Le Coq, “Phase retrieval procedure for microwave linear arrays,” Antennas and Propagation (EuCAP), 2015 9th European Conference on pp. 1–4 (2015).
  15. B. Fuchs and L. Le Coq, “Excitation retrieval of microwave linear arrays from phaseless far-field data,” Antennas and Propagation, IEEE Transactions on 63, 748–754 (2015).
    [Crossref]
  16. K. Wei, “Solving systems of phaseless equations via kaczmarz methods: a proof of concept study,” Inverse Problems 31, 125008 (2015).
    [Crossref]
  17. S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).
  18. E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” Information Theory, IEEE Transactions on 61, 1985–2007 (2015).
    [Crossref]
  19. E. J. Candès and X. Li, “Solving quadratic equations via phaselift when there are about as many equations as unknowns,” Foundations of Computational Mathematics 14, 1017–1026 (2014).
    [Crossref]
  20. D. L. Marks, J. Gollub, and D. R. Smith, “Spatially resolving antenna arrays using frequency diversity,” J. Opt. Soc. Am. A 33, 899–912 (2016).
    [Crossref]
  21. T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
    [Crossref]
  22. T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
    [Crossref]
  23. T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Applied Physics Letters 107, 204104 (2015).
    [Crossref]
  24. T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
    [Crossref]
  25. G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
    [Crossref] [PubMed]
  26. O. Yurduseven, V. R. Gowda, J. N. Gollub, and D. R. Smith, “Printed aperiodic cavity for computational and microwave imaging,” IEEE Microwave and Wireless Components Letters 26, 367–369 (2016).
    [Crossref]
  27. T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).
  28. G. Montaldo, D. Palacio, M. Tanter, and M. Fink, “Building three-dimensional images using a time-reversal chaotic cavity,” Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 52, 1489–1497 (2005).
    [Crossref]

2016 (4)

T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
[Crossref]

O. Yurduseven, V. R. Gowda, J. N. Gollub, and D. R. Smith, “Printed aperiodic cavity for computational and microwave imaging,” IEEE Microwave and Wireless Components Letters 26, 367–369 (2016).
[Crossref]

D. L. Marks, J. Gollub, and D. R. Smith, “Spatially resolving antenna arrays using frequency diversity,” J. Opt. Soc. Am. A 33, 899–912 (2016).
[Crossref]

2015 (8)

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Applied Physics Letters 107, 204104 (2015).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Review 57, 225–251 (2015).
[Crossref]

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” Antennas and Propagation, IEEE Transactions on 63, 593–600 (2015).
[Crossref]

B. Fuchs and L. Le Coq, “Excitation retrieval of microwave linear arrays from phaseless far-field data,” Antennas and Propagation, IEEE Transactions on 63, 748–754 (2015).
[Crossref]

K. Wei, “Solving systems of phaseless equations via kaczmarz methods: a proof of concept study,” Inverse Problems 31, 125008 (2015).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” Information Theory, IEEE Transactions on 61, 1985–2007 (2015).
[Crossref]

2014 (1)

E. J. Candès and X. Li, “Solving quadratic equations via phaselift when there are about as many equations as unknowns,” Foundations of Computational Mathematics 14, 1017–1026 (2014).
[Crossref]

2013 (2)

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

2010 (2)

A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems 27, 015005 (2010).
[Crossref]

X. Zhuge, A. G. Yarovoy, T. Savelyev, and L. Ligthart, “Modified kirchhoff migration for uwb mimo array-based radar imaging,” Geoscience and Remote Sensing, IEEE Transactions on 48, 2692–2703 (2010).
[Crossref]

2005 (1)

G. Montaldo, D. Palacio, M. Tanter, and M. Fink, “Building three-dimensional images using a time-reversal chaotic cavity,” Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 52, 1489–1497 (2005).
[Crossref]

2000 (1)

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-d radar imaging using range migration techniques,” Antennas and Propagation, IEEE Transactions on 48, 728–737 (2000).
[Crossref]

1982 (1)

1978 (1)

1972 (1)

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

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

1937 (1)

S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

Boyarsky, M.

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
[Crossref]

Boyd, S.

M. Grant, S. Boyd, and Y. Ye, “Cvx: Matlab software for disciplined convex programming,” (2008).

Brady, D.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

Candes, E.

Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Advances in Neural Information Processing Systems pp. 739–747 (2015).

Candès, E. J.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” Information Theory, IEEE Transactions on 61, 1985–2007 (2015).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Review 57, 225–251 (2015).
[Crossref]

E. J. Candès and X. Li, “Solving quadratic equations via phaselift when there are about as many equations as unknowns,” Foundations of Computational Mathematics 14, 1017–1026 (2014).
[Crossref]

Carsenat, D.

T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” Antennas and Propagation, IEEE Transactions on 63, 593–600 (2015).
[Crossref]

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Chai, A.

A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems 27, 015005 (2010).
[Crossref]

Chen, Y.

Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Advances in Neural Information Processing Systems pp. 739–747 (2015).

Decroze, C.

T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” Antennas and Propagation, IEEE Transactions on 63, 593–600 (2015).
[Crossref]

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Driscoll, T.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

Edgar, M. P.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

Eldar, Y. C.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Review 57, 225–251 (2015).
[Crossref]

Fienup, J. R.

Fink, M.

G. Montaldo, D. Palacio, M. Tanter, and M. Fink, “Building three-dimensional images using a time-reversal chaotic cavity,” Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 52, 1489–1497 (2005).
[Crossref]

Fortuny-Guasch, J.

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-d radar imaging using range migration techniques,” Antennas and Propagation, IEEE Transactions on 48, 728–737 (2000).
[Crossref]

Fromenteze, T.

T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” Antennas and Propagation, IEEE Transactions on 63, 593–600 (2015).
[Crossref]

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Fuchs, B.

B. Fuchs and L. Le Coq, “Excitation retrieval of microwave linear arrays from phaseless far-field data,” Antennas and Propagation, IEEE Transactions on 63, 748–754 (2015).
[Crossref]

B. Fuchs and L. Le Coq, “Phase retrieval procedure for microwave linear arrays,” Antennas and Propagation (EuCAP), 2015 9th European Conference on pp. 1–4 (2015).

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

Gerchberg, R.

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

Gollub, J.

D. L. Marks, J. Gollub, and D. R. Smith, “Spatially resolving antenna arrays using frequency diversity,” J. Opt. Soc. Am. A 33, 899–912 (2016).
[Crossref]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Gollub, J. N.

O. Yurduseven, V. R. Gowda, J. N. Gollub, and D. R. Smith, “Printed aperiodic cavity for computational and microwave imaging,” IEEE Microwave and Wireless Components Letters 26, 367–369 (2016).
[Crossref]

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Applied Physics Letters 107, 204104 (2015).
[Crossref]

Gowda, V. R.

O. Yurduseven, V. R. Gowda, J. N. Gollub, and D. R. Smith, “Printed aperiodic cavity for computational and microwave imaging,” IEEE Microwave and Wireless Components Letters 26, 367–369 (2016).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

Grant, M.

M. Grant, S. Boyd, and Y. Ye, “Cvx: Matlab software for disciplined convex programming,” (2008).

Hunt, J.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

Imani, M.

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Imani, M. F.

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Applied Physics Letters 107, 204104 (2015).
[Crossref]

Kaczmarz, S.

S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

Kpre, E. L.

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Kpré, E.

T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

Le Coq, L.

B. Fuchs and L. Le Coq, “Excitation retrieval of microwave linear arrays from phaseless far-field data,” Antennas and Propagation, IEEE Transactions on 63, 748–754 (2015).
[Crossref]

B. Fuchs and L. Le Coq, “Phase retrieval procedure for microwave linear arrays,” Antennas and Propagation (EuCAP), 2015 9th European Conference on pp. 1–4 (2015).

Li, X.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” Information Theory, IEEE Transactions on 61, 1985–2007 (2015).
[Crossref]

E. J. Candès and X. Li, “Solving quadratic equations via phaselift when there are about as many equations as unknowns,” Foundations of Computational Mathematics 14, 1017–1026 (2014).
[Crossref]

Ligthart, L.

X. Zhuge, A. G. Yarovoy, T. Savelyev, and L. Ligthart, “Modified kirchhoff migration for uwb mimo array-based radar imaging,” Geoscience and Remote Sensing, IEEE Transactions on 48, 2692–2703 (2010).
[Crossref]

Lipworth, G.

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

Lopez-Sanchez, J. M.

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-d radar imaging using range migration techniques,” Antennas and Propagation, IEEE Transactions on 48, 728–737 (2000).
[Crossref]

Marks, D. L.

Montaldo, G.

G. Montaldo, D. Palacio, M. Tanter, and M. Fink, “Building three-dimensional images using a time-reversal chaotic cavity,” Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 52, 1489–1497 (2005).
[Crossref]

Moscoso, M.

A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems 27, 015005 (2010).
[Crossref]

Mrozack, A.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

Odabasi, H.

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

Padgett, M.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

Palacio, D.

G. Montaldo, D. Palacio, M. Tanter, and M. Fink, “Building three-dimensional images using a time-reversal chaotic cavity,” Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 52, 1489–1497 (2005).
[Crossref]

Papanicolaou, G.

A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems 27, 015005 (2010).
[Crossref]

Reynolds, M.

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

Rose, A.

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

Sakamoto, T.

T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

Savelyev, T.

X. Zhuge, A. G. Yarovoy, T. Savelyev, and L. Ligthart, “Modified kirchhoff migration for uwb mimo array-based radar imaging,” Geoscience and Remote Sensing, IEEE Transactions on 48, 2692–2703 (2010).
[Crossref]

Saxton, W.

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

Sleasman, T.

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Applied Physics Letters 107, 204104 (2015).
[Crossref]

Smith, D. R.

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
[Crossref]

D. L. Marks, J. Gollub, and D. R. Smith, “Spatially resolving antenna arrays using frequency diversity,” J. Opt. Soc. Am. A 33, 899–912 (2016).
[Crossref]

O. Yurduseven, V. R. Gowda, J. N. Gollub, and D. R. Smith, “Printed aperiodic cavity for computational and microwave imaging,” IEEE Microwave and Wireless Components Letters 26, 367–369 (2016).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Applied Physics Letters 107, 204104 (2015).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Soltanolkotabi, M.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” Information Theory, IEEE Transactions on 61, 1985–2007 (2015).
[Crossref]

Strohmer, T.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Review 57, 225–251 (2015).
[Crossref]

Sun, B.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

Tanter, M.

G. Montaldo, D. Palacio, M. Tanter, and M. Fink, “Building three-dimensional images using a time-reversal chaotic cavity,” Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 52, 1489–1497 (2005).
[Crossref]

Trofatter, P.

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

Vittert, L. E.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

Voroninski, V.

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Review 57, 225–251 (2015).
[Crossref]

Wei, K.

K. Wei, “Solving systems of phaseless equations via kaczmarz methods: a proof of concept study,” Inverse Problems 31, 125008 (2015).
[Crossref]

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

Yarovoy, A. G.

X. Zhuge, A. G. Yarovoy, T. Savelyev, and L. Ligthart, “Modified kirchhoff migration for uwb mimo array-based radar imaging,” Geoscience and Remote Sensing, IEEE Transactions on 48, 2692–2703 (2010).
[Crossref]

Ye, Y.

M. Grant, S. Boyd, and Y. Ye, “Cvx: Matlab software for disciplined convex programming,” (2008).

Yurduseven, O.

O. Yurduseven, V. R. Gowda, J. N. Gollub, and D. R. Smith, “Printed aperiodic cavity for computational and microwave imaging,” IEEE Microwave and Wireless Components Letters 26, 367–369 (2016).
[Crossref]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Zhuge, X.

X. Zhuge, A. G. Yarovoy, T. Savelyev, and L. Ligthart, “Modified kirchhoff migration for uwb mimo array-based radar imaging,” Geoscience and Remote Sensing, IEEE Transactions on 48, 2692–2703 (2010).
[Crossref]

Antennas and Propagation, IEEE Transactions on (3)

J. M. Lopez-Sanchez and J. Fortuny-Guasch, “3-d radar imaging using range migration techniques,” Antennas and Propagation, IEEE Transactions on 48, 728–737 (2000).
[Crossref]

T. Fromenteze, C. Decroze, and D. Carsenat, “Waveform coding for passive multiplexing: Application to microwave imaging,” Antennas and Propagation, IEEE Transactions on 63, 593–600 (2015).
[Crossref]

B. Fuchs and L. Le Coq, “Excitation retrieval of microwave linear arrays from phaseless far-field data,” Antennas and Propagation, IEEE Transactions on 63, 748–754 (2015).
[Crossref]

Appl. Opt. (1)

Applied optics (1)

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Applied optics 54, 9343–9353 (2015).
[Crossref] [PubMed]

Applied Physics Letters (2)

T. Fromenteze, O. Yurduseven, M. F. Imani, J. Gollub, C. Decroze, D. Carsenat, and D. R. Smith, “Computational imaging using a mode-mixing cavity at microwave frequencies,” Applied Physics Letters 106, 194104 (2015).
[Crossref]

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Applied Physics Letters 107, 204104 (2015).
[Crossref]

Bulletin International de l’Académie Polonaise des Sciences et des Lettres (1)

S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

Foundations of Computational Mathematics (1)

E. J. Candès and X. Li, “Solving quadratic equations via phaselift when there are about as many equations as unknowns,” Foundations of Computational Mathematics 14, 1017–1026 (2014).
[Crossref]

Geoscience and Remote Sensing, IEEE Transactions on (1)

X. Zhuge, A. G. Yarovoy, T. Savelyev, and L. Ligthart, “Modified kirchhoff migration for uwb mimo array-based radar imaging,” Geoscience and Remote Sensing, IEEE Transactions on 48, 2692–2703 (2010).
[Crossref]

IEEE Access (1)

T. Fromenteze, E. Kpré, D. Carsenat, C. Decroze, and T. Sakamoto, “Single-shot compressive multiple-inputs multiple-outputs radar imaging using a two-port passive device,” IEEE Access 4, 1050–1060 (2016).
[Crossref]

IEEE Microwave and Wireless Components Letters (1)

O. Yurduseven, V. R. Gowda, J. N. Gollub, and D. R. Smith, “Printed aperiodic cavity for computational and microwave imaging,” IEEE Microwave and Wireless Components Letters 26, 367–369 (2016).
[Crossref]

Information Theory, IEEE Transactions on (1)

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” Information Theory, IEEE Transactions on 61, 1985–2007 (2015).
[Crossref]

Inverse Problems (2)

K. Wei, “Solving systems of phaseless equations via kaczmarz methods: a proof of concept study,” Inverse Problems 31, 125008 (2015).
[Crossref]

A. Chai, M. Moscoso, and G. Papanicolaou, “Array imaging using intensity-only measurements,” Inverse Problems 27, 015005 (2010).
[Crossref]

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

JOSA B (1)

T. Sleasman, M. Boyarsky, M. F. Imani, J. N. Gollub, and D. R. Smith, “Design considerations for a dynamic metamaterial aperture for computational imaging at microwave frequencies,” JOSA B 33, 1098–1111 (2016).
[Crossref]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref] [PubMed]

Opt. Lett. (1)

Optik (1)

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

Science (2)

J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, and D. R. Smith, “Metamaterial apertures for computational imaging,” Science 339, 310–313 (2013).
[Crossref] [PubMed]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. Padgett, “3d computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref] [PubMed]

SIAM Review (1)

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM Review 57, 225–251 (2015).
[Crossref]

Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on (1)

G. Montaldo, D. Palacio, M. Tanter, and M. Fink, “Building three-dimensional images using a time-reversal chaotic cavity,” Ultrasonics, Ferroelectrics, and Frequency Control, IEEE Transactions on 52, 1489–1497 (2005).
[Crossref]

Other (4)

T. Fromenteze, E. L. Kpre, C. Decroze, D. Carsenat, O. Yurduseven, M. Imani, J. Gollub, and D. R. Smith, “Unification of compressed imaging techniques in the microwave range and deconvolution strategy,” European Microwave Week 2015-Eurad (2015).

Y. Chen and E. Candes, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” Advances in Neural Information Processing Systems pp. 739–747 (2015).

M. Grant, S. Boyd, and Y. Ye, “Cvx: Matlab software for disciplined convex programming,” (2008).

B. Fuchs and L. Le Coq, “Phase retrieval procedure for microwave linear arrays,” Antennas and Propagation (EuCAP), 2015 9th European Conference on pp. 1–4 (2015).

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Figures (18)

Fig. 1
Fig. 1 The coded diffraction pattern measured from the illumination of a molecule by an X-ray source.
Fig. 2
Fig. 2 Computational imaging system used for the experimental demonstration. A metasurface radiating frequency diverse patterns is applied to the localization of field sources from the intensity measurement of a compressed frequency domain waveform.
Fig. 3
Fig. 3 Impact of quality factor Q on the damping time of the metasurface random responses.
Fig. 4
Fig. 4 Numerical simulation of the phaseless computational system applied to the localization of a field source. The radiated field is propagated to the receiving structure plane, coded by the response of this metasurface and compressed into a unique frequency domain signal.
Fig. 5
Fig. 5 Empirical probability of successful recovery. 100 trials are simulated for each pair of parameters m/n and αt.
Fig. 6
Fig. 6 Empirical success rate according to the sampling m/n. αt must be larger than 1/π to ensure that there is at least as many measured modes m as orthogonal channels available in the sensing matrix H to reconstruct n voxels.
Fig. 7
Fig. 7 Convergence of the phase retrieval algorithm according to the SNR for m/n = 6 and αt = 2. 100 trials are simulated for each SNR value.
Fig. 8
Fig. 8 Statistical distribution of the relative errors according to the SNR. In each case, the average μ of the relative error converges to the normalized noise floor 1 / SNR. The standard deviation of each distribution is given by σd.
Fig. 9
Fig. 9 Statistical study of the convergence of the algorithm according to the factor αt. The results are gathered in the right-hand side graphic, presenting the average μ of each distribution and the standard deviation σd. 1000 trials are computed for each value of αt.
Fig. 10
Fig. 10 Radiating metasurface implemented for the validation of the proposed phaseless computational technique.
Fig. 11
Fig. 11 Radiating metasurface implemented for the validation of the proposed phaseless computational technique.
Fig. 12
Fig. 12 Comparison of the near-field distributions Φ1(rr, ν) and Φ2(rr, ν) measured for the independent excitation of ports 1 and 2. The results are depicted for two consecutive frequency ν1 = 23 GHz and ν2 = 23.002 GHz of the frequency vector ν.
Fig. 13
Fig. 13 Localization of a field source on a domain of 10 × 10 × 10 voxels, with and without the phase information. The blue square represents the array of equivalent dipoles constituting the radiating metasurface.
Fig. 14
Fig. 14 Comparison of the x, y, and z-cuts extracted at the maximum value of the reconstructed fields and I. The orange solid lines correspond to the phaseless results I, and are compared to the dashed blue lines standing for the reconstructions from complex measurements .
Fig. 15
Fig. 15 Localization of a field source set in front of the radiating metasurface in a domain of 20 × 20 × 10 voxels, with and without the phase information. The blue square represents the array of equivalent dipoles constituting the radiating metasurface.
Fig. 16
Fig. 16 Comparison of the x, y, and z-cuts extracted at the maximum value of the reconstructed fields and I. The orange solid lines correspond to the phaseless results I, and are compared to the dashed blue lines standing for the reconstructions from complex measurements .
Fig. 17
Fig. 17 Localization of a field source shifted from the center in a domain of 20 × 20 × 10 voxels, with and without the phase information. The blue square represents the array of equivalent dipoles constituting the radiating metasurface.
Fig. 18
Fig. 18 Comparison of the x, y, and z-cuts extracted at the maximum value of the reconstructed fields and I for a source field shifted from the center of the imaging domain. The orange solid lines correspond to the phaseless results I, and are compared to the dashed blue lines standing for the reconstructions from complex measurements .

Equations (26)

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

y = | Ax | 2
y ( l ) = | FD ( l ) x | 2
y i = | a i , x | 2 = Tr ( x a i a i x ) = Tr ( a i a i x x ) = Tr ( a i a i X )
X 0 , rank ( X ) = 1 , y i = Tr ( a i a i X ) for i = 1 , , m
minimize X Tr ( X ) subject to X 0 , Tr ( a k a k X ) = y k , k = 1 , , m
x ( t + 1 ) = x ( t ) + μ t m i S t + 1 m l i ( x ( t ) )
l i ( x ( t ) ) = 2 y i | a i * x ( t ) | 2 x ( t ) * a i
a i * x ( t ) x ( t )
y i | a i * x ( t ) | 2 a i * x ( t ) 1 m | y i | a i * x ( t ) | 2 | x ( t )
ρ ( ν ) = r r r ϕ ( r r , ν ) g ( r r , r , ν ) f ( r ) d 3 r d 2 r r
H n ( ν k ) = G n , n r ( ν k ) ϕ n r ( ν k )
ρ = Hf
| ρ | 2 = | Hf | 2
ϕ ( r r , ν ) = 𝔉 [ d ( r r , t ) exp ( t π ν 0 / Q ) ]
ε = min θ [ 0 , 2 π ] f ^ I f ^ e j θ f ^
τ = α t d ν
Q = α t π ν 0 d ν
α t = Q d ν π ν 0 1 π
n Q B 5 ν 0
| ρ | 2 = | H f + η | 2
ρ = [ ρ 1 , ρ 2 ]
H = [ H 1 , H 2 ]
τ = Q π ν 0
m ν < π τ B
< Q B ν 0 4900
α t = Q d ν π ν 0 = Q B π ν 0 m ν 4.3

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