Abstract

A spectral camera based on ghost imaging via sparsity constraints (GISC) acquires a three-dimensional (3D) spatial-spectral data cube of the target through a two-dimensional (2D) detector in a single snapshot. However, the spectral and spatial resolution are interrelated because both of them are modulated by the same spatial random phase modulator. In this paper, we theoretically and experimentally demonstrate a system by equipping the GISC spectral camera with a flat-field grating to disperse the light fields before the spatial random phase modulator, hence consequently decoupling the spatial and spectral resolution. By theoretical derivation of the imaging process we obtain the spectral resolution 1nm and spatial resolution 50μm about the new system which are verified by the experiment. The new system can not only modulate the spatial and spectral resolution separately, but also provide a possibility of optimizing the light field fluctuations of different wavelengths according to the imaging scene.

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

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References

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    [Crossref]
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  13. G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
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  15. J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).
  16. Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
    [Crossref]
  17. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g (r),” Phys. Rev. Lett. 85(7), 1416 (2000).
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  21. Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University, 2012).
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  22. M. Elad, “Optimized projections for compressed sensing,” IEEE T. Signal Process. 55(12), 5695–5702(2007).
    [Crossref]
  23. M. Chen, E. Li, and S. Han, “Application of multi-correlation-scale measurement matrices in ghost imaging via sparsity constraints,” Appl. Optics 53(13), 2924–2928(2014).
    [Crossref]
  24. J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE T. Image Process. 18(7), 1395–1408(2009).
    [Crossref]
  25. X. Xu, E. Li, X. Shen, and S. Han, “Optimization of speckle patterns in ghost imaging via sparse constraints by mutual coherence minimization,” Chin. Opt. Lett. 13(7), 071101(2015).
    [Crossref]
  26. J. M. Lerner, R. J. Chambers, and G. Passereau, “Flat field imaging spectroscopy using aberration corrected holographic gratings,” Proc. SPIE 268, 122–128 (1981).
    [Crossref]
  27. E. Sokolova, “Holographic diffraction gratings for flat-field spectrometers,” J. Mod. Optic. 47(13), 2377–2389 (2000).
    [Crossref]
  28. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93(9), 093602 (2004).
    [Crossref] [PubMed]
  29. B. Luo, Z. Wen, Z. Wen, and T. Zeng, “Design of concave grating for ultraviolet-spectrum,” Spectrosc. Spect. Anal. 32(6), 1717–1721 (2012).
  30. S. Tan, Z. Liu, E. Li, and S. Han, “Hyperspectral compressed sensing based on prior images constrained,” Acta Optica Sinica 35(8), 0811003 (2015).
    [Crossref]
  31. A. F. Goetz, G. Vane, J. E. Solomon, and B. N. Rock, “Imaging spectrometry for earth remote sensing,” Science 228(4704), 1147–1153 (1985).
    [Crossref] [PubMed]
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  33. T. Zimmermann, J. Rietdorf, and R. Pepperkok, “Spectral imaging and its applications in live cell microscopy,” FEBS letters 546(1), 87–92 (2003).
    [Crossref] [PubMed]

2017 (1)

2016 (2)

L. Gao and L. V. Wang, “A review of snapshot multidimensional optical imaging: measuring photon tags in parallel,” Phys. Reports 616, 1–37 (2016).
[Crossref]

Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
[Crossref]

2015 (2)

X. Xu, E. Li, X. Shen, and S. Han, “Optimization of speckle patterns in ghost imaging via sparse constraints by mutual coherence minimization,” Chin. Opt. Lett. 13(7), 071101(2015).
[Crossref]

S. Tan, Z. Liu, E. Li, and S. Han, “Hyperspectral compressed sensing based on prior images constrained,” Acta Optica Sinica 35(8), 0811003 (2015).
[Crossref]

2014 (4)

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

M. Chen, E. Li, and S. Han, “Application of multi-correlation-scale measurement matrices in ghost imaging via sparsity constraints,” Appl. Optics 53(13), 2924–2928(2014).
[Crossref]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
[Crossref]

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516(7529), 74–77 (2014).
[Crossref] [PubMed]

2013 (1)

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

2012 (2)

M. W. Kudenov, J. M. Craven-Jones, C. J. Vandervlugt, E. L. Dereniak, and R. W. Aumiller, “Faceted grating prism for a computed tomographic imaging spectrometer,” Opt. Eng. 51(4), 044002 (2012).
[Crossref]

B. Luo, Z. Wen, Z. Wen, and T. Zeng, “Design of concave grating for ultraviolet-spectrum,” Spectrosc. Spect. Anal. 32(6), 1717–1721 (2012).

2009 (3)

2008 (2)

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[Crossref]

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

2007 (1)

M. Elad, “Optimized projections for compressed sensing,” IEEE T. Signal Process. 55(12), 5695–5702(2007).
[Crossref]

2006 (2)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E.J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

2004 (1)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

2003 (1)

T. Zimmermann, J. Rietdorf, and R. Pepperkok, “Spectral imaging and its applications in live cell microscopy,” FEBS letters 546(1), 87–92 (2003).
[Crossref] [PubMed]

2000 (2)

E. Sokolova, “Holographic diffraction gratings for flat-field spectrometers,” J. Mod. Optic. 47(13), 2377–2389 (2000).
[Crossref]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g (r),” Phys. Rev. Lett. 85(7), 1416 (2000).
[Crossref] [PubMed]

1998 (1)

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

1994 (1)

E. Herrala, J. T. Okkonen, T. S. Hyvarinen, M. Aikio, and J. Lammasniemi, “Imaging spectrometer for process industry applications,” Proc. SPIE 2248(33), 33–40 (1994).
[Crossref]

1985 (1)

A. F. Goetz, G. Vane, J. E. Solomon, and B. N. Rock, “Imaging spectrometry for earth remote sensing,” Science 228(4704), 1147–1153 (1985).
[Crossref] [PubMed]

1981 (1)

J. M. Lerner, R. J. Chambers, and G. Passereau, “Flat field imaging spectroscopy using aberration corrected holographic gratings,” Proc. SPIE 268, 122–128 (1981).
[Crossref]

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell system technical journal 27(3), 379–423 (1948).
[Crossref]

Aikio, M.

E. Herrala, J. T. Okkonen, T. S. Hyvarinen, M. Aikio, and J. Lammasniemi, “Imaging spectrometer for process industry applications,” Proc. SPIE 2248(33), 33–40 (1994).
[Crossref]

Arce, G. R.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
[Crossref]

Arguello, H.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
[Crossref]

Aronsson, M.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Aumiller, R. W.

M. W. Kudenov, J. M. Craven-Jones, C. J. Vandervlugt, E. L. Dereniak, and R. W. Aumiller, “Faceted grating prism for a computed tomographic imaging spectrometer,” Opt. Eng. 51(4), 044002 (2012).
[Crossref]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Bösecke, P.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

Brady, D. J.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
[Crossref]

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[Crossref]

Brady, D.J.

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Candès, E.J.

E.J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Carin, L.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
[Crossref]

Carpineti, M.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g (r),” Phys. Rev. Lett. 85(7), 1416 (2000).
[Crossref] [PubMed]

Cerbino, R.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

Chambers, R. J.

J. M. Lerner, R. J. Chambers, and G. Passereau, “Flat field imaging spectroscopy using aberration corrected holographic gratings,” Proc. SPIE 268, 122–128 (1981).
[Crossref]

Chen, M.

M. Chen, E. Li, and S. Han, “Application of multi-correlation-scale measurement matrices in ghost imaging via sparsity constraints,” Appl. Optics 53(13), 2924–2928(2014).
[Crossref]

Chen, Z.

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Chippendale, B. J.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Chovita, C. J.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Chrien, T. G.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, 2012).

Craven-Jones, J. M.

M. W. Kudenov, J. M. Craven-Jones, C. J. Vandervlugt, E. L. Dereniak, and R. W. Aumiller, “Faceted grating prism for a computed tomographic imaging spectrometer,” Opt. Eng. 51(4), 044002 (2012).
[Crossref]

Dang, C.

Dereniak, E. L.

M. W. Kudenov, J. M. Craven-Jones, C. J. Vandervlugt, E. L. Dereniak, and R. W. Aumiller, “Faceted grating prism for a computed tomographic imaging spectrometer,” Opt. Eng. 51(4), 044002 (2012).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Duarte-Carvajalino, J. M.

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE T. Image Process. 18(7), 1395–1408(2009).
[Crossref]

Eastwood, M. L.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Elad, M.

M. Elad, “Optimized projections for compressed sensing,” IEEE T. Signal Process. 55(12), 5695–5702(2007).
[Crossref]

Eldar, Y. C.

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University, 2012).
[Crossref]

Fausta, J. A.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Gao, L.

L. Gao and L. V. Wang, “A review of snapshot multidimensional optical imaging: measuring photon tags in parallel,” Phys. Reports 616, 1–37 (2016).
[Crossref]

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516(7529), 74–77 (2014).
[Crossref] [PubMed]

L. Gao, R. T. Kester, and T. S. Tkaczyk, “Compact Image Slicing Spectrometer (ISS) for hyperspectral fluorescence microscopy,” Opt. Express 17(15), 12293–12308 (2009).
[Crossref] [PubMed]

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Giglio, M.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g (r),” Phys. Rev. Lett. 85(7), 1416 (2000).
[Crossref] [PubMed]

Goetz, A. F.

A. F. Goetz, G. Vane, J. E. Solomon, and B. N. Rock, “Imaging spectrometry for earth remote sensing,” Science 228(4704), 1147–1153 (1985).
[Crossref] [PubMed]

Green, R. O.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Hagen, N. A.

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

Han, S.

S. Tan, Z. Liu, E. Li, and S. Han, “Hyperspectral compressed sensing based on prior images constrained,” Acta Optica Sinica 35(8), 0811003 (2015).
[Crossref]

X. Xu, E. Li, X. Shen, and S. Han, “Optimization of speckle patterns in ghost imaging via sparse constraints by mutual coherence minimization,” Chin. Opt. Lett. 13(7), 071101(2015).
[Crossref]

M. Chen, E. Li, and S. Han, “Application of multi-correlation-scale measurement matrices in ghost imaging via sparsity constraints,” Appl. Optics 53(13), 2924–2928(2014).
[Crossref]

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Han, X. S.

Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
[Crossref]

Herrala, E.

E. Herrala, J. T. Okkonen, T. S. Hyvarinen, M. Aikio, and J. Lammasniemi, “Imaging spectrometer for process industry applications,” Proc. SPIE 2248(33), 33–40 (1994).
[Crossref]

Hsieh, J.

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIEBellingham, WA, 2014).

Hyvarinen, T. S.

E. Herrala, J. T. Okkonen, T. S. Hyvarinen, M. Aikio, and J. Lammasniemi, “Imaging spectrometer for process industry applications,” Proc. SPIE 2248(33), 33–40 (1994).
[Crossref]

Kester, R. T.

Kittle, D. S.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
[Crossref]

Kudenov, M. W.

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

M. W. Kudenov, J. M. Craven-Jones, C. J. Vandervlugt, E. L. Dereniak, and R. W. Aumiller, “Faceted grating prism for a computed tomographic imaging spectrometer,” Opt. Eng. 51(4), 044002 (2012).
[Crossref]

Kutyniok, G.

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University, 2012).
[Crossref]

Lammasniemi, J.

E. Herrala, J. T. Okkonen, T. S. Hyvarinen, M. Aikio, and J. Lammasniemi, “Imaging spectrometer for process industry applications,” Proc. SPIE 2248(33), 33–40 (1994).
[Crossref]

Lerner, J. M.

J. M. Lerner, R. J. Chambers, and G. Passereau, “Flat field imaging spectroscopy using aberration corrected holographic gratings,” Proc. SPIE 268, 122–128 (1981).
[Crossref]

Li, C.

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516(7529), 74–77 (2014).
[Crossref] [PubMed]

Li, E.

Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
[Crossref]

X. Xu, E. Li, X. Shen, and S. Han, “Optimization of speckle patterns in ghost imaging via sparse constraints by mutual coherence minimization,” Chin. Opt. Lett. 13(7), 071101(2015).
[Crossref]

S. Tan, Z. Liu, E. Li, and S. Han, “Hyperspectral compressed sensing based on prior images constrained,” Acta Optica Sinica 35(8), 0811003 (2015).
[Crossref]

M. Chen, E. Li, and S. Han, “Application of multi-correlation-scale measurement matrices in ghost imaging via sparsity constraints,” Appl. Optics 53(13), 2924–2928(2014).
[Crossref]

Liang, J.

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516(7529), 74–77 (2014).
[Crossref] [PubMed]

Liu, Z.

Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
[Crossref]

S. Tan, Z. Liu, E. Li, and S. Han, “Hyperspectral compressed sensing based on prior images constrained,” Acta Optica Sinica 35(8), 0811003 (2015).
[Crossref]

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Luo, B.

B. Luo, Z. Wen, Z. Wen, and T. Zeng, “Design of concave grating for ultraviolet-spectrum,” Spectrosc. Spect. Anal. 32(6), 1717–1721 (2012).

Okkonen, J. T.

E. Herrala, J. T. Okkonen, T. S. Hyvarinen, M. Aikio, and J. Lammasniemi, “Imaging spectrometer for process industry applications,” Proc. SPIE 2248(33), 33–40 (1994).
[Crossref]

Olaha, M. R.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Passereau, G.

J. M. Lerner, R. J. Chambers, and G. Passereau, “Flat field imaging spectroscopy using aberration corrected holographic gratings,” Proc. SPIE 268, 122–128 (1981).
[Crossref]

Pavria, B. E.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Pepperkok, R.

T. Zimmermann, J. Rietdorf, and R. Pepperkok, “Spectral imaging and its applications in live cell microscopy,” FEBS letters 546(1), 87–92 (2003).
[Crossref] [PubMed]

Peverini, L.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

Pitsianis, N. P.

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D.J. Brady, “Compressive coded aperture spectral imaging: An introduction,” Opt. Express 17(8), 6368–6388 (2009).
[Crossref] [PubMed]

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[Crossref]

Potenza, M. A. C.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

Rietdorf, J.

T. Zimmermann, J. Rietdorf, and R. Pepperkok, “Spectral imaging and its applications in live cell microscopy,” FEBS letters 546(1), 87–92 (2003).
[Crossref] [PubMed]

Robert, A.

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

Rock, B. N.

A. F. Goetz, G. Vane, J. E. Solomon, and B. N. Rock, “Imaging spectrometry for earth remote sensing,” Science 228(4704), 1147–1153 (1985).
[Crossref] [PubMed]

Romberg, J.

E.J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Sabins, F. F.

F. F. Sabins, Remote Sensing: Principles and Applications (Waveland, 2007).

Sahoo, S. K.

Sapiro, G.

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE T. Image Process. 18(7), 1395–1408(2009).
[Crossref]

Sarture, C. M.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell system technical journal 27(3), 379–423 (1948).
[Crossref]

Shen, X.

X. Xu, E. Li, X. Shen, and S. Han, “Optimization of speckle patterns in ghost imaging via sparse constraints by mutual coherence minimization,” Chin. Opt. Lett. 13(7), 071101(2015).
[Crossref]

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Sokolova, E.

E. Sokolova, “Holographic diffraction gratings for flat-field spectrometers,” J. Mod. Optic. 47(13), 2377–2389 (2000).
[Crossref]

Solisa, M.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Solomon, J. E.

A. F. Goetz, G. Vane, J. E. Solomon, and B. N. Rock, “Imaging spectrometry for earth remote sensing,” Science 228(4704), 1147–1153 (1985).
[Crossref] [PubMed]

Sun, X.

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D.J. Brady, “Compressive coded aperture spectral imaging: An introduction,” Opt. Express 17(8), 6368–6388 (2009).
[Crossref] [PubMed]

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[Crossref]

Tan, S.

Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
[Crossref]

S. Tan, Z. Liu, E. Li, and S. Han, “Hyperspectral compressed sensing based on prior images constrained,” Acta Optica Sinica 35(8), 0811003 (2015).
[Crossref]

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Tang, D.

Tao, T.

E.J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Tao, Z.

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, 2012).

Tkaczyk, T. S.

Vailati, A.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g (r),” Phys. Rev. Lett. 85(7), 1416 (2000).
[Crossref] [PubMed]

Vandervlugt, C. J.

M. W. Kudenov, J. M. Craven-Jones, C. J. Vandervlugt, E. L. Dereniak, and R. W. Aumiller, “Faceted grating prism for a computed tomographic imaging spectrometer,” Opt. Eng. 51(4), 044002 (2012).
[Crossref]

Vane, G.

A. F. Goetz, G. Vane, J. E. Solomon, and B. N. Rock, “Imaging spectrometry for earth remote sensing,” Science 228(4704), 1147–1153 (1985).
[Crossref] [PubMed]

Wagadarikar, A. A.

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D.J. Brady, “Compressive coded aperture spectral imaging: An introduction,” Opt. Express 17(8), 6368–6388 (2009).
[Crossref] [PubMed]

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[Crossref]

Wang, L. V.

L. Gao and L. V. Wang, “A review of snapshot multidimensional optical imaging: measuring photon tags in parallel,” Phys. Reports 616, 1–37 (2016).
[Crossref]

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516(7529), 74–77 (2014).
[Crossref] [PubMed]

Wen, Z.

B. Luo, Z. Wen, Z. Wen, and T. Zeng, “Design of concave grating for ultraviolet-spectrum,” Spectrosc. Spect. Anal. 32(6), 1717–1721 (2012).

B. Luo, Z. Wen, Z. Wen, and T. Zeng, “Design of concave grating for ultraviolet-spectrum,” Spectrosc. Spect. Anal. 32(6), 1717–1721 (2012).

Williamsa, O.

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Wu, J.

Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
[Crossref]

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Xu, X.

Yu, H.

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Zeng, T.

B. Luo, Z. Wen, Z. Wen, and T. Zeng, “Design of concave grating for ultraviolet-spectrum,” Spectrosc. Spect. Anal. 32(6), 1717–1721 (2012).

Zimmermann, T.

T. Zimmermann, J. Rietdorf, and R. Pepperkok, “Spectral imaging and its applications in live cell microscopy,” FEBS letters 546(1), 87–92 (2003).
[Crossref] [PubMed]

Acta Optica Sinica (1)

S. Tan, Z. Liu, E. Li, and S. Han, “Hyperspectral compressed sensing based on prior images constrained,” Acta Optica Sinica 35(8), 0811003 (2015).
[Crossref]

Acta Phys. Sin-CH ED (1)

J. Wu, X. Shen, H. Yu, Z. Chen, Z. Tao, S. Tan, and S. Han, “Snapshot compressive imaging by phase modulation,” Acta Phys. Sin-CH ED 34(10), 1011005 (2014).

Appl. Optics (1)

M. Chen, E. Li, and S. Han, “Application of multi-correlation-scale measurement matrices in ghost imaging via sparsity constraints,” Appl. Optics 53(13), 2924–2928(2014).
[Crossref]

Bell system technical journal (1)

C. E. Shannon, “A mathematical theory of communication,” Bell system technical journal 27(3), 379–423 (1948).
[Crossref]

Chin. Opt. Lett. (1)

FEBS letters (1)

T. Zimmermann, J. Rietdorf, and R. Pepperkok, “Spectral imaging and its applications in live cell microscopy,” FEBS letters 546(1), 87–92 (2003).
[Crossref] [PubMed]

IEEE Signal Process. Mag. (1)

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An introduction,” IEEE Signal Process. Mag. 31(1), 105–115 (2014).
[Crossref]

IEEE T. Image Process. (1)

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE T. Image Process. 18(7), 1395–1408(2009).
[Crossref]

IEEE T. Signal Process. (1)

M. Elad, “Optimized projections for compressed sensing,” IEEE T. Signal Process. 55(12), 5695–5702(2007).
[Crossref]

IEEE Trans. Inf. Theory (2)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E.J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

J. Mod. Optic. (1)

E. Sokolova, “Holographic diffraction gratings for flat-field spectrometers,” J. Mod. Optic. 47(13), 2377–2389 (2000).
[Crossref]

Nature (1)

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516(7529), 74–77 (2014).
[Crossref] [PubMed]

Nature Phys. (1)

R. Cerbino, L. Peverini, M. A. C. Potenza, A. Robert, P. Bösecke, and M. Giglio, “X-ray-scattering information obtained from near-field speckle,” Nature Phys. 4(3), 238–243 (2008).
[Crossref]

Opt. Eng. (2)

N. A. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52(9), 090901 (2013).
[Crossref]

M. W. Kudenov, J. M. Craven-Jones, C. J. Vandervlugt, E. L. Dereniak, and R. W. Aumiller, “Faceted grating prism for a computed tomographic imaging spectrometer,” Opt. Eng. 51(4), 044002 (2012).
[Crossref]

Opt. Express (2)

Optica (1)

Phys. Reports (1)

L. Gao and L. V. Wang, “A review of snapshot multidimensional optical imaging: measuring photon tags in parallel,” Phys. Reports 616, 1–37 (2016).
[Crossref]

Phys. Rev. Lett. (2)

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g (r),” Phys. Rev. Lett. 85(7), 1416 (2000).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Proc. SPIE (3)

J. M. Lerner, R. J. Chambers, and G. Passereau, “Flat field imaging spectroscopy using aberration corrected holographic gratings,” Proc. SPIE 268, 122–128 (1981).
[Crossref]

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Spectral image estimation for coded aperture snapshot spectral imagers,” Proc. SPIE 7076, 707602 (2008).
[Crossref]

E. Herrala, J. T. Okkonen, T. S. Hyvarinen, M. Aikio, and J. Lammasniemi, “Imaging spectrometer for process industry applications,” Proc. SPIE 2248(33), 33–40 (1994).
[Crossref]

Remote Sens. Environ. (1)

R. O. Green, M. L. Eastwood, C. M. Sarture, T. G. Chrien, M. Aronsson, B. J. Chippendale, J. A. Fausta, B. E. Pavria, C. J. Chovita, M. Solisa, M. R. Olaha, and O. Williamsa, “Imaging spectrometer for process industry applications,” Remote Sens. Environ. 65(3), 227–248 (1998).
[Crossref]

Sci. Reports (1)

Z. Liu, S. Tan, J. Wu, E. Li, and X. S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Reports 6, 25718 (2016).
[Crossref]

Science (1)

A. F. Goetz, G. Vane, J. E. Solomon, and B. N. Rock, “Imaging spectrometry for earth remote sensing,” Science 228(4704), 1147–1153 (1985).
[Crossref] [PubMed]

Spectrosc. Spect. Anal. (1)

B. Luo, Z. Wen, Z. Wen, and T. Zeng, “Design of concave grating for ultraviolet-spectrum,” Spectrosc. Spect. Anal. 32(6), 1717–1721 (2012).

Other (4)

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University, 2012).
[Crossref]

F. F. Sabins, Remote Sensing: Principles and Applications (Waveland, 2007).

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIEBellingham, WA, 2014).

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, 2012).

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Figures (6)

Fig. 1
Fig. 1 Schematic of GISC hyperspectral camera based on a flat-field grating. (a) The object plane; (b) the first imaging plane; (c) the diffractive plane; (d) the speckles plane; (1) an objective lens; (2) a flat-field grating; (3) a spatial random phase modulator; (4) a microscope objective; (5) a Charge Coupled Device (CCD).
Fig. 2
Fig. 2 Schematic diagram of coordinate conversion relations.
Fig. 3
Fig. 3 The simple sketches of the calibration. (a) Point sources with different wavelengths are at the same pixel of field-of-view (FoV) and illuminate the flat-field grating, and then are spatially dispersed and imaged to the diffractive plane at different places with a certain adjacent distance determined by the dispersion coefficient. (b) Point sources with the same wavelength are at different pixels of FoV and illuminate the flat-field grating, and then are spatially dispersed and imaged to the diffraction plane at different places with a certain adjacent distance determined by the distance between two pixels.
Fig. 4
Fig. 4 The experimental setup of GISC hyperspectral camera based on a flat-field grating. During the calibration, a calibration setup is put in front of the objective lens to acquire incoherent intensity impulse response functions of the system. After the calibration, an object is put before the objective lens to obtain the detected signal which is the overlay of the speckles from different pixels and wavelengths of the object.
Fig. 5
Fig. 5 Experimentally determined resolution. (a) Two normalized second-order correlation functions of light fields at a pixel in FoV with two different wavelengths, g d r ( 2 ) (x0, y0, λ; x0, y0, λ′), and at two different pixels with the same wavelength, g d r ( 2 ) (x0, y0, λ′; x0, y0, λ′), whose half width respectively determine the spectral resolution (1nm) and spacial resolution (50μm). The experiment result (the blue line) is consist with the theoretical result (red dotted line). (b) Tow point light sources with two different wavelengths, 539 nm and 540 nm, whose distance is 40μm. (c) The analysis of spectral and spatial resolution (according to the reconstructed image of two points). The 3D schematic about the spectral and spatial distribution (upper). The red line and green line (lower left) are respectively the spectral distributions about two points with 539nm and 540nm center wavelength and the red line (lower right) is the spatial distributions about two points. The spectrum and position of the points are considered resolved if they are separated by a dip of at least 20%. The result verifies the resolution determined by the normalized second-order correlation function g d r ( 2 ).
Fig. 6
Fig. 6 Experimentally imaging result. (a) A little girl taken by conventional camera. (b,c) A little girl & institute logo passing through a 536–545 nm narrowed band pass filter detected by CCD1. (d) The reconstructed spectral images of institute logo & little girl of spectral 3D data-cube, displaying all the channels from 536 to 545 nm.

Equations (22)

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I d ( x 2 , y 2 ) = I b ( x 0 , y 0 , λ ) h I ( x 2 , y 2 ; x 0 , y 0 , λ ) d x 0 d y 0 d λ ,
I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) = I b r ( x 0 , y 0 , λ ; x 0 , y 0 , λ ) h I ( x 2 , y 2 , ; x 0 , y 0 , λ ) d x 0 d y 0 d λ = δ ( x 0 x 0 , y 0 y 0 , λ λ ) h I ( x 2 , y 2 , ; x 0 , y 0 , λ ) d x 0 d y 0 d λ = h I ( x 2 , y 2 ; x 0 , y 0 , λ ) .
I d t ( x 2 , y 2 ) = T 0 ( x 0 , y 0 , λ ) I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) d x 0 d y 0 d λ ,
G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d t ) = E d r * ( x 2 , y 2 ; x 0 , y 0 , λ ) E d t * ( x 2 , y 2 ) E d t ( x 2 , y 2 ) E d r ( x 2 , y 2 ; x 0 , y 0 , λ ) ,
G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d r ) = I d r ( x 2 , y 2 ; x 0 y 0 , λ ) T ( x 0 , y 0 , λ ) I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) d x 0 d y 0 d λ = T ( x 0 , y 0 , λ ) G d r ( 2 ) ( x 0 y 0 , λ ; x 0 , y 0 , λ ) d x 0 d y 0 d λ ,
G d r ( 2 ) ( x 0 y 0 , λ ; x 0 , y 0 , λ ) = E d r * ( x 2 , y 2 ; x 0 y 0 , λ ) E d r * ( x 2 , y 2 ; x 0 , y 0 , λ ) E d r ( x 2 , y 2 ; x 0 , y 0 , λ ) E d r ( x 2 , y 2 ; x 0 y 0 , λ ) ,
G d r ( 2 ) ( x 0 y 0 , λ ; x 0 , y 0 , λ ) = ( I d r ( x 2 , y 2 ; x 0 y 0 , λ ) I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) ) × ( 1 + g d r ( 2 ) ( x 0 y 0 , λ ; x 0 , y 0 , λ ) ) ,
g d r ( 2 ) ( x 0 y 0 , λ ; x 0 , y 0 , λ ) = | J ( x 0 y 0 , λ ; x 0 , y 0 , λ ) | 2 I d r ( x 2 , y 2 ; x 0 y 0 , λ ) I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) ,
J ( x 0 y 0 , λ ; x 0 , y 0 , λ ) = E d r * ( x 2 , y 2 ; x 0 y 0 , λ ) E d r ( x 2 , y 2 ; x 0 , y 0 , λ ) .
t g ( x , y ) = ( rect ( x a ) e j ϕ ) 1 d eff comb ( x d eff ) ,
t p ( x m , y m , λ ) = exp [ j 2 π ( n 1 ) h ( x m , y m ) λ ] ,
R h ( x m , y m ; x m y m ) = h ( x m , y m ) h ( x m y m ) = ω 2 exp [ ( x m x m ) 2 + ( y m y m ) 2 ζ 2 ] ,
E d r ( x 2 , y 2 ; x 0 y 0 , λ ) = e j ϕ a λ z 1 λ 2 z 2 z 3 exp [ j k ( z 0 + z 1 ) ] j λ z 0 z 1 exp ( j 2 π λ x 0 2 + y 0 2 2 z 0 ) × exp ( j 2 π λ x 1 2 + y 1 2 2 z 1 ) sinc ( a λ z 1 f x ) 1 d eff n = δ ( f x λ z 1 d eff n ) × exp { j 2 π λ [ ( z 2 + z 3 ) + ( X 2 X 1 ) 2 + ( y 2 y 1 ) 2 2 ( z 2 + z 3 ) ] } t p ( X m , y m , λ ) × exp { j π λ z 2 + z 3 z 2 z 3 [ ( X m z 3 X 1 + z 2 X 2 z 2 + z 3 ) 2 + ( y m z 3 y 1 + z 2 y 2 z 2 + z 3 ) 2 ] } d X m d y m ,
g d r ( 2 ) ( x 0 , y 0 ; x 0 y 0 , λ ) = ( z 2 + z 3 ) 4 λ 4 z 2 4 z 3 4 × | t p ( X m , y m , λ ) t * ( X m y m , λ ) × exp { j π ( z 2 + z 3 ) z 2 z 3 [ 1 λ ( α 2 + α 2 ) 1 λ ( β 2 + β 2 ) ] } d X m d y m d X m d y m | 2 ,
t p ( X m , y m , λ ) t p * ( X m , y m , λ ) = exp { j 2 π ( n 1 ) [ h ( X m , y m ) λ h ( X m , y m , ) λ ] } = M ˜ H ( X m , y m ) H ( X m , y m ) ( 2 π ( n 1 ) λ , 2 π ( n 1 ) λ ) ,
M ˜ H ( X m , y m ) H ( X m , y m ) ( 2 π ( n 1 ) λ 2 π ( n 1 ) λ ) = exp { 1 2 [ 2 π ( n 1 ) ] 2 [ ( 1 λ 2 + 1 λ 2 ) ω 2 2 R h ( X m , y m , X m , y m ) λ λ ] } .
g d r ( 2 ) ( x 0 , y 0 , λ ; x 0 , y 0 , λ ) exp { [ 2 π ω ( n 1 ) ] 2 [ ( 1 λ 1 λ ) 2 + 2 λ λ ] } × exp ( 2 [ 2 π ω ( n 1 ) ] 2 λ λ exp { ( z 3 z 1 sec β H ) 2 [ ( λ λ ) d eff ( x 0 x 0 ) z 0 ] 2 z 3 2 ( y 0 y 0 ) 2 ( z 2 + z 3 ) 2 ζ 2 } ) .
Δ G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d t ) = G ( 2 ) ( ( x 2 , y 2 ) d r , ( x 2 , y 2 ) d t ) I d r ( x 2 , y 2 ; x 0 , y 0 , λ ) I d t ( x 2 , y 2 ) k 2 a 4 ( z 2 + z 3 ) 4 z 0 4 d eff 4 [ sinc ( a d eff ) ] 4 T ( x 0 , y 0 , k ) exp { 2 [ 2 π ω ( n 1 ) ] 2 × exp [ z 3 2 z 1 2 ( sec β H ) 2 ( z 2 + z 3 ) 2 ζ 2 z 0 2 x 0 2 z 3 2 y 0 2 ( z 2 + z 3 ) 2 ζ 2 ] × [ k 2 2 z 3 2 z 1 2 ( sec β H ) 2 ( z 2 + z 3 ) 2 ζ 2 d eff z 0 k x 0 ] 3 [ 2 π ω ( n 1 ) ] 2 k 2 } .
Y = AX ,
A = ( A 1 , 1 λ 1 A 1 , N λ 1 A 1 , 1 λ 2 A 1 , N λ 2 A 1 , 1 λ L A 1 , N λ L A 2 , 1 λ 1 A 2 , N λ 1 A 2 , 1 λ 2 A 2 , N λ 2 A 2 , 1 λ L A 2 , N λ L A M , 1 λ 1 A M , N λ 1 A M , 1 λ 2 A M , N λ 2 A M , 1 λ L A M , N λ L )
y p = λ = 1 L q = 1 N A p , q λ x q λ ,
X = arg min X 0 Y A X 2 2 + μ 1 Φ X 1 + μ 2 X * ,