Abstract

We present a new type of compressive spectroscopy technique employing a liquid crystal (LC) phase retarder. A tunable LC cell is used in a manner compliant with the compressive sensing (CS) framework to significantly reduce the spectral scanning effort. The presented optical spectrometer consists of a single LC phase retarder combined with a single photo detector, where the LC phase retarder is used to modulate the input spectrum and the photodiode is used to measure the transmitted spectral signal. Sequences of measurements are taken, where each measurement is done with a different state of the retarder. Then, the set of photodiode measurements is used as input data to a CS solver algorithm. We demonstrate numerally compressive spectral sensing with approximately ten times fewer measurements than with an equivalent conventional spectrometer.

© 2013 Optical Society of America

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  1. R. Richards-Kortum and E. Sevick-Muraca, Annu. Rev. Phys. Chem. 47, 555 (1996).
    [CrossRef]
  2. M. T. Eismann, Hyperspectral Remote Sensing (SPIE, 2012).
  3. T. A. Warner, M. D. Nellis, and G. M. Foody, The Sage Handbook of Remote Sensing (Sage, 2009).
  4. P. S. Thenkabail, J. G. Lyon, and A. Huete, Hyperspectral Remote Sensing of Vegetation (CRC Press, 2012).
  5. D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
  6. E. J. Candes and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).
  7. Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University, 2012).
  8. M. Elad, Sparse and Redundant Representations (Springer, 2010).
  9. R. Baraniuk, IEEE Signal Process. Mag. 24, 118 (2007).
  10. J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
  11. M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, IEEE_J. Sel. Topics Signal Process. 1, 586 (2007).
  12. A. Wagadarikar, R. John, R. Willett, and D. Brady, Appl. Opt. 47, B44 (2008).
    [CrossRef]
  13. Y. August, C. Vachman, Y. Rivenson, and A. Stern, Appl. Opt. 52, D46 (2013).
    [CrossRef]
  14. T. Sun and K. Kelly, in Computational Optical Sensing and Imaging (Optical Society of America, 2009), pp. TuA5.
  15. H. Arguello and G. Arce, in Imaging Systems Applications (Optical Society of America, 2011), pp. TuA4.
  16. F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).
  17. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  18. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).
  19. J. Beeckman, K. Neyts, and P. J. Vanbrabant, Opt. Eng. 50, 081202 (2011).
    [CrossRef]
  20. O. Aharon and I. Abdulhalim, Opt. Express 17, 11426 (2009).
    [CrossRef]
  21. J. W. Evans, J. Opt. Soc. Am. 39, 229 (1949).
    [CrossRef]
  22. J. W. Evans, J. Opt. Soc Am. 48, 142 (1958).
  23. R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.
  24. J. Romberg, SIAM J. Imaging Sci. 2, 1098 (2009).
  25. A. Karnieli, G. J. Kidron, C. Glaesser, and E. Ben-Dor, Remote Sens. Environ. 69, 67 (1999).
    [CrossRef]
  26. B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
    [CrossRef]

2013

2011

J. Beeckman, K. Neyts, and P. J. Vanbrabant, Opt. Eng. 50, 081202 (2011).
[CrossRef]

2009

O. Aharon and I. Abdulhalim, Opt. Express 17, 11426 (2009).
[CrossRef]

J. Romberg, SIAM J. Imaging Sci. 2, 1098 (2009).

2008

E. J. Candes and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

A. Wagadarikar, R. John, R. Willett, and D. Brady, Appl. Opt. 47, B44 (2008).
[CrossRef]

2007

R. Baraniuk, IEEE Signal Process. Mag. 24, 118 (2007).

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, IEEE_J. Sel. Topics Signal Process. 1, 586 (2007).

2006

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).

1999

A. Karnieli, G. J. Kidron, C. Glaesser, and E. Ben-Dor, Remote Sens. Environ. 69, 67 (1999).
[CrossRef]

1996

R. Richards-Kortum and E. Sevick-Muraca, Annu. Rev. Phys. Chem. 47, 555 (1996).
[CrossRef]

1958

J. W. Evans, J. Opt. Soc Am. 48, 142 (1958).

1949

Abdulhalim, I.

Abed, D.

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

Aharon, O.

Arce, G.

H. Arguello and G. Arce, in Imaging Systems Applications (Optical Society of America, 2011), pp. TuA4.

Arguello, H.

H. Arguello and G. Arce, in Imaging Systems Applications (Optical Society of America, 2011), pp. TuA4.

August, Y.

Baraniuk, R.

R. Baraniuk, IEEE Signal Process. Mag. 24, 118 (2007).

Beeckman, J.

J. Beeckman, K. Neyts, and P. J. Vanbrabant, Opt. Eng. 50, 081202 (2011).
[CrossRef]

Ben-Dor, E.

A. Karnieli, G. J. Kidron, C. Glaesser, and E. Ben-Dor, Remote Sens. Environ. 69, 67 (1999).
[CrossRef]

Benkelfat, B.

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).

Boumakh, Y.

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

Brady, D.

Büdel, B.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Candes, E. J.

E. J. Candes and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).

Clemente, P.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

Deutschewitz, K.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Donoho, D. L.

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).

Durán, V.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

Eismann, M. T.

M. T. Eismann, Hyperspectral Remote Sensing (SPIE, 2012).

Eitel, B.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Elad, M.

M. Elad, Sparse and Redundant Representations (Springer, 2010).

Eldar, Y. C.

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

Evans, J. W.

J. W. Evans, J. Opt. Soc Am. 48, 142 (1958).

J. W. Evans, J. Opt. Soc. Am. 39, 229 (1949).
[CrossRef]

Farha, R. M.

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

Fernández-Alonso, M.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, IEEE_J. Sel. Topics Signal Process. 1, 586 (2007).

Foody, G. M.

T. A. Warner, M. D. Nellis, and G. M. Foody, The Sage Handbook of Remote Sensing (Sage, 2009).

Glaesser, C.

A. Karnieli, G. J. Kidron, C. Glaesser, and E. Ben-Dor, Remote Sens. Environ. 69, 67 (1999).
[CrossRef]

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

Halassi, A.

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

Hamdi, R.

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

Hill, J.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Huete, A.

P. S. Thenkabail, J. G. Lyon, and A. Huete, Hyperspectral Remote Sensing of Vegetation (CRC Press, 2012).

Irles, E.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

John, R.

Karnieli, A.

A. Karnieli, G. J. Kidron, C. Glaesser, and E. Ben-Dor, Remote Sens. Environ. 69, 67 (1999).
[CrossRef]

Kelly, K.

T. Sun and K. Kelly, in Computational Optical Sensing and Imaging (Optical Society of America, 2009), pp. TuA5.

Kidron, G. J.

A. Karnieli, G. J. Kidron, C. Glaesser, and E. Ben-Dor, Remote Sens. Environ. 69, 67 (1999).
[CrossRef]

Knerr, T.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Kutyniok, G.

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

Lancis, J.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

Lyon, J. G.

P. S. Thenkabail, J. G. Lyon, and A. Huete, Hyperspectral Remote Sensing of Vegetation (CRC Press, 2012).

Nellis, M. D.

T. A. Warner, M. D. Nellis, and G. M. Foody, The Sage Handbook of Remote Sensing (Sage, 2009).

Neyts, K.

J. Beeckman, K. Neyts, and P. J. Vanbrabant, Opt. Eng. 50, 081202 (2011).
[CrossRef]

Nowak, R. D.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, IEEE_J. Sel. Topics Signal Process. 1, 586 (2007).

Olehowski, C.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Redadaa, S.

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

Richards-Kortum, R.

R. Richards-Kortum and E. Sevick-Muraca, Annu. Rev. Phys. Chem. 47, 555 (1996).
[CrossRef]

Rivenson, Y.

Romberg, J.

J. Romberg, SIAM J. Imaging Sci. 2, 1098 (2009).

Sevick-Muraca, E.

R. Richards-Kortum and E. Sevick-Muraca, Annu. Rev. Phys. Chem. 47, 555 (1996).
[CrossRef]

Soldevila, F.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

Stern, A.

Sun, T.

T. Sun and K. Kelly, in Computational Optical Sensing and Imaging (Optical Society of America, 2009), pp. TuA5.

Tajahuerce, E.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

Thenkabail, P. S.

P. S. Thenkabail, J. G. Lyon, and A. Huete, Hyperspectral Remote Sensing of Vegetation (CRC Press, 2012).

Vachman, C.

Vanbrabant, P. J.

J. Beeckman, K. Neyts, and P. J. Vanbrabant, Opt. Eng. 50, 081202 (2011).
[CrossRef]

Wagadarikar, A.

Wakin, M. B.

E. J. Candes and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).

Warner, T. A.

T. A. Warner, M. D. Nellis, and G. M. Foody, The Sage Handbook of Remote Sensing (Sage, 2009).

Weber, B.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Wessels, D. C. J.

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

Willett, R.

Wright, S. J.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, IEEE_J. Sel. Topics Signal Process. 1, 586 (2007).

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

Annu. Rev. Phys. Chem.

R. Richards-Kortum and E. Sevick-Muraca, Annu. Rev. Phys. Chem. 47, 555 (1996).
[CrossRef]

Appl. Opt.

IEEE Signal Process. Mag.

E. J. Candes and M. B. Wakin, IEEE Signal Process. Mag. 25(2), 21 (2008).

R. Baraniuk, IEEE Signal Process. Mag. 24, 118 (2007).

IEEE Trans. Image Process.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).

IEEE Trans. Inf. Theory

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).

IEEE_J. Sel. Topics Signal Process.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, IEEE_J. Sel. Topics Signal Process. 1, 586 (2007).

J. Opt. Soc Am.

J. W. Evans, J. Opt. Soc Am. 48, 142 (1958).

J. Opt. Soc. Am.

Opt. Eng.

J. Beeckman, K. Neyts, and P. J. Vanbrabant, Opt. Eng. 50, 081202 (2011).
[CrossRef]

Opt. Express

Remote Sens. Environ.

A. Karnieli, G. J. Kidron, C. Glaesser, and E. Ben-Dor, Remote Sens. Environ. 69, 67 (1999).
[CrossRef]

B. Weber, C. Olehowski, T. Knerr, J. Hill, K. Deutschewitz, D. C. J. Wessels, B. Eitel, and B. Büdel, Remote Sens. Environ. 112, 2187 (2008).
[CrossRef]

SIAM J. Imaging Sci.

J. Romberg, SIAM J. Imaging Sci. 2, 1098 (2009).

Other

R. Hamdi, R. M. Farha, S. Redadaa, B. Benkelfat, D. Abed, A. Halassi, and Y. Boumakh, ICT Telecommunications, 19th International Conference (2012), pp. 1–4.

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

M. Elad, Sparse and Redundant Representations (Springer, 2010).

M. T. Eismann, Hyperspectral Remote Sensing (SPIE, 2012).

T. A. Warner, M. D. Nellis, and G. M. Foody, The Sage Handbook of Remote Sensing (Sage, 2009).

P. S. Thenkabail, J. G. Lyon, and A. Huete, Hyperspectral Remote Sensing of Vegetation (CRC Press, 2012).

T. Sun and K. Kelly, in Computational Optical Sensing and Imaging (Optical Society of America, 2009), pp. TuA5.

H. Arguello and G. Arce, in Imaging Systems Applications (Optical Society of America, 2011), pp. TuA4.

F. Soldevila, E. Irles, V. Durán, P. Clemente, M. Fernández-Alonso, E. Tajahuerce, and J. Lancis, Appl. Phys. B, 1 (2013).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 2009).

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Figures (5)

Fig. 1.
Fig. 1.

Figurative description of spectral modulation by passing through the structure of a LC-based retarder.

Fig. 2.
Fig. 2.

Liquid crystal variable retarder used as a spectral modulator and a photo sensor that integrates the modulated spectrum. The spectral modulation at the output is a function of the birefringence, which can be controlled by the applied voltage over the cell.

Fig. 3.
Fig. 3.

Three different spectral responses obtained by changing the electrical potential over the LC retarder cell (theoretical model).

Fig. 4.
Fig. 4.

Example for spectroscopic transmission measurements of three different spectral responses of LC cell. These spectral responses were obtained by changing the electrical potential (1.05, 0.95, and 0.85 V) over the experimental LC retarder cell.

Fig. 5.
Fig. 5.

Directly measured (continuous line) and reconstructed (dashed line) biogenic crust-like spectra. The spectral data has N=1024 samples and the CS spectrum is recovered from M=108 measurements (“coif3” sparsifying operator). Peak SNR values for the reconstructions are 39.6, 33.8, 23.6 and 22.6 dB for spectra A-D, respectively.

Equations (5)

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

gi=[ϕi(λ1)ϕi(λk)ϕi(λN)]·[f(λ1)f(λk)f(λN)],
f^=Ψα^;α^=minα^{gΦΨα22+γα1},
gi=f(λ)·hi(λ)dλ,
ϕi(λ)=Ii(λ)/I0(λ)sin2(δi(λ)/2).
Φ=(sin2(12δ1(λ1))sin2(12δ1(λN))sin2(12δi(λj))sin2(12δM(λ1))sin2(12δM(λN)))NM

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