Abstract

Inspection and imaging through scattering layers is a decades-old problem in optics. Unlike x-ray and ultrasonic imaging techniques, time-domain spectroscopy methods can provide detailed chemical and structural information of subsurfaces along with their depth. Although high-resolution time-of-flight measurement in time-domain spectroscopy provides 3D information, unfortunately it also induces an unwanted sensitivity to misalignments of the system and distortion of the layers themselves. Such high sensitivity to alignment and sample surface is a well known problem in time-domain and interferometric imaging, and is a major concern when the alignment error is comparable to the pulse wavelength. Here, we propose and implement an algorithmic framework based on low-rank matrix recovery and alternating minimization to remove such unwanted distortions from time-domain images. The method allows for recovery of the original sample texture in spite of the presence of temporal-spatial distortions. We address a blind-demodulation problem where, based on several observations of the sample texture modulated by undesired sweep distortions, the two classes of signals are separated with minimal damage to the main features. The performance of the method is examined in both synthetic and real data in the case of a terahertz time-domain system, and the successful reconstructions are demonstrated. The proposed general scheme can be implemented to advance inspection and imaging applications in THz and other time-resolved spectral imaging modalities.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
  4. A. Redo-Sanchez, N. Laman, B. Schulkin, and T. Tongue, “Review of terahertz technology readiness assessment and applications,” J. Infrared Millimeter Terahertz Waves 34, 500–518 (2013).
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  5. B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
    [Crossref]
  6. B. Heshmat, H. Pahlevaninezhad, and T. Darcie, “Carbon nanotube-based photoconductive switches for THz detection: an assessment of capabilities and limitations,” IEEE Photon. J. 4, 970–985 (2012).
    [Crossref]
  7. B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
    [Crossref]
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    [Crossref]
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    [Crossref]
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  22. B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
    [Crossref]
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    [Crossref]
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    [Crossref]
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2015 (1)

A. Aghasi and J. Romberg, “Convex cardinal shape composition,” SIAM J. Imaging Sci. 8, 2887–2950 (2015).
[Crossref]

2014 (1)

W. Withayachumnankul and D. Abbott, “Terahertz imaging: compressing onto a single pixel,” Nat. Photonics 8, 593–594 (2014).
[Crossref]

2013 (6)

A. Aghasi, I. Mendoza-Sanchez, E. L. Miller, C. A. Ramsburg, and L. M. Abriola, “A geometric approach to joint inversion with applications to contaminant source zone characterization,” Inverse Prob. 29, 115014 (2013).
[Crossref]

A. Redo-Sanchez, N. Laman, B. Schulkin, and T. Tongue, “Review of terahertz technology readiness assessment and applications,” J. Infrared Millimeter Terahertz Waves 34, 500–518 (2013).
[Crossref]

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

A. Aghasi and J. Romberg, “Sparse shape reconstruction,” SIAM J. Imaging Sci. 6, 2075–2108 (2013).
[Crossref]

C. Seco-Martorell, V. López-Domnguez, G. Arauz-Garofalo, A. Redo-Sanchez, J. Palacios, and J. Tejada, “Goya’s artwork imaging with terahertz waves,” Opt. Express 21, 17800–17805 (2013).
[Crossref]

C. Li, J. Grant, J. Wang, and D. R. Cumming, “A nipkow disk integrated with Fresnel lenses for terahertz single pixel imaging,” Opt. Express 21, 24452–24459 (2013).
[Crossref]

2012 (4)

H. Shen, L. Gan, N. Newman, Y. Dong, C. Li, Y. Huang, and Y. Shen, “Spinning disk for compressive imaging,” Opt. Lett. 37, 46–48 (2012).
[Crossref]

E. L. Miller, L. M. Abriola, and A. Aghasi, “Environmental remediation and restoration: hydrological and geophysical processing methods,” IEEE Signal Process. Mag. 29(4), 16–26 (2012).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, and T. Darcie, “Carbon nanotube-based photoconductive switches for THz detection: an assessment of capabilities and limitations,” IEEE Photon. J. 4, 970–985 (2012).
[Crossref]

2011 (2)

A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. 4, 618–650 (2011).
[Crossref]

Y.-C. Shen, “Terahertz pulsed spectroscopy and imaging for pharmaceutical applications: a review,” Int. J. Pharm. 417, 48–60 (2011).
[Crossref]

2010 (2)

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Y. Chen, S. Huang, and E. Pickwell-MacPherson, “Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy,” Opt. Express 18, 1177–1190 (2010).
[Crossref]

2009 (1)

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

2008 (1)

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

2007 (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[Crossref]

2005 (1)

2003 (1)

1992 (1)

R. W. Scharstein, “Transient electromagnetic plane wave reflection from a dielectric slab,” IEEE Trans. Educ. 35, 170–175 (1992).
[Crossref]

Abbott, D.

W. Withayachumnankul and D. Abbott, “Terahertz imaging: compressing onto a single pixel,” Nat. Photonics 8, 593–594 (2014).
[Crossref]

X. Yin, B. W.-H. Ng, and D. Abbott, Terahertz Imaging for Biomedical Applications: Pattern Recognition and Tomographic Reconstruction (Springer, 2012).

Abriola, L. M.

A. Aghasi, I. Mendoza-Sanchez, E. L. Miller, C. A. Ramsburg, and L. M. Abriola, “A geometric approach to joint inversion with applications to contaminant source zone characterization,” Inverse Prob. 29, 115014 (2013).
[Crossref]

E. L. Miller, L. M. Abriola, and A. Aghasi, “Environmental remediation and restoration: hydrological and geophysical processing methods,” IEEE Signal Process. Mag. 29(4), 16–26 (2012).
[Crossref]

Aghasi, A.

A. Aghasi and J. Romberg, “Convex cardinal shape composition,” SIAM J. Imaging Sci. 8, 2887–2950 (2015).
[Crossref]

A. Aghasi, I. Mendoza-Sanchez, E. L. Miller, C. A. Ramsburg, and L. M. Abriola, “A geometric approach to joint inversion with applications to contaminant source zone characterization,” Inverse Prob. 29, 115014 (2013).
[Crossref]

A. Aghasi and J. Romberg, “Sparse shape reconstruction,” SIAM J. Imaging Sci. 6, 2075–2108 (2013).
[Crossref]

E. L. Miller, L. M. Abriola, and A. Aghasi, “Environmental remediation and restoration: hydrological and geophysical processing methods,” IEEE Signal Process. Mag. 29(4), 16–26 (2012).
[Crossref]

A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. 4, 618–650 (2011).
[Crossref]

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

A. Aghasi and J. Romberg, “Learning shapes by convex composition,” arXiv:1602.07613 (2016).

Arauz-Garofalo, G.

Baraniuk, R. G.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

Bowen, J.

Bresler, Y.

K. Lee, Y. Wu, and Y. Bresler, “Near optimal compressed sensing of sparse rank-one matrices via sparse power factorization,” arXiv:1312.0525 (2013).

Burnett, A. D.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Burton Lewis, R.

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Chan, W. L.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

Charan, K.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

Chen, Y.

Coelho, C.

Cumming, D. R.

Cunningham, J. E.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Darcie, T.

B. Heshmat, H. Pahlevaninezhad, and T. Darcie, “Carbon nanotube-based photoconductive switches for THz detection: an assessment of capabilities and limitations,” IEEE Photon. J. 4, 970–985 (2012).
[Crossref]

Darcie, T. E.

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Davies, A. G.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Deng, Y.

Y. Deng, Q. Sun, F. Liu, C. Wang, and Q. Xing, “Terahertz time-resolved spectroscopy with wavelet-transform,” in 3rd International Congress on Image and Signal Processing (CISP) (IEEE, 2010), Vol. 7, pp. 3462–3464.

Dong, Y.

Edwards, H. G.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Fan, W.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Fazel, M.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Galvão, R.

Gan, L.

Gordon, R.

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Grant, J.

Hadjiloucas, S.

Hargreaves, M. D.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Heshmat, B.

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, and T. Darcie, “Carbon nanotube-based photoconductive switches for THz detection: an assessment of capabilities and limitations,” IEEE Photon. J. 4, 970–985 (2012).
[Crossref]

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

Hu, Q.

Huang, S.

Huang, Y.

Jain, P.

P. Jain, P. Netrapalli, and S. Sanghavi, “Low-rank matrix completion using alternating minimization,” in Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing (ACM, 2013), pp. 665–674.

Kelly, K. F.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

Kilmer, M.

A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. 4, 618–650 (2011).
[Crossref]

Laman, N.

A. Redo-Sanchez, N. Laman, B. Schulkin, and T. Tongue, “Review of terahertz technology readiness assessment and applications,” J. Infrared Millimeter Terahertz Waves 34, 500–518 (2013).
[Crossref]

Lee, A. W.

Lee, K.

K. Lee, Y. Wu, and Y. Bresler, “Near optimal compressed sensing of sparse rank-one matrices via sparse power factorization,” arXiv:1312.0525 (2013).

Lewis, R. B.

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

Li, C.

Linfield, E. H.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Liu, F.

Y. Deng, Q. Sun, F. Liu, C. Wang, and Q. Xing, “Terahertz time-resolved spectroscopy with wavelet-transform,” in 3rd International Congress on Image and Signal Processing (CISP) (IEEE, 2010), Vol. 7, pp. 3462–3464.

López-Domnguez, V.

Masnadi-Shirazi, M.

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Mendoza-Sanchez, I.

A. Aghasi, I. Mendoza-Sanchez, E. L. Miller, C. A. Ramsburg, and L. M. Abriola, “A geometric approach to joint inversion with applications to contaminant source zone characterization,” Inverse Prob. 29, 115014 (2013).
[Crossref]

Miller, E. L.

A. Aghasi, I. Mendoza-Sanchez, E. L. Miller, C. A. Ramsburg, and L. M. Abriola, “A geometric approach to joint inversion with applications to contaminant source zone characterization,” Inverse Prob. 29, 115014 (2013).
[Crossref]

E. L. Miller, L. M. Abriola, and A. Aghasi, “Environmental remediation and restoration: hydrological and geophysical processing methods,” IEEE Signal Process. Mag. 29(4), 16–26 (2012).
[Crossref]

A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. 4, 618–650 (2011).
[Crossref]

Mittleman, D. M.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

Munshi, T.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Naqvi, S.

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

Netrapalli, P.

P. Jain, P. Netrapalli, and S. Sanghavi, “Low-rank matrix completion using alternating minimization,” in Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing (ACM, 2013), pp. 665–674.

Newman, N.

Ng, B. W.-H.

X. Yin, B. W.-H. Ng, and D. Abbott, Terahertz Imaging for Biomedical Applications: Pattern Recognition and Tomographic Reconstruction (Springer, 2012).

Nguyen, T.

G. Strang and T. Nguyen, Wavelets and Filter Banks (SIAM, 1996).

Pahlevaninezhad, H.

B. Heshmat, H. Pahlevaninezhad, and T. Darcie, “Carbon nanotube-based photoconductive switches for THz detection: an assessment of capabilities and limitations,” IEEE Photon. J. 4, 970–985 (2012).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Palacios, J.

Pang, Y.

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Parrilo, P. A.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Pickwell-MacPherson, E.

Ramsburg, C. A.

A. Aghasi, I. Mendoza-Sanchez, E. L. Miller, C. A. Ramsburg, and L. M. Abriola, “A geometric approach to joint inversion with applications to contaminant source zone characterization,” Inverse Prob. 29, 115014 (2013).
[Crossref]

Raskar, R.

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

Recht, B.

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Redo-Sanchez, A.

C. Seco-Martorell, V. López-Domnguez, G. Arauz-Garofalo, A. Redo-Sanchez, J. Palacios, and J. Tejada, “Goya’s artwork imaging with terahertz waves,” Opt. Express 21, 17800–17805 (2013).
[Crossref]

A. Redo-Sanchez, N. Laman, B. Schulkin, and T. Tongue, “Review of terahertz technology readiness assessment and applications,” J. Infrared Millimeter Terahertz Waves 34, 500–518 (2013).
[Crossref]

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

Romberg, J.

A. Aghasi and J. Romberg, “Convex cardinal shape composition,” SIAM J. Imaging Sci. 8, 2887–2950 (2015).
[Crossref]

A. Aghasi and J. Romberg, “Sparse shape reconstruction,” SIAM J. Imaging Sci. 6, 2075–2108 (2013).
[Crossref]

A. Aghasi and J. Romberg, “Learning shapes by convex composition,” arXiv:1602.07613 (2016).

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

Sanghavi, S.

P. Jain, P. Netrapalli, and S. Sanghavi, “Low-rank matrix completion using alternating minimization,” in Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing (ACM, 2013), pp. 665–674.

Scharstein, R. W.

R. W. Scharstein, “Transient electromagnetic plane wave reflection from a dielectric slab,” IEEE Trans. Educ. 35, 170–175 (1992).
[Crossref]

Schulkin, B.

A. Redo-Sanchez, N. Laman, B. Schulkin, and T. Tongue, “Review of terahertz technology readiness assessment and applications,” J. Infrared Millimeter Terahertz Waves 34, 500–518 (2013).
[Crossref]

Seco-Martorell, C.

Shen, H.

Shen, Y.

Shen, Y.-C.

Y.-C. Shen, “Terahertz pulsed spectroscopy and imaging for pharmaceutical applications: a review,” Int. J. Pharm. 417, 48–60 (2011).
[Crossref]

Strang, G.

G. Strang and T. Nguyen, Wavelets and Filter Banks (SIAM, 1996).

Sun, Q.

Y. Deng, Q. Sun, F. Liu, C. Wang, and Q. Xing, “Terahertz time-resolved spectroscopy with wavelet-transform,” in 3rd International Congress on Image and Signal Processing (CISP) (IEEE, 2010), Vol. 7, pp. 3462–3464.

Takhar, D.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

Tejada, J.

Tiedje, T.

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Tongue, T.

A. Redo-Sanchez, N. Laman, B. Schulkin, and T. Tongue, “Review of terahertz technology readiness assessment and applications,” J. Infrared Millimeter Terahertz Waves 34, 500–518 (2013).
[Crossref]

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[Crossref]

Upadhya, P. C.

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Wang, C.

Y. Deng, Q. Sun, F. Liu, C. Wang, and Q. Xing, “Terahertz time-resolved spectroscopy with wavelet-transform,” in 3rd International Congress on Image and Signal Processing (CISP) (IEEE, 2010), Vol. 7, pp. 3462–3464.

Wang, J.

Withayachumnankul, W.

W. Withayachumnankul and D. Abbott, “Terahertz imaging: compressing onto a single pixel,” Nat. Photonics 8, 593–594 (2014).
[Crossref]

Wu, Y.

K. Lee, Y. Wu, and Y. Bresler, “Near optimal compressed sensing of sparse rank-one matrices via sparse power factorization,” arXiv:1312.0525 (2013).

Xing, Q.

Y. Deng, Q. Sun, F. Liu, C. Wang, and Q. Xing, “Terahertz time-resolved spectroscopy with wavelet-transform,” in 3rd International Congress on Image and Signal Processing (CISP) (IEEE, 2010), Vol. 7, pp. 3462–3464.

Yin, X.

X. Yin, B. W.-H. Ng, and D. Abbott, Terahertz Imaging for Biomedical Applications: Pattern Recognition and Tomographic Reconstruction (Springer, 2012).

Zhang, J.

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

Zhang, M.

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

Adv. Opt. Mater. (1)

B. Heshmat, M. Masnadi-Shirazi, R. B. Lewis, J. Zhang, T. Tiedje, R. Gordon, and T. E. Darcie, “Enhanced terahertz bandwidth and power from GaAsBi-based sources,” Adv. Opt. Mater. 1, 714–719 (2013).
[Crossref]

Analyst (1)

A. D. Burnett, W. Fan, P. C. Upadhya, J. E. Cunningham, M. D. Hargreaves, T. Munshi, H. G. Edwards, E. H. Linfield, and A. G. Davies, “Broadband terahertz time-domain spectroscopy of drugs-of-abuse and the use of principal component analysis,” Analyst 134, 1658–1668 (2009).
[Crossref]

Appl. Phys. Lett. (1)

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[Crossref]

IEEE Photon. J. (1)

B. Heshmat, H. Pahlevaninezhad, and T. Darcie, “Carbon nanotube-based photoconductive switches for THz detection: an assessment of capabilities and limitations,” IEEE Photon. J. 4, 970–985 (2012).
[Crossref]

IEEE Signal Process. Mag. (1)

E. L. Miller, L. M. Abriola, and A. Aghasi, “Environmental remediation and restoration: hydrological and geophysical processing methods,” IEEE Signal Process. Mag. 29(4), 16–26 (2012).
[Crossref]

IEEE Trans. Educ. (1)

R. W. Scharstein, “Transient electromagnetic plane wave reflection from a dielectric slab,” IEEE Trans. Educ. 35, 170–175 (1992).
[Crossref]

Int. J. Pharm. (1)

Y.-C. Shen, “Terahertz pulsed spectroscopy and imaging for pharmaceutical applications: a review,” Int. J. Pharm. 417, 48–60 (2011).
[Crossref]

Inverse Prob. (1)

A. Aghasi, I. Mendoza-Sanchez, E. L. Miller, C. A. Ramsburg, and L. M. Abriola, “A geometric approach to joint inversion with applications to contaminant source zone characterization,” Inverse Prob. 29, 115014 (2013).
[Crossref]

J. Infrared Millimeter Terahertz Waves (1)

A. Redo-Sanchez, N. Laman, B. Schulkin, and T. Tongue, “Review of terahertz technology readiness assessment and applications,” J. Infrared Millimeter Terahertz Waves 34, 500–518 (2013).
[Crossref]

Nano Lett. (1)

B. Heshmat, H. Pahlevaninezhad, Y. Pang, M. Masnadi-Shirazi, R. Burton Lewis, T. Tiedje, R. Gordon, and T. E. Darcie, “Nanoplasmonic terahertz photoconductive switch on GaAs,” Nano Lett. 12, 6255–6259 (2012).
[Crossref]

Nat. Photonics (2)

W. Withayachumnankul and D. Abbott, “Terahertz imaging: compressing onto a single pixel,” Nat. Photonics 8, 593–594 (2014).
[Crossref]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

SIAM J. Imaging Sci. (3)

A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. 4, 618–650 (2011).
[Crossref]

A. Aghasi and J. Romberg, “Convex cardinal shape composition,” SIAM J. Imaging Sci. 8, 2887–2950 (2015).
[Crossref]

A. Aghasi and J. Romberg, “Sparse shape reconstruction,” SIAM J. Imaging Sci. 6, 2075–2108 (2013).
[Crossref]

SIAM Rev. (1)

B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Rev. 52, 471–501 (2010).
[Crossref]

Other (7)

P. Jain, P. Netrapalli, and S. Sanghavi, “Low-rank matrix completion using alternating minimization,” in Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing (ACM, 2013), pp. 665–674.

K. Lee, Y. Wu, and Y. Bresler, “Near optimal compressed sensing of sparse rank-one matrices via sparse power factorization,” arXiv:1312.0525 (2013).

G. Strang and T. Nguyen, Wavelets and Filter Banks (SIAM, 1996).

A. Redo-Sanchez, B. Heshmat, A. Aghasi, S. Naqvi, M. Zhang, J. Romberg, and R. Raskar, “Terahertz time-gated spectral imaging for content extraction through layered structures,” Nat. Commun. (to be published).

A. Aghasi and J. Romberg, “Learning shapes by convex composition,” arXiv:1602.07613 (2016).

Y. Deng, Q. Sun, F. Liu, C. Wang, and Q. Xing, “Terahertz time-resolved spectroscopy with wavelet-transform,” in 3rd International Congress on Image and Signal Processing (CISP) (IEEE, 2010), Vol. 7, pp. 3462–3464.

X. Yin, B. W.-H. Ng, and D. Abbott, Terahertz Imaging for Biomedical Applications: Pattern Recognition and Tomographic Reconstruction (Springer, 2012).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Dielectric slab, emitted and reflected electrical fields. (b) Setup schematics; blue is the emitted THz field and the red waveform is indicative of a typical reflected signal in one pixel. (c, d) Induction of sweep distortion: observation of the returned field at two different time instances.

Fig. 2.
Fig. 2.

Distributions for f ( ρ | C = 0 ) and f ( ρ | C = 1 ) as truncated normal distributions.

Fig. 3.
Fig. 3.

Demodulation of simulated data. (a) Reference profile of ρ ( x ) . (b.1)–(b.3) Observed reflections modulated by horizontal sweep profiles. (c.1) Demodulated image using the proposed algorithm using σ 0 = σ 1 = 10 5 . (c.2) Binary approximation of the result in (c.1). (d.1) Recovered image using the nuclear norm minimization. (d.2) Binary approximation of the result in (d.1).

Fig. 4.
Fig. 4.

MSE as a function of available number of frames, M , for various scenarios: a total of 20 frames using uniform sampling in time is made available through the simulation. For each MSE report, M frames are randomly selected from the 20 frames and passed to the algorithm. The process is performed multiple times for each M and the average MSE is reported.

Fig. 5.
Fig. 5.

MSE as a function of SNR for a fixed number of available frames ( M = 10 ).

Fig. 6.
Fig. 6.

Experimental demonstration of sweep decoupling for a metallic surface. (a.1)–(a.3) Three time instances of the recorded image from the sample; vertical sweep distortions are dominant over the cross pattern. (b.1)–(b.3) Recovered distortion profiles corresponding to the observations shown. (c) Recovered binary profile.

Fig. 7.
Fig. 7.

Experimental demonstration of sweep decoupling for a paper stack sample. (a.1)–(a.3) Three time instances of the recorded image from the first page. (b.1)–(b.3) Second page. (c.1)–(c.3) Third page; irregular distortions are dominantly present. (a.4)–(a.6) The algorithm has demodulated the distortion from the first page; (b.4)–(b.6) for the second page; (c.4)–(c.6) third page. (a.7) Recovered letter “M” on the first page. (b.7) Recovered letter “I” on the second page. (c.7) Recovered letter “T” on the third page.

Tables (2)

Tables Icon

Algorithm 1: MAP Alternation Scheme

Tables Icon

Algorithm 2: Decoupling algorithm

Equations (31)

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E⃗ 0 + ( z , t ) = χ ( t z c ) a⃗ x
E⃗ 0 ( z , t ) = ρ u ( t + z c ) a⃗ x ,
u ( τ ) = χ ( τ ) * ( δ ( τ τ ρ ) 1 ρ 2 ρ 2 m = 1 ρ 2 m δ ( τ τ ρ 2 m ) ) ,
ρ ( x , y , z ) = { ρ 0 ( x , y ) D 0 ρ 1 ( x , y ) D 1
y j ( x ) = ρ ( x ) u j ( x ) + n j ( x ) , j = 1 , , M , x D ,
min ρ , u 1 , , u M j = 1 M y j ρ u j 2 2 s.t.    u j span ( s 1 j , , s N j j ) .
min β , α 1 , , α M j = 1 M y j ( Q β ) ( S j α j ) 2 2 ,
( ρ u j ) i = ( Q β ) i ( S j α j ) i = ( Q i , : β ) S i , : j P j α = ( Q i , : β ) T S i , : j P j α = tr ( α β T Q i , : T S i , : j P j ) = β α T , Q i , : T S i , : j P j .
( A ( X ) ) i , j = X , Q i , : T S i , : j P j ,
min X rank ( X ) s.t.    Y A ( X ) F ε ,
min X X *    s.t. Y A ( X ) F ϵ .
{ ρ MAP , { u j MAP } j = 1 M } = argmax ρ , u 1 , , u M f ( ρ , { u j } j = 1 M | { y j } j = 1 M ) .
α j ( k ) = argmin α y j diag ( ρ ( k ) ) S j α 2 2 .
α j ( k ) = ( diag ( ρ ( k ) ) S j ) y j ,
y j ( x ) = w c 0 , w φ 0 , w ( x ) + = 0 w d , w ψ , w ( x ) ,
max ρ , C G ( ρ , C ) ,
G ( ρ , C ) f ( ρ , C | { u j ( k ) , y j } j = 1 M ) .
f ( ρ , C | y , u ) = f ( y | ρ , C , u ) f ( ρ , C , u ) f ( y | u ) f ( u ) = f ( y | ρ , C , u ) f ( ρ | C , u ) f ( y | u ) P ( C | u ) ,
G ( ρ , C ) f ( { y j } j = 1 M | ρ , C ) f ( ρ | C ) P ( C ) ,
G ( ρ , C ) i = 1 P f ( ρ i | C i ) P ( C i ) j = 1 M f ( y j , i | ρ i , C i ) ,
P ( C i ) = p C i = { p 0 if    C i = 0 p 1 if    C i = 1 ,
f N + ( ρ ; μ ˜ , σ ˜ ) = { γ σ ˜ exp ( ( ρ μ ˜ ) 2 2 σ ˜ 2 ) ρ 0 0 ρ < 0 .
f ( ρ i | C i ) = f N + ( ρ i ; ρ C i , σ C i 2 ) .
g ( ρ , C ) = log G ( ρ , C ) .
g ( ρ , C ) = { K + i = 1 P g ( ρ i , C i ) ρ i 0 + ρ i < 0 ,
g ( ρ i , C i ) log σ C i p C i + ( ρ i ρ C i ) 2 2 σ C i 2 + j = 1 M ( y j , i ρ i u j , i ( k ) ) 2 2 σ 2 ,
min C , ρ 0 g ( ρ , C ) = K + i = 1 P min C i , ρ i 0 g ( ρ i , C i ) = K + i = 1 P min C i min ρ i 0 g ( ρ i , C i ) .
argmin ρ i g ( ρ i , C ) = σ 2 ρ C + σ C 2 j = 1 M y j , i u j , i ( k ) σ 2 + σ C 2 j = 1 M u j , i ( k ) 2 .
ρ i * argmin ρ i 0 g ( ρ i , C ) = max ( σ 2 ρ C + σ C 2 j = 1 M y j , i u j , i ( k ) σ 2 + σ C 2 j = 1 M u j , i ( k ) 2 , 0 ) .
C i * argmin C i { 0,1 } g ( ρ i * , C i ) = { 1 if    g ( ρ i * , 1 ) < g ( ρ i * , 0 ) 0 if    g ( ρ i * , 1 ) g ( ρ i * , 0 ) .
χ ( t ) = ( t 0 t ) exp ( ( t t 0 ) 2 2 T 2 ) t 0 ,

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