Abstract

Terahertz (THz) time-domain imaging is an emerging modality and has attracted a lot of interest. However, existing THz imaging systems often require a long scan time and sophisticated system design. Recently, a new design incorporating compressed sensing (CS) leads to a lower detector cost and shorter scan time, in exchange for computation in an image reconstruction step. In this paper, we develop two reconstruction algorithms that can estimate the underlying scene as accurately as possible. First is a single-band CS reconstruction method, where we show that by making use of prior information about the phase and the correlation between the spatial distributions of the amplitude and phase, the reconstruction quality can be significantly improved over previously published methods. Second, we develop a method that uses the multi-frequency nature of the THz pulse. Through effective use of the spatial sparsity, spectroscopic phase information, and correlations across the hyperspectral bands, our method can further enhance the recovered image quality. This is demonstrated by computation on a set of experimental THz data captured in a single-pixel THz system.

© 2010 Optical Society of America

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2010 (1)

Z. Xu and E. Y. Lam, “Hyperspectral reconstruction in biomedical imaging using terahertz systems,” in IEEE International Symposium on Circuits and Systems (IEEE, 2010), pp. 2079–2082.

2009 (5)

Z. Xu, W. L. Chan, D. M. Mittleman, and E. Y. Lam, “Sparse reconstruction of complex signals in compressed sensing terahertz imaging,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA4.

G. R. Brady, M. Guizar-Sicairos, and J. R. Fienup, “Optical wavefront measurement using phase retrieval with transverse translation diversity,” Opt. Express 17, 624–639 (2009).
[CrossRef] [PubMed]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[CrossRef] [PubMed]

E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48, H113–H119 (2009).
[CrossRef] [PubMed]

L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia, and P. Ciuciu, “Minimization of a sparsity promoting criterion for the recovery of complex-valued signals,” presented at SPARS’09: Signal Processing with Adaptive Sparse Structured Representation, Saint Malo, France, 2009.

2008 (5)

X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express 16, 17215–17226 (2008).
[CrossRef] [PubMed]

E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[CrossRef]

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. (USA) 31, 890–912 (2008).
[CrossRef]

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef] [PubMed]

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 (6)

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70, 325–1379 (2007).
[CrossRef]

J. J. Fuchs, “Convergence of a sparse representations algorithm applicable to real or complex data,” IEEE J. Sel. Top. Signal Process. 1, 598–605 (2007).
[CrossRef]

R. M. Willett and R. D. Nowak, “Multiscale poisson intensity and density estimation,” IEEE Trans. Inf. Theory 53, 3171–3187 (2007).
[CrossRef]

K. Krishnamurthy and R. M. Willett, “Multiscale reconstruction of photon-limited hyperspectral data,” in IEEE/SP 14th Workshop on Statistical Signal Processing, (IEEE, 2007), pp. 596–600.
[CrossRef]

2006 (6)

J. Ma and M. Fenn, “Combined complex ridgelet shrinkage and total variation minimization,” SIAM J. Sci. Comput. (USA) 28, 984–1000 (2006).
[CrossRef]

M. C. Kemp, A. Glauser, and C. Baker, “Recent developments in people screening using terahertz technology: seeing the world through terahertz eyes,” Proc. SPIE 6212, 62120T (2006).
[CrossRef]

I. Atkinson, F. Kamalabadi, S. Mohan, and D. L. Jones, “Asymptotically optimal blind estimation of multichannel images,” IEEE Trans. Image Process. 15, 992–1007 (2006).
[CrossRef] [PubMed]

E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (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, 489–509 (2006).
[CrossRef]

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

2005 (4)

R. M. Woodward, “Terahertz technology in global homeland security,” Proc. SPIE 5781, 22–31 (2005).
[CrossRef]

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

D. Zimdars, “High speed terahertz reflection imaging,” Proc. SPIE 5692, 255–259 (2005).
[CrossRef]

A. Luukanen, A. J. Miller, and E. N. Grossman, “Passive hyperspectral terahertz imagery for security screening using a cryogenic microbolometer,” Proc. SPIE 5789, 127–134 (2005).
[CrossRef]

2003 (2)

I. Atkinson, F. Kamalabadi, and D. L. Jones, “Wavelet-based hyperspectral image estimation,” in IEEE International Geoscience and Remote Sensing SymposiumIEEE, (2003), pp. 743–745.

K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express 11, 2549–2554 (2003).
[CrossRef] [PubMed]

2001 (1)

D. L. Donoho and X. Huo, “Beamlets and multiscale image analysis,” in Multiscale and Multiresolution Methods: Theory and Applications, T.J.Barth, T.Chan, and R.Haimes, eds., Lecture Notes in Computational Science and Engineering (Springer, 2001), pp. 149–196.

2000 (1)

1999 (1)

Atkinson, I.

I. Atkinson, F. Kamalabadi, S. Mohan, and D. L. Jones, “Asymptotically optimal blind estimation of multichannel images,” IEEE Trans. Image Process. 15, 992–1007 (2006).
[CrossRef] [PubMed]

I. Atkinson, F. Kamalabadi, and D. L. Jones, “Wavelet-based hyperspectral image estimation,” in IEEE International Geoscience and Remote Sensing SymposiumIEEE, (2003), pp. 743–745.

Baker, C.

M. C. Kemp, A. Glauser, and C. Baker, “Recent developments in people screening using terahertz technology: seeing the world through terahertz eyes,” Proc. SPIE 6212, 62120T (2006).
[CrossRef]

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]

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef] [PubMed]

Bastian, M.

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

Baudrit, B.

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

Benazza-Benyahia, A.

L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia, and P. Ciuciu, “Minimization of a sparsity promoting criterion for the recovery of complex-valued signals,” presented at SPARS’09: Signal Processing with Adaptive Sparse Structured Representation, Saint Malo, France, 2009.

Boyd, S. P.

E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[CrossRef]

Brady, D. J.

Brady, G. R.

Candès, E. J.

E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[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, 489–509 (2006).
[CrossRef]

E. J. Candès and J. Romberg, “l1-magic: Recovery of sparse signals via convex programming,” http://www.acm.caltech.edu/l1magic.

Chaâri, L.

L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia, and P. Ciuciu, “Minimization of a sparsity promoting criterion for the recovery of complex-valued signals,” presented at SPARS’09: Signal Processing with Adaptive Sparse Structured Representation, Saint Malo, France, 2009.

Chan, W. L.

Z. Xu, W. L. Chan, D. M. Mittleman, and E. Y. Lam, “Sparse reconstruction of complex signals in compressed sensing terahertz imaging,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA4.

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]

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef] [PubMed]

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70, 325–1379 (2007).
[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]

Choi, K.

Ciuciu, P.

L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia, and P. Ciuciu, “Minimization of a sparsity promoting criterion for the recovery of complex-valued signals,” presented at SPARS’09: Signal Processing with Adaptive Sparse Structured Representation, Saint Malo, France, 2009.

Coutaz, J.-L.

Deibel, J.

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70, 325–1379 (2007).
[CrossRef]

Donoho, D. L.

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

D. L. Donoho and X. Huo, “Beamlets and multiscale image analysis,” in Multiscale and Multiresolution Methods: Theory and Applications, T.J.Barth, T.Chan, and R.Haimes, eds., Lecture Notes in Computational Science and Engineering (Springer, 2001), pp. 149–196.

Duvillaret, L.

Fenn, M.

J. Ma and M. Fenn, “Combined complex ridgelet shrinkage and total variation minimization,” SIAM J. Sci. Comput. (USA) 28, 984–1000 (2006).
[CrossRef]

Fienup, J. R.

Figueiredo, M. A. T.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

Friedlander, M. P.

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. (USA) 31, 890–912 (2008).
[CrossRef]

E. van den Berg and M. P. Friedlander, “SPGL1: A solver for large-scale sparse reconstruction,” http://www.cs.ubc.ca/labs/scl/spgl1.

Fuchs, J. J.

J. J. Fuchs, “Convergence of a sparse representations algorithm applicable to real or complex data,” IEEE J. Sel. Top. Signal Process. 1, 598–605 (2007).
[CrossRef]

Garet, F.

Glauser, A.

M. C. Kemp, A. Glauser, and C. Baker, “Recent developments in people screening using terahertz technology: seeing the world through terahertz eyes,” Proc. SPIE 6212, 62120T (2006).
[CrossRef]

Grossman, E. N.

A. Luukanen, A. J. Miller, and E. N. Grossman, “Passive hyperspectral terahertz imagery for security screening using a cryogenic microbolometer,” Proc. SPIE 5789, 127–134 (2005).
[CrossRef]

Guizar-Sicairos, M.

Horisaki, R.

Huo, X.

D. L. Donoho and X. Huo, “Beamlets and multiscale image analysis,” in Multiscale and Multiresolution Methods: Theory and Applications, T.J.Barth, T.Chan, and R.Haimes, eds., Lecture Notes in Computational Science and Engineering (Springer, 2001), pp. 149–196.

Hwang, J.-S.

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

Indebetouw, G.

Inoue, H.

Jöerdens, C.

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

Jones, D. L.

I. Atkinson, F. Kamalabadi, S. Mohan, and D. L. Jones, “Asymptotically optimal blind estimation of multichannel images,” IEEE Trans. Image Process. 15, 992–1007 (2006).
[CrossRef] [PubMed]

I. Atkinson, F. Kamalabadi, and D. L. Jones, “Wavelet-based hyperspectral image estimation,” in IEEE International Geoscience and Remote Sensing SymposiumIEEE, (2003), pp. 743–745.

Kamalabadi, F.

I. Atkinson, F. Kamalabadi, S. Mohan, and D. L. Jones, “Asymptotically optimal blind estimation of multichannel images,” IEEE Trans. Image Process. 15, 992–1007 (2006).
[CrossRef] [PubMed]

I. Atkinson, F. Kamalabadi, and D. L. Jones, “Wavelet-based hyperspectral image estimation,” in IEEE International Geoscience and Remote Sensing SymposiumIEEE, (2003), pp. 743–745.

Karpowicz, N.

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

Kawase, K.

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]

Kemp, M. C.

M. C. Kemp, A. Glauser, and C. Baker, “Recent developments in people screening using terahertz technology: seeing the world through terahertz eyes,” Proc. SPIE 6212, 62120T (2006).
[CrossRef]

Koch, M.

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

Krishnamurthy, K.

K. Krishnamurthy and R. M. Willett, “Multiscale reconstruction of photon-limited hyperspectral data,” in IEEE/SP 14th Workshop on Statistical Signal Processing, (IEEE, 2007), pp. 596–600.
[CrossRef]

Krumbholz, N.

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

Lam, E. Y.

Z. Xu and E. Y. Lam, “Hyperspectral reconstruction in biomedical imaging using terahertz systems,” in IEEE International Symposium on Circuits and Systems (IEEE, 2010), pp. 2079–2082.

Z. Xu, W. L. Chan, D. M. Mittleman, and E. Y. Lam, “Sparse reconstruction of complex signals in compressed sensing terahertz imaging,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA4.

E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48, H113–H119 (2009).
[CrossRef] [PubMed]

X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express 16, 17215–17226 (2008).
[CrossRef] [PubMed]

Lim, S.

Lin, K.-I.

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

Luukanen, A.

A. Luukanen, A. J. Miller, and E. N. Grossman, “Passive hyperspectral terahertz imagery for security screening using a cryogenic microbolometer,” Proc. SPIE 5789, 127–134 (2005).
[CrossRef]

Ma, J.

J. Ma and M. Fenn, “Combined complex ridgelet shrinkage and total variation minimization,” SIAM J. Sci. Comput. (USA) 28, 984–1000 (2006).
[CrossRef]

Marks, D. L.

Marroquin, J. L.

Miller, A. J.

A. Luukanen, A. J. Miller, and E. N. Grossman, “Passive hyperspectral terahertz imagery for security screening using a cryogenic microbolometer,” Proc. SPIE 5789, 127–134 (2005).
[CrossRef]

Mittleman, D. M.

Z. Xu, W. L. Chan, D. M. Mittleman, and E. Y. Lam, “Sparse reconstruction of complex signals in compressed sensing terahertz imaging,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA4.

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]

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef] [PubMed]

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70, 325–1379 (2007).
[CrossRef]

Mohan, S.

I. Atkinson, F. Kamalabadi, S. Mohan, and D. L. Jones, “Asymptotically optimal blind estimation of multichannel images,” IEEE Trans. Image Process. 15, 992–1007 (2006).
[CrossRef] [PubMed]

Moravec, M. L.

Nowak, R. D.

R. M. Willett and R. D. Nowak, “Multiscale poisson intensity and density estimation,” IEEE Trans. Inf. Theory 53, 3171–3187 (2007).
[CrossRef]

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

Ogawa, Y.

Pesquet, J.-C.

L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia, and P. Ciuciu, “Minimization of a sparsity promoting criterion for the recovery of complex-valued signals,” presented at SPARS’09: Signal Processing with Adaptive Sparse Structured Representation, Saint Malo, France, 2009.

Pickwell, E.

E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006).
[CrossRef]

Poon, T.-C.

Quiroga, J. A.

Rodriguez-Vera, R.

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, 489–509 (2006).
[CrossRef]

E. J. Candès and J. Romberg, “l1-magic: Recovery of sparse signals via convex programming,” http://www.acm.caltech.edu/l1magic.

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]

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, 489–509 (2006).
[CrossRef]

van den Berg, E.

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. (USA) 31, 890–912 (2008).
[CrossRef]

E. van den Berg and M. P. Friedlander, “SPGL1: A solver for large-scale sparse reconstruction,” http://www.cs.ubc.ca/labs/scl/spgl1.

Vo, H.

Wakin, M. B.

E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[CrossRef]

Wallace, V. P.

E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006).
[CrossRef]

Watanabe, Y.

Wietzke, S.

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

Willett, R. M.

R. M. Willett and R. D. Nowak, “Multiscale poisson intensity and density estimation,” IEEE Trans. Inf. Theory 53, 3171–3187 (2007).
[CrossRef]

K. Krishnamurthy and R. M. Willett, “Multiscale reconstruction of photon-limited hyperspectral data,” in IEEE/SP 14th Workshop on Statistical Signal Processing, (IEEE, 2007), pp. 596–600.
[CrossRef]

Woodward, R. M.

R. M. Woodward, “Terahertz technology in global homeland security,” Proc. SPIE 5781, 22–31 (2005).
[CrossRef]

Wright, S. J.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

Xu, J.

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

Xu, Z.

Z. Xu and E. Y. Lam, “Hyperspectral reconstruction in biomedical imaging using terahertz systems,” in IEEE International Symposium on Circuits and Systems (IEEE, 2010), pp. 2079–2082.

Z. Xu, W. L. Chan, D. M. Mittleman, and E. Y. Lam, “Sparse reconstruction of complex signals in compressed sensing terahertz imaging,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA4.

Zhang, C.

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

Zhang, X.

Zhang, X.-C.

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

Zhong, H.

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

Zimdars, D.

D. Zimdars, “High speed terahertz reflection imaging,” Proc. SPIE 5692, 255–259 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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]

N. Karpowicz, H. Zhong, C. Zhang, K.-I. Lin, J.-S. Hwang, J. Xu, and X.-C. Zhang, “Compact continuous-wave subterahertz system for inspection applications,” Appl. Phys. Lett. 86, 054105 (2005).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (2)

J. J. Fuchs, “Convergence of a sparse representations algorithm applicable to real or complex data,” IEEE J. Sel. Top. Signal Process. 1, 598–605 (2007).
[CrossRef]

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Signal Process. 1, 586–597 (2007).
[CrossRef]

IEEE Trans. Image Process. (1)

I. Atkinson, F. Kamalabadi, S. Mohan, and D. L. Jones, “Asymptotically optimal blind estimation of multichannel images,” IEEE Trans. Image Process. 15, 992–1007 (2006).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (3)

R. M. Willett and R. D. Nowak, “Multiscale poisson intensity and density estimation,” IEEE Trans. Inf. Theory 53, 3171–3187 (2007).
[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, 489–509 (2006).
[CrossRef]

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

J. Eur. Opt. Soc. Rapid Publ. (1)

S. Wietzke, C. Jöerdens, N. Krumbholz, B. Baudrit, M. Bastian, and M. Koch, “Terahertz imaging: a new non-destructive technique for the quality control of plastic weld joints,” J. Eur. Opt. Soc. Rapid Publ. 2, 07013 (2007).
[CrossRef]

J. Fourier Anal. Appl. (1)

E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2008).
[CrossRef]

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

J. Phys. D: Appl. Phys. (1)

E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (4)

R. M. Woodward, “Terahertz technology in global homeland security,” Proc. SPIE 5781, 22–31 (2005).
[CrossRef]

D. Zimdars, “High speed terahertz reflection imaging,” Proc. SPIE 5692, 255–259 (2005).
[CrossRef]

A. Luukanen, A. J. Miller, and E. N. Grossman, “Passive hyperspectral terahertz imagery for security screening using a cryogenic microbolometer,” Proc. SPIE 5789, 127–134 (2005).
[CrossRef]

M. C. Kemp, A. Glauser, and C. Baker, “Recent developments in people screening using terahertz technology: seeing the world through terahertz eyes,” Proc. SPIE 6212, 62120T (2006).
[CrossRef]

Rep. Prog. Phys. (1)

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70, 325–1379 (2007).
[CrossRef]

SIAM J. Sci. Comput. (USA) (2)

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput. (USA) 31, 890–912 (2008).
[CrossRef]

J. Ma and M. Fenn, “Combined complex ridgelet shrinkage and total variation minimization,” SIAM J. Sci. Comput. (USA) 28, 984–1000 (2006).
[CrossRef]

Other (8)

I. Atkinson, F. Kamalabadi, and D. L. Jones, “Wavelet-based hyperspectral image estimation,” in IEEE International Geoscience and Remote Sensing SymposiumIEEE, (2003), pp. 743–745.

K. Krishnamurthy and R. M. Willett, “Multiscale reconstruction of photon-limited hyperspectral data,” in IEEE/SP 14th Workshop on Statistical Signal Processing, (IEEE, 2007), pp. 596–600.
[CrossRef]

D. L. Donoho and X. Huo, “Beamlets and multiscale image analysis,” in Multiscale and Multiresolution Methods: Theory and Applications, T.J.Barth, T.Chan, and R.Haimes, eds., Lecture Notes in Computational Science and Engineering (Springer, 2001), pp. 149–196.

E. van den Berg and M. P. Friedlander, “SPGL1: A solver for large-scale sparse reconstruction,” http://www.cs.ubc.ca/labs/scl/spgl1.

E. J. Candès and J. Romberg, “l1-magic: Recovery of sparse signals via convex programming,” http://www.acm.caltech.edu/l1magic.

Z. Xu and E. Y. Lam, “Hyperspectral reconstruction in biomedical imaging using terahertz systems,” in IEEE International Symposium on Circuits and Systems (IEEE, 2010), pp. 2079–2082.

Z. Xu, W. L. Chan, D. M. Mittleman, and E. Y. Lam, “Sparse reconstruction of complex signals in compressed sensing terahertz imaging,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA4.

L. Chaâri, J.-C. Pesquet, A. Benazza-Benyahia, and P. Ciuciu, “Minimization of a sparsity promoting criterion for the recovery of complex-valued signals,” presented at SPARS’09: Signal Processing with Adaptive Sparse Structured Representation, Saint Malo, France, 2009.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of a single-pixel, pulsed THz imaging system based on that in [8].

Fig. 2
Fig. 2

(a) A Example of an incomplete RDP and its quad-tree. (b) Sample partition of a two-dimensional image.

Fig. 3
Fig. 3

Hyperspectral image reconstruction algorithm flow chart.

Fig. 4
Fig. 4

Rectangular object mask [17].

Fig. 5
Fig. 5

CS reconstruction results on the single-band THz data at the frequency of 0.1 THz . Each image is of size 32 × 32 . The left column shows the amplitude images, and the phase images are in the right column. (a), (b) Results obtained by directly minimizing the l 1 norm of the underlying signal itself. (c), (d) Results reconstructed by minimizing the l 1 norm of the Daubechies-4 wavelet coefficients. (e), (f) Results obtained by using Daubechies-8 wavelet transform for sparsification. (g), (h) Results obtained by applying our single-band CS reconstruction method with Eq. (7) as the complex TV norm. (i), (j) Results obtained by using our single-band approach with the TV norm in the form of Eq. (8).

Fig. 6
Fig. 6

Hyperspectral reconstruction results with the practical THz data of size 600 × 16 . (a), (b) Amplitude and phase obtained by minimizing the l 1 norm of the Daubechies-8 wavelet coefficients only with the measurements at 0.1 THz . (c), (d) Amplitude and phase reconstructed by using our single-band reconstruction method only with the measurements at 0.1 THz . (e), (f) Amplitude and phase obtained by performing our proposed hyperspectral algorithm with data across all 16 spectral bands, displayed at 0.1 THz .

Fig. 7
Fig. 7

Hyperspectral reconstruction results with the practical THz data of size 400 × 16 . (a),(b) Amplitude and phase obtained by minimizing the l 1 norm of the Daubechies-8 wavelet coefficients only with the measurements at 0.1 THz . (c), (d) Amplitude and phase reconstructed by using our single-band reconstruction method only with the measurements at 0.1 THz . (e), (f) Amplitude and phase obtained by performing our proposed hyperspectral algorithm with data across all 16 spectral bands, displayed at 0.1 THz .

Equations (20)

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minimize x 0 subject to Ψ x = b ,
minimize x 1 subject to Ψ x = b ,
b ( f k ) = Ψ x ( f k ) ,
minimize x ( f k ) 1 subject to Ψ x ( f k ) b ( f k ) 2 ϵ ,
ϕ ( x ) ϕ ( x ) ¯ 2 = ( i = 1 N 2 [ ϕ ( x i ) ϕ ( Ω i ) ¯ ] 2 ) 1 2 σ ,
ϕ ( x i ) = { j log x i | x i | , if | x i | T 0 , otherwise } [ π , π ) .
i Δ i h x 0 + Δ i v x 0 ,
x TV i | Δ i h x | + | Δ i v x | .
x TV i ( | Δ i h x | 2 + | Δ i v x | 2 ) 1 2 .
minimize x ( f k ) TV subject to Ψ x ( f k ) b ( f k ) 2 ϵ , ϕ ( x ( f k ) ) ϕ ( x ( f k ) ) ¯ 2 σ ,
x ̂ ( f k ) = arg min x ( f k ) 1 2 Ψ x ( f k ) b ( f k ) 2 2 + λ x ( f k ) TV + μ ϕ ( x ( f k ) ) ϕ ( x ( f k ) ) ¯ 2 .
x ̂ A ( c a ) = arg min x A ( c a ) L A ( c a ) = arg min x A ( c a ) { log p ( y A ( c a ) | x A ( c a ) ) } ,
x ̂ ϕ ( c ϕ ) = arg min x ϕ ( c ϕ ) L ϕ ( c ϕ ) = arg min x ϕ ( c ϕ ) { log p ( y ϕ ( c ϕ ) | x ϕ ( c ϕ ) ) } .
p ( y A ( c a ) | x A ( c a ) ) = ( i , j ) c a k = 0 M 1 1 2 π σ A 2 exp { [ y A ( i , j , k ) x A ( i , j , k ) ] 2 2 σ A 2 } ,
p ( y ϕ ( c ϕ ) | x ϕ ( c ϕ ) ) = ( i , j ) c ϕ k = 0 M 1 1 2 π σ ϕ 2 exp { [ y ϕ ( i , j , k ) x ϕ ( i , j , k ) ] 2 2 σ ϕ 2 } .
P ̂ A = arg min P A { c a P A L A ( c a ) + η A ( P A ) } ,
x ̂ A = { c a P ̂ A x ̂ A ( c a ) } ,
P ̂ ϕ = arg min P ϕ { c ϕ P ϕ L ϕ ( c ϕ ) + η ϕ ( P ϕ ) } ,
x ̂ ϕ = { c ϕ P ̂ ϕ x ̂ ϕ ( c ϕ ) } .
x ̂ ( t + 1 ) x ̂ ( t ) 1 x ̂ ( t ) 1

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