Abstract

We describe a novel, high-speed pulsed terahertz (THz) Fourier imaging system based on compressed sensing (CS), a new signal processing theory, which allows image reconstruction with fewer samples than traditionally required. Using CS, we successfully reconstruct a 64×64 image of an object with pixel size 1.4mm using a randomly chosen subset of the 4096 pixels, which defines the image in the Fourier plane, and observe improved reconstruction quality when we apply phase correction. For our chosen image, only about 12% of the pixels are required for reassembling the image. In combination with phase retrieval, our system has the capability to reconstruct images with only a small subset of Fourier amplitude measurements and thus has potential application in THz imaging with cw sources.

© 2008 Optical Society of America

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2007 (4)

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[CrossRef]

M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, Opt. Express 15, 14013 (2007).
[CrossRef] [PubMed]

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, Proc. SPIE 6701, 670120 (2007).
[CrossRef]

2006 (5)

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

A. Bandyopadhyay, A. Stepanov, B. Schulkin, M. D. Federici, A. Sengupta, D. Gary, J. F. Federici, R. Barat, Z.-H. Michalopoulou, and D. Zimdars, J. Opt. Soc. Am. A 23, 1168 (2006).
[CrossRef]

D. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

2005 (2)

2004 (1)

S. Wang and X.-C. Zhang, J. Phys. D : Appl. Phys. 37, R1 (2004).
[CrossRef]

2003 (1)

1978 (1)

Bandyopadhyay, A.

Baraniuk, R.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Baraniuk, R. G.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, Proc. SPIE 6701, 670120 (2007).
[CrossRef]

Barat, R.

Baron, D.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Brady, D. J.

Candes, E.

E. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Chan, W. L.

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[CrossRef]

Cheville, R. A.

Choi, H.

Deibel, J.

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[CrossRef]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

D. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Duarte, M.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Federici, J. F.

Federici, M. D.

Fienup, J. R.

Friedlander, M. P.

E. van den Berg and M. P. Friedlander, SPGL1: a solver for large-scale sparse reconstruction, http://www.cs.ubc.ca/labs/scl/spgl1 (2007).

Gary, D.

Gehm, M. E.

Harmon, S. A.

Hu, Q.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

John, R.

Kelly, K.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Kumar, S.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

Laska, J.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Lee, A. W. M.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

Michalopoulou, Z.-H.

Mittleman, D. M.

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[CrossRef]

J. Pearce, H. Choi, and D. M. Mittleman, Opt. Lett. 30, 1653 (2005).
[CrossRef] [PubMed]

Moravec, M. L.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, Proc. SPIE 6701, 670120 (2007).
[CrossRef]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

Pearce, J.

Qin, Q.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

Reiten, M. T.

Reno, J. L.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Romberg, J. K.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, Proc. SPIE 6701, 670120 (2007).
[CrossRef]

Sarvotham, S.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Schulkin, B.

Schulz, T. J.

Sengupta, A.

Stepanov, A.

Takhar, D.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Tao, T.

E. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

van den Berg, E.

E. van den Berg and M. P. Friedlander, SPGL1: a solver for large-scale sparse reconstruction, http://www.cs.ubc.ca/labs/scl/spgl1 (2007).

Wakin, M.

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Wang, S.

S. Wang and X.-C. Zhang, J. Phys. D : Appl. Phys. 37, R1 (2004).
[CrossRef]

Willett, R. M.

Williams, B. S.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

Zhang, X.-C.

S. Wang and X.-C. Zhang, J. Phys. D : Appl. Phys. 37, R1 (2004).
[CrossRef]

Zimdars, D.

Appl. Phys. Lett. (1)

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, Appl. Phys. Lett. 89, 141125 (2006).
[CrossRef]

IEEE Trans. Inf. Theory (2)

D. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

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

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

J. Phys. D (1)

S. Wang and X.-C. Zhang, J. Phys. D : Appl. Phys. 37, R1 (2004).
[CrossRef]

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (3)

D. Zimdars, Proc. SPIE 5692, 255 (2005).
[CrossRef]

D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, Proc. SPIE 6065, 606509 (2006).
[CrossRef]

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, Proc. SPIE 6701, 670120 (2007).
[CrossRef]

Rep. Prog. Phys. (1)

W. L. Chan, J. Deibel, and D. M. Mittleman, Rep. Prog. Phys. 70, 1325 (2007).
[CrossRef]

Other (1)

E. van den Berg and M. P. Friedlander, SPGL1: a solver for large-scale sparse reconstruction, http://www.cs.ubc.ca/labs/scl/spgl1 (2007).

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Figures (3)

Fig. 1
Fig. 1

THz Fourier imaging setup. An approximately collimated beam from the THz transmitter illuminates an object mask, placed one focal length away from the focusing lens. The THz receiver raster scans and samples the Fourier transform of the object on the focal plane.

Fig. 2
Fig. 2

Compressed sensing imaging results. (a) Magnitude of image reconstructed by inverse Fourier transform using the full dataset (4096 uniformly sampled measurements) and (d) its phase. Note the phase distortion inherent in the THz beam in (d). Compressed sensing reconstruction result using 500 measurements (12%) from the full dataset: (b) magnitude and (e) phase. Compressed sensing with phase correction improves image quality and eliminates phase distortion [see (c) and (f)]. All figures show a zoom-in view on a 40 × 40 grid centered on the object.

Fig. 3
Fig. 3

Image reconstruction results using (a) CPR with the full dataset (4096 magnitude measurements) and (b) CSPR with a subset of 1500 measurements from the dataset used in (a).

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arg min x 1 subject to constraint y = Φ x ,

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