American Institute of Mathematical Sciences

Advanced
November  2013, 7(4): 1295-1305. doi: 10.3934/ipi.2013.7.1295

Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate

Jiying Liu 1, , Jubo Zhu 1, , Fengxia Yan 1, and  Zenghui Zhang 1,

1. 

Department of Mathematics and Systems Science, National University of Defense Technology, Changsha 410073, China, China, China, China

Received  February 2012 Revised  January 2013 Published  November 2013

This paper presents a new Synthetic Aperture Radar (SAR) imaging system based on compressive sampling scheme and $l_1$ minimization. The compressive sampling scheme comprises of randomization and integration of radar echoes which slows down the analog-to-digital converters (ADC) rate significantly without an aliasing in image formation. Numerical experiments indicate that the resolution of SAR images retrieved by our method outperform that obtained by conventional methods. The results also reveal that the new SAR imaging system can still retrieve non-ambiguous images even when the data rate is $\frac{1}{10}$ of the original one. Finally, we applied the new method on raw data of RADARSAT-1 to testify its practicability.
Keywords: sampling rate, $l_1$ minimization., Synthetic aperture radar, compressive sensing.
Mathematics Subject Classification: 94A20, 4A1.
Citation: Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems and Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295
References:
[1]

B. Le, T. Rondeau, J. Reed and C. Bostian, Analog-to-digital converters, IEEE Signal Proc. Mag., 22 (2005), 69-77. doi: 10.1109/4.173093.

[2]

M. Vetterli, P. Marziliano and T. Blu, Sampling signals with finite rate of innovation, IEEE Trans. Signal Process., 50 (2002), 1417-1428. doi: 10.1109/TSP.2002.1003065.

[3]

I. Maravic and M. Vetterli, Sampling and reconstruction of signals with finite rate of innovation in the presence of noise, IEEE Transactions on Signal Processing, 53 (2004), 2788-2805. doi: 10.1109/TSP.2005.850321.

[4]

E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

[5]

E. Candes, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. on Pure and Applied Math., 59 (2006), 1207-1223. doi: 10.1002/cpa.20124.

[6]

E. Candes and T. Tao, Near-optimal signal recovery from random projections and universal encoding strategies? IEEE Trans. on Information Theory, 52 (2006), 5406-5425. doi: 10.1109/TIT.2006.885507.

[7]

D. Donoho, Compressed sensing, IEEE Trans. on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[8]

T. Ragheb, S. Kirolos, J. Laska, A. Gilbert, M. Strauss, R. Baraniuk and Y. Massound, Implementation models for analog-to-information conversion via random sampling, Proc.of 50th Midwest Symposium on Circuits and Systems, (2007), 325-328. doi: 10.1109/MWSCAS.2007.4488599.

[9]

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. Baraniuk and Y. Massound, Theory and implementation of an analog-to-information converter using random demodulation, Proc. of IEEE International Symposium on Circuits and Systems, (2007), 1959-1962. doi: 10.1109/ISCAS.2007.378360.

[10]

R. Baraniuk and P. Steeghs, Compressive radar imaging, Proc. of 2007 IEEE Radar Conference, (2007), 128-133. doi: 10.1109/RADAR.2007.374203.

[11]

M. Herman and T. Strohmer, Compressed sensing radar, Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 1509-1512.

[12]

M. Herman and T. Strohmer, High-resolution radar via compressed sensing, IEEE Trans. on Signal Processing, 57 (2009), 2275-2284. doi: 10.1109/TSP.2009.2014277.

[13]

M. Cetin, Feature-Enhanced Synthetic Aperture Radar Imaging, College of Engineering, Boston University, Ph.D. Thesis, 2001. doi: 10.1109/83.913596.

[14]

M. Cetin and W. C. Karl, Feature-enhanced synthetic aperture radar imaging formation based on non-quadratic regularization, IEEE Trans. Image Process, 10 (2001), 623-631.

[15]

S. Bhattacharya, T. Blumensath, B. Mulgrew and M. Davies, Synthetic Aperture Radar raw data encoding using compressed sensing, Proc. of Radar Conference, pp. 1-5, May 2008.

[16]

S. Bhattacharya, T. Blumensath, B. Mulgrew, and M. Davies, Fast encoding of synthetic aperture radar raw data using compressed sensing, Proc. of IEEE/SP 14th Workshop on Statistical Signal Processing, (2007), 448-452. doi: 10.1109/SSP.2007.4301298.

[17]

G. Rilling, M. Davies and Bernard, Compressed sensing based compression of SAR raw data, Signal Processing with Adaptive Sparse Structured Representaition, 2009.

[18]

A. FannJiang, Compressive inverse scattering I. High frequency SIMO measurements,, , ().  doi: 10.1088/0266-5611/26/3/035008.

[19]

A. FannJiang, Compressive inverse scattering II. SISO measurements with born scatterers,, , ().  doi: 10.1088/0266-5611/26/3/035009.

[20]

J. H. G. Ender, On compressive sensing applied to radar, Signal Processing, 90 (2010), 1402-1414. doi: 10.1016/j.sigpro.2009.11.009.

[21]

L. Zhang, M. Xing, C. Qiun, J. Li and Z. Bao, Achieving higher resolution ISAR imaging with limited pulses via compressed sampling, IEEE Geoscience and Remote Sensing Letters, 6 (2009), 567-571.

[22]

J. Fowler, Compressive-projection principal component analysis, IEEE Trans. Image Process., 18 (2009), 2230-2242. doi: 10.1109/TIP.2009.2025089.

[23]

S. Kirolos, J. Laska, M. Wakin, M. Duarte, D. Baron, T. Ragheb, Y. Massoud and R. Baraniuk, Analog-to-information conversion via random demodulation, Proc. IEEE Dallas/CAS Workshop on Design, Applications, Integration, and Software, (2006), 71-74. doi: 10.1109/DCAS.2006.321036.

[24]

J. A. Tropp, J. N. Laska, M. F. Duarte, J. Romberg and R. G. Baraniuk, Beyond nyquist: Effecient sampling of sparse bandlimited signals, Submitted to IEEE. Trans. Inform. Theory, (2009). doi: 10.1109/TIT.2009.2034811.

[25]

J. Romgerg, Compressive sensing by random convolution, SIAM J. Imaging Sci., 2 (2009), 1098-1128. doi: 10.1137/08072975X.

[26]

G. E. Smith, T. Diethe, Z. Hussain, J. Shawe-Taylor and D. R. Hardoon, Compressed sampling for pulse Doppler radar, Proc. the IEEE International Radar Conference, (2010). doi: 10.1109/RADAR.2010.5494496.

[27]

Z. Bao, M. Xing and T. Wang, Radar Imaging Technology, Beijing, China: Publishing House of Electronics Industry, 2006.

[28]

W. G. Carrara, R. S. Goodman and R. M. Majewski, Spotlight Synthetic Aperture Radar, Boston, MA: Artech House, 1995.

[29]

M. Aharon, M. Elad and A. Bruckstein, K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 54 (2006), 4311-4322. doi: 10.1109/TSP.2006.881199.

[30]

R. Baraniuk, M. Davenport, R. DeVore and M. Wakin, A simple proof of the restricted isometry property for random matrices,, Constr. Approx., 28 (): 253.  doi: 10.1007/s00365-007-9003-x.

[31]

E. Van Den Berg and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientic Computing, 31 (2008), 890-912. doi: 10.1137/080714488.

[32]

J. Li, and P. Stoica, An Adaptive Filtering Approach to Spectral Estimation and SAR Imaging, IEEE Trans. on Signal Processing, 44 (1996), 1469-1484. doi: 10.1117/12.210835.

[33]

J. Pdendaal, E. Barnard and C. Pistorius, Two-dimensional superresolution radar imaging using the MUSIC algorithm, IEEE Trans. Antennas Propag., 42 (1994), 1386-1391.

[34]

Z. Bi, J. Li and Z.-S. Liu, Super resolution SAR imaging via parametric spectral estimation methods, IEEE Trans. Aerosp. Electron. Syst., 35 (1999), 267-281.

[35]

H. Rauhut, Stability results for random sampling of sparse trigonometric polynomials, IEEE Trans. on Information Theory, 54 (2008), 5661-5670. doi: 10.1109/TIT.2008.2006382.

[36]

I. Cumming and F. Wong, Digital Processing of Synthetic Aperture Radar Data: Algorithm and Implementaion, Artech Hourse, 656 Canton Street, Norwood, MA, 2005.

show all references

References:
[1]

B. Le, T. Rondeau, J. Reed and C. Bostian, Analog-to-digital converters, IEEE Signal Proc. Mag., 22 (2005), 69-77. doi: 10.1109/4.173093.

[2]

M. Vetterli, P. Marziliano and T. Blu, Sampling signals with finite rate of innovation, IEEE Trans. Signal Process., 50 (2002), 1417-1428. doi: 10.1109/TSP.2002.1003065.

[3]

I. Maravic and M. Vetterli, Sampling and reconstruction of signals with finite rate of innovation in the presence of noise, IEEE Transactions on Signal Processing, 53 (2004), 2788-2805. doi: 10.1109/TSP.2005.850321.

[4]

E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

[5]

E. Candes, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. on Pure and Applied Math., 59 (2006), 1207-1223. doi: 10.1002/cpa.20124.

[6]

E. Candes and T. Tao, Near-optimal signal recovery from random projections and universal encoding strategies? IEEE Trans. on Information Theory, 52 (2006), 5406-5425. doi: 10.1109/TIT.2006.885507.

[7]

D. Donoho, Compressed sensing, IEEE Trans. on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[8]

T. Ragheb, S. Kirolos, J. Laska, A. Gilbert, M. Strauss, R. Baraniuk and Y. Massound, Implementation models for analog-to-information conversion via random sampling, Proc.of 50th Midwest Symposium on Circuits and Systems, (2007), 325-328. doi: 10.1109/MWSCAS.2007.4488599.

[9]

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. Baraniuk and Y. Massound, Theory and implementation of an analog-to-information converter using random demodulation, Proc. of IEEE International Symposium on Circuits and Systems, (2007), 1959-1962. doi: 10.1109/ISCAS.2007.378360.

[10]

R. Baraniuk and P. Steeghs, Compressive radar imaging, Proc. of 2007 IEEE Radar Conference, (2007), 128-133. doi: 10.1109/RADAR.2007.374203.

[11]

M. Herman and T. Strohmer, Compressed sensing radar, Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 1509-1512.

[12]

M. Herman and T. Strohmer, High-resolution radar via compressed sensing, IEEE Trans. on Signal Processing, 57 (2009), 2275-2284. doi: 10.1109/TSP.2009.2014277.

[13]

M. Cetin, Feature-Enhanced Synthetic Aperture Radar Imaging, College of Engineering, Boston University, Ph.D. Thesis, 2001. doi: 10.1109/83.913596.

[14]

M. Cetin and W. C. Karl, Feature-enhanced synthetic aperture radar imaging formation based on non-quadratic regularization, IEEE Trans. Image Process, 10 (2001), 623-631.

[15]

S. Bhattacharya, T. Blumensath, B. Mulgrew and M. Davies, Synthetic Aperture Radar raw data encoding using compressed sensing, Proc. of Radar Conference, pp. 1-5, May 2008.

[16]

S. Bhattacharya, T. Blumensath, B. Mulgrew, and M. Davies, Fast encoding of synthetic aperture radar raw data using compressed sensing, Proc. of IEEE/SP 14th Workshop on Statistical Signal Processing, (2007), 448-452. doi: 10.1109/SSP.2007.4301298.

[17]

G. Rilling, M. Davies and Bernard, Compressed sensing based compression of SAR raw data, Signal Processing with Adaptive Sparse Structured Representaition, 2009.

[18]

A. FannJiang, Compressive inverse scattering I. High frequency SIMO measurements,, , ().  doi: 10.1088/0266-5611/26/3/035008.

[19]

A. FannJiang, Compressive inverse scattering II. SISO measurements with born scatterers,, , ().  doi: 10.1088/0266-5611/26/3/035009.

[20]

J. H. G. Ender, On compressive sensing applied to radar, Signal Processing, 90 (2010), 1402-1414. doi: 10.1016/j.sigpro.2009.11.009.

[21]

L. Zhang, M. Xing, C. Qiun, J. Li and Z. Bao, Achieving higher resolution ISAR imaging with limited pulses via compressed sampling, IEEE Geoscience and Remote Sensing Letters, 6 (2009), 567-571.

[22]

J. Fowler, Compressive-projection principal component analysis, IEEE Trans. Image Process., 18 (2009), 2230-2242. doi: 10.1109/TIP.2009.2025089.

[23]

S. Kirolos, J. Laska, M. Wakin, M. Duarte, D. Baron, T. Ragheb, Y. Massoud and R. Baraniuk, Analog-to-information conversion via random demodulation, Proc. IEEE Dallas/CAS Workshop on Design, Applications, Integration, and Software, (2006), 71-74. doi: 10.1109/DCAS.2006.321036.

[24]

J. A. Tropp, J. N. Laska, M. F. Duarte, J. Romberg and R. G. Baraniuk, Beyond nyquist: Effecient sampling of sparse bandlimited signals, Submitted to IEEE. Trans. Inform. Theory, (2009). doi: 10.1109/TIT.2009.2034811.

[25]

J. Romgerg, Compressive sensing by random convolution, SIAM J. Imaging Sci., 2 (2009), 1098-1128. doi: 10.1137/08072975X.

[26]

G. E. Smith, T. Diethe, Z. Hussain, J. Shawe-Taylor and D. R. Hardoon, Compressed sampling for pulse Doppler radar, Proc. the IEEE International Radar Conference, (2010). doi: 10.1109/RADAR.2010.5494496.

[27]

Z. Bao, M. Xing and T. Wang, Radar Imaging Technology, Beijing, China: Publishing House of Electronics Industry, 2006.

[28]

W. G. Carrara, R. S. Goodman and R. M. Majewski, Spotlight Synthetic Aperture Radar, Boston, MA: Artech House, 1995.

[29]

M. Aharon, M. Elad and A. Bruckstein, K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process., 54 (2006), 4311-4322. doi: 10.1109/TSP.2006.881199.

[30]

R. Baraniuk, M. Davenport, R. DeVore and M. Wakin, A simple proof of the restricted isometry property for random matrices,, Constr. Approx., 28 (): 253.  doi: 10.1007/s00365-007-9003-x.

[31]

E. Van Den Berg and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientic Computing, 31 (2008), 890-912. doi: 10.1137/080714488.

[32]

J. Li, and P. Stoica, An Adaptive Filtering Approach to Spectral Estimation and SAR Imaging, IEEE Trans. on Signal Processing, 44 (1996), 1469-1484. doi: 10.1117/12.210835.

[33]

J. Pdendaal, E. Barnard and C. Pistorius, Two-dimensional superresolution radar imaging using the MUSIC algorithm, IEEE Trans. Antennas Propag., 42 (1994), 1386-1391.

[34]

Z. Bi, J. Li and Z.-S. Liu, Super resolution SAR imaging via parametric spectral estimation methods, IEEE Trans. Aerosp. Electron. Syst., 35 (1999), 267-281.

[35]

H. Rauhut, Stability results for random sampling of sparse trigonometric polynomials, IEEE Trans. on Information Theory, 54 (2008), 5661-5670. doi: 10.1109/TIT.2008.2006382.

[36]

I. Cumming and F. Wong, Digital Processing of Synthetic Aperture Radar Data: Algorithm and Implementaion, Artech Hourse, 656 Canton Street, Norwood, MA, 2005.

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