June  2015, 8(3): 547-561. doi: 10.3934/dcdss.2015.8.547

Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method

1. 

Faculty of Science, South University of Science and Technology of China, Shenzhen, 518055

2. 

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

3. 

Department of Computing Sciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

Received  December 2013 Revised  February 2014 Published  October 2014

Some effective imaging schemes for inverse scattering problems were recently proposed in [13,14] for locating multiple multiscale electromagnetic (EM) scatterers, namely a combination of components of possible small size and regular size compared to the detecting EM wavelength. In this paper, instead of using a single far-field measurement, we relax the assumption of one fixed frequency to multiple ones, and develop efficient numerical techniques to speed up those imaging schemes by adopting multi-frequency and Multilevel ideas in a two-stage manner. Numerical tests are presented to demonstrate the efficiency and the salient features of the proposed fast imaging scheme.
Citation: Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547
References:
[1]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, 1846, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245.

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Springer, New York, 2007.

[3]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, NewYork, 1965. doi: 10.1119/1.1972842.

[4]

H. Ammari and J.-C. Nédélec, Low-frequency electromagnetic scattering, SIAM J. Math. Anal., 31 (2000), 836-861. doi: 10.1137/S0036141098343604.

[5]

H. Ammari, H. Kang, E. Kim and J. Lee, The generalized polarization tensors for resolved imaging. Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements, Math. Comp., 81 (2012), 839-860. doi: 10.1090/S0025-5718-2011-02534-2.

[6]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer, 2006.

[7]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, Philadelphia: SIAM, 2011. doi: 10.1137/1.9780898719406.

[8]

X. Chen and Y. Zhong, MUSIC electromagnetic imaging with enhanced resolution for small inclusions, Inverse Problems, 25 (2009), 12 pp. doi: 10.1088/0266-5611/25/1/015008.

[9]

D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Rev., 42 (2000), 369-414. doi: 10.1137/S0036144500367337.

[10]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.

[11]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983.

[12]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Cambridge: Cambridge University Press, 1998. doi: 10.1007/978-3-662-03537-5.

[13]

J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Mat., 73 (2013), 1721-1746. doi: 10.1137/130907690.

[14]

J. Li, H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356.

[15]

J. Li, H. Liu and Q. Wang, Enhanced multilevel linear sampling methods for inverse scattering problems, J. Comput. Phys., 22 (2014), 554-571. doi: 10.1016/j.jcp.2013.09.048.

[16]

J. Li, H. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM J. Sci. Comp., 30 (2008),1228-1250. doi: 10.1137/060674247.

[17]

J-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, volume 144, Springer, 2001. doi: 10.1007/978-1-4757-4393-7.

[18]

C. G. Someda, Electromagnetic Waves, Boca Raton, FL: CRC Press, 2nd edition, 2006.

[19]

J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems & Imaging, 7 (2013). doi: 10.3934/ipi.2013.7.757.

[20]

M. Ikehata, Reconstruction of obstacles from boundary measurements, Wave Motion, 3 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2.

[21]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.

[22]

R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1-R47. doi: 10.1088/0266-5611/22/2/R01.

[23]

G. Uhlmann, Inside Out: Inverse Problems and Applications, MSRI Publications, 47, Cambridge University Press, 2003. doi: 10.1090/conm/333.

show all references

References:
[1]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, 1846, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245.

[2]

H. Ammari and H. Kang, Polarization and Moment Tensors. With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Springer, New York, 2007.

[3]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, NewYork, 1965. doi: 10.1119/1.1972842.

[4]

H. Ammari and J.-C. Nédélec, Low-frequency electromagnetic scattering, SIAM J. Math. Anal., 31 (2000), 836-861. doi: 10.1137/S0036141098343604.

[5]

H. Ammari, H. Kang, E. Kim and J. Lee, The generalized polarization tensors for resolved imaging. Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements, Math. Comp., 81 (2012), 839-860. doi: 10.1090/S0025-5718-2011-02534-2.

[6]

F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory: An Introduction, Springer, 2006.

[7]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, Philadelphia: SIAM, 2011. doi: 10.1137/1.9780898719406.

[8]

X. Chen and Y. Zhong, MUSIC electromagnetic imaging with enhanced resolution for small inclusions, Inverse Problems, 25 (2009), 12 pp. doi: 10.1088/0266-5611/25/1/015008.

[9]

D. Colton, J. Coyle and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Rev., 42 (2000), 369-414. doi: 10.1137/S0036144500367337.

[10]

D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.

[11]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983.

[12]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Cambridge: Cambridge University Press, 1998. doi: 10.1007/978-3-662-03537-5.

[13]

J. Li, H. Liu, Z. Shang and H. Sun, Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Mat., 73 (2013), 1721-1746. doi: 10.1137/130907690.

[14]

J. Li, H. Liu and Q. Wang, Locating multiple multiscale electromagnetic scatterers by a single far-field measurement, SIAM J. Imaging Sci., 6 (2013), 2285-2309. doi: 10.1137/130920356.

[15]

J. Li, H. Liu and Q. Wang, Enhanced multilevel linear sampling methods for inverse scattering problems, J. Comput. Phys., 22 (2014), 554-571. doi: 10.1016/j.jcp.2013.09.048.

[16]

J. Li, H. Liu and J. Zou, Multilevel linear sampling method for inverse scattering problems, SIAM J. Sci. Comp., 30 (2008),1228-1250. doi: 10.1137/060674247.

[17]

J-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, volume 144, Springer, 2001. doi: 10.1007/978-1-4757-4393-7.

[18]

C. G. Someda, Electromagnetic Waves, Boca Raton, FL: CRC Press, 2nd edition, 2006.

[19]

J. Li and J. Zou, A direct sampling method for inverse scattering using far-field data, Inverse Problems & Imaging, 7 (2013). doi: 10.3934/ipi.2013.7.757.

[20]

M. Ikehata, Reconstruction of obstacles from boundary measurements, Wave Motion, 3 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2.

[21]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.

[22]

R. Potthast, A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), R1-R47. doi: 10.1088/0266-5611/22/2/R01.

[23]

G. Uhlmann, Inside Out: Inverse Problems and Applications, MSRI Publications, 47, Cambridge University Press, 2003. doi: 10.1090/conm/333.

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