American Institute of Mathematical Sciences

Advanced
June  2015, 8(3): 389-417. doi: 10.3934/dcdss.2015.8.389

Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging

Habib Ammari 1, , Josselin Garnier 2, and  Vincent Jugnon 3,

1. 

Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d'Ulm, 75005 Paris, France
2. 

Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris Diderot, 75205 Paris Cedex 13
3. 

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, United States

Received  June 2013 Revised  January 2014 Published  October 2014

The detection, localization, and characterization of a collection of targets embedded in a medium is an important problem in multistatic wave imaging. The responses between each pair of source and receiver are collected and assembled in the form of a response matrix, known as the multi-static response matrix. When the data are corrupted by measurement or instrument noise, the structure of the response matrix is studied by using random matrix theory. It is shown how the targets can be efficiently detected, localized and characterized. Both the case of a collection of point reflectors in which the singular vectors have all the same form and the case of small-volume electromagnetic inclusions in which the singular vectors may have different forms depending on their magnetic or dielectric type are addressed.
Keywords: random matrices, detection and reconstruction algorithms, Multistatic wave imaging, noisy data, Helmholtz equation., measurement noise.
Mathematics Subject Classification: Primary: 78M35, 78A46; Secondary: 15B5.
Citation: Habib Ammari, Josselin Garnier, Vincent Jugnon. Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 389-417. doi: 10.3934/dcdss.2015.8.389
References:
[1]

M. Alam, V. Cevher, J. H. McClellan, G. D. Larson, and W. R. Scott, Jr., Optimal maneuvering of seismic sensors for localization of subsurface targets,, IEEE Trans. Geo. Remote Sensing, 45 (2007), 1247.  doi: 10.1109/TGRS.2007.894551.  Google Scholar

[2]

H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging,, Mathematics & Applications, (2008).   Google Scholar

[3]

H. Ammari, P. Calmon and E. Iakovleva, Direct elastic imaging of a small inclusion,, SIAM J. Imaging Sci., 1 (2008), 169.  doi: 10.1137/070696076.  Google Scholar

[4]

H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volokov, Target detection and characterization from electromagnetic induction data,, J. Math. Pures Appl., 101 (2014), 54.  doi: 10.1016/j.matpur.2013.05.002.  Google Scholar

[5]

H. Ammari, P. Garapon, L. Guadarrama Bustos and H. Kang, Transient anomaly imaging by the acoustic radiation force,, J. Diff. Equat., 249 (2010), 1579.  doi: 10.1016/j.jde.2010.07.012.  Google Scholar

[6]

H. Ammari, J. Garnier, H. Kang, M. Lim and K. Sølna, Multistatic imaging of extended targets,, SIAM J. Imag. Sci., 5 (2012), 564.  doi: 10.1137/10080631X.  Google Scholar

[7]

H. Ammari, J. Garnier and K. Sølna, A statistical approach to target detection and localization in the presence of noise,, Waves Random Complex Media, 22 (2012), 40.  doi: 10.1080/17455030.2010.532518.  Google Scholar

[8]

H. Ammari, J. Garnier and K. Sølna, Limited view resolving power of linearized conductivity imaging from boundary measurements,, SIAM J. Math. Anal., 45 (2013), 1704.  doi: 10.1137/120861849.  Google Scholar

[9]

H. Ammari, J. Garnier and K. Sølna, Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,, Proc. Amer. Math. Soc., 141 (2013), 3431.  doi: 10.1090/S0002-9939-2013-11590-X.  Google Scholar

[10]

H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Sølna, Imaging schemes for cracks and inclusions,, SIAM J. Appl. Math., 71 (2011), 68.  doi: 10.1137/100800130.  Google Scholar

[11]

H. Ammari, E. Iakovleva and D. Lesselier, A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,, Multiscale Model. Simul., 3 (2005), 597.  doi: 10.1137/040610854.  Google Scholar

[12]

H. Ammari, E. Iakovleva and D. Lesselier, Two numerical methods for recovering small inclusions from the scattering amplitude at a fixed frequency,, SIAM J. Sci. Comput., 27 (2005), 130.  doi: 10.1137/040612518.  Google Scholar

[13]

H. Ammari, E. Iakovleva, D. Lesselier and G. Perrusson, A MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions,, SIAM J. Sci. Comput., 29 (2007), 674.  doi: 10.1137/050640655.  Google Scholar

[14]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements,, Lecture Notes in Mathematics, (1846).  doi: 10.1007/b98245.  Google Scholar

[15]

H. Ammari and H. Kang, Polarization and Moment Tensors: with Applications to Inverse Problems and Effective Medium Theory,, Applied Mathematical Sciences, (2007).   Google Scholar

[16]

H. Ammari, H. Kang, E. Kim and J.-Y. 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.  doi: 10.1090/S0025-5718-2011-02534-2.  Google Scholar

[17]

H. Ammari, H. Kang, H. Lee and W. K. Park, Asymptotic imaging of perfectly conducting cracks,, SIAM J. Sci. Comput., 32 (2010), 894.  doi: 10.1137/090749013.  Google Scholar

[18]

A. Aubry and A. Derode, Random matrix theory applied to acoustic backscattering and imaging in complex media,, Phys. Rev. Lett., 102 (2009).  doi: 10.1103/PhysRevLett.102.084301.  Google Scholar

[19]

A. Aubry and A. Derode, Singular value distribution of the propagation matrix in random scattering media,, Waves Random Complex Media, 20 (2010), 333.  doi: 10.1080/17455030903499698.  Google Scholar

[20]

A. Aubry and A. Derode, Detection and imaging in a random medium: A matrix method to overcome multiple scattering and aberration,, J. Appl. Physics, 106 (2009).  doi: 10.1063/1.3200962.  Google Scholar

[21]

J. Baik and J. W. Silverstein, Eigenvalues of large sample covariance matrices of spiked population models,, Journal of Multivariate Analysis, 97 (2006), 1382.  doi: 10.1016/j.jmva.2005.08.003.  Google Scholar

[22]

G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm,, J. Comp. Phys., 227 (2007), 755.  doi: 10.1016/j.jcp.2007.08.020.  Google Scholar

[23]

F. Benaych-Georges and R. R. Nadakuditi, The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices,, Advances in Mathematics, 227 (2011), 494.  doi: 10.1016/j.aim.2011.02.007.  Google Scholar

[24]

J. Byrnes, Advances in Sensing with Security Applications,, Springer-Verlag, (2006).  doi: 10.1007/1-4020-4295-7.  Google Scholar

[25]

D. H. Chambers, Target characterization using time-reversal symmetry of wave propagation,, Int. J. Modern Phys. B, 21 (2007), 3511.  doi: 10.1142/S0217979207037521.  Google Scholar

[26]

D. H. Chambers and J. G. Berryman, Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,, IEEE Trans. Antennas Propagat., 52 (2004), 1729.  doi: 10.1109/TAP.2004.831323.  Google Scholar

[27]

D. H. Chambers and J. G. Berryman, Time-reversal analysis for scatterer characterization,, Phys. Rev. Lett., 92 (2004).  doi: 10.1103/PhysRevLett.92.023902.  Google Scholar

[28]

A. J. Devaney, Time reversal imaging of obscured targets from multistatic data,, IEEE Trans. Antennas Propagat., 523 (2005), 1600.  doi: 10.1109/TAP.2005.846723.  Google Scholar

[29]

A. J. Devaney, E. A. Marengo and F. K. Gruber, Time-reversal-based imaging and inverse scattering of multiply scattering point targets,, J. Acoust. Soc. Am., 118 (2005), 3129.  doi: 10.1121/1.2042987.  Google Scholar

[30]

J. Garnier, Use of random matrix theory for target detection, localization, and reconstruction,, in Proceedings of the Conference Mathematical and Statistical Methods for Imaging, 548 (2011), 151.  doi: 10.1090/conm/548/10832.  Google Scholar

[31]

D. J. Hansen and M. S. Vogelius, High frequency perturbation formulas for the effect of small inhomogeneities,, J. Phys.: Conf. Ser., 135 (2008).  doi: 10.1088/1742-6596/135/1/012106.  Google Scholar

[32]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985).  doi: 10.1017/CBO9780511810817.  Google Scholar

[33]

I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis,, Ann. Statist., 29 (2001), 295.  doi: 10.1214/aos/1009210543.  Google Scholar

[34]

S. Lee, F. Zou and F. A. Wright, Convergence and prediction of principal component scores in high-dimensional settings,, Ann. Stat., 38 (2010), 3605.  doi: 10.1214/10-AOS821.  Google Scholar

[35]

J. G. Minonzio, D. Clorennec, A. Aubry, T. Folégot, T. Pélican, C. Prada, J. de Rosny and M. Fink, Application of the DORT method to the detection and characterization of two targets in a shallow water wave-guide,, IEEE Oceans 2005 Eur., 2 (2005), 1001.  doi: 10.1109/OCEANSE.2005.1513193.  Google Scholar

[36]

M. Oristaglio and H. Blok, Wavefield Imaging and Inversion in Electromagnetics and Acoustics,, Cambridge University Press, (2004).   Google Scholar

[37]

D. Paul, Asymptotics of sample eigenstructure for a large dimensional spiked covariance model,, Statist. Sinica, 17 (2007), 1617.   Google Scholar

[38]

A. Shabalin and A. Nobel, Reconstruction of a low-rank matrix in the presence of Gaussian noise,, Journal of Multivariate Analysis, 118 (2013), 67.  doi: 10.1016/j.jmva.2013.03.005.  Google Scholar

[39]

G. W. Stewart, Perturbation theory for the singular value decomposition,, in SVD and Signal Processing, (1990), 99.   Google Scholar

[40]

M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities,, Math. Model. Numer. Anal., 34 (2000), 723.  doi: 10.1051/m2an:2000101.  Google Scholar

[41]

D. Volkov, Numerical methods for locating small dielectric inhomogeneities,, Wave Motion, 38 (2003), 189.  doi: 10.1016/S0165-2125(03)00047-7.  Google Scholar

[42]

P.-Å. Wedin, Perturbation bounds in connection with singular value decomposition,, BIT Numerical Mathematics, 12 (1972), 99.   Google Scholar

[43]

X. Yao, G. Bin, X. Luzhou, L. Jian and P. Stoica, Multistatic adaptive microwave imaging for early breast cancer detection,, IEEE Trans. Biomedical Eng., 53 (2006), 1647.   Google Scholar

show all references

References:
[1]

M. Alam, V. Cevher, J. H. McClellan, G. D. Larson, and W. R. Scott, Jr., Optimal maneuvering of seismic sensors for localization of subsurface targets,, IEEE Trans. Geo. Remote Sensing, 45 (2007), 1247.  doi: 10.1109/TGRS.2007.894551.  Google Scholar

[2]

H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging,, Mathematics & Applications, (2008).   Google Scholar

[3]

H. Ammari, P. Calmon and E. Iakovleva, Direct elastic imaging of a small inclusion,, SIAM J. Imaging Sci., 1 (2008), 169.  doi: 10.1137/070696076.  Google Scholar

[4]

H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volokov, Target detection and characterization from electromagnetic induction data,, J. Math. Pures Appl., 101 (2014), 54.  doi: 10.1016/j.matpur.2013.05.002.  Google Scholar

[5]

H. Ammari, P. Garapon, L. Guadarrama Bustos and H. Kang, Transient anomaly imaging by the acoustic radiation force,, J. Diff. Equat., 249 (2010), 1579.  doi: 10.1016/j.jde.2010.07.012.  Google Scholar

[6]

H. Ammari, J. Garnier, H. Kang, M. Lim and K. Sølna, Multistatic imaging of extended targets,, SIAM J. Imag. Sci., 5 (2012), 564.  doi: 10.1137/10080631X.  Google Scholar

[7]

H. Ammari, J. Garnier and K. Sølna, A statistical approach to target detection and localization in the presence of noise,, Waves Random Complex Media, 22 (2012), 40.  doi: 10.1080/17455030.2010.532518.  Google Scholar

[8]

H. Ammari, J. Garnier and K. Sølna, Limited view resolving power of linearized conductivity imaging from boundary measurements,, SIAM J. Math. Anal., 45 (2013), 1704.  doi: 10.1137/120861849.  Google Scholar

[9]

H. Ammari, J. Garnier and K. Sølna, Resolution and stability analysis in full-aperture, linearized conductivity and wave imaging,, Proc. Amer. Math. Soc., 141 (2013), 3431.  doi: 10.1090/S0002-9939-2013-11590-X.  Google Scholar

[10]

H. Ammari, J. Garnier, H. Kang, W. K. Park and K. Sølna, Imaging schemes for cracks and inclusions,, SIAM J. Appl. Math., 71 (2011), 68.  doi: 10.1137/100800130.  Google Scholar

[11]

H. Ammari, E. Iakovleva and D. Lesselier, A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency,, Multiscale Model. Simul., 3 (2005), 597.  doi: 10.1137/040610854.  Google Scholar

[12]

H. Ammari, E. Iakovleva and D. Lesselier, Two numerical methods for recovering small inclusions from the scattering amplitude at a fixed frequency,, SIAM J. Sci. Comput., 27 (2005), 130.  doi: 10.1137/040612518.  Google Scholar

[13]

H. Ammari, E. Iakovleva, D. Lesselier and G. Perrusson, A MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions,, SIAM J. Sci. Comput., 29 (2007), 674.  doi: 10.1137/050640655.  Google Scholar

[14]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements,, Lecture Notes in Mathematics, (1846).  doi: 10.1007/b98245.  Google Scholar

[15]

H. Ammari and H. Kang, Polarization and Moment Tensors: with Applications to Inverse Problems and Effective Medium Theory,, Applied Mathematical Sciences, (2007).   Google Scholar

[16]

H. Ammari, H. Kang, E. Kim and J.-Y. 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.  doi: 10.1090/S0025-5718-2011-02534-2.  Google Scholar

[17]

H. Ammari, H. Kang, H. Lee and W. K. Park, Asymptotic imaging of perfectly conducting cracks,, SIAM J. Sci. Comput., 32 (2010), 894.  doi: 10.1137/090749013.  Google Scholar

[18]

A. Aubry and A. Derode, Random matrix theory applied to acoustic backscattering and imaging in complex media,, Phys. Rev. Lett., 102 (2009).  doi: 10.1103/PhysRevLett.102.084301.  Google Scholar

[19]

A. Aubry and A. Derode, Singular value distribution of the propagation matrix in random scattering media,, Waves Random Complex Media, 20 (2010), 333.  doi: 10.1080/17455030903499698.  Google Scholar

[20]

A. Aubry and A. Derode, Detection and imaging in a random medium: A matrix method to overcome multiple scattering and aberration,, J. Appl. Physics, 106 (2009).  doi: 10.1063/1.3200962.  Google Scholar

[21]

J. Baik and J. W. Silverstein, Eigenvalues of large sample covariance matrices of spiked population models,, Journal of Multivariate Analysis, 97 (2006), 1382.  doi: 10.1016/j.jmva.2005.08.003.  Google Scholar

[22]

G. Bao, S. Hou and P. Li, Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm,, J. Comp. Phys., 227 (2007), 755.  doi: 10.1016/j.jcp.2007.08.020.  Google Scholar

[23]

F. Benaych-Georges and R. R. Nadakuditi, The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices,, Advances in Mathematics, 227 (2011), 494.  doi: 10.1016/j.aim.2011.02.007.  Google Scholar

[24]

J. Byrnes, Advances in Sensing with Security Applications,, Springer-Verlag, (2006).  doi: 10.1007/1-4020-4295-7.  Google Scholar

[25]

D. H. Chambers, Target characterization using time-reversal symmetry of wave propagation,, Int. J. Modern Phys. B, 21 (2007), 3511.  doi: 10.1142/S0217979207037521.  Google Scholar

[26]

D. H. Chambers and J. G. Berryman, Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field,, IEEE Trans. Antennas Propagat., 52 (2004), 1729.  doi: 10.1109/TAP.2004.831323.  Google Scholar

[27]

D. H. Chambers and J. G. Berryman, Time-reversal analysis for scatterer characterization,, Phys. Rev. Lett., 92 (2004).  doi: 10.1103/PhysRevLett.92.023902.  Google Scholar

[28]

A. J. Devaney, Time reversal imaging of obscured targets from multistatic data,, IEEE Trans. Antennas Propagat., 523 (2005), 1600.  doi: 10.1109/TAP.2005.846723.  Google Scholar

[29]

A. J. Devaney, E. A. Marengo and F. K. Gruber, Time-reversal-based imaging and inverse scattering of multiply scattering point targets,, J. Acoust. Soc. Am., 118 (2005), 3129.  doi: 10.1121/1.2042987.  Google Scholar

[30]

J. Garnier, Use of random matrix theory for target detection, localization, and reconstruction,, in Proceedings of the Conference Mathematical and Statistical Methods for Imaging, 548 (2011), 151.  doi: 10.1090/conm/548/10832.  Google Scholar

[31]

D. J. Hansen and M. S. Vogelius, High frequency perturbation formulas for the effect of small inhomogeneities,, J. Phys.: Conf. Ser., 135 (2008).  doi: 10.1088/1742-6596/135/1/012106.  Google Scholar

[32]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985).  doi: 10.1017/CBO9780511810817.  Google Scholar

[33]

I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis,, Ann. Statist., 29 (2001), 295.  doi: 10.1214/aos/1009210543.  Google Scholar

[34]

S. Lee, F. Zou and F. A. Wright, Convergence and prediction of principal component scores in high-dimensional settings,, Ann. Stat., 38 (2010), 3605.  doi: 10.1214/10-AOS821.  Google Scholar

[35]

J. G. Minonzio, D. Clorennec, A. Aubry, T. Folégot, T. Pélican, C. Prada, J. de Rosny and M. Fink, Application of the DORT method to the detection and characterization of two targets in a shallow water wave-guide,, IEEE Oceans 2005 Eur., 2 (2005), 1001.  doi: 10.1109/OCEANSE.2005.1513193.  Google Scholar

[36]

M. Oristaglio and H. Blok, Wavefield Imaging and Inversion in Electromagnetics and Acoustics,, Cambridge University Press, (2004).   Google Scholar

[37]

D. Paul, Asymptotics of sample eigenstructure for a large dimensional spiked covariance model,, Statist. Sinica, 17 (2007), 1617.   Google Scholar

[38]

A. Shabalin and A. Nobel, Reconstruction of a low-rank matrix in the presence of Gaussian noise,, Journal of Multivariate Analysis, 118 (2013), 67.  doi: 10.1016/j.jmva.2013.03.005.  Google Scholar

[39]

G. W. Stewart, Perturbation theory for the singular value decomposition,, in SVD and Signal Processing, (1990), 99.   Google Scholar

[40]

M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities,, Math. Model. Numer. Anal., 34 (2000), 723.  doi: 10.1051/m2an:2000101.  Google Scholar

[41]

D. Volkov, Numerical methods for locating small dielectric inhomogeneities,, Wave Motion, 38 (2003), 189.  doi: 10.1016/S0165-2125(03)00047-7.  Google Scholar

[42]

P.-Å. Wedin, Perturbation bounds in connection with singular value decomposition,, BIT Numerical Mathematics, 12 (1972), 99.   Google Scholar

[43]

X. Yao, G. Bin, X. Luzhou, L. Jian and P. Stoica, Multistatic adaptive microwave imaging for early breast cancer detection,, IEEE Trans. Biomedical Eng., 53 (2006), 1647.   Google Scholar

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