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
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show all references

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[1]

IEEE Trans. Geo. Remote Sensing, 45 (2007), 1247-1257. doi: 10.1109/TGRS.2007.894551.  Google Scholar

[2]

Mathematics & Applications, 62, Springer-Verlag, Berlin, 2008.  Google Scholar

[3]

SIAM J. Imaging Sci., 1 (2008), 169-187. doi: 10.1137/070696076.  Google Scholar

[4]

J. Math. Pures Appl., 101 (2014), 54-75. doi: 10.1016/j.matpur.2013.05.002.  Google Scholar

[5]

J. Diff. Equat., 249 (2010), 1579-1595. doi: 10.1016/j.jde.2010.07.012.  Google Scholar

[6]

SIAM J. Imag. Sci., 5 (2012), 564-600. doi: 10.1137/10080631X.  Google Scholar

[7]

Waves Random Complex Media, 22 (2012), 40-65. doi: 10.1080/17455030.2010.532518.  Google Scholar

[8]

SIAM J. Math. Anal., 45 (2013), 1704-1722. doi: 10.1137/120861849.  Google Scholar

[9]

Proc. Amer. Math. Soc., 141 (2013), 3431-3446. doi: 10.1090/S0002-9939-2013-11590-X.  Google Scholar

[10]

SIAM J. Appl. Math., 71 (2011), 68-91. doi: 10.1137/100800130.  Google Scholar

[11]

Multiscale Model. Simul., 3 (2005), 597-628. doi: 10.1137/040610854.  Google Scholar

[12]

SIAM J. Sci. Comput., 27 (2005), 130-158. doi: 10.1137/040612518.  Google Scholar

[13]

SIAM J. Sci. Comput., 29 (2007), 674-709. doi: 10.1137/050640655.  Google Scholar

[14]

Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245.  Google Scholar

[15]

Applied Mathematical Sciences, 162, Springer-Verlag, New York, 2007.  Google Scholar

[16]

Math. Comp., 81 (2012), 839-860. doi: 10.1090/S0025-5718-2011-02534-2.  Google Scholar

[17]

SIAM J. Sci. Comput., 32 (2010), 894-922. doi: 10.1137/090749013.  Google Scholar

[18]

Phys. Rev. Lett., 102 (2009), 084301. doi: 10.1103/PhysRevLett.102.084301.  Google Scholar

[19]

Waves Random Complex Media, 20 (2010), 333-363. doi: 10.1080/17455030903499698.  Google Scholar

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[22]

J. Comp. Phys., 227 (2007), 755-762. doi: 10.1016/j.jcp.2007.08.020.  Google Scholar

[23]

Advances in Mathematics, 227 (2011), 494-521. doi: 10.1016/j.aim.2011.02.007.  Google Scholar

[24]

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[28]

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[29]

J. Acoust. Soc. Am., 118 (2005), 3129-3138. doi: 10.1121/1.2042987.  Google Scholar

[30]

in Proceedings of the Conference Mathematical and Statistical Methods for Imaging, (eds. H. Ammari, J. Garnier, H. Kang and K. Sølna), Contemporary Mathematics Series, American Mathematical Society, 548 (2011), 151-163. doi: 10.1090/conm/548/10832.  Google Scholar

[31]

J. Phys.: Conf. Ser., 135 (2008), 012106. doi: 10.1088/1742-6596/135/1/012106.  Google Scholar

[32]

Cambridge University Press, Cambridge, 1985. doi: 10.1017/CBO9780511810817.  Google Scholar

[33]

Ann. Statist., 29 (2001), 295-327. doi: 10.1214/aos/1009210543.  Google Scholar

[34]

Ann. Stat., 38 (2010), 3605-3629. doi: 10.1214/10-AOS821.  Google Scholar

[35]

IEEE Oceans 2005 Eur., 2 (2005), 1001-1006. doi: 10.1109/OCEANSE.2005.1513193.  Google Scholar

[36]

Cambridge University Press, 2004. Google Scholar

[37]

Statist. Sinica, 17 (2007), 1617-1642.  Google Scholar

[38]

Journal of Multivariate Analysis, 118 (2013), 67-76. doi: 10.1016/j.jmva.2013.03.005.  Google Scholar

[39]

in SVD and Signal Processing, II: Algorithms, Analysis and Applications, Elsevier, 1990, pp. 99-109. Google Scholar

[40]

Math. Model. Numer. Anal., 34 (2000), 723-748. doi: 10.1051/m2an:2000101.  Google Scholar

[41]

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[42]

BIT Numerical Mathematics, 12 (1972), 99-111.  Google Scholar

[43]

IEEE Trans. Biomedical Eng., 53 (2006), 1647-1657. Google Scholar

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