The MUSIC algorithm for impedance tomography of small inclusions from discrete data

We consider a point-electrode model for electrical impedance tomography and show that current-to-voltage measurements from finitely many electrodes are sufficient to characterize the positions of a finite number of point-like inclusions. More precisely, we consider an asymptotic expansion with respect to the size of the small inclusions of the relative Neumann-to-Dirichlet operator in the framework of the point electrode model. This operator is naturally finite-dimensional and models difference measurements by finitely many small electrodes of the electric potential with and without the small inclusions. Moreover, its leading-order term explicitly characterizes the centers of the small inclusions if the (finite) number of point electrodes is large enough. This characterization is based on finite-dimensional test vectors and leads naturally to a MUSIC algorithm for imaging the inclusion centers. We show both the feasibility and limitations of this imaging technique via two-dimensional numerical experiments, considering in particular the influence of the number of point electrodes on the algorithm's images.

[1]  Nuutti Hyvönen,et al.  Point Measurements for a Neumann-to-Dirichlet Map and the Calderón Problem in the Plane , 2012, SIAM J. Math. Anal..

[2]  Richard M. Leahy,et al.  Source localization using recursively applied and projected (RAP) MUSIC , 1997 .

[3]  Ekaterina Iakovleva,et al.  A MUSIC Algorithm for Locating Small Inclusions Buried in a Half-Space from the Scattering Amplitude at a Fixed Frequency , 2005, Multiscale Model. Simul..

[4]  Michael Vogelius,et al.  A direct impedance tomography algorithm for locating small inhomogeneities , 2003, Numerische Mathematik.

[5]  Bastian Harrach,et al.  JUSTIFICATION OF POINT ELECTRODE MODELS IN ELECTRICAL IMPEDANCE TOMOGRAPHY , 2011 .

[6]  Otto Seiskari,et al.  Detection of multiple inclusions from sweep data of electrical impedance tomography , 2012 .

[7]  Mourad Sini,et al.  Detection of point-like scatterers using one type of scattered elastic waves , 2012, J. Comput. Appl. Math..

[8]  Seick Kim,et al.  Neumann functions for second order elliptic systems with measurable coefficients , 2011, 1112.2436.

[9]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[10]  A. Kirsch.  The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media , 2002 .

[11]  Ekaterina Iakovleva,et al.  Direct Elastic Imaging of a Small Inclusion , 2008, SIAM J. Imaging Sci..

[12]  Ren-Cang Li,et al.  Relative Perturbation Theory: II. Eigenspace and Singular Subspace Variations , 1996, SIAM J. Matrix Anal. Appl..

[13]  Michael Vogelius,et al.  Identification of conductivity imperfections of small diameter by boundary measurements. Continuous , 1998 .

[14]  Otto Seiskari.  Locating multiple inclusions from sweep data of electrical impedance tomography , 2011 .

[15]  Construction of invisible conductivity perturbations for the point electrode model in electrical impedance tomography , 2014, 1412.6768.

[16]  Otto Seiskari,et al.  Point Electrode Problems in Piecewise Smooth Plane Domains , 2012, SIAM J. Math. Anal..

[17]  Habib Ammari,et al.  Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion , 2002 .

[18]  J. Devaney,et al.  1 Super-resolution Processing of Multi-static Data Using Time Reversal and MUSIC A , 2000 .

[19]  Michael Stewart,et al.  Perturbation of the SVD in the presence of small singular values , 2006 .

[20]  Nuutti Hyvönen,et al.  Convex backscattering support in electric impedance tomography , 2011, Numerische Mathematik.

[21]  J. Hayashi.  [Sampling methods]. , 1982, Josanpu zasshi = The Japanese journal for midwife.

[22]  Habib Ammari,et al.  Identification of small inhomogeneities: Asymptotic factorization , 2007, Math. Comput..

[23]  N I Grinberg,et al.  The Factorization Method for Inverse Problems , 2007 .

[24]  R. O. Schmidt,et al.  Multiple emitter location and signal Parameter estimation , 1986 .

[25]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[26]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[27]  Christophe Hazard,et al.  Selective Acoustic Focusing Using Time-Harmonic Reversal Mirrors , 2004, SIAM J. Appl. Math..

[28]  Roland Griesmaier,et al.  An Asymptotic Factorization Method for Inverse Electromagnetic Scattering in Layered Media , 2008, SIAM J. Appl. Math..