An Inverse Backscatter Problem for Electric Impedance Tomography

We consider a variant of (two dimensional) electric impedance tomography with very sparse data that resemble so-called backscatter data in inverse scattering. Such data arise in practice if the same single pair of electrodes is used to drive currents and measure voltage differences, subsequently at various neighboring locations on the boundary of the object to be illuminated. We prove that these data uniquely determine an insulating cavity within the object.

[1]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[2]  Inverse backscattering for the Schrödinger equation in 2D , 2007, 1209.2779.

[3]  Avner Friedman,et al.  On the Uniqueness in the Inverse Conductivity Problem with One Measurement , 1988 .

[4]  Elena Beretta,et al.  Stable determination of boundaries from Cauchy data , 1999 .

[5]  W. Hackbusch.  Elliptic Differential Equations , 1992 .

[6]  Rainer Kress,et al.  Inverse Dirichlet problem and conformal mapping , 2004, Math. Comput. Simul..

[7]  Jin Keun Seo,et al.  The inverse conductivity problem with one measurement: uniqueness for convex polyhedra , 1994 .

[8]  David Isaacson,et al.  Effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms , 1991 .

[9]  A. Calderón,et al.  On an inverse boundary value problem , 2006 .

[10]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[11]  G. Uhlmann,et al.  Generalized backscattering and the Lax-Phillips transform , 2007, 0712.4236.

[12]  Gennadi Vainikko,et al.  Periodic Integral and Pseudodifferential Equations with Numerical Approximation , 2001 .

[13]  Houssem Haddar,et al.  The Convex Back-Scattering Support , 2005, SIAM J. Appl. Math..

[14]  RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS , 2001 .

[15]  E. Hille,et al.  Ordinary di?erential equations in the complex domain , 1976 .

[16]  C. Pommerenke.  Boundary Behaviour of Conformal Maps , 1992 .

[17]  Jin Keun Seo,et al.  On the Uniqueness in the Inverse Conductivity Problem , 1995 .

[18]  R. Kress.  Linear Integral Equations , 1989 .

[19]  Roland Schinzinger,et al.  Conformal Mapping: Methods and Applications , 1991 .

[20]  William Rundell,et al.  Inverse Obstacle Scattering Using Reduced Data , 1998, SIAM J. Appl. Math..

[21]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[22]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .