Imaging with thermal noise induced currents

We use thermal noise induced currents to image the real and imaginary parts of the conductivity of a body. Covariances of the thermal noise currents measured at a few electrodes are shown to be related to a deterministic problem. We use the covariances obtained while selectively heating the body to recover the real power density in the body under known boundary conditions and at a known frequency. The resulting inverse problem is related to acousto-electric tomography, but where the conductivity is complex and only the real power is measured. We study the local solvability of this problem by determining where its linearization is elliptic. Numerical experiments illustrating this inverse problem are included.

[1]  Albert Y. Zomaya,et al.  Partial Differential Equations , 2007, Explorations in Numerical Analysis.

[2]  Wesley C. Sanders.  Conductive Atomic Force Microscopy , 2019, Atomic Force Microscopy.

[3]  U. Feige,et al.  Spectral graph theory , 2019, Zeta and 𝐿-functions in Number Theory and Combinatorics.

[4]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[5]  Simon R. Arridge,et al.  Gradient-Based Quantitative Image Reconstruction in Ultrasound-Modulated Optical Tomography: First Harmonic Measurement Type in a Linearised Diffusion Formulation , 2014, IEEE Transactions on Medical Imaging.

[6]  E. M. Lifshitz,et al.  Course in Theoretical Physics , 2013 .

[7]  G. Bal,et al.  Hybrid inverse problems for a system of Maxwell’s equations , 2013, 1308.5439.

[8]  G. Bal,et al.  Linearized internal functionals for anisotropic conductivities , 2013, 1302.3354.

[9]  O. Scherzer,et al.  The Levenberg-Marquardt Iteration for Numerical Inversion of the Power Density Operator , 2012, 1211.6034.

[10]  G. Bal.  Hybrid inverse problems and redundant systems of partial differential equations , 2012, 1210.0265.

[11]  G. Bal,et al.  Inverse Anisotropic Conductivity from Power Densities in Dimension n ≥ 3 , 2012, 1208.6029.

[12]  G. Bal.  Cauchy problem for Ultrasound Modulated EIT , 2012, 1201.0972.

[13]  Habib Ammari,et al.  Microwave Imaging by Elastic Deformation , 2011, SIAM J. Appl. Math..

[14]  Guillaume Bal,et al.  Reconstruction of Coefficients in Scalar Second‐Order Elliptic Equations from Knowledge of Their Solutions , 2011, 1111.5051.

[15]  Guillaume Bal,et al.  Hybrid inverse problems and internal functionals , 2011, 1110.4733.

[16]  G. Bal,et al.  Inverse anisotropic diffusion from power density measurements in two dimensions , 2011, 1110.4606.

[17]  G. Bal,et al.  Inverse diffusion from knowledge of power densities , 2011, 1110.4577.

[18]  Peter Kuchment,et al.  Stabilizing inverse problems by internal data , 2011, 1110.1819.

[19]  G. Bal,et al.  Inverse diffusion problems with redundant internal information , 2011, 1106.4277.

[20]  Peter Kuchment,et al.  2D and 3D reconstructions in acousto-electric tomography , 2010, 1011.3059.

[21]  G. Bal,et al.  Inverse scattering and acousto-optic imaging. , 2009, Physical review letters.

[22]  Jérôme Fehrenbach,et al.  Imaging by Modification: Numerical Reconstruction of Local Conductivities from Corresponding Power Density Measurements , 2009, SIAM J. Imaging Sci..

[23]  Peter Kuchment,et al.  Synthetic focusing in ultrasound modulated tomography , 2009, 0901.2552.

[24]  Otmar Scherzer,et al.  Impedance-Acoustic Tomography , 2008, SIAM J. Appl. Math..

[25]  Eric Bonnetier,et al.  Electrical Impedance Tomography by Elastic Deformation , 2008, SIAM J. Appl. Math..

[26]  Habib Ammari,et al.  An Introduction to Mathematics of Emerging Biomedical Imaging , 2008 .

[27]  David Isaacson,et al.  Electrical Impedance Tomography , 1999, SIAM Rev..

[28]  T. R. Anthony,et al.  Heat treating and melting material with a scanning laser or electron beam , 1977 .

[29]  Louis Nirenberg,et al.  Interior estimates for elliptic systems of partial differential equations , 1955 .

[30]  H. Nyquist.  Thermal Agitation of Electric Charge in Conductors , 1928 .

[31]  J. Johnson.  Thermal Agitation of Electricity in Conductors , 1927, Nature.

[32]  B. Bhatia,et al.  HEATED ATOMIC FORCE MICROSCOPE CANTILEVERS AND THEIR APPLICATIONS , 2013 .

[33]  M. Bellac,et al.  Nonequilibrium statistical mechanics , 2007, Physics Subject Headings (PhySH).

[34]  S. M. Rytov,et al.  Principles of statistical radiophysics , 1987 .

[35]  R. Kubo.  The fluctuation-dissipation theorem , 1966 .