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Magneto-Acousto-Electric Tomography with Magnetic Field Measurements: Modeling, Inversion and Stability

Lingyun Qiu, Siqin Zheng

Abstract

Magneto-acousto-electric tomography (MAET) combines ultrasound with a static magnetic field to infer the electrical conductivity of an object. In this paper, we present a rigorous quasi-static mathematical model for MAET with magnetic field measurements and introduce an adjoint problem to decouple the resulting hybrid inverse problem. This yields a two-step inversion procedure: solving an acoustic inverse source problem and then recovering the conductivity from an internal current density. We prove that, under explicit smallness assumptions on the conductivity and coil geometry, the conductivity can be recovered with Lipschitz stability in both bounded and half-space geometries.

Magneto-Acousto-Electric Tomography with Magnetic Field Measurements: Modeling, Inversion and Stability

Abstract

Magneto-acousto-electric tomography (MAET) combines ultrasound with a static magnetic field to infer the electrical conductivity of an object. In this paper, we present a rigorous quasi-static mathematical model for MAET with magnetic field measurements and introduce an adjoint problem to decouple the resulting hybrid inverse problem. This yields a two-step inversion procedure: solving an acoustic inverse source problem and then recovering the conductivity from an internal current density. We prove that, under explicit smallness assumptions on the conductivity and coil geometry, the conductivity can be recovered with Lipschitz stability in both bounded and half-space geometries.
Paper Structure (12 sections, 9 theorems, 103 equations, 4 figures)

This paper contains 12 sections, 9 theorems, 103 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Notations
  4. Preliminaries on the adjoint problem
  5. Inversion procedure
  6. Recovering the total effect of an acoustic source
  7. Recovering the adjoint current
  8. Recovering the conductivity
  9. Stability of recovering the conductivity from an internal current density
  10. Nonzero constraint and stability in bounded domain
  11. Nonzero constraint and stability in half-space
  12. Concluding remarks

Key Result

proposition 2.2

Let $\Omega$ be a bounded domain in $\mathbb{R}^3$. If $\sigma\in \Sigma(\lambda)$, then

Figures (4)

  • Figure 1: A simple experimental setup of MAET with magnetic field measurements
  • Figure 2: Diagram of inversion procedure
  • Figure 3: Rotationally symmetric domain and coil with the same axis
  • Figure 4: Assumptions of conductivity distribution in half-space

Theorems & Definitions (21)

  • definition 2.1
  • proposition 2.2
  • proof
  • lemma 2.4
  • corollary 2.5
  • proof : Proof of \ref{['eq:vec_pot']}
  • proposition 4.1
  • proof
  • remark 4.2
  • example 4.3
  • ...and 11 more