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Acousto-electric tomography by the convergence of Kaczamrz two-point gradient-$Θ$ method

Kai Zhu, Jijun Liu, Min Zhong

TL;DR

The paper addresses reconstructing conductivity in acousto-electric tomography from interior power density data. It introduces a Kaczmarz-type algorithm called Two-Point-Gradient-Theta (TPG-Theta) with a general convex penalty to promote sparse and discontinuous conductivities. It establishes convergence of the iterative regularized method and demonstrates feasibility and effectiveness through extensive numerical experiments. The work offers a flexible framework for recovering sparse/discontinuous conductivities from interior measurements, with potential impact on high-resolution imaging in AET.

Abstract

We study the numerical reconstruction problem in acousto-electric tomography (AET) of recovering the conductivity distribution in a bounded domain from multiple interior power density data. The Two-Point-Gradient-$Θ$ (TPG-$Θ$) in Kaczmarz type is proposed, with a general convex penalty term $Θ$, the algorithm can be utilized in AET problem for recovering sparse and discontinuous conductivity distributions. We establish the convergence of such iterative regularized method. Extensive numerical experiments are presented to illustrate the feasibility and effectiveness of the proposed approach.

Acousto-electric tomography by the convergence of Kaczamrz two-point gradient-$Θ$ method

TL;DR

The paper addresses reconstructing conductivity in acousto-electric tomography from interior power density data. It introduces a Kaczmarz-type algorithm called Two-Point-Gradient-Theta (TPG-Theta) with a general convex penalty to promote sparse and discontinuous conductivities. It establishes convergence of the iterative regularized method and demonstrates feasibility and effectiveness through extensive numerical experiments. The work offers a flexible framework for recovering sparse/discontinuous conductivities from interior measurements, with potential impact on high-resolution imaging in AET.

Abstract

We study the numerical reconstruction problem in acousto-electric tomography (AET) of recovering the conductivity distribution in a bounded domain from multiple interior power density data. The Two-Point-Gradient- (TPG-) in Kaczmarz type is proposed, with a general convex penalty term , the algorithm can be utilized in AET problem for recovering sparse and discontinuous conductivity distributions. We establish the convergence of such iterative regularized method. Extensive numerical experiments are presented to illustrate the feasibility and effectiveness of the proposed approach.
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem