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Eigenstructure of the linearized electrical impedance tomography problem under radial perturbations

Markus Hirvensalo

TL;DR

This work characterizes the linearization of electrical impedance tomography near a homogeneous conductivity on the unit ball by analyzing the Fréchet derivative $F$ of the Neumann-to-Dirichlet map. For rotationally symmetric perturbations, $F\eta$ is diagonal in the spherical-harmonic basis with explicitly computable eigenvalues $\lambda_\ell(\eta)$, and these eigenvalues decay like $\ell^{-1/2}$ uniformly in the perturbation norm. The authors show that $F$ restricted to radial perturbations is compact and can be well approximated by finite-rank operators, and they extend these results to $L^2(B)$ perturbations. The work leverages a separation of variables approach using Jacobi polynomials for the radial part and spherical harmonics for the angular part, providing a robust framework for numerical linearization in EIT.

Abstract

We analyze the Fréchet derivative $F$, that maps a perturbation in conductivity to the linearized change in boundary measurements governed by the conductivity equation. The domain is taken to be the unit ball $B \subset \mathbb{R}^d$ with $d \geq 2$, and we choose perturbations $η$ from the Hilbert space $L^2(B)$. Under the condition that the perturbations are rotationally symmetric, we show that the eigenfunctions of the linear approximation $F η$ correspond to the spherical harmonics. Furthermore, we establish an explicit formula for the associated eigenvalues and show that for perturbations from any bounded subset, the decay of these eigenvalues is uniform with respect to the degree of the spherical harmonics. The established structure of $F η$ enables us to show that the Fréchet derivative $F$ can be approximated by finite-rank operators when restricted to rotationally symmetric perturbations. Both the extension to $L^2(B)$ perturbations and the approximability by finite-rank operators are favorable properties for further analysis of $F$ in numerical algorithms.

Eigenstructure of the linearized electrical impedance tomography problem under radial perturbations

TL;DR

This work characterizes the linearization of electrical impedance tomography near a homogeneous conductivity on the unit ball by analyzing the Fréchet derivative of the Neumann-to-Dirichlet map. For rotationally symmetric perturbations, is diagonal in the spherical-harmonic basis with explicitly computable eigenvalues , and these eigenvalues decay like uniformly in the perturbation norm. The authors show that restricted to radial perturbations is compact and can be well approximated by finite-rank operators, and they extend these results to perturbations. The work leverages a separation of variables approach using Jacobi polynomials for the radial part and spherical harmonics for the angular part, providing a robust framework for numerical linearization in EIT.

Abstract

We analyze the Fréchet derivative , that maps a perturbation in conductivity to the linearized change in boundary measurements governed by the conductivity equation. The domain is taken to be the unit ball with , and we choose perturbations from the Hilbert space . Under the condition that the perturbations are rotationally symmetric, we show that the eigenfunctions of the linear approximation correspond to the spherical harmonics. Furthermore, we establish an explicit formula for the associated eigenvalues and show that for perturbations from any bounded subset, the decay of these eigenvalues is uniform with respect to the degree of the spherical harmonics. The established structure of enables us to show that the Fréchet derivative can be approximated by finite-rank operators when restricted to rotationally symmetric perturbations. Both the extension to perturbations and the approximability by finite-rank operators are favorable properties for further analysis of in numerical algorithms.

Paper Structure

This paper contains 6 sections, 4 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. On the basis elements
  4. Spherical harmonics
  5. Jacobi polynomials
  6. Proof of Theorem \ref{['main_thm']}

Key Result

Theorem 2.1

Let $\eta \in \mathcal{R}$ be expanded as in eq:radial_expa and let $f_{\ell,m}$ denote the spherical harmonics on the sphere ${\partial B}$ with degree $\ell \geq 1$ and order $m$. The operator $F \eta : L^2_\diamond({\partial B}) \to L^2_\diamond({\partial B})$ expanded in the spherical harmonic b where The eigenvalues $\lambda_\ell(\eta)$ are independent of the index $m$ and are bounded from a

Theorems & Definitions (7)

  • Theorem 2.1
  • Corollary 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof