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Reconstruction in the Calderón problem on a fixed partition from finite and partial boundary data

Henrik Garde

TL;DR

The paper tackles the Calderón problem for piecewise constant conductivities on a fixed partition using partial boundary data. It reformulates the reconstruction as a monotonicity-based, layer-by-layer procedure with local Neumann-to-Dirichlet maps and reduces it to a finite-data setting via projections onto a finite basis, enabling 1D optimizations to recover each piece. A key result is that finite boundary measurements suffice to approximate the infinite-data tests to any prescribed accuracy ε, by choosing a sufficiently large basis size M_m. The approach avoids a priori bounds on conductivity and supports ROI-focused reconstructions, contrasting with semidefinite optimization methods that require additional coefficient information.

Abstract

This short note modifies a reconstruction method by the author (Comm. PDE, 45(9):1118-1133, 2020), for reconstructing piecewise constant conductivities in the Calderón problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.

Reconstruction in the Calderón problem on a fixed partition from finite and partial boundary data

TL;DR

The paper tackles the Calderón problem for piecewise constant conductivities on a fixed partition using partial boundary data. It reformulates the reconstruction as a monotonicity-based, layer-by-layer procedure with local Neumann-to-Dirichlet maps and reduces it to a finite-data setting via projections onto a finite basis, enabling 1D optimizations to recover each piece. A key result is that finite boundary measurements suffice to approximate the infinite-data tests to any prescribed accuracy ε, by choosing a sufficiently large basis size M_m. The approach avoids a priori bounds on conductivity and supports ROI-focused reconstructions, contrasting with semidefinite optimization methods that require additional coefficient information.

Abstract

This short note modifies a reconstruction method by the author (Comm. PDE, 45(9):1118-1133, 2020), for reconstructing piecewise constant conductivities in the Calderón problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.

Paper Structure

This paper contains 4 sections, 1 theorem, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The setting
  3. The reconstruction method for infinite measurements
  4. The reconstruction method for finite measurements

Key Result

Theorem 4.1

For any $\epsilon>0$, there exists $M_m\in\mathbb{N}$ such that for all $M\geq M_m$, in the setting of Section sec:infinite,

Figures (1)

  • Figure 2.1: Examples of orderings of the $P_j$ sets. Left: Reconstructing the outermost pixels first, then proceeding to the next "layer" of pixels. Right: Example of a ROI (blue dashed outline), where the reconstruction is adapted to this.

Theorems & Definitions (4)

  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • proof