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Linearization-based direct reconstruction for EIT using triangular Zernike decompositions

Antti Autio, Henrik Garde, Markus Hirvensalo, Nuutti Hyvönen

TL;DR

The paper addresses rapid, direct reconstruction in 2D EIT by exploiting the Fréchet derivative $F = D\Lambda(1)$ of the forward map and expanding perturbations in Zernike polynomials, which decouples the problem into independent triangular subsystems by angular frequency $|j|$. Two regularization strategies are developed: (i) a truncated-SVD approach on the block-diagonalized system and (ii) a forward-substitution-based truncation of the triangular subsystems, both guided by Morozov discrepancy. Numerical experiments demonstrate that the SVD-based method yields robust reconstructions for CM and CEM data on disks and polygons and can even handle real water-tank measurements, while the triangular-truncation method can struggle without careful parameter tuning. The results highlight the practical viability of fast, angular-frequency-aware direct reconstruction for EIT, with stability enhanced by exploiting the explicit triangular structure and conformal-domain extensions. Overall, the work shows that linearization-based direct reconstruction can deliver accurate qualitative and quantitative results in realistic EIT scenarios, despite modeling and discretization errors.

Abstract

This work implements and numerically tests the direct reconstruction algorithm introduced in [Garde & Hyvönen, SIAM J. Math. Anal., 2024] for two-dimensional linearized electrical impedance tomography. Although the algorithm was originally designed for a linearized setting, we numerically demonstrate its functionality when the input data is the corresponding change in the current-to-voltage boundary operator. Both idealized continuum model and practical complete electrode model measurements are considered in the numerical studies, with the examined domain being either the unit disk or a convex polygon. Special attention is paid to regularizing the algorithm and its connections to the singular value decomposition of a truncated linearized forward map, as well as to the explicit triangular structures originating from the properties of the employed Zernike polynomial basis for the conductivity.

Linearization-based direct reconstruction for EIT using triangular Zernike decompositions

TL;DR

The paper addresses rapid, direct reconstruction in 2D EIT by exploiting the Fréchet derivative of the forward map and expanding perturbations in Zernike polynomials, which decouples the problem into independent triangular subsystems by angular frequency . Two regularization strategies are developed: (i) a truncated-SVD approach on the block-diagonalized system and (ii) a forward-substitution-based truncation of the triangular subsystems, both guided by Morozov discrepancy. Numerical experiments demonstrate that the SVD-based method yields robust reconstructions for CM and CEM data on disks and polygons and can even handle real water-tank measurements, while the triangular-truncation method can struggle without careful parameter tuning. The results highlight the practical viability of fast, angular-frequency-aware direct reconstruction for EIT, with stability enhanced by exploiting the explicit triangular structure and conformal-domain extensions. Overall, the work shows that linearization-based direct reconstruction can deliver accurate qualitative and quantitative results in realistic EIT scenarios, despite modeling and discretization errors.

Abstract

This work implements and numerically tests the direct reconstruction algorithm introduced in [Garde & Hyvönen, SIAM J. Math. Anal., 2024] for two-dimensional linearized electrical impedance tomography. Although the algorithm was originally designed for a linearized setting, we numerically demonstrate its functionality when the input data is the corresponding change in the current-to-voltage boundary operator. Both idealized continuum model and practical complete electrode model measurements are considered in the numerical studies, with the examined domain being either the unit disk or a convex polygon. Special attention is paid to regularizing the algorithm and its connections to the singular value decomposition of a truncated linearized forward map, as well as to the explicit triangular structures originating from the properties of the employed Zernike polynomial basis for the conductivity.
Paper Structure (20 sections, 44 equations, 9 figures, 4 tables)

This paper contains 20 sections, 44 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Article structure
  3. Linearization of the CM of EIT
  4. Explicit representation for $F$ and its inverse
  5. Zernike polynomials and Fourier basis
  6. Matrix representation of the linearized model
  7. Inversion of the infinite triangular systems
  8. Regularization for truncated systems
  9. Truncated SVD
  10. Truncation of the triangular subsystems
  11. Simulating data on piecewise smooth domains and with the CEM
  12. Other planar domains with CM
  13. CEM measurements
  14. Numerical experiments
  15. Visualization of right singular vectors of $F^M$
  16. ...and 5 more sections

Figures (9)

  • Figure 4.1: The structure of $F^M$, for $M=10$, as a block diagonal matrix with lower triangular blocks.
  • Figure 6.1: The radial components of certain right singular functions and the first hundred singular values of $F^M$ for $M=32$.
  • Figure 6.2: Reconstructions of a discoidal inclusion inside the unit disk from highly accurate data generated using a Möbius transformation with $M=32$.
  • Figure 6.3: The accepted Zernike indices for the reconstructions in Fig .
  • Figure 6.4: Reconstructions of a wave-like conductivity perturbation in the unit disk with $M=16$.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Remark 3.1
  • Remark 5.1