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Total variation regularization with reduced basis in electrical impedance tomography

A. Hannukainen, N. Hyvönen, V. Toresen

TL;DR

This work addresses fast, edge-preserving reconstruction in electrical impedance tomography by coupling a smoothened total variation prior with reduced-basis methods. An offline POD-based reduced basis for the interior potentials is used in conjunction with a projected lagged diffusivity algorithm, enabling 3D reconstructions with online times of a few seconds on a standard laptop. The approach integrates a projection that eliminates the influence of contact resistances, a sequential linearization framework, and LSQR with priorconditioning to solve TV-regularized subproblems efficiently. Numerical experiments on simulated and experimental data demonstrate that the reduced-basis implementation yields reconstructions of comparable quality to the full, non-reduced method while providing substantial speedups, thereby moving towards real-time EIT on commodity hardware.

Abstract

This work considers using reduced basis techniques in connection to (smoothened) total variation regularization in electrical impedance tomography, but analogous ideas can also be used for other inverse elliptic boundary value problems. It is demonstrated that resorting to reduced bases can speed up a reconstruction algorithm based on combining the lagged diffusivity algorithm with sequential linearizations and preconditioned LSQR iteration without any significant loss of reconstruction quality or of the edge-enhancing nature of total variation regularization. The ideas are numerically tested in three dimensions on unstructured finite element meshes with both simulated and experimental data, resulting in online reconstruction times of only a few seconds on a standard laptop computer.

Total variation regularization with reduced basis in electrical impedance tomography

TL;DR

This work addresses fast, edge-preserving reconstruction in electrical impedance tomography by coupling a smoothened total variation prior with reduced-basis methods. An offline POD-based reduced basis for the interior potentials is used in conjunction with a projected lagged diffusivity algorithm, enabling 3D reconstructions with online times of a few seconds on a standard laptop. The approach integrates a projection that eliminates the influence of contact resistances, a sequential linearization framework, and LSQR with priorconditioning to solve TV-regularized subproblems efficiently. Numerical experiments on simulated and experimental data demonstrate that the reduced-basis implementation yields reconstructions of comparable quality to the full, non-reduced method while providing substantial speedups, thereby moving towards real-time EIT on commodity hardware.

Abstract

This work considers using reduced basis techniques in connection to (smoothened) total variation regularization in electrical impedance tomography, but analogous ideas can also be used for other inverse elliptic boundary value problems. It is demonstrated that resorting to reduced bases can speed up a reconstruction algorithm based on combining the lagged diffusivity algorithm with sequential linearizations and preconditioned LSQR iteration without any significant loss of reconstruction quality or of the edge-enhancing nature of total variation regularization. The ideas are numerically tested in three dimensions on unstructured finite element meshes with both simulated and experimental data, resulting in online reconstruction times of only a few seconds on a standard laptop computer.
Paper Structure (24 sections, 56 equations, 6 figures)

This paper contains 24 sections, 56 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Complete electrode model
  3. Reduced basis
  4. Review of subspace surrogate methods
  5. Basis generation
  6. Proper orthogonal decomposition
  7. Sampling
  8. Bayesian framework and prior
  9. Projected lagged diffusivity algorithm
  10. Sequential Linearization
  11. Projected equations
  12. MAP estimate
  13. Lagged diffusivity
  14. Priorconditioning
  15. Implementation
  16. ...and 9 more sections

Figures (6)

  • Figure 1: The measurement configurations and the target conductivities for Case 1 (left) and Case 2 (right).
  • Figure 2: Case 1: The target is a cylinder of unit height and radius with two embedded cylindrical inclusions; see the left image of Figure \ref{['fig:cases']}. The reconstruction and random draw cross sections are at heights $0.9$, $0.6$, $0.4$, and $0.1$.
  • Figure 3: Case 2: The target is a ball of radius 5 with two embedded inclusions; see the right image of Figure \ref{['fig:cases']}. The reconstruction and random draw cross sections are at heights $3$, $1$, $-2.5$, and $-4$.
  • Figure 4: Case 3: Thorax-shaped water tank with two embedded cylindrical inclusions. The reconstruction and random draw cross sections are at height $2.5\,$cm and the unit of conductivity is mS/cm.
  • Figure 5: Case 4: Left: The $L = 24$ current patterns used for the cylindrical tank in Figure \ref{['fig:case4Afalse_and_true']}. The vertical axis corresponds to electrode numbers and the horizontal axis to different current patterns. Red color indicates the source and blue color the sink of an 1 mA current injection. Right: Random draw from \ref{['eq:sample_sigma']}.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Remark 4.1
  • Remark 6.1
  • Remark 6.2