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Deep D-bar: Real time Electrical Impedance Tomography Imaging with Deep Neural Networks

Sarah Jane Hamilton, Andreas Hauptmann

arXiv:1711.03180v2math.NAcs.NEmath.AP

TL;DR

EIT is a nonlinear, ill-posed inverse problem, and D-bar methods stabilize direct reconstruction but blur sharp features through low-pass filtering. The paper adds a U-Net CNN trained only on simulated data to deblur D-bar reconstructions, producing sharper images that transfer to experimental data without experimental training. The approach also yielded lower simulated relative ℓ2-error and is reported as real-time capable.

  • Problem

    EIT image recovery is a highly nonlinear ill-posed inverse problem, while D-bar low-pass filtering produces reconstructions lacking sharp features such as clear organ boundaries.

  • Method

    The paper combines D-bar reconstruction with a U-Net CNN trained on simulated data and applies the trained network directly to experimental data without experimental training.

  • Results

    The CNN produced visual and quantitative improvements, reducing relative ℓ2-error from 28.05% to 9.92% for ACT4 simulations and from 16.82% to 9.12% for KIT4 simulations.

  • Takeaways & Limitations

    Deep D-bar can learn deblurring from simulated data and transition to experimental absolute EIT images without experimental training or transfer adaptation.

Abstract

from arXiv · show

The mathematical problem for Electrical Impedance Tomography (EIT) is a highly nonlinear ill-posed inverse problem requiring carefully designed reconstruction procedures to ensure reliable image generation. D-bar methods are based on a rigorous mathematical analysis and provide robust direct reconstructions by using a low-pass filtering of the associated nonlinear Fourier data. Similarly to low-pass filtering of linear Fourier data, only using low frequencies in the image recovery process results in blurred images lacking sharp features such as clear organ boundaries. Convolutional Neural Networks provide a powerful framework for post-processing such convolved direct reconstructions. In this study, we demonstrate that these CNN techniques lead to sharp and reliable reconstructions even for the highly nonlinear inverse problem of EIT. The network is trained on data sets of simulated examples and then applied to experimental data without the need to perform an additional transfer training. Results for absolute EIT images are presented using experimental EIT data from the ACT4 and KIT4 EIT systems.

I. INTRODUCTION

EIT reconstructs interior conductivity from boundary measurements but is a severely ill-posed nonlinear inverse problem. The paper combines robust, fast D-bar reconstruction with CNN post-processing to recover sharper absolute images and applies a simulation-trained network directly to experimental data.

  • The D-bar method stabilizes direct reconstruction against measurement noise through low-pass filtering of nonlinear Fourier data, but this removes sharp image features.
  • CNN post-processing is proposed to learn and undo D-bar image blurring while retaining a real-time-capable reconstruction pipeline.
  • The paper presents a real-time reconstruction algorithm for sharp, robust absolute EIT images by combining the D-bar algorithm with a CNN.
  • The method uses a U-Net adapted to D-bar EIT image structures, trained on simulated data and applied to experimental data without training on experimental pairs.
  • EIT determines interior conductivity from boundary current-to-voltage measurements, but image recovery is a severely ill-posed nonlinear inverse problem requiring noise-robust regularization.

A. Real-time reconstructions using an approximate D-bar method

The approximate D-bar method transforms EIT data into nonlinear scattering data, solves a D-bar equation, and directly recovers conductivity. Its low-frequency truncation supports fast, noise-robust reconstruction, while the Deep D-bar network receives the resulting 64 × 64 image for sharpening.

  • The method uses scattering data as nonlinear Fourier data and applies it to a ∂k, or D-bar, equation whose solution yields the conductivity.
  • The D-bar approach transforms the physical conductivity equation into a nonphysical Schrödinger equation and then transforms the solution back to recover conductivity.
  • The approximate Born scattering data is used because it supports fast, noise-robust D-bar reconstruction suitable for real-time imaging.
  • Deep D-bar takes a 64 × 64 D-bar reconstruction as input and produces a sharpened output through multilevel convolutional processing with ReLU nonlinearities.
  • The scattering data is low-pass filtered by retaining values for 0 < |k| ≤ R and setting them to zero for |k| > R.

B. Robustness of D-bar Methods for EIT

Prior studies report that D-bar reconstructions are robust to incorrect electrode locations and boundary shape in two-dimensional EIT. This robustness applies to both absolute and time-difference imaging, although anisotropic conductivity is not uniquely recoverable.

  • D-bar reconstruction methods for 2D EIT are reported to be robust to incorrect electrode locations and boundary-shape information.
  • The reported robustness applies to both absolute and time-difference imaging, with both behaving similarly under incorrect boundary shape and electrode locations.
  • Incorrect domain modeling can produce data corresponding to anisotropic conductivity even when the true conductivity is isotropic.
  • Anisotropic conductivity cannot be recovered uniquely, but a unique isotropic representative can be recovered through the determinant of the anisotropic conductivity tensor.

III. DEEP D-BAR

Deep D-bar combines a D-bar reconstruction with a CNN to produce sharp absolute EIT images suitable for real-time reconstruction. The network uses a modified U-Net, is trained on simulated data, and is applied to experimental data without transfer training.

  • The method combines the D-bar algorithm with a CNN to reconstruct sharp and robust absolute EIT images in real time.
  • The CNN uses a modified U-Net architecture whose multilevel structure addresses nonlinear reconstructions and sharpening across large image areas.Pooling layers support translational invariance, while 5 × 5 filters were used instead of 3 × 3 filters.
  • The network outputs a single sharpened version of the D-bar input rather than learning a residual update.
  • D-bar reconstructions and ground-truth conductivities are represented as 64 × 64 square images on [−1, 1]2 for network processing.
  • The network is trained by minimizing the ℓ2-error between its output and the phantom conductivity.
  • Training takes 4 hours on a single Titan XP GPU, and the trained network is applied to experimental data without transfer training.Optimization uses Adam for 1,000 epochs in batches of 16 with an initial learning rate of 10−4.

IV. EXPERIMENTAL SETUP AND COMPUTATIONAL NOTES

The study evaluates Deep D-bar on experimental ACT4 and KIT4 EIT systems using chest-like and conductive/resistive phantoms. The reconstruction setup accounts for discrete electrode measurements through basis changes and matrix approximations.

  • Experiments use ACT4 data from RPI and KIT4 data from UEF to evaluate the reconstruction method on two EIT machines.
  • ACT4 chest phantoms contain agar/graphite targets modeling the heart, lungs, aorta, and spine, with three right-lung injury configurations.The injuries simulate a pleural effusion, a pneumothorax-like low-conductivity region, and a metal-tube region.
  • KIT4 experiments place conductive metal and resistive plastic targets in a circular tank with 16 electrodes and apply adjacent 2mA current patterns at 1kHz.The data is included for illustrative purposes and potential industrial applications because it may not satisfy human-imaging safety standards.
  • The simulated boundary model uses a continuum electrode model to project continuum traces onto electrode-associated boundary subsets.
  • The ND map is approximated in an orthonormal boundary basis, with ACT4 using L = 32 electrodes and KIT4 using L = 16.The DN matrix Lσ is formed by inverting the ND matrix Rσ and applying radius and boundary-conductivity scaling when needed.

B. Simulation of Training Data

Training data are generated entirely by simulation, with separate procedures for ACT4 chest phantoms and KIT4 circular inclusions. Simulated measurements are reconstructed using D-bar before CNN training.

  • Separate simulated training datasets are created for the ACT4 and KIT4 experiments.
  • ACT4 training data: ACT4 simulations derive approximate organ boundaries from the healthy phantom and randomly include targets with specified inclusion probabilities.White Gaussian noise is added to approximate boundary points before constructing conductivity phantoms.
  • ACT4 training data: The ACT4 forward problem uses a 65,536-element FEM mesh, adds relative voltage noise with variance 10−4, and reconstructs the data with a low-pass D-bar method.
  • KIT4 training data: KIT4 simulations contain one to three non-overlapping circular inclusions with random radii, centers, angles, conductivities, and conductive or resistive labels.
  • KIT4 training data: KIT4 experimental targets are infinite conductors or resistors that violate D-bar’s theoretical conductivity bounds, but the method still provides useful conductor/resistor information.The scattering-data cutoff is reduced from 24 to 8 to reduce noise-related oscillatory artefacts at higher frequencies.

C. Computational Notes for D-bar Reconstructions from Experimental EIT Data

Experimental D-bar reconstructions adapt discrete ACT4 and KIT4 measurements into the matrix and scattering-data representations required by the reconstruction pipeline. Changes of basis synthesize normalized trigonometric current patterns from the systems’ physical measurements.

  • Experimental D-bar reconstructions follow the simulated-data procedure, except that DN matrices are formed from discrete rather than continuous measurements.
  • For both systems, a change of basis synthesizes measurements corresponding to orthonormal trigonometric current patterns.This adapts the physical measurement patterns to the reconstruction formulation.
  • The normalized current patterns are defined on the electrodes, and their orthonormality enables the solution method used for the DN representation.
  • The experimental ND matrix Rσ is formed from discrete inner products of voltage responses, with electrode areas |eℓ| entering the formula.The formulation applies to L electrodes with L − 1 linearly independent current patterns.
  • The experimental DN matrices for ACT4 and KIT4 use L = 32 and L = 16 electrodes, respectively, with the reference matrix L1 computed at σ = 1 on FEM meshes.The ACT4 and KIT4 meshes contain 4,149 and 3,493 triangular elements, respectively.
  • Experimental scattering data are evaluated using a Simpson’s-rule approximation before solving the D-bar equation with Fast Fourier Transforms.

V. RESULTS

The study evaluates Deep D-bar on simulated and experimental data for absolute EIT imaging.

  • Deep D-bar is evaluated on simulated data.
  • Deep D-bar is also evaluated on experimental data.
  • The evaluation concerns absolute EIT imaging.

A. Reconstructions from Simulated Data

Simulated ACT4 and KIT4 experiments compare low-pass D-bar reconstructions with CNN-processed Deep D-bar outputs, including cases outside the ACT4 training distribution. Deep D-bar improves reported reconstruction quality and is designed for efficient application to experimental data.

  • Simulated ACT4 reconstructions: ACT4 simulations include one held-out training-consistent case and two pathologies not supported by the training data.The unsupported cases contain three horizontal divisions or a vertical division in the left lung.
  • Reconstruction comparison: The CNN receives full-square low-pass D-bar images and produces Deep D-bar images displayed on the tank’s circular geometry.The circular display is for presentation only, and each row uses its own scale.
  • Quantitative evaluation: 28.05% to 9.92% relative ℓ2-error was reported for ACT4 simulations, while KIT4 test data improved from 16.82% to 9.12%.These mean relative errors were computed on test sets drawn from the training distribution.
  • Quantitative evaluation: Significant SSIM increases were observed for Deep D-bar versus Low-pass D-bar across the reported simulated and experimental comparisons.SSIM measurements are shown for ACT4 and KIT4 simulated reconstructions, with additional experimental ACT4 measurements.
  • Simulated KIT4 reconstructions: KIT4 simulated test phantoms are all drawn from the same distribution as the training data.
  • Experimental transfer: The network was trained only on simulated data and applied to experimental data without experimental truth-reconstruction pairs or transfer training.Application required adaptation only to the number of electrodes in the experimental system.
  • Discussion and limitations: The study used simplified modeling, including a continuum electrode model, untuned FEM solvers, and an unoptimized D-bar solver.These choices were used to examine tolerance to noise and modeling errors at multiple reconstruction stages.
  • Computational performance: GPU CNN evaluation averaged 7.65ms per sample, supporting the expectation that Deep D-bar can operate in real time.The authors propose combining D-bar reconstruction and CNN application in a unified framework to reduce data-transmission overhead.

A. Generalization

Deep D-bar generalizes across experimental conditions, but shape complexity and input consistency remain important boundaries. Variable cutoff-radius training improves localization and recovered conductivity while preserving SSIM consistency.

  • A. Generalization: Deep D-bar localized KIT4 triangular inclusions, but did not recover their sharp angular boundaries when trained only on circular inclusions.The authors suggest including substantial triangular-inclusion training could improve this limitation.
  • A. Generalization: More complex simulated injuries may further improve ACT4 reconstructions, whose training examples used elementary horizontal dividing lines rather than true diagonal cuts or incomplete regional replacements.For human targets, the authors propose expanding training data using anatomical atlases or CT/MR collections with and without abnormalities.
  • A. Generalization: Training with scattering data from varying cutoff radii improved injury localization and recovered conductivity while SSIMs remained consistent.The ACT4 network used cutoff radii randomized over R ∈ [4, 5].
  • A. Generalization: Alternative reconstruction methods could provide the CNN input because the study selected D-bar for its robustness and the convolved nature of its reconstructions.

VII. CONCLUSIONS

The conclusions pair reliable but blurred D-bar reconstructions with CNN deblurring to produce sharper absolute EIT images. The approach transitions from simulated training to experimental ACT4 and KIT4 data with minimal added processing time, while 3D extension remains future work.

  • VII. CONCLUSIONS: Deep D-bar couples reliable low-pass D-bar reconstructions with a CNN that learns deblurring from simulated data and applies it to experimental data.
  • VII. CONCLUSIONS: KIT4 results compare low-pass D-bar input images with Deep D-bar outputs across phantoms containing conductive and/or resistive targets.Images are displayed on circular tank geometry for presentation, while each row uses its own scale.
  • VII. CONCLUSIONS: ACT4 comparisons include the original fixed-radius CNN and a newer variable-radius CNN evaluated with R = 4 and R = 4.5 inputs.The figure caption states that SSIMs remained consistent.
  • VII. CONCLUSIONS: The demonstrated work is two-dimensional, and extension to three dimensions is expected only after further development of the D-bar computational framework.
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