A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography (1312.5523v2)

Abstract: In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known \emph{a priori} that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called {\em CGO sinogram}. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.

• The paper integrates a D-bar approach with an Ambrosio-Tortorelli edge-preserving algorithm to enhance sharp conductivity contrasts in noisy EIT data.
• It demonstrates that incorporating a CGO sinogram with regularized inversion significantly improves spatial resolution and stability in reconstructions.
• The method achieves promising results in simulated tests including heart-and-lungs phantoms and industrial pipe examples, indicating broad application potential.

A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography

The development of robust numerical methods for solving inverse problems in Electrical Impedance Tomography (EIT) is crucial given its relevance in various fields, including medical imaging and industrial monitoring. The paper "A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography" introduces an innovative D-bar method that integrates edge-preserving algorithms and regularized inversion techniques to improve the spatial resolution of EIT reconstructions while effectively managing measurement noise.

Technical Overview

EIT involves recovering an object's internal conductivity distribution by analyzing current and voltage measurements captured at its surface. This task is markedly ill-posed and necessitates regularization to combat noise sensitivity. The D-bar method, rooted in complex geometrical optics (CGO) theory, is essential here due to its established regularization properties. However, the low-pass filtering inherent in regularization leads to blurred images, diluting high-frequency details crucial in applications like medical imaging, where organ boundaries must be discerned.

The proposed technique addresses this challenge by coupling the D-bar approach with an Ambrosio-Tortorelli (AT) functional minimization, which iteratively sharpens image features based on edge-preserving diffusion. An additional innovation is the incorporation of a "CGO sinogram," which extracts fundamental geometric information from the object, offering a stable measure for comparison throughout the reconstruction procedure.

Numerical Results and Observations

The paper showcases the efficacy of this method through simulated EIT data, particularly focusing on two test cases: a heart-and-lungs phantom and an industrial pipe example. The results demonstrate the method's capacity to recover sharp contrasts and precise boundaries, even amidst significant measurement noise.

A pivotal aspect of this approach is its reliance on minimizing discrepancies in the CGO sinogram rather than the classical Dirichlet-to-Neumann map. The CGO sinogram, representing traces of CGO solutions, is computationally stable and enriched with robust geometrical insights into the conductivity distribution, aiding in more informed and accurate image corrections.

Implications and Future Directions

This edge-preserving methodology signifies a noteworthy contribution to EIT, notably enhancing image clarity and providing theoretically grounded noise resilience. The integration of image processing algorithms in the inverse problem context uniquely bridges computational inverse problems and digital image processing.

Future research can extend these findings by exploring alternative contrast enhancement strategies to refine image properties without compromising numerical accuracy. Additionally, further theoretical investigation into the stability and robustness of the CGO sinogram will be valuable, potentially steering advancements in partial EIT data applications. Adaptation of this method to three-dimensional scenarios offers exciting potential, especially in complex medical diagnostics and geophysical explorations.

Overall, this paper marks a significant step forward in refining EIT techniques, aiming to harness the full dimensional bandwidth of nonlinear Fourier imaging to vastly benefit practical applications. The method’s innovative merger of abstract mathematical theory and applied computational techniques could inspire fresh approaches in tackling inverse conductivity problems across various domains.