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Register a new userPositive-energy D-bar method for acoustic tomography: a computational study
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A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -- in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between $10^{-5}$ and $5$, smaller values giving better results.
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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.
A method for including a priori information in the 2-D D-bar algorithm is presented. Two methods of assigning conductivity values to the prior are presented, each corresponding to a different scenario on applications. The method is tested on several numerical examples with and without noise and is demonstrated to be highly effective in improving the spatial resolution of the D-bar method.
Objective: To develop, and demonstrate the feasibility of, a novel image reconstruction method for absolute Electrical Impedance Tomography (a-EIT) that pairs deep learning techniques with real-time robust D-bar methods. Approach: A D-bar method is paired with a trained Convolutional Neural Network (CNN) as a post-processing step. Training data is simulated for the network using no knowledge of the boundary shape by using an associated nonphysical Beltrami equation rather than simulating the traditional current and voltage data specific to a given domain. This allows the training data to be boundary shape independent. The method is tested on experimental data from two EIT systems (ACT4 and KIT4). Main Results: Post processing the D-bar images with a CNN produces significant improvements in image quality measured by Structural SIMilarity indices (SSIMs) as well as relative $ell_2$ and $ell_1$ image errors. Significance: This work demonstrates that more general networks can be trained without being specific about boundary shape, a key challenge in EIT image reconstruction. The work is promising for future studies involving databases of anatomical atlases.
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.
This study suggests a fast computational method for crack propagation, which is based on the extended finite element method (X-FEM). It is well known that the X-FEM might be the most popular numerical method for crack propagation. However, with the increase of complexity of the given problem, the size of FE model and the number of iterative steps are increased correspondingly. To improve the efficiency of X-FEM, an efficient computational method termed decomposed updating reanalysis (DUR) method is suggested. For most of X-FEM simulation procedures, the change of each iterative step is small and it will only lead a local change of stiffness matrix. Therefore, the DUR method is proposed to predict the modified response by only calculating the changed part of equilibrium equations. Compared with other fast computational methods, the distinctive characteristic of the proposed method is to update the modified stiffness matrix with a local updating strategy, which only the changed part of stiffness matrix needs to be updated. To verify the performance of the DUR method, several typical numerical examples have been analyzed and the results demonstrate that this method is a highly efficient method with high accuracy.
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