Imaging conductivity from current density magnitude using neural networks*

Abstract

Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by formulating the problem as a relaxed weighted least-gradient problem, and then approximating its minimizer by standard fully connected feedforward neural networks. We derive bounds on two components of the generalization error, i.e., approximation error and statistical error, explicitly in terms of properties of the neural networks (e.g., depth, total number of parameters, and the bound of the network parameters). We illustrate the performance and distinct features of the approach on several numerical experiments. Numerically, it is observed that the approach enjoys remarkable robustness with respect to the presence of data noise.

Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks

“…nonconvexity of the loss landscape, the optimizer may fail to find a global minimizer of the empirical loss \widehat J \bfitgam but instead only an approximate local minimizer. This phenomenon has been observed across a broad range of neural solvers based on DNNs [37,16,25]. Table 3(b) and (c) shows that the L 2 (\Omega ) error e(\q) of the reconstruction \q does not vary much with different DNN architectures and numbers of sampling points.…”

“…In recent years, the use of DNNs for solving direct and inverse problems for PDEs has received a lot of attention; see [15,43] for overviews. Existing neural inverse schemes using DNNs roughly fall into two groups: supervised approaches (see, e.g., [40,28,22]) and unsupervised approaches (see, e.g., [7,8,34,49,25,35]). Supervised methods exploit the availability of (abundant) paired training data to extract problem-specific features and are concerned with learning approximate inverse operators.…”

“…The works [7] and [8] investigated EIT reconstruction using the weak and strong formulations (also with the L \infty norm consistency for the latter), respectively; see also [35] for further developments of the approach [8] using energy-based models. The work [25] applied the deep Ritz method to a least-gradient reformulation for current density impedance imaging and derived a generalization error for the loss function. The approach performs reasonably well for both full and partial interior current density data and shows remarkable robustness against data noise.…”

“…Note that in (3.2), we have imposed a mild regularity condition on the noisy data z \delta , which may be obtained by presmoothing the raw noisy data beforehand, e.g., via filtering [33] or denoising via DNNs [44,25]. This assumption is used in, e.g., Kohn--Lowe [30] and equation error [3] formulations.…”

Divide-and-conquer DNN approach for the inverse point source problem using a few single frequency measurements

“…Recently, DNNs became very popular for inverse problems and attracted many researchers (see, e.g. [1,6,8,12,13,17,19,24,25] and the references therein). These methods usually build a single DNN.…”