Problem Of Impedance Tomography Research Articles

An optimal transport approach for 3D electrical impedance tomography

Abstract This work solves the three-dimensional inverse boundary value problem with the quadratic Wasserstein distance (W 2), which originates from the optimal transportation (OT) theory. The computation of the W 2 distance on the manifold surface is boiled down to solving the generalized Monge–Ampère equation, whose solution is directly related to the gradient of the W 2 distance. An efficient first-order method based on iteratively solving Poisson’s equation is introduced to solve the fully nonlinear elliptic equation. Combining with the adjoint-state technique, the optimization framework based on the W 2 distance is developed to solve the three-dimensional electrical impedance tomography problem. The proposed method is especially suitable for severely ill-posed and highly nonlinear inverse problems. Numerical experiments demonstrate that our method improves the stability and outperforms the traditional regularization methods.

Oracle-Net for Nonlinear Compressed Sensing in Electrical Impedance Tomography Reconstruction Problems

Sparse recovery principles play an important role in solving many nonlinear ill-posed inverse problems. We investigate a variational framework with learned support estimation for compressed sensing sparse reconstructions, where the available measurements are nonlinear and possibly corrupted by noise. A graph neural network, named Oracle-Net, is proposed to predict the support from the nonlinear measurements and is integrated into a regularized recovery model to enforce sparsity. The derived nonsmooth optimization problem is then efficiently solved through a constrained proximal gradient method. Error bounds on the approximate solution of the proposed Oracle-based optimization are provided in the context of the ill-posed Electrical Impedance Tomography problem (EIT). Numerical solutions of the EIT nonlinear inverse reconstruction problem confirm the potential of the proposed method which improves the reconstruction quality from undersampled measurements, under sparsity assumptions.

Deep unrolling networks with recurrent momentum acceleration for nonlinear inverse problems

Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. Although DuNets have been successfully applied to many linear inverse problems, their performance tends to be impaired by nonlinear problems. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets—the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA, respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

Optimal transportation for electrical impedance tomography

This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance ( W 2 W_{2} ). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on S 1 \mathbb {S}^{1} is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to O ( N ) O(N) from O ( N 3 ) O(N^{3}) of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.

On the electric impedance tomography problem for nonorientable surfaces with internal holes

Let ( M , g ) (M,g) be a compact (in general, nonorientable) surface with boundary ∂ M \partial M and let Γ 0 \Gamma _0 , …, Γ m − 1 \Gamma _{m-1} be connected components of ∂ M \partial M . Let u = u f ( x ) u=u^{f}(x) be a solution to the problem Δ g u = 0 \Delta _{g}u=0 in M M , u | Γ 0 = f u\big |_{\Gamma _0}=f , u | Γ j = 0 u\big |_{\Gamma _j}=0 , j = 1 j=1 , …, m ′ m’ , ∂ ν u | Γ j = 0 \partial _{\nu }u\big |_{\Gamma _j}=0 , j = m ′ + 1 j=m’+1 , …, m − 1 m-1 , where ν \nu is the outward normal. With this problem, one associates the DN map Λ : f ↦ ∂ ν u f | Γ 0 \Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0} . The purpose is to determine M M from Λ \Lambda . To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra A \mathfrak {A} of functions holomorphic on the appropriate orientable double cover of M M . It is proved that A \mathfrak {A} is determined by Λ \Lambda up to isometric isomorphism. The spectrum of the algebra A \mathfrak {A} provides a relevant copy M ′ M’ of M M . This copy is conformally equivalent to M M while its DN map coincides with Λ \Lambda .

Level Set and Optimal Control for the Inverse Inclusion Reconstruction in Electrical Impedance Tomography Modeling

In this paper, we address the identification of an unknown inclusion in an elliptic equation, which arises in the context of electrical impedance tomography and multiphase problems. We present a novel approach by formulating the overdetermined system as an optimal design problem based on the unknown inclusion and two state solutions, which is derived from by introducing a least square functional. We establish the existence results and first optimality conditions, and employ the finite element method to discretize the involved variational equations. Furthermore, we update the unknown inclusion using the level-set method. To demonstrate the effectiveness and validity of our proposed algorithm, we investigate simple and complex geometries numerically. Our results showcase the efficiency and accuracy of the method, making it a promising tool for solving similar inverse problems in various engineering applications.

A modified Levenberg–Marquardt scheme for solving a class of parameter identification problems

Parameter identification problems in PDEs are special class of nonlinear inverse problems which has many applications in science and technology. One such application is the Electrical Impedance Tomography (EIT) problem. Although many methods are available in literature to tackle nonlinear problems, the computation of Fréchet derivative is often a bottle neck for deriving the solution. Moreover, many assumptions are required to establish the convergence of such methods. In this paper, we propose a modified form of Levenberg–Marquardt scheme which does not require the knowledge of exact Fréchet derivative, instead, uses an approximate form of it and at the same time, no additional assumptions are required to establish the convergence of the scheme. We illustrate the theoretical result through numerical examples. In order to ensure that the proposed scheme can be applied to practical problems, we have applied the scheme to EIT problem and the reconstruction process clearly demonstrates that the method can be successfully applied to practical problems.

Crack identification using smart paint and machine learning

Information on the presence and location of cracks in civil structures can be precious to support operators in making decisions related to structural management and scheduling informed maintenance. This paper investigates the efficacy of supervised machine learning to solve the inverse electrical impedance tomography problem and to reconstruct the conductivity distribution of a piezoresistive sensing film. This film consists of a conductive paint applied onto structural components, and operators can use its conductivity distribution to identify crack sizes and locations in the underlying structure. A deep neural network is employed to reconstruct a dense conductivity distribution within the painted area by using only voltage measurements collected at sparse boundary locations. Since one of the most challenging aspects of using supervised learning tools for real-world applications is generating a representative training dataset, this paper presents a new approach to test the suitability of synthetic datasets built using a finite element model of the sensing film. Results are reported for four sensing specimens fabricated with two different techniques (i.e., using carbon nanotubes and graphene nanosheets, respectively). Crack-like damage is induced to the substrate of the sensing film and identified using the proposed machine learning technique. Promising results are obtained as compared to conventional methods.

A path integral Monte Carlo (PIMC) method based on Feynman-Kac formula for electrical impedance tomography

A path integral Monte Carlo method (PIMC) based on a Feynman-Kac formula for the Laplace equation with mixed boundary conditions is proposed to solve the forward problem of the electrical impedance tomography (EIT). The forward problem is an important part of iterative algorithms of the inverse EIT problem, and the proposed PIMC provides a local solution to find the potentials and currents on individual electrodes. Improved techniques are proposed to compute with better accuracy both the local time of reflecting Brownian motions (RBMs) and the Feynman-Kac formula for mixed boundary problems of the Laplace equation. Accurate voltage-to-current maps on the electrodes of a model 3-D EIT problem with eight electrodes are obtained by solving a mixed boundary problem with the proposed PIMC method.

Recovery algorithm of electrical conductance distribution in the volume of the object of research in electrical impedance tomography

An efficient, rigorously substantiated algorithm for solving electrical impedance tomography problems is proposed and tested. The features of the algorithm are the use of a methodology based on solving conditionally correct inverse problems and the presence of a search stage for the optimal current configuration that provides the maximum sensitivity of the problem functionals to the desired parameters, which allows to speed up the iterative process and improve the accuracy of the problem solution. The algorithm can be used in medicine and industry to solve diagnostic problems.

A range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt–Kaczmarz method for solving systems of non-linear ill-posed equations: Application to EIT-CEM with real data

Abstract In this article we propose and analyze a Levenberg–Marquardt–Kaczmarz-type (LMK) method for obtaining stable approximate solutions to systems of ill-posed equations modeled by non-linear operators acting between Hilbert spaces. We extend to the LMK iteration the strategy proposed in [A. Leitão, F. Margotti and B. F. Svaiter, Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method, IMA J. Numer. Anal. 41 2021, 4, 2962–2989] for choosing the Lagrange multipliers in the Levenberg–Marquardt (LM) method. Our main goal is to devise a simple (and easy to implement) strategy for computing the multiplier in each iterative step, such that the resulting LMK iteration is both stable and numerically efficient. Convergence analysis for the proposed LMK type method is provided, including convergence for exact data, stability and semi-convergence. Numerical experiments using real data are presented for a 2D parameter identification problem, namely the Electrical Impedance Tomography (EIT) problem. The mathematical model known as complete electrode model (EIT-CEM) is considered. The obtained numerical results validate the efficiency of the proposed LMK-type method.

Parametric derivatives in inverse conductivity problems with total variation regularization

The focus of this paper is an improved differentiability result for the forward map in inverse problems involving elliptic partial differential equations, and examination of its significance in the context of the electrical impedance tomography (EIT) problem with total variation (TV) regularization. We base our analysis on the Fréchet derivative of the mapping which takes a given conductivity function (spatially varying) in an electrostatic model to a corresponding elliptic PDE solution, and we develop the implications of a certain compactness property of the parameter space. By following this approach, we show Fréchet differentiability with a weaker norm (the L1 norm) for the parameter space than is usually used (the L∞ norm), thus improving the Fréchet differentiability result. The EIT problem with TV regularization is well studied in the literature, and several authors have addressed the Fréchet differentiability question. However, to the best of our knowledge and as we argue, our result is the strongest analytical result in this context. Many derivative-based methods such as Gauss–Newton and Levenburg-Marquardt lie at the heart of many proposed methods for EIT, and the results described herein for these derivative calculations provide a firm theoretical footing for them.

Bridging and Improving Theoretical and Computational Electrical Impedance Tomography via Data Completion

In computational PDE-based inverse problems, a finite amount of data is collected to infer unknown parameters in the PDE. In order to obtain accurate inferences, the collected data must be informative about the unknown parameters. How to decide which data is most informative and how to efficiently sample it is the notoriously challenging task of optimal experimental design (OED). In this context, the best, and often infeasible, scenario is when the full input-to-output (ItO) map, i.e., an infinite amount of data, is available: This is the typical setting in many theoretical inverse problems, which is used to guarantee the unique parameter reconstruction. These two different settings have created a gap between computational and theoretical inverse problems, where finite and infinite amounts of data are used, respectively. In this article we aim to bridge this gap while circumventing the OED task. This is achieved by exploiting the structures of the ItO data from the underlying inverse problem, using the electrical impedance tomography (EIT) problem as an example. To accomplish our goal, we leverage the rank structure of the EIT model and formulate the ItO matrix\textemdash the discretized ItO map\textemdash as an $\mathcal{H}$-matrix whose off-diagonal blocks are low rank. This suggests that, when equipped with the matrix completion technique, one can recover the full ItO matrix, with high probability, from a subset of its entries sampled following the rank structure: The data in the diagonal blocks is informative and should be fully sampled, while data in the off-diagonal blocks can be subsampled. This recovered ItO matrix is then utilized to present the full ItO map up to a discretization error, paving the way to connect with the problem in the theoretical setting where the unique reconstruction of parameters is guaranteed. This strategy achieves two goals: (I) it bridges the gap between the finite- and infinite-dimensional settings for numerical and theoretical inverse problems and (II) it improves the quality of computational inverse solutions. We detail the theory for the EIT model and provide numerical verification to both EIT and optical tomography problems.

On the tangential cone condition for electrical impedance tomography

We state some sufficient criteria for the tangential cone conditions to hold for the electrical impedance tomography problem. The results are based on Löwner convexity of the forward operator. As a consequence, we show that for conductivities that satisfy various properties, such as Hölder source conditions, finite-dimensionality, or certain monotonicity criteria, the tangential cone condition is verified.

Using Feedback Strategies in Simulated Annealing with Crystallization Heuristic and Applications

This paper represents how typical advanced engineering design can be structured using a set of parameters and objective functions corresponding to the nature of the problem. The set of parameters can be in different types, including integer, real, cyclic, combinatorial, interval, etc. Similarly, the objective function can be presented in various types including integer (discrete), float, and interval. The simulated annealing with crystallization heuristic can deal with all these combinations of parameters and objective functions when the crystallization heuristic presents a sensibility for real parameters. Herein, simulated annealing with the crystallization heuristic is enhanced by combining Bates and Gaussian distributions and by incorporating feedback strategies to emphasize exploration or refinement, or a combination of the two. The problems that are studied include solving an electrical impedance tomography problem with float parameters and a partially evaluated objective function represented by an interval requiring the solution of 32 sparse linear systems defined by the finite element method, as well as an airplane design problem with several parameters and constraints used to reduce the explored domain. The combination of the proposed feedback strategies and simulated annealing with the crystallization heuristic is compared with existing simulated annealing algorithms and their benchmark results are shown. The enhanced simulated annealing approach proposed herein showed better results for the majority of the studied cases.

Electric impedance tomography problem for surfaces with internal holes * *Supported by RFBR Grant 20-01 627A.

Let (M, g) be a smooth compact Riemann surface with the multicomponent boundary . Let u = u f obey Δu = 0 in M, (the grounded holes) and v = v h obey Δv = 0 in M, (the isolated holes). Let and be the corresponding Dirichlet-to-Neumann map. The electric impedance tomography problem is to determine M from or . To solve it, an algebraic variant of the boundary control method is applied. The central role is played by the algebra of functions holomorphic on the manifold obtained by gluing two examples of M along . We show that is determined by (or ) up to isometric isomorphism. A relevant copy (M′, g′, Γ0′) of (M, g, Γ0) is constructed from the Gelfand spectrum of . By construction, this copy turns out to be conformally equivalent to (M, g, Γ0), obeys and provides a solution of the problem.

A Numerical Method for Determining the Inhomogeneity Boundary in the Electrical Impedance Tomography Problem in the Case of Piecewise-Constant Conductivity

A Numerical Method for Determining the Inhomogeneity Boundary in the Electrical Impedance Tomography Problem in the Case of Piecewise-Constant Conductivity

Sparsity-based nonlinear reconstruction of optical parameters in two-photon photoacoustic computed tomography

We present a new nonlinear optimization approach for the sparse reconstruction of single-photon absorption and two-photon absorption coefficients in the photoacoustic computed tomography (PACT). This framework comprises of minimizing an objective functional involving a least squares fit of the interior pressure field data corresponding to two boundary source functions, where the absorption coefficients and the photon density are related through a semi-linear elliptic partial differential equation (PDE) arising in photoacoustic tomography. Further, the objective functional consists of an L 1 regularization term that promotes sparsity patterns in absorption coefficients. The motivation for this framework primarily comes from some recent works related to solving inverse problems in acousto-electric tomography and current density impedance tomography. We provide a new proof of existence and uniqueness of a solution to the semi-linear PDE. Further, a proximal method, involving a Picard solver for the semi-linear PDE and its adjoint, is used to solve the optimization problem. Several numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis.

The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the $ m $ electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from $ m $ up to $ m! $ while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from $ m $ to $ m^2 $ through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.

On Determination of Nonorientable Surface via its Diriclet-to-Neumann Operator

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 2 November 2020Accepted: 02 June 2021Published online: 21 September 2021Keywordselectric impedance tomography problem for nonorientable two-dimensional Riemannian manifolds, determination of manifold from DN mapAMS Subject Headings35R30, 46J15, 46J20, 30F15Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah