Multifrequency electrical impedance tomography with total variation regularization

Multifrequency electrical impedance tomography (MFEIT) reconstructs the distribution of conductivity by exploiting the dependence of tissue conductivity on frequency. MFEIT can be performed on a single instance of data, making it promising for applications such as stroke and cancer imaging, where it is not possible to obtain a baseline measurement of healthy tissue. A nonlinear MFEIT algorithm able to reconstruct the volume fraction distribution of tissue rather than conductivities has been developed previously. For each volume, the fraction of a certain tissue should be either 1 or 0; this implies that the sharp changes of the fractions, representing the boundaries of tissue, contain all the relevant information. However, these boundaries are blurred by traditional regularization methods using l2 norm. The total variation (TV) regularization can overcome this problem, but it is difficult to solve due to its non-differentiability. Because the fraction must be between 0 and 1, this imposes a constraint on the MFEIT method based on the fraction model. Therefore, a constrained optimization method capable of dealing with non-differentiable problems is required. Based on the primal and dual interior point method, we propose a new constrained TV regularized method to solve the fraction reconstruction problem. The noise performance of the new MFEIT method is analysed using simulations on a 2D cylindrical mesh. Convergence performance is also analysed through experiments using a cylindrical tank. Finally, simulations on an anatomically realistic head-shaped mesh are demonstrated. The proposed MFEIT method with TV regularization shows higher spatial resolution, particularly at the edges of the perturbation, and stronger noise robustness, and its image noise and shape error are 20% to 30% lower than the traditional fraction method

A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem

AbstractThe Generalized Fermat Problem (in the plane) is: given n≥3 destination points find the point x̄∗ which minimizes the sum of Euclidean distances from x̄∗ to each of the destination points.The Weiszfeld iterative algorithm for this problem is globally convergent, independent of the initial guess. Also, a test is available, à priori, to determine when x̄∗ a destination point. This paper generalizes earlier work by the first author by introducing an asymmetric Euclidean distance in which, at each destination, the x-component is weighted differently from the y-component. A Weiszfeld algorithm is studied to compute x̄∗ and is shown to be a descent method which is globally convergent (except possibly for a denumerable number of starting points). Local convergence properties are characterized. When x̄∗ is not a destination point the iteration matrix at x̄∗ is shown to be convergent and local convergence is always linear. When x̄∗ is a destination point, local convergence can be linear, sub-linear or super-linear, depending upon a computable criterion. A test, which does not require iteration, for x̄∗ to be a destination, is derived. Comparisons are made between the symmetric and asymmetric problems. Numerical examples are given

A cell-based smoothed finite element method for kinematic limit analysis

This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged

Comparison of total variation algorithms for electrical impedance tomography

The applications of total variation (TV) algorithms for electrical impedance tomography (EIT) have been investigated. The use of the TV regularisation technique helps to preserve discontinuities in reconstruction, such as the boundaries of perturbations and sharp changes in conductivity, which are unintentionally smoothed by traditional l2 norm regularisation. However, the non-differentiability of TV regularisation has led to the use of different algorithms. Recent advances in TV algorithms such as the primal dual interior point method (PDIPM), the linearised alternating direction method of multipliers (LADMM) and the spilt Bregman (SB) method have all been demonstrated successful EIT applications, but no direct comparison of the techniques has been made. Their noise performance, spatial resolution and convergence rate applied to time difference EIT were studied in simulations on 2D cylindrical meshes with different noise levels, 2D cylindrical tank and 3D anatomically head-shaped phantoms containing vegetable material with complex conductivity. LADMM had the fastest calculation speed but worst resolution due to the exclusion of the second-derivative; PDIPM reconstructed the sharpest change in conductivity but with lower contrast than SB; SB had a faster convergence rate than PDIPM and the lowest image errors

Depth analysis of planar array for 3D electrical impedance tomography

Electrical impedance tomography (EIT) imaging modality has great potential on industrial applications with the advantages of being high temporal resolution. It is especially useful in cases, such as geophysical detection, landmine detection, and detections on non-transparent region, where the measurement data are only available from single surface, for data acquisition. Instead of the circular EIT model that uses the traditional circular electrode model, in this paper, planar array EIT is implemented, aiming to visualize a pipeline transporting a two-phase flow. The planar array can explore spatial information within its detectable region by producing 3D images, which have a higher spatial resolution in the axis direction than a traditional EIT with a dual-plane electrode sensor. However, in solving the inverse problem of a 3D subsurface EIT using a planar array, the images may be degraded, especially in cases where the location of the target is relatively deep. The total variation (TV) algorithm as block prior assumption-based regularization method has the potential to improve the image quality, and some works have shown that TV reconstructs sharper images, which provides an advantage when representing spatial information. In this paper, the performance of subsurface EIT using the TV algorithm for 3D visualization is presented based on simulations and experiments, and the results of quantitative measurement of depth are discussed

An interior-point method for the single-facility location problem with mixed norms using a conic formulation

Abstract We consider the single-facility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in R n , where each distance can be measured according to a different p-norm. We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving three-dimensional cones. Using the availability of a self-concordant barrier for these cones, we present a polynomial-time algorithm (a long-step path-following interior-point scheme) to solve the problem up to a given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem

A primal-dual active set algorithm for three-dimensional contact problems with coulomb friction

International audienceIn this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth com-plementarity function for the three-dimensional friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization are applied, and superlinear convergence can be observed locally. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples