Estimating shapes and free surfaces with electrical impedance tomography

Abstract

Electrical impedance tomography (EIT) is a diffuse imaging modality in which the resistivity distribution inside the object is estimated based on electrical measurements made on the boundary. Several applications can be found in geophysics, medicine and industry. Image reconstruction is an iterative procedure in which the norm between the computed and measured voltages is minimized. Also, an additional regularization term is included in the minimized functional due to the ill-posedness of the problem. In the reconstruction process, the geometry of the object is assumed to be known. Geometry is known in many cases but there are various situations in which the shape of the domain is unknown. For example, in medical applications the shape of the domain, a part of the human body on which the measurement electrodes are attached, is unknown unless some other imaging modality is used for receiving the shape. In industrial applications, such as in the imaging of a stirrer vessel for detecting air distribution or detecting large air bubbles in pipelines, the free surface between the liquid and air is unknown and should be estimated. Within the domain we may also have ‘voids’ having zero conductivity and we might be interested in detecting the shapes and locations of the voids. An example could be the detection of corrosion faults in metallic plates. In this paper, two approaches for shape and free surface estimation are proposed. The approaches taken here are based on the idea used in shape optimization problems. In the first approach, the unknown shape is parametrized using the mesh nodes as parameters. In the second approach, we define new ‘design variables’ which are used as parameters. These design variables are the coefficients of a Bézier curve that defines the shape of the surface. In this paper, we show results of the free surface estimation from both computer simulations and tank measurements. Also, results of simultaneous reconstruction of the resistivity distribution and the free surface are shown. Comparison of the results between these two approaches will be given. The comparison shows better performance in the Bézier curve approach.

Boundary Estimation Techniques in Two-phase Flows using Electrical Impedance Tomography: A Review

“…The Fourier coefficients are estimated using boundary element method approach with Levenberg-Marquardt method. Several approaches are applied to elliptic PDE problems where the boundary is recovered based on the data from the exterior boundary using certain boundary conditions [8][9][10][11][12][13][14][15]. The methods presented in the above papers are based on specific elliptic PDE problems and the boundary is recovered by using certain boundary conditions and the data from the exterior domain boundary.…”

“…The boundary is parameterized using Chebyshev polynomials and the coefficients are the unknowns to be estimated using the regularization methods, the Marquardt method and a combination of the two techniques. In another approach developed by Tossavainen et al [10], a partially filled pipe with water that has a void region of zero conductivity is considered. The unknown boundary surface is parameterized with mesh nodes and the coefficients of Bezier curve.…”

“…This leads to a nonlinear and ill-posed inverse problem in which the coefficients representing the boundary are the unknowns to be estimated. Therefore, the forward solver has to be modified as a set of coefficients representing the boundary shapes to the data on ∂Ω [10,11]. Two types of boundary problems are considered: closed boundary and open boundary (also known as free surface) as illustrated in Fig.…”

Boundary element method to estimate the time-varying interfacial boundary in horizontal immiscible liquids flow using electrical resistance tomography

“…In open boundary problem, the flow domain is divided into two disjoint simply connected regions that are separated by an open interfacial boundary (for example, transport of immiscible liquids in a stratified flow). For more information on closed boundary estimation with ERT, please see [7][8][9][10][11][12][13]. Several methods have been proposed in literature to represent the shape of an open boundary using known functions [9,[14][15][16][17].…”

EIT image reconstruction with four dimensional regularization

“…Previously, shape-based inverse solutions were investigated in EIT applications. These methods modelled spatial priors according to geometrical knowledge of the target surface, e.g., boundary element method (BEM) based 3-D solution [6] and spline-based 2-D solution [30].…”