Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes

Abstract

The inverse conductivity problem in electrical impedance tomography involves the solving of a nonlinear and under-determined system of equations. This paper presents a new approach, which leads to a quadratic and overdetermined system of equations. The aim of the paper is to establish new research directions in handling of the inverse conductivity problem. The basis of the proposed method is that the material, which can be considered as an isotropic continuum, is modeled as a linear network with concentrated parameters. The weights of the obtained graph represent the properties of the discretized continuum. Further, the application of the developed procedure allows for the dielectric constant to be used in the multi-frequency approach, as a result of which the optimized system of equations always remains overdetermined. Through case studies, the efficacy of the reconstruction method by changing the mesh resolution applied for discretizing is presented and evaluated. The presented results show, that, due to the application of discrete, symmetric mathematical structures, the new approach even at coarse mesh resolution is capable of localizing the inhomogeneities of the material.

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

“…The fundamental recursive formula is obtained by adding Equation (16) and Equation (18), and using the equalities…”

“…Further, the research group is ambitious in studying other types of non-linear ODEs, which are still intensively researched today [15][16][17]. The second research area is the inverse problems involving second-order differential equations, which are also non-linear mathematical problems [18]. Still, compared to the classical differential equation solvers, they require a completely different handling, since the functions to be determined are the coefficients of the differential equation [19][20][21].…”

Continuous Electrode Models and Application of Exact Schemes in Modeling of Electrical Impedance Measurements

“…Moreover, great attention is paid to the application of the most efficient computational procedures for modeling EI measurements. Vizvari et al [5,25] have established a completely new basis for the mathematical modeling procedures required for their interpretation. One of the achievements is the absolutely new mathematical approach, where Vizvari et al [25] introduced the exact scheme for second-order ordinary differential equations (ODEs) using arbitrary spatial discretization.…”

“…The combination of all these methods offers interesting and useful implementation possibilities. For example, combining EIS and EIT can contribute to overcoming difficulties in the image reconstruction with the EIT method [4,5]. An example of such a technology is the multi-frequency EIT (mfEIT), where the application-specific prototypes have been developed beyond the basic research [5,6].…”

“…For example, combining EIS and EIT can contribute to overcoming difficulties in the image reconstruction with the EIT method [4,5]. An example of such a technology is the multi-frequency EIT (mfEIT), where the application-specific prototypes have been developed beyond the basic research [5,6]. The motivation of this research is to develop a completely new mathematical method in order to implement the mfEIT.…”