Scrivanti, Gabriele Luca Giovanni
(2020)

*Nonsmooth Nonconvex Variational
Reconstruction for Electrical Impedance Tomography.*
[Laurea magistrale], Università di Bologna, Corso di Studio in

Matematica [LM-DM270]

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## Abstract

Electrical Impedance Tomography is an imaging technique that aims to reconstruct the inner conductivity distribution of a medium starting from a set of measured voltages registered by a series of electrodes that are positioned on the surface of the medium. Such technique was used for the first time in geological studies in 1930 and then applied to industrial procedures. The first clinical use of EIT dates back to 1987. In 2018 EIT was validated in tissue engineering as a noninvasive and label-free imaging and monitoring tool for cell distribution (cell growth, differentiation and tissue formation) in 3D scaffolds. EIT problem can be split into a Forward and an Inverse problem. The aim of Forward EIT is to define the set of measured voltages starting from a known conductivity distribution. If the forward problem is characterized by a nonlinear mapping, called Forward Operator, from the conductivity distribution to the measured voltages, inverse EIT consists of inverting the Forward Operator. This leads to an ill-posed problem which requires regularization, either in the model or in the numerical method that is applied to define the solution. The inverse problem is modelled as a Nonlinear Least Squares problem, where one seeks to minimize the mismatch beetween the measured voltages and the ones generated by the reconstructed conductivity. Reconstruction techniques require the introduction of a regularization term which forces the reconstructed data to stick to certain prior information. In this dissertation, some state-of-the-art regularization methods are analyzed and compared via EIDORS, a specific software for EIT problems. The aim is to reconstruct the variation in conductivity within a 2D section of a 3D scaffold. Furthermore a variational formulation on a 2D mesh for a space-variant regularization is proposed, based on a combination of high order and nonconvex operators, which respectively seek to recover piecewise inhomogeneous and piecewise linear regions.

Abstract

Electrical Impedance Tomography is an imaging technique that aims to reconstruct the inner conductivity distribution of a medium starting from a set of measured voltages registered by a series of electrodes that are positioned on the surface of the medium. Such technique was used for the first time in geological studies in 1930 and then applied to industrial procedures. The first clinical use of EIT dates back to 1987. In 2018 EIT was validated in tissue engineering as a noninvasive and label-free imaging and monitoring tool for cell distribution (cell growth, differentiation and tissue formation) in 3D scaffolds. EIT problem can be split into a Forward and an Inverse problem. The aim of Forward EIT is to define the set of measured voltages starting from a known conductivity distribution. If the forward problem is characterized by a nonlinear mapping, called Forward Operator, from the conductivity distribution to the measured voltages, inverse EIT consists of inverting the Forward Operator. This leads to an ill-posed problem which requires regularization, either in the model or in the numerical method that is applied to define the solution. The inverse problem is modelled as a Nonlinear Least Squares problem, where one seeks to minimize the mismatch beetween the measured voltages and the ones generated by the reconstructed conductivity. Reconstruction techniques require the introduction of a regularization term which forces the reconstructed data to stick to certain prior information. In this dissertation, some state-of-the-art regularization methods are analyzed and compared via EIDORS, a specific software for EIT problems. The aim is to reconstruct the variation in conductivity within a 2D section of a 3D scaffold. Furthermore a variational formulation on a 2D mesh for a space-variant regularization is proposed, based on a combination of high order and nonconvex operators, which respectively seek to recover piecewise inhomogeneous and piecewise linear regions.

Tipologia del documento

Tesi di laurea
(Laurea magistrale)

Autore della tesi

Scrivanti, Gabriele Luca Giovanni

Relatore della tesi

Correlatore della tesi

Scuola

Corso di studio

Indirizzo

Curriculum A: Generale e applicativo

Ordinamento Cds

DM270

Parole chiave

Electrical Impedance Tomography Inverse Conductivity Problem Finite Elements Regularization Methods Total Variation Medical Imaging Tissue Engineering EIDORS Alternating Direction Method of Multipliers

Data di discussione della Tesi

27 Marzo 2020

URI

## Altri metadati

Tipologia del documento

Tesi di laurea
(NON SPECIFICATO)

Autore della tesi

Scrivanti, Gabriele Luca Giovanni

Relatore della tesi

Correlatore della tesi

Scuola

Corso di studio

Indirizzo

Curriculum A: Generale e applicativo

Ordinamento Cds

DM270

Parole chiave

Electrical Impedance Tomography Inverse Conductivity Problem Finite Elements Regularization Methods Total Variation Medical Imaging Tissue Engineering EIDORS Alternating Direction Method of Multipliers

Data di discussione della Tesi

27 Marzo 2020

URI

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