Regularization of ill-posed inverse problem in ultrasound tomography

doctoral thesis

• Title (croatian): Regularizacija loše uvjetovanog inverznog problema u ultrazvučnoj tomografiji
• Author: Anita Carević
• Mentor: Ivan Slapničar (mentor)
• Mentor: Mohamed Almekkawy https://orcid.org/0000-0002-9222-3003 (komentor)
• Committee member: Luka Grubišić (predsjednik povjerenstva)
• Committee member: Zoran Tomljanović (član povjerenstva)
• Committee member: Jesse Barlow (član povjerenstva); VIAF: 56178110
• Committee member: Zvonimir Bujanović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-12-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Ultrasound tomography (UT) is a medical imaging modality which can be used for the detection of breast cancer. One of the ways to solve the problem of UT is to use the distorted Born iterative method. In order for this method to provide a good solution, an ill-posed inverse problem must be solved within each iteration. In this dissertation, we regularize the inverse problem using direct spectral filtering methods. We show the advantage of using them in general form instead of standard, since employing modification of discrete version of first order derivative operator will provide additional regularization. The goal of the aforementioned methods is to minimize the influence of smaller (generalized) singular values, so the selection of the regularization parameter that is determining which of the values will be omitted is crucial to these methods. For this purpose we develop a new algorithm for choosing the regularization parameter that is, minimizing the residual and the error resulted from the noise of the measured data. In addition, we regularize the inverse problem using new forms of regularized total least squares where the existing problem is projected onto lower dimensional subspace. The dimension reduction is achieved by employing a generalized Krylov subspace expansion which results in significant decrease of computational time. In addition, the problem associated with finding the regularization parameter is avoided since an integrated parameter search inside the method is provided.

Abstract (croatian)

Ultrazvučna tomografija (UT) je dijagnostička radiološka metoda koja se može koristiti za otkrivanje karcinoma dojke. Jedan od načina za rješavanje problema UT-a je uporaba perturbirane Bornove iterativne metode. Ova metoda može dati dobro rješenje samo ako se unutar svake njene iteracije riješi loše uvjetovan inverzni problem. U ovoj disertaciji regulariziramo loše uvjetovani inverzni problem u ultrazvučnoj tomografiji koristeći direktne spektralne metode kod kojih ćemo pokazati prednost korištenja njihovog generaliziranog oblika nad standardnim budući da se dodatna regularizacija postiže upotrebom modificirane diskretizirane verzije operatora prve derivacije. Ideja prije spomenutih metoda je smanjiti utjecaj najmanjih (generaliziranih) singularnih vrijednosti pa je odabir regularizacijskog parametra, koji će odrediti to smanjenje, od iznimne važnosti. U tu svrhu razvijamo novi algoritam za odabir parametra koji će, uz minimizaciju reziduala, minimizirati i grešku koja je posljedica šuma na vektoru izmjerenih podataka. Također, regularizaciju ćemo provesti koristeći novi ubrzani oblik metode potpunih najmanjih kvadrata gdje će se postojeći problem projicirati na potprostor niže dimenzije. Ovo rezultira smanjenjem vremena izvršavanja metode. Manja dimenzija potprostora je postignuta koristeći generalizirane Krylovljeve potprostore. Budući da metoda ima integriran algoritam za traženje parametara, izbjegnut je problem korištenja vanjske metode u tu svrhu.

• Keywords: Ill posed inverse problem; regularization methods; regularization parameters; ultrasound tomography; distorted Born iterative method; regularized total least squares method
• Keywords (croatian): Loša uvjetovnost; inverzni problem; regularizacijske metode; regularizacijski parametri; ultrazvučna tomografija; preturbirana Bornova iterativna metoda; regularizirana metoda potpunih najmanjih kvadrata
• Language: english
• URN:NBN: urn:nbn:hr:217:364338
• Promotion: 2021
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: xiv, 109 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-03-03 10:52:34