The Radon transform, its generalization and their applications in PET and SPECT medical imaging

Protonotarios, Nicholas; Πρωτονοτάριος, Νικόλαος

dc.contributor.author	Protonotarios, Nicholas	el
dc.contributor.author	Πρωτονοτάριος, Νικόλαος	en
dc.date.accessioned	2020-03-09T14:31:00Z
dc.date.available	2020-03-09T14:31:00Z
dc.date.issued	2020-03-09	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/49900
dc.identifier.uri	http://dx.doi.org/10.26240/heal.ntua.17598
dc.rights	Default License
dc.subject	Μετασχηματισμός Radon	el
dc.subject	Radon transform	en
dc.subject	Εξασθενημένος μετασχηματισμός Radon	el
dc.subject	Attenuated Radon transform	en
dc.subject	PET	el
dc.subject	PET	en
dc.subject	SPECT	el
dc.subject	SPECT	en
dc.subject	Ανακατασκευή ιατρικής εικόνας	el
dc.subject	medical image reconstruction	en
dc.title	The Radon transform, its generalization and their applications in PET and SPECT medical imaging	en
dc.title	O μετασχηματισμός Radon, οι γενικεύσεις του και εφαρμογές αυτών στις ιατρικές απεικονίσεις PET και SPECT	el
dc.contributor.department	Τομέας Μαθηματικών	el
heal.type	doctoralThesis
heal.classification	Μαθηματικά	el
heal.classification	Mathematics	en
heal.language	el
heal.language	en
heal.access	free
heal.recordProvider	ntua	el
heal.publicationDate	2019-12-04
heal.abstract	In the present thesis, the following three different mathematical problems are solved: (a) the problem of edge detection in the Radon $(\rho, \theta)$-space, (b) the problem of deblurring in the attenuated Radon $(\rho, \theta)$-space, and (c) the problem of the inversion of the attenuated Radon transform via a new analytic formula, following the pioneering work of Novikov and Fokas, and the associated numerical implementation, referred to as the attenuated spline reconstruction technique (aSRT). The above mathematical problems involve the inversion of the celebrated Radon transform of a function, defined as the set of all its line integrals, as well as the inversion of a certain generalization of the Radon transform of a function, the so-called attenuated Radon transform, defined as the set of all its attenuated line integrals. The non-attenuated and attenuated versions of the Radon transform provide the mathematical foundation of two of the most important available medical imaging techniques, namely positron emission tomography (PET), and single-photon emission computed tomography (SPECT). Although Radon himself derived in 1917 the inversion of the transform bearing his name, seventy four years later Novikov and Fokas rederived this well-known formula by considering two classical problems in complex analysis known as the $\bar{d}$-problem and the scalar Riemann-Hilbert problem. Although the inversion can be obtained in a simpler manner by the use of the Fourier transform, the derivation of Novikov and Fokas allowed Novikov to invert the attenuated Radon transform in 2002. It took Fokas, Iserles and Marinakis four more years to establish a more straightforward derivation of this inversion. Following their work, one of the main results of the present thesis involves the formulation of an equivalent inversion for the attenuated Radon transform. It is not suprising that even today, more than a century after its seminal publication, Radon's work is still highly influential. In Chapter 1, the Radon transform in $\mathbb{R}^2$ and its attenuated generalization are presented, whereas in Chapter 2 their inversions, especially in the context of non-Fourier analysis, are constructed. Chapter 3 deals with the problem of sinogram edge detection in the context of inverse problems. Furthermore, in Chapter 4, aSRT is analyzed, which provides a novel analytic inversion formula for the attenuated Radon transform. This new inversion formula involves the computation of the Hilbert transforms of the linear attenuation function and of two sinusoidal functions of the attenuated data. Finally, in Chapter 5, the problem of deblurring in the attenuated Radon $(\rho, \theta)$-space is solved. The mathematical problems solved in this thesis are quite different from one another, however they bear several intrinsic similarities. They are all key elements of a large class of mathematical problems, associated with the mathematical foundations of emission tomography. The inversion of the Radon transform and of its attenuated generalization constitute fundamental problems in the mathematical core of medical imaging.	en
heal.advisorName	Χαραλαμπόπουλος, Αντώνιος	el
heal.advisorName	Καστής, Γεώργιος	el
heal.committeeMemberName	Φωκάς, Αθανάσιος	el
heal.committeeMemberName	Δούκα, Ευανθία	el
heal.committeeMemberName	Γιαννακάκης, Νικόλαος	el
heal.committeeMemberName	Νικήτα, Κωνσταντίνα	el
heal.committeeMemberName	Στρατής, Ιωάννης	el
heal.academicPublisher	Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών	el
heal.academicPublisherID	ntua
heal.numberOfPages	192
heal.fullTextAvailability	false