Image reconstruction from fan-beam and cone-beam projections [Elektronische Ressource] = Bildrekonstruktion aus Fächerstrahl- und Kegelstrahlprojektionen / of Frank Dennerlein

Sujets

• Informatik
• Gesundheit

Informations

• Publié par: friedrich-alexander-universitat_erlangen-nurnberg
• Publié le: 01 janvier 2009
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 8 Mo

Extrait

Image Reconstruction from Fan-Beam and
Cone-Beam Projections
Bildrekonstruktion aus Facherstrahl- und˜
Kegelstrahlprojektionen
Submitted to
Technische Fakultat der˜
Universitat Erlangen-Nurnberg˜ ˜
in partial fulflllment of the requirements for
the degree of
DOKTOR-INGENIEUR
of
Frank Dennerlein
Erlangen | 2008As dissertation accepted by
Technische Fakultat der˜
Universit˜at Erlangen-Nurn˜ berg
Date of submission: September 8, 2008
Date of doctorate: December 4, 2008
Dean: Prof. Dr.-Ing. habil. Johannes Huber
Reviewer: Prof. Joachim Hornegger
Fr¶ed¶ericNoo,PhD,AssociateProfessorMeinen ElternAbstract
This thesis addresses the problem of reconstructing static objects in 2D and 3D transmission com-
puted tomography (CT). After reviewing the classical CT reconstruction theory, we discuss and
thoroughly evaluate various novel reconstruction methods, two of which are original.
Ourflrstoriginalapproachisfor2DCTreconstructionfromfull-scanfan-beamdata,i.e.,for2D
imaginginthegeometryofdiagnosticmedicalCTscanners.Comparedtoconventionalmethods,our
approach is computationally more e–cient and also yields results with an overall reduction of image
noise at comparable spatial resolution, as demonstrated in detailed evaluations based on simulated
fan-beam data and on data collected with a Siemens Somatom CT scanner. Part two of this thesis
discusses the problem of 3D reconstruction in the short-scan circular cone-beam (CB) geometry,
i.e., the geometry of medical C-arm systems. We flrst present a detailed comparative evaluation of
innovative methods recently suggested in the literature for reconstruction in this geometry and of
the approach applied on many existing systems. This evaluation involves various quantitative and
qualitative flgures-of-merit to assess image quality. We then derive an original short-scan CB recon-
struction method that is based on a novel, theoretically-exact factorization of the 3D reconstruction
problem into a set of independent 2D inversion problems, each of which is solved iteratively and
yieldstheobjectdensityonasingleplane.Incontrasttothestate-of-the-artmethodsdiscussedear-
lier in this thesis, our factorization approach does not involve any geometric approximations during
its derivation and enforces all reconstructed values to be positive; it thus provides quantitatively
veryaccurateresultsandefiectivelyreducesCBartifactsinthereconstructions,asillustratedinthe
numerical evaluations based on computer-simulated CB data and also real CB data acquired with
a Siemens Axiom Artis C-arm system.
Kurzfassung
Diese Arbeit behandelt das Problem der Rekonstruktion statischer Objekte in der 2D und 3D
˜Transmissions-Computertomographie (CT). Wir geben einen Uberblick der klassischen CT Rekon-
struktionstheorie und diskutieren und evaluieren anschliessend mehrere neue CT Rekonstruktions-
methoden; zwei dieser Methoden sind originar.˜
Unser erstes originares Verfahren ist fur die 2D CT Rekonstruktion aus Vollkreis-Facherstrahl-˜ ˜ ˜
daten, das heisst, fur 2D Bildgebung in der Geometrie diagnositischer medizinischer CT Scanner.˜
Unser Verfahren ist im Vergleich zu herkommlichen Methoden rechene–zienter und liefert aus-˜
serdem Ergebnisse mit reduziertem Bildrauschen bei vergleichbarerer Ortsau osung, was anhand˜
ausfuhrlicher Untersuchungen mit simulierten Daten und mit Daten eines Siemens Somatom CT˜
Scanners demonstriert wird. Teil zwei dieser Arbeit behandelt das 3D Rekonstruktionsproblem in
der Teilkreis-Kegelstrahlgeometrie, das heisst, in der Geometrie medizinischer C-Bogen Systeme.
Wir prasen˜ tieren eine detaillierte Vergleichsstudie innovativer Methoden, die kurzlic˜ h in der Li-
teratur fur˜ die Rekonstruktion in dieser Geometrie vorgeschlagen wurden, sowie des Verfahrens,
das in vielen existierenden Systemen Anwendung flndet. Unser Vergleich basiert auf quantitati-
ven sowie qualitativen Bildqualitatsk˜ enngr˜ossen. Wir leiten anschliessend eine origin˜are Teilkreis-
Kegelstrahlrekonstruktionsmethode her, die auf einer neuen, theoretisch exakten Faktorisierung des
3D Rekonstruktionsproblems in eine Menge unabh˜angiger 2D Inversionsprobleme beruht. Jedes In-
versionsproblem wird iterativ gelost˜ und liefert die Objektdichte auf einer einzelnen Ebene. Unser
Faktorisierungsverfahren bezieht im Gegensatz zu den vorher untersuchten Methoden keinerlei geo-
metrische Annaherungen˜ wahrend˜ seiner Herleitung mit ein und erzwingt, dass alle rekonstruierten
Wertepositivsind.EsliefertdaherquantitativsehrakkurateErgebnisseundeineefiektiveReduzie-
rung von Kegelstrahlartefakten, was in den numerischen Auswertungen mit simulierten Daten und
mit Daten eines Siemens Axiom Artis C-Bogen Systems veranschaulicht wird.Acknowledgement
This thesis covers some of the topics I was dealing with as a research associate at the
Utah Center for Advanced Imaging Research (UCAIR) in Salt Lake City, USA. My stay
abroad,from2005to2008,wasfacilitatedinthecontextofaresearchcollaborationbetween
the University of Utah, the University of Erlangen-Nurnberg and Siemens AG, Medical˜
Solutions and I would like to express my gratitude to everyone who was involved in this
project for the support and for valuable discussions over the years, in particular to
Prof. Dr. Fr¶ed¶eric Noo, for his excellent supervision of my research at UCAIR and for
sharing his great expertise about image reconstruction theory, and to
Prof. Dr.-Ing. Joachim Hornegger, for his guidance and motivating in uence on my work
despite the large distance that separated our o–ces for most of the time as well as for
placing emphasis on real-data evaluations, and to
Dr. Gun˜ ter Lauritsch, for flrst introducing me to the exciting world of CT, for his advice
and the discussions during my semi-annual visits in Forchheim and for supporting C-arm
data acquisition.
Furthermore,Iwouldliketothankmyco-workersattheUniversitiesofUtahandErlangen-
Nurn˜ berg and everyone I had the chance to work with over the years, for the fruitful time
wespenttogetherduringandoutsideworking-hours. IwanttothankAdamWunderlichfor
acquiring and providing the 2D CT data sets that were used for the evaluation in chapter 4
as well as Stefan Hoppe, Marcus Prummer and Christopher Rohkohl for their assistance in˜
converting and preparing the real C-arm data for the evaluations in chapter 7.
Finally, I would like to thank the Siemens AG and the NIH for providing flnancial support
for my research.Contents
1 Introduction 1
1.1 Computed Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 CT Image Reconstruction in the Medical Environment . . . . . . . . . . . . 3
1.3 Scope and Original Contribution of this Thesis . . . . . . . . . . . . . . . . 4
1.4 Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 The Data Model in CT 7
3 Classical Theory for Image Reconstruction in Two Dimensions 11
3.1 2D Radon Transform and Its Inversion . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 2D Parallel-beam Geometry . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 The 2D Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.3 Concept of Backprojection. . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.4 Classical Inversion Formula for the 2D Radon Transform. . . . . . . 17
3.1.5 Parallel Beam Reconstruction Formula for Redundant Data . . . . . 19
3.1.6 Numerical Algorithm . . . . . . . . . . . . . . . . . . 20
3.2 2D Fan-Beam Transform and Its Classical Inversion Formula . . . . . . . . 22
3.2.1 The 2D Fan-Beam Geometry . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 The 2D F Transform . . . . . . . . . . . . . . . . . . . . . . 23
3.2.3 Classical Inversion Formula for the 2D Fan-Beam Transform . . . . 24
3.2.4 Numerical Reconstruction Algorithm . . . . . . . . . . . . . . . . . . 27
4 Fan-Beam Reconstruction without Backprojection Weight 29
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Alternative Inversion Formula for the 2D Fan-Beam Transform . . . . . . . 30
4.3 Fan-Beam Reconstruction with No Backprojection Weight . . . . . . . . . . 32
4.4 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.1 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.2 Evaluation of Spatial Resolution . . . . . . . . . . . . . . . . . . . . 35
4.4.3 Ev of Image Noise . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4.4 Evaluation of Computational E–ciency . . . . . . . . . . . . . . . . 39
4.5 Reconstruction from Real CT Data . . . . . . . . . . . . . . . . . . . . . . . 43
4.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 General Theory for Image Reconstruction in Three Dimensions 49
5.1 3D Radon Transform and Its Inversion . . . . . . . . . . . . . . . . . . . . . 49
5.1.1 3D Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.2 Analytical 3D Radon Inversion Formula . . . . . . . . . . . . . . . . 51
i5.2 Image Reconstruction in the Cone-Beam Geometry . . . . . . . . . . . . . . 53
5.2.1 General Cone-beam Acquisition . . . . . . . . . . . . . . . 53
5.2.2 Cone-Beam Reconstruction in General . . . . . . . . . . . . . . . . . 54
5.2.3 The Issue of CB Data Su–ciency . . . . . . . . . . . . . . . . . . . . 56
5.2.4 Reconstruction via Filtered Backprojection