Reconstructions in limited angle x-ray tomography

122 Pages·2013·2.51 MB·English

TechnischeUniversitätMünchen
FakultätfürMathematik
Reconstructions in limited angle x-ray tomography:
Characterization of classical reconstructions and
adapted curvelet sparse regularization
JürgenFrikel
VollständigerAbdruckdervonderFakultätfürMathematikderTechnischenUniversitätMünchen
zurErlangungdesakademischenGradeseines
DoktorsderNaturwissenschaften(Dr.rer.nat.)
genehmigtenDissertation.
Vorsitzender: Univ.-Prof.Dr. PeterGritzmann
PrüferderDissertation: 1. Univ.-Prof.Dr. BrigitteForster-Heinlein,
UniversitätPassau
2. Univ.-Prof.Dr. RupertLasser
3. Prof. SamuliSiltanen,Ph.D.
UniversityofHelsinki/Finland
DieDissertationwurdeam01.10.2012beiderTechnischenUniversitätMüncheneingereichtund
durchdieFakultätfürMathematikam12.03.2013angenommen.

Abstract
Inthisthesisweinvestigatethereconstructionproblemoflimitedangletomography. Suchprob-
lemsarisenaturallyinpracticalapplicationslikedigitalbreasttomosynthesis,dentaltomography
orelectronmicroscopy. Manyofthesemodalitiesstillemploythefilteredbackprojection(FBP)
algorithmforpracticalreconstructions. However,astheFBPalgorithmimplementsaninversion
formulafortheRadontransform,anessentialrequirementforitsapplicationisthecompleteness
of tomographic data. Consequently, the use of the FBP algorithm is theoretically not justified
in limited angle tomography. Another issue that arises in limited angle tomography is that only
specificfeaturesoftheoriginalobjectcanbereconstructedreliablyandadditionalartifactsmight
becreatedinthereconstruction.
The first part of this work is devoted to a detailed analysis of classical reconstructions at a
limited angular range. For this purpose, we derive an exact formula of filtered backprojection
reconstructionsatalimitedangularrangeandinterprettheseresultsinthecontextofmicrolocal
analysis. Inparticular,weshowthatameaningfula-prioriinformationcanbeextractedfromthe
limitedangledata. Moreover,weproveacharacterizationoflimitedangleartifactsthataregen-
eratedbythefilteredbackprojectionalgorithmanddevelopanartifactreductionstrategy. Wealso
illustrate the performance of the proposed method in some numerical experiments. Finally, we
study the ill-posedness of the limited angle reconstruction problem. We show that the existence
of invisible singularities at a limited angular range entails severe ill-posedness of limited angle
tomography. Owingtothisobservation,wederiveastabilizationstrategyandjustifytheapplica-
tion of the filtered backprojection algorithm to limited angle data under certain assumptions on
thetargetfunctions.
In the second part of this work, we develop curvelet sparse regularization (CSR) as a novel
reconstruction method for limited angle tomography and argue that this method is stable and
edge-preserving. We also formulate an adapted version of curvelet sparse regularization (A-
CSR)byapplyingthestabilizationstrategyofthefirstpart,andprovideathoroughmathematical
analysis of this method. To this end, we prove a characterization of the kernel of the limited
angleRadontransformintermsofcurveletsandderiveacharacterizationofCSRreconstructions
at a limited angular range. Finally, we show in a series of numerical experiments that the theo-
reticalresultsdirectlytranslateintopracticeandthattheproposedmethodoutperformsclassical
reconstructions.
3

Acknowledgements
Atthisplacemygratitudegoestoallwhocontributedtothecompletionofthiswork.
Firstandforemost,IwishtothankmyPhDadvisorProf. Dr. BrigitteForster-Heinleinforher
help, her advice and the freedom she gave me during the time I spent on this thesis. I have also
enjoyedhersupportedthroughoutthetimeofmystudentprojectandmydiplomathesisatTUM,
where I already had the opportunity to work in the field of tomographic image reconstruction.
Furthermore,IalsothankProf. Dr. RupertLasserandProf. Dr. SamuliSiltanenfortheirinterest
inmyworkandforhavingagreedtowriteareportforthisthesis.
The first two and a half years of my research to the topic of this thesis were carried out at
GE Healthcare, Image Diagnost International, Munich, whose financial support I gratefully ac-
knowledge. Inparticular,IammuchobligedtoDr. PeterHeinlein,myadvisoratGEHealthcare,
ImageDiagnostInternational,forgivingmethechancetodoresearchinapracticalenvironment.
Iappreciateverymuchhishelpandhispatienceduringthistime.
I sincerely thank Dr. Frank Filbir for giving me the opportunity to work on my dissertation
while I was employed in his research group at the Institute of Biomathematics and Biometry,
Helmholtz Zentrum München. I have profited much from his advice and his support during this
time. Inparticular,Iwanttothankhimforgivingmethechancetotravelandtogetintouchwith
distinguished scientists. During that time I received financial support from DFG (grant no. FI
883/3/3-1)whichIherewithgratefullyacknowledge.
It is my pleasure to express my gratitude to Prof. Dr. Eric Todd Quinto (Tufts University,
Medford,USA),formanyfruitfulandinspiringdiscussionsonthetopicoflimitedangletomog-
raphyandforhishospitalityduringmyvisits, wheresomeideasofSection3.4weredeveloped.
Furthermore, I wish to thank PD Dr. Peter Massopust, Prof. Dr. Wolodymyr R. Madych (Uni-
versity of Connecticut, Storrs, USA) and Prof. Dr. Hrushikesh Mhaskar (Caltech, USA) for
discussingmathematicswithmeandforproofreadingpartsofmywork.
Especially, I would like to thank my colleague and friend Martin Storath, with whom I have
shared an office over the past years, for many inspiring and valuable discussions as well as for
proofreading all versions of this work. Moreover, I am very grateful to Prof. Dr. Stefan Kunis
forcarefullyproofreadingthisworkandmanyvaluableremarks. Ihavealsobenefitedalotfrom
brainstorming ideas and mathematical discussions with Dr. Laurent Demaret and Dr. Thomas
Amler.
Finally,andmostimportantly,IwouldliketoexpressmydeepgratitudetomywifeElenaand
tomyparentsfortheirconstantsupport,encouragement,patienceandlove.
5

Contents
Abstract 3
Acknowledgements 5
1 Introduction 9
2 Anoverviewofcomputedtomography 13
2.1 Theprincipleofx-raytomography . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Mathematicsofcomputedtomography . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Radontransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Inversionformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.3 Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Reconstructionmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Analyticreconstructionmethods . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Algebraicreconstructionmethods . . . . . . . . . . . . . . . . . . . . . 24
3 Characterizationoflimitedanglereconstructionsandartifactreduction 27
3.1 Notationsandbasicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Characterizationsoflimitedanglereconstructions . . . . . . . . . . . . . . . . . 29
3.3 Microlocalcharacterizationoflimitedanglereconstructions . . . . . . . . . . . 38
3.3.1 Basicfactsfrommicrolocalanalysis . . . . . . . . . . . . . . . . . . . . 38
3.3.2 Microlocalcharacterizationsoflimitedanglereconstructions . . . . . . . 41
3.4 Characterizationandreductionoflimitedangleartifacts . . . . . . . . . . . . . . 44
3.4.1 Characterizationoflimitedangleartifacts . . . . . . . . . . . . . . . . . 44
3.4.2 Reductionoflimitedangleartifacts . . . . . . . . . . . . . . . . . . . . 47
3.5 Amicrolocalapproachtostabilizationoflimitedangletomography . . . . . . . 51
3.5.1 Amicrolocalprinciplebehindthesevereill-posedness . . . . . . . . . . 51
3.5.2 Astabilizationstrategyforlimitedangletomography . . . . . . . . . . . 54
3.6 Summaryandconcludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Anadaptedandedge-preservingreconstructionmethodforlimitedangle
tomography 57
4.1 Astableandedge-preservingreconstructionmethod . . . . . . . . . . . . . . . . 58
4.1.1 Regularizationbysparsityconstraints . . . . . . . . . . . . . . . . . . . 59
7
8 Contents
4.1.2 Amultiscaledictionaryforsparseandedge-preservingrepresentations. . 61
4.1.3 Curveletsparseregularization(CSR) . . . . . . . . . . . . . . . . . . . 66
4.2 Adaptedcurveletsparseregularization(A-CSR) . . . . . . . . . . . . . . . . . . 67
4.3 Characterizationofcurveletsparseregularizations . . . . . . . . . . . . . . . . . 70
4.4 Relationtobiorthogonalcurveletdecomposition(BCD)fortheRadontransform 75
4.4.1 AclosedformformulaforCSRatafullangularrange . . . . . . . . . . 76
4.4.2 CSRasageneralizationoftheBCDapproach . . . . . . . . . . . . . . . 78
4.5 Summaryandconcludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Implementationofcurveletsparseregularizationandnumericalexperiments 81
5.1 Implementationofcurveletsparseregularization . . . . . . . . . . . . . . . . . . 81
5.2 Visibilityofcurveletsanddimensionalityreduction . . . . . . . . . . . . . . . . 83
5.3 Executiontimesandreconstructionquality . . . . . . . . . . . . . . . . . . . . . 87
5.3.1 Experimentalsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.2 Executiontimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.3 Reconstructionquality . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Artifactreductionincurveletsparseregularizations . . . . . . . . . . . . . . . . 93
5.5 Summaryandconcludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . 93
A TheFouriertransformandSobolevspaces 99
A.1 TheFouriertransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.2 Sobolevspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B Singularvaluedecompositionofcompactoperators 105
B.1 Somepropertiesofcompactoperators . . . . . . . . . . . . . . . . . . . . . . . 105
B.2 Spectraltheoremandsingularvaluedecompositionofcompactoperators. . . . . 106
C Basicfactsaboutinverseproblems 109
C.1 Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C.2 Classificationofill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
C.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Bibliography 115
hapter
C 1
Introduction
Computed tomography (CT) has revolutionized diagnostic radiology and is by now one of the
standardmodalitiesinmedicalimaging. Itsgoalconsistsinimagingcrosssectionsofthehuman
bodybymeasuringandprocessingtheattenuationofx-raysalongalargenumberoflinesthrough
thesection. Inthisprocess,afundamentalfeatureofCTisthemathematicalreconstructionofan
imagethroughtheapplicationofasuitablealgorithm. Thecorrespondingmathematicalproblem
consistsinthereconstructionofafunction f : R2 Rfromtheknowledgeofitslineintegrals
→
(cid:90)
f(θ,s) = f(x)dS(x),
R
L(θ,s)
where L(θ,s) = x R2 : x cosθ+ x sinθ = s . The purely mathematical problem of recon-
1 2
{ ∈ }
structingafunctionfromitslineintegralswasfirststudiedandsolvedbytheAustrianmathemati-
cianJohannRadonin1917inhispioneeringpaper“ÜberdieBestimmungvonFunktionendurch
ihre Integralwerte längs gewisser Mannigfaltigkeiten”, cf. [Rad17]. In that work, he derived an
explicit inversion formula for the integral transform : f f(θ,s) under the assumption
R (cid:55)→ R
that the data f(θ,s) is complete, i.e., available for all possible values (θ,s) [ π/2,π/2) R.
R ∈ − ×
Morethan50yearslater,whenAllanM.Cormack,[Cor63,Cor64],andGodfreyN.Hounsfield,
[Hou73], invented computer assisted tomography1, this mathematical problem has finally be-
come relevant for practical applications. This breakthrough has attracted very much attention
in engineering sciences as well as in the mathematical community. Especially, the tomographic
reconstruction problem has been studied extensively, and many different reconstruction algo-
rithms were developed for the case of complete tomographic data, cf. [Nat86, NW01], [KS88],
[RK96],[Eps08],[Her09].
Thesuccessofcomputedtomographyhasalsoinitiatedthedevelopmentofnewimagingtech-
niques. In some of these applications the tomographic data can be acquired only at a limited
angular range, i.e., the integral values f(θ,s) are merely available for (θ,s) [ Φ,Φ] R,
R ∈ − ×
where Φ < π/2. According to that, the reconstruction problem of limited angle tomography is
equivalent to the inversion of the limited angle Radon transform Φ : f f [ Φ,Φ] R, rather
R (cid:55)→ R |− ×
than the full Radon transform . Typical examples of modalities where such problems arise are
R + +
digital breast tomosynthesis (DBT) [N 97], [III09], dental tomography [HKL 10], [MS12], or
1Allan M. Cormack and Godfrey N. Hounsfield were jointly awarded with the Nobel prize in Physiology and
Medicinein1979forthedevelopmentofcomputerassistedtomography.
9
10 Chapter1 Introduction
electronmicroscopy[DRK68]. Inthesecases,theavailableangularrange[ Φ,Φ]mightbevery
−
small. In DBT, for example, the angular range parameter usually varies between Φ = 10 and
◦
Φ = 25 , [ZCS+06], [III09]. In such situations, the reconstruction methods for the full angu-
◦
lar problem are no longer applicable. In [Lou79], [Per79], [Grü80], [Ino79], [Ram91, Ram92],
dedicated inversion methods for the limited angle tomography problem were developed under
the assumption that the target functions have compact support. Essentially, all of these methods
utilizeanalyticcontinuationintheFourierdomainoremploythecompletionofthelimitedangle
data. Inparticular,thesemethodsshowthataperfectreconstructionistheoreticallypossibleeven
from a very small angular range. Nevertheless, there are still substantial problems that arise in
practice:
Severe ill-posedness. Since the limited angle reconstruction problem is severely ill-posed
[Nat86], [Dav83], all reconstruction procedures are extremely unstable, i.e., small measurement
errors can cause huge reconstruction errors. This is a serious drawback since, in practice, the
acquired data is (to some extent) always corrupted by noise. As a consequence, only specific
featuresoftheoriginalobjectcanbereconstructedstably,[Qui93].
Filtered backprojection in limited angle tomography. In practice, the filtered backprojec-
tion (FBP) algorithm is usually preferred over those inversion techniques mentioned above, cf.
[PSV09], [LMKH08], [SMB03], [IG03], [SFS06], [III09]. However, as the FBP algorithm im-
plementsaninversionformulafortheRadontransform,cf. Section2.4,anessentialrequirement
foritsapplicationisthecompletenessoftomographicdata. Consequently,thereisnotheoretical
justificationfortheapplicationoftheFBPalgorithminlimitedangletomography.
Limitedangleartifacts. Besidesthefactthatonlyspecificfeaturesoftheoriginalobjectcan
be reconstructed reliably, the application of the FBP algorithm to limited angle data can also
create additional artifacts in reconstructions, cf. Figure 3.1. Thus, even the reliable information
canbedistortedbytheseartifacts.
Edge-preservingreconstruction. Thefilteredbackprojectionalgorithmdoesnotperformwell
in the presence of noise, [NW01], [Her09], [MS12]. Depending on the choice of the regulari-
zation parameter, the reconstructions may appear either noisy or tend to oversmooth the edges,
cf. Figure4.1. However,edge-preservingreconstructionsareofparticularimportanceformedical
imagingpurposes.
In this thesis, we solve the problems listed above. To this end, we prove different characteri-
zations of classical reconstructions at a limited angular range, justify the application of filtered
backprojectiontothelimitedangledata,andprovideanartifactreductionstrategyfortheFBPal-
gorithm. Furthermore,weaddressthesevereill-posednessbydevelopinganoveledge-preserving
reconstruction method that is adapted to the limited angle geometry, and show its practical rele-
vance.
The thesis is organized as follows. Chapter 2 has an introductory character. It is devoted to
the presentation of basic concepts of computed tomography at a full angular range. We begin
with a brief overview of the imaging principle in x-ray tomography and introduce the Radon
transformasamathematicalmodelofthemeasurementprocessinx-rayCT.Moreover, wenote
some fundamental properties of the Radon transform. The main part of this chapter is devoted