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ON
TW
ERP K
AFT
: NAT
AC
HA H
OEV
ENA
EGEL
- N
IEU
WE M
EDIA
DIE
NST
PROMOTOREN
prof. dr. Jan Sijbersprof. dr. Joost Batenburg
Faculteit WetenschappenDepartement Fysica
Antwerpen 2015
Proefschrift voorgelegd tot het behalen van de graad van doctor in de wetenschappen aan de Universiteit Antwerpen te verdedigen door
Andrei Dabravolski
Towards In Loco X-ray Computed Tomography
Andrei D
abravolskiTow
ards In Loco X-ray Computed Tom
ography
.
Faculteit Wetenschappen
Departement Fysica
Towards In Loco X-ray Computed Tomography
Een Aanzet tot In Loco X-stralenComputertomogra�e
Proefschrift voorgelegd tot het behalen van de graad van
Doctor in de Wetenschappen
aan de Universiteit Antwerpen, te verdedigen door
Andrei Dabravolski
Promotoren
prof. dr. Jan Sijbers
prof. dr. Joost Batenburg Antwerpen, 2015
Members of the Jury
prof. dr. Joost Batenburg1
dr. ir. Matthieu Boone2
prof. dr. Koen Janssens3
dr. Xuan Liu4
prof. dr. Jan Sijbers3
prof. dr. Wim Wenseleers3
1Centrum Wiskunde & Informatica, Amsterdam, The Netherlands2Ghent University, Ghent, Belgium3University of Antwerp, Antwerp, Belgium4Bruker microCT, Kontich, Belgium
Contact Information
Andrei Dabravolski
B iMinds-Vision Lab, Department of Physics
University of Antwerp (CDE)
Universiteitsplein 1, N-1.14
2610 Wilrijk, Belgium
T +32 (0) 3 265 28 40
v +32 (0) 3 265 22 45
m http://visielab.uantwerpen.be/people/andrei-dabravolski
Preface
Four years ago I started my doctoral studies, and they are now almost completed.
These four years were not easy for me, but rather interesting, challenging and
enjoyable. In this preface, I would like to thank people that have been supporting
and inspiring me, helping me either directly or indirectly.
My �rst thanks go to my supervisors, Jan and Joost. I would like to thank Jan
for giving me the chance to start my PhD studies, for his constant support and
patience, as well as his guidance and advice. I am grateful to Joost for informing
me about the position at the Vision lab, for sharing his ideas, for his constant
support and patience.
I would like to thank the members of the doctoral jury for their time, attention
and e�ort they have put into the reading of the draft of the thesis, for their
questions, comments and suggestions that helped me to improve the text.
Next, I am grateful to all current and former members of the Vision lab, you
are nice colleagues to work next to and you give the Vision lab its pleasant, cosy
and creative atmosphere.
My special thanks go to my parents and to my brother, for their constant and
unconditional support, help and encouragement, which I feel even at a distance.
And my biggest thanks go to my wife Katya, for the support, encouragement,
inspiration and love she gives me. Thank you!
Andrei Dabravolski
Wilrijk, 2015
3
Contents
Preface 3
Table of Contents 5
Common Abbreviations 7
Samenvatting 9
Summary 13
1 Introduction 17
1.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Work�ow of Computed Tomography . . . . . . . . . . . . . . . . . 19
1.2.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Prospects of In Loco Tomography . . . . . . . . . . . . . . . . . . . 21
1.4 Current Developments and Challenges in In Loco Tomography . . 22
1.4.1 Existing Devices . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Technical Developments . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Challenges in Data Acquisition and Reconstruction . . . . . 23
1.5 Proposed steps towards In Loco Tomography . . . . . . . . . . . . 26
1.6 Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Variable Distance Approach 33
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Noiseless simulations in two dimensions . . . . . . . . . . . 37
2.3.2 Noiseless simulations in three dimensions . . . . . . . . . . 40
2.3.3 Simulations with noise . . . . . . . . . . . . . . . . . . . . . 44
2.3.4 Real experiment . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Dynamic Angle Selection 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5
CONTENTS
3.2.1 Information gain . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Upper bound for the diameter of the solution set . . . . . . 54
3.2.3 Surrogate solutions . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.4 Candidate angles . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.5 Dynamic angle selection algorithm . . . . . . . . . . . . . . 55
3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 Randomly oriented bars . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Wood phantoms: noiseless simulations . . . . . . . . . . . . 59
3.3.3 Wood phantom: simulations with noise . . . . . . . . . . . 61
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Multiresolution DART 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Motivation and approach . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Noiseless simulations . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Simulations with noise . . . . . . . . . . . . . . . . . . . . . 81
4.3.3 Real experiments . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Conclusions 93
Acknowledgements 97
A Scienti�c Contributions 101
A.1 Journal Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.2 Conference Proceedings (full paper) . . . . . . . . . . . . . . . . . 101
A.3 Conference Proceedings (abstract) . . . . . . . . . . . . . . . . . . 102
6
Common Abbreviations
ARM Algebraic Reconstruction Method
ART Algebraic Reconstruction Technique
CAD Computer-Aided Design
CGLS Conjugate Gradient Least Squares
CT Computed Tomography
CTA Circular Trajectory Approach
DT Discrete Tomography
DART Discrete Algebraic Reconstruction Technique
FBP Filtered Backprojection
FDK Feldkamp, Davis, and Kress
GPU Graphics Processing Unit
MDART Multiresolution Discrete Algebraic Reconstruction Technique
MDART q MDART operating on q reconstruction grids
MSE Mean Squared Error
MTF Modulation Transfer Function
RNMP Relative Number of Misclassi�ed Pixels
SART Simultaneous Algebraic Reconstruction Technique
SIRT Simultaneous Iterative Reconstruction Technique
SPECT Single-Photon Emission Computed Tomography
SSIRT Segmented Simultaneous Iterative Reconstruction Technique
TV Total Variation
VDA Variable Distance Approach
7
Samenvatting
Inleiding
Computertomogra�e (CT) is een niet-invasieve beeldvormingstechniek die het in-
wendige van een object in beeld brengt door een reeks van projecties te combi-
neren die opgenomen werden vanuit verschillende richtingen. Tegenwoordig heeft
CT zeer diverse toepassingen in o.m. de geneeskunde, preklinisch onderzoek, niet-
destructief onderzoek, en materiaalkunde.
Een algemeen kenmerk van de tomogra�sche setups die in de meeste toepas-
singsgebieden gebruikt worden, is de eis om een object in een scanner te positio-
neren. Het eerste belangrijke nadeel van deze eis is de beperking opgelegd aan de
afmetingen van het object dat gescand dient te worden. Het tweede nadeel is de
noodzaak voor het verplaatsen van het object wat moeilijk te realiseren kan zijn of
wat onwenselijke veranderingen in het object kan veroorzaken. Een mogelijkheid
om in loco, i. e. 'ter plaatse', tomogra�e uit te voeren, zou talrijke toepassingen
voor tomogra�e in niet-destructief onderzoek, veiligheid, geneeskunde, archeolo-
gie en diergeneeskunde kunnen openen en toelaten om objecten te scannen die
te groot, zwaar, breekbaar of gevaarlijk zijn om in bestaande scanners te zetten.
Een mobiel tomogra�sch toestel met de X-stralenbron en de detector op aparte
gerobotiseerde platformen, zou toelaten om de beperkingen van de conventionele
CT setups te overwinnen en in loco tomogra�e te realiseren in de praktijk.
De huidige prestaties en de veelbelovende resultaten in de ontwikkeling van
mobiele robots, X-stralenbronnen en detectoren maken de ontwikkeling van een
mobiel gerobotiseerd tomogra�sch toestel technisch haalbaar in de komende ja-
ren. Echter, de opname en de reconstructie van de datasets a.h.v. mobiele tomo-
gra�sche toestellen zullen waarschijnlijk een aantal moeilijkheden opleveren voor
de conventionele algoritmen. Ten eerste kunnen cirkelvormige of spiraalvormige
bron-detector-banen die gebruikt worden in de meeste tomogra�sche setups, niet
beschikbaar of onpraktisch zijn vanwege de obstakels in de scanscene, wat het pro-
bleem van baanselectie vormt. Ten tweede kan de con�guratie van de scanscene
dermate zijn dat de opname van projecties in bepaalde richtingen niet mogelijk
is, wat in een beperkt hoekbereik resulteert. Ook kan herpositionering van de bron
en de detector tijdrovend zijn, waardoor mogelijkheid een beperkt aantal projecties
opgenomen kan worden in een bepaalde tijdsspanne. Bovendien kan het, vanuit be-
paalde richtingen, moeilijk zijn om projecties van het volledige object op te nemen,
wat het nodig maakt om met getrunceerde projecties om te gaan. Ten slotte kan
nauwkeurige bepaling van de positie en de oriëntatie van de bron en de detector
een uitdaging zijn, waardoor projectie uitlijning noodzakelijk is.
In deze thesis worden drie technieken voorgesteld die bijdragen tot de ont-
9
Samenvatting
wikkeling van de opname- en reconstructiealgoritmen voor mobiele tomogra�sche
toestellen die in staat zijn tot in loco tomogra�e. Slechts drie van de genoemde
problemen worden aangepakt en de voorgestelde technieken zijn niet bedoeld om
de complete oplossingen te zijn, maar we hopen te hebben bijgedragen aan de
oplossingen die op termijn in-loco tomogra�e mogelijk maken. Waar de variabele-
afstand-opname en het dynamische hoekselectie algoritme naar de verbetering van
de opname streven, is het multiresolutie DART-algoritme een reconstructiealgo-
ritme dat de datasets kan behandelen met een klein aantal projecties die verworven
zijn van een beperkt hoekbereik, wat aanzienlijk de bijbehorende artefacten ver-
mindert en sterk verbeterde reconstructies produceert.
Variabele Afstand Opname
De variabele-afstand-opname (Hoofdstuk 2) werd ontwikkeld om het gebruik van
de detector te optimaliseren en meer gedetailleerde informatie te verwerven bij het
scannen van langwerpige objecten, wat in nauwkeurigere reconstructies resulteert.
De variabele-afstand-opname maakt gebruik van voorkennis over het convexe
omhulsel van een object om de bron zo dicht mogelijk bij het object te plaatsen en
daarbij de detector optimaal te benutten. Dit maakt het mogelijk om meer gede-
tailleerde informatie over het object uit elke projectierichting te behalen waardoor
de kwaliteit van de reconstructies wordt verbeterd. Een geschat convex omhulsel
van het object kan worden verkregen uit een voorbereidende scan, beelden van
het object in optisch bereik of een CAD-model. De voorgestelde benadering kan
eenvoudig aangepast worden aan verschillende setups en is de eerste stap naar een
automatische baanselectie van de X-stralenbron.
De voorgestelde variabele-afstand-opname wordt eerst toegepast op een aantal
fantoombeelden in 2D en 3D, wat het vermogen aantoont om nauwkeurigere re-
constructies te bekomen. Vervolgens wordt er een experiment uitgevoerd die de
gewenste setup nabootst met een aantal microCT-scans, wat het vermogen van
de voorgestelde aanpak bevestigt om meer details in het object te onthullen in
vergelijking met de conventionele cirkelbaan.
Dynamische Hoekselectie
Het dynamische hoekselectie algoritme (Hoofdstuk 3) is een opnamealgoritme tel-
kens dat een nieuwe projectiehoek selecteert op basis van de reeds verworven pro-
jectiedata, zodat de meeste informatie over het object wordt verkregen.
Het dynamische hoekselectie algoritme kiest een nieuwe projectierichting zo-
danig dat de nieuw verworven opname zoveel mogelijk informatie over het object
10
Samenvatting
toevoegt. Om de informatiehoeveelheid te kwanti�ceren wordt er een begrip infor-
matiewinst geïntroduceerd dat gebaseerd is op de diameter van de oplossingsverza-
meling van het reconstructieprobleem die gede�nieerd is voor de reeds verworven
projectiedata. Er worden een aantal benaderingsstappen geïntroduceerd om een
praktisch berekenbare maat te verkrijgen. De diameter van de oplossingsverzame-
ling is door een bovengrens vervangen die wordt berekend voor een verzameling
van surrogaatoplossingen die de echte oplossingverzameling vertegenwoordigen. Er
is hierbij geen voorkennis over het object vereist.
In de experimenten wordt het voorgestelde algoritme vergeleken met drie con-
ventionele hoekselectie strategieën door middel van een aantal fantomen. De expe-
rimenten tonen het vermogen van het dynamische hoekselectie algoritme aan om
projectiehoeken te selecteren die leiden tot nauwkeurigere reconstructies vanuit
mindere projecties, en dit in vergelijking met de eerder beschreven projectiehoek-
selectie algoritmen. De grootste verbetering in de kwaliteit van de reconstructies
wordt bereikt voor objecten met een klein aantal voorkeursrichtingen en een klein
aantal projectiehoeken.
Multiresolutie DART
Het multiresolutie DART-algoritme (Hoofdstuk 4) is een reconstructiealgoritme
voor discrete tomogra�e op basis van Discrete Algebraïsche Reconstructietechniek
(DART) die het mogelijk maakt om de kwaliteit van reconstructies aanzienlijk te
verbeteren en rekentijd te verlagen.
In discrete tomogra�e wordt een object verondersteld te bestaan uit een klein
aantal verschillende materialen. De voorkennis van het aantal materialen in het
object en hun absorpties laat toe om, via DART, nauwkeurigere reconstructies te
berekenen vanuit beperkte data in vergelijking met conventionele reconstructieal-
goritmen. Net zoals de meeste iteratieve reconstructiealgoritmen, is de rekentijd
van DART relatief lang. De multiresolutie versie van DART die in deze thesis is
voorgesteld (Hoofdstuk 4), maakt het mogelijk om de computationele e�ciëntie en
de kwaliteit van reconstucties te verbeteren in vergelijking met DART. Dit wordt
bereikt door het starten van de reconstructie op een grof reconstructierooster. De
resulterende reconstructie wordt dan opnieuw herbemonsterd en als uitgangspunt
gebruikt voor een daaropvolgende reconstructie op een �jner raster. Dit wordt
iteratief herhaald totdat de gewenste pixelgrootte bereikt wordt. Het gebruik van
grove rasters maakt het reconstructieprobleem minder slecht gesteld omdat het
aantal onbekenden verminderd wordt en het aantal vergelijkingen gelijk blijft. Dit
resulteert in een nauwkeurigere initiële reconstructie voor de volgende reconstructie
op een �jner rooster waardoor het sneller convergeert.
Het voorgestelde multiresolutie DART-algoritme wordt eerst op een aantal fan-
11
Samenvatting
toombeelden toegepast om zijn eigenschappen te onderzoeken. Daarna wordt het
toegepast op twee microCT-datasets en een synchrotron dataset. De resultaten
tonen het vermogen van de multiresolutie DART aan om nauwkeurigere recon-
structies te berekenen in een fractie van tijd die DART vereist. De grootste verbe-
tering wordt bereikt voor de datasets met een zeer klein aantal projecties die uit
een beperkt hoekbereik verworven zijn.
12
Summary
Introduction
Computed tomography (CT) is a non-invasive imaging technique that allows to
reveal the inner structure of an object by combining a series of projection images
that were acquired from di�erent directions. CT nowadays has a broad range
of applications, including those in medicine, preclinical research, nondestructive
testing, materials science, etc.
One common feature of the tomographic setups used in most applications is
the requirement to put an object into a scanner. The �rst major disadvantage of
such a requirement is the constraint imposed on the size of the object that can be
scanned. The second one is the need to move the object which might be di�cult
or might cause undesirable changes in the object. A possibility to perform in loco,
i. e. on site, tomography will open up numerous applications for tomography in
nondestructive testing, security, medicine, archaeology and veterinary, allowing to
scan objects that are too large, heavy, fragile or dangerous to put into existing
scanners. A mobile tomographic device with the X-ray source and the detector
mounted on separate robotized platforms will allow to overcome the limitations of
the conventional CT setups and provide a means of performing in loco tomography.
The current achievements and promising results in the development of mobile
robots, X-ray sources and detectors make the appearance of a mobile robotized
tomographic device technically feasible in the coming years. However, the acquisi-
tion and the reconstruction of the datasets using mobile tomographic devices are
likely to present a number of di�culties for the conventional algorithms. Firstly,
circular or helical source-detector trajectories, used nowadays in the majority of
tomographic setups, might be unavailable or impractical due to the obstacles in
the scanning scene, constituting the trajectory selection problem. Secondly, the
con�guration of the scanning scene might render certain projection directions un-
available, resulting in a limited angular range. Next, repositioning of the source
and the detector might be time-consuming, leading to a possibility to only acquire
a limited number of projections in a reasonable time. Furthermore, it might be im-
possible to acquire projections of the complete object from certain directions, thus
requiring to deal with projection truncation. Finally, accurate determination of the
position and the orientation of the source and the detector might be challenging,
resulting in the need for projection alignment.
In this thesis, three techniques are proposed that contribute towards the de-
velopment of acquisition and reconstruction algorithms for mobile tomographic
devices capable of in loco tomography. Only three of the mentioned issues are
addressed and the proposed techniques are not supposed to be the complete so-
13
Summary
lutions, but we hope to have contributed to the solutions yet to be found. While
the variable distance approach and the dynamic angle selection algorithm aim at
the improvement of the acquisition, making the �rst steps towards the trajectory
selection, the multiresolution Discrete Algebraic Reconstruction Technique (mul-
tiresolution DART, MDART) algorithm is a reconstruction algorithm that can
handle the datasets with a small number of projections acquired from a limited
angular range, signi�cantly reducing the related artefacts and producing accurate
reconstructions.
Variable Distance Approach
The variable distance approach (Chapter 2) was developed to optimize the detector
usage and to acquire more detailed information when scanning elongated objects,
resulting in more accurate reconstructions.
The variable distance approach uses prior knowledge about the convex hull
of an object to place the source as close as possible to the object while avoiding
truncation. The position of the source for every projection direction is calculated
based on the convex hull of the object such that the full width of the detector is
exploited. This allows to obtain more detailed information about the object from
each projection angle thus increasing the reconstruction quality. An approximate
convex hull of the object can be obtained from a preparatory scan, images of the
object in optical range or a CAD model. The proposed approach can easily be
adapted to various setups and is the �rst step towards an automatic trajectory
selection.
The proposed variable distance approach is �rst applied to a number of phan-
tom images in 2D and 3D, demonstrating its ability to provide more accurate re-
constructions with lower errors. Next, a real experiment, mimicking the required
setup with a number of micro-CT scans, is conducted, con�rming the ability of
the proposed approach to reveal more details in the object compared to the con-
ventional circular trajectory.
Dynamic Angle Selection
The dynamic angle selection algorithm (Chapter 3) is an online acquisition algo-
rithm that selects a new projection angle based on the already acquired projection
data so that the most information about the object is gained.
The dynamic angle selection algorithm selects a new projection direction in
such a way that the newly acquired projection will add as much information about
the object as possible. A concept of information gain is used to quantify the
14
Summary
amount of information, which is based on the diameter of the solution set of
the reconstruction problem de�ned for the already acquired projection data. A
number of approximation steps are introduced to obtain a practically computable
measure. The diameter of the solution set is substituted with an upper bound that
is computed for a set of surrogate solutions, which represent the true solution set.
No prior knowledge about the object is required.
In the experiments, the proposed algorithm is compared with three conven-
tional angle selection strategies using a number of phantoms. The experiments
demonstrate the ability of the dynamic angle selection algorithm to select pro-
jection angles that lead to more accurate reconstructions from fewer projections
compared to the widely used angle selection algorithms. The biggest improvement
in the reconstruction quality is achieved for the objects with a small number of
preferential directions in the shape and interior and for a small number of projec-
tion angles.
Multiresolution DART
The multiresolution DART algorithm (Chapter 4) is a reconstruction algorithm
for discrete tomography based on Discrete Algebraic Reconstruction Technique
(DART) that allows to signi�cantly improve the reconstruction quality and de-
crease computation time.
In discrete tomography, an object is assumed to consist of a small number of
di�erent materials. The prior knowledge of the number and the attenuations of
the materials in the object allows DART to provide more accurate reconstructions
from limited data compared to conventional reconstruction algorithms. Being an
iterative reconstruction algorithm, DART can su�er from long computation times.
The multiresolution version of DART, proposed in this thesis (Chapter 4), allows
to improve the computational e�ciency and the reconstruction quality compared
to DART. This is achieved by starting the reconstruction on a coarse reconstruc-
tion grid with big pixel size. The resulting reconstruction is then resampled and
used as a starting point for a subsequent reconstruction on a �ner grid. This is re-
peated iteratively until the target pixel size is reached. The use of the coarse grids
makes the reconstruction problem less ill-posed, since the number of unknowns is
decreased and the number of equations remains the same. This results in providing
a more accurate initial reconstruction for the following reconstruction on a �ner
grid, allowing it to converge faster.
The proposed multiresolution DART algorithm is �rst applied to a number of
phantom images to investigate its properties. The algorithm is then applied to
two micro-CT datasets and to a synchrotron dataset. The results demonstrate the
ability of the multiresolution DART to provide more accurate reconstructions in
15
Summary
a fraction of time compared to DART. The biggest improvement is achieved for
the datasets with a very small number of projections and acquired from a limited
angular range.
16
1Introduction
Contents
1.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Work�ow of Computed Tomography . . . . . . . . . . . . . . . 19
1.2.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . 19
1.2.2 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Prospects of In Loco Tomography . . . . . . . . . . . . . . . . 21
1.4 Current Developments and Challenges in In Loco Tomography . 22
1.4.1 Existing Devices . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Technical Developments . . . . . . . . . . . . . . . . . . 23
1.4.3 Challenges in Data Acquisition and Reconstruction . . . 23
1.5 Proposed steps towards In Loco Tomography . . . . . . . . . . 26
1.6 Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.1 Historical Overview
Computed tomography is an imaging technique that is capable of revealing the
inner structure of an object by combining projection images acquired from di�erent
directions. The word tomography is derived from the Greek words τoµoσ, �slice�,
and γραφω, �to write�, explaining the essential feature of tomography to provide
sectional imaging of the object.
The history of CT started in 1895, when Wilhelm Röntgen (1845�1923) discov-
ered penetrating rays which he named X-rays. In 1901 he received the Nobel prize
in Physics for this discovery. In 1915 Carlo Baese patented an imaging method
based on the simultaneous movement of an X-ray tube and a �lm cassette and
17
CHAPTER 1. INTRODUCTION
in 1916 Karol Mayer (1882�1946) obtained X-ray images of the heart using a sta-
tionary �lm and a moving X-ray tube. André Bocage (1892�1953) described the
basic principles of a device for moving an X-ray tube and a �lm in a patent speci�-
cation in 1921. Furthermore, he proposed using multidirectional (circular or spiral)
tube-�lm movement and pointed out the need to eliminate scattered photons and
the importance of providing a small focal spot for the X-ray beam.
The further steps from an idea towards a practical system were made by
Alessandro Vallebona (1899�1987) and Bernard Ziedses des Plantes (1902�1993)
who independently designed and built working prototypes of the tomographic sys-
tems. In 1934 Gusztáv Grossmann (1878�1957) developed and patented the �rst
commercially produced body-section machine, which was based on the pendular
motion with the tube describing an arc in the vertical plane and the �lm mov-
ing horizontally. He also coined the term tomography and called his device the
tomograph.
It should be clari�ed here that all above mentioned systems were based on the
principles of the analog geometric tomography, where points only in one plane
through the object are imaged sharply and other points are blurred. To allow
the development of computed tomography an inverse problem of reconstructing
a function from its surface integrals had to be solved. The �rst solution to this
problem for a three-dimensional function to be reconstructed from two-dimensional
surface integrals belongs to Hendrik Antoon Lorentz (1853�1928). Unfortunately,
the result was not published and the context of this work is still unknown. The
result is associated with Lorentz by his doctoral student, H. Bockwinkel, who used
Lorentz's solution in a 1906 publication on light propagation in crystals.
The way for computed tomography was paved by Johann Radon (1887�1956),
who published a thorough solution to the problem of reconstructing a two-dimen-
sional function from its line integrals and considered generalisations for curves
on non-Euclidean planes and for higher dimensions [1] (English translations are
available in [2] and [3]). However, there were no practical applications mentioned
in the paper itself and its importance for tomography was realised much later. In
1956, the �rst application for the above-mentioned problem, often referred to as
a tomographic reconstruction problem, was found in radio-astronomy by Ronald
Bracewell (1921�2007), who used Fourier techniques to improve the reconstruction
of the distribution of the celestial sources of radio waves [4].
The medical relevance of computed tomography was identi�ed by Allan Cor-
mack (1924�1998), who published the solution to the tomographic reconstruction
problem, pointed out its applications in radiology and radiotherapy and con�rmed
the results experimentally [5, 6]. Sir Godfrey Houns�eld (1919�2004) indepen-
dently built a head scanner and obtained the �rst clinical results [7]. Later he
developed the �rst whole-body scanner. It is important to note that Houns�eld
18
1.2. WORKFLOW OF COMPUTED TOMOGRAPHY
considered the reconstruction problem as a system of linear equations and devel-
oped an iterative method to solve it using a computer. In 1979 Cormack and
Houns�eld won the Nobel Prize in Physiology or Medicine for their developments
in X-ray computed tomography.
A more detailed overview of the history of tomography can be found in [3, 8�11].
1.2 Work�ow of Computed Tomography
In this section, the work�ow of computed tomography is brie�y reviewed. The
major steps in tomographic imaging are data acquisition and reconstruction. Af-
ter that the data is ready for analysis, which depends entirely on the speci�c
application.
1.2.1 Data Acquisition
In order to reveal the inner structure of an object, penetrating radiation needs to
pass through the object, where part of that radiation is absorbed or scattered.
The remaining part of radiation then hits a detector forming a projection image
containing information about the object. A number of such projection images
acquired from the di�erent directions around the object form the input of the
tomographic reconstruction methods (Section 1.2.2). In 2D case, projection images
are usually arranged in a single image where each row represents a projection from
one direction. Such an image is called a sinogram for its characteristic appearance
(Fig. 1.1b).
Di�erent types of radiation are used for tomographic imaging, such as X-rays,
gamma rays [12], electrons [13], neutrons [14], muons [15, 16]. Since this thesis
focuses on X-ray tomography, the data acquisition process for X-ray imaging is
brie�y described below.
X-rays used in tomography are produced by either X-ray tubes, where photons
are emitted in a cone beam as a result of collision of electrons from the cathode
with the anode material, or by synchrotron light sources, where the energy of the
electrons travelling near the speed of light is converted into a parallel beam of
photons by the strong magnetic �elds. The X-ray beam then passes through the
object and hits a detector, resulting in a projection image. Typically, either the
object or the source-detector system is rotated to acquire images from di�erent
directions. In the systems with X-ray tubes, circular or helical (spiral) source-
detector trajectories or the equivalent object movements are typically used, while
in the synchrotron-based systems the object is rotated.
19
CHAPTER 1. INTRODUCTION
1.2.2 Reconstruction
There are two main approaches to the tomographic reconstruction. The analytical
reconstruction methods use approximations of Radon's solution. These methods
are computationally e�cient and not very �exible, since an analytical reconstruc-
tion method can be derived only for a particular acquisition geometry and adding
prior knowledge to such a method is cumbersome. Examples of analytical methods
are Filtered Backprojection (FBP) [8], algorithms of Feldkamp, Davis, and Kress
(FDK) [17], Grangeat [18], Katsevich [19] and Kudo [20].
(a) Phantom (b) Sinogram
(c) FBP reconstruction (d) SIRT reconstruction
Figure 1.1: (a) Phantom image. (b) Sinogram from 500 projections. (c) FBP reconstruction.(d) SIRT reconstruction.
The algebraic (or iterative) reconstruction methods consider the reconstruction
problem as a system of linear equations
Wx = p, (1.1)
where W = (wij) ∈ Rm×n is a matrix determining the projection geometry (a pro-
jection matrix) with the elements wij representing the contribution of the pixel j
to the detector element i, which can be computed in a number of ways [8, 21, 22];
20
1.3. PROSPECTS OF IN LOCO TOMOGRAPHY
x = (xj) ∈ Rn denotes the unknown image and p = (pi) ∈ Rm corresponds
to a measured projection data. Algebraic methods are more computationally in-
tensive compared to analytical ones, but they are much more �exible, i. e. can be
readily adapted for any acquisition geometry (see Chapter 2) and allow to incorpo-
rate prior knowledge easier (see Chapter 4). Algebraic Reconstruction Technique
(ART) [23], Simultaneous Algebraic Reconstruction Technique (SART) [24] and
Simultaneous Iterative Reconstruction Technique (SIRT) [25] are examples of such
reconstruction algorithms. Throughout the thesis, SIRT is used extensively. The
update expression for SIRT is given by [26]
xt+1 = xt + CWTR(p−Wxt
), (1.2)
where C ∈ Rn×n and R ∈ Rm×m are diagonal matrices with cjj = 1/∑i wij and
rii = 1/∑j wij , and xt =
(xtj)∈ Rn is the reconstruction at iteration t, with the
initial reconstruction x0 typically being a zero matrix, i. e. x0 = 0. The element-
wise version of Eq. (1.2), which is more suitable for implementation, is given by
xt+1j = xtj +
1∑mi=1 wij
m∑i=1
wij (pi −∑nh=1 wihx
th)∑n
h=1 wih. (1.3)
More details about e�cient implementations of SIRT and other algebraic methods
can be found in [27�30].
Figure 1.1 shows a phantom representing a fragment of foam (Fig. 1.1a), the
sinogram containing 500 equiangular projections (Fig. 1.1b) and the corresponding
FBP (Fig. 1.1c) and SIRT (Fig. 1.1d) reconstructions.
1.3 Prospects of In Loco Tomography
CT nowadays has a wide variety of applications including those in medicine and
preclinical research, nondestructive testing and materials science. One common
trait in the tomographic setups used in most of these applications is the necessity
to put an object into a scanner for imaging. Firstly, this immediately limits the
size of the object that can be scanned. Secondly, the object needs to be moved
which might result in undesirable changes in the object or might be di�cult due to
the weight, fragility or other properties of the object. While there exist "portable"
medical CT scanners, they are mainly designed to make tomography available in an
operating room without the need to move the patient to the radiology department.
Existing handheld X-ray devices are capable of acquiring a single image or a series
of images and do not allow for CT.
A mobile tomographic device, in which the X-ray source and the detector are
mounted on two separate robotized platforms, will allow to overcome the limita-
21
CHAPTER 1. INTRODUCTION
tions of the conventional CT, having the �exibility with respect to the size and
the position of the object, the acquisition geometry and eliminating the need to
move the object and therefore allowing for in loco, i. e. on site, tomography. The
possible application �elds of such a system include:
� Archaeology: studying ancient artefacts that are too fragile to be moved or
that are parts of bigger objects;
� Medicine: scanning overweight patients and patients in any pose, not only
lying, which might provide an insight into the bone or muscle behaviour
under load;
� Nondestructive testing: evaluating (parts of) buildings, bridges or other
structures;
� Security: checking suspicious objects in public places, such as unattended
bags at the airports;
� Veterinary: scanning cattle and horses.
1.4 Current Developments and Challenges in In Loco
Tomography
The key components of a tomographic system, related recent developments and
the challenges for in loco tomography are discussed in this section.
1.4.1 Existing Devices
Nowadays attempts are made to overcome the limitations of conventional CT and
expand its applications. In medicine, C-arm-mounted CT systems are successfully
used for many years [31], allowing to move a tomographic system into an operating
room and to leave a patient stationary. A design of a portable CT scanner with
interleaved detectors and emitters was proposed [32], which can be wrapped around
an object. In nondestructive testing, a setup combining stationary detector and a
source moving linearly was proposed for imaging of the objects located in a corner
of a building [33]. In [34], a dedicated linac X-ray source and the �rst results of
on site inspection performed at a chemical plant are presented. Several robotized
platforms are being developed for pipe inspection [35�37], which can potentially
carry an X-ray source and a detector.
While aforementioned developments expand (or will potentially expand) the
use of CT, they all are very dedicated and can successfully solve only the problem
for which they were developed, resulting in a need to seek a di�erent solution
22
1.4. CURRENT DEVELOPMENTS AND CHALLENGES IN IN LOCO
TOMOGRAPHY
for every type of object or scanning setup. A concept of in loco tomography
provides a framework for developing general solutions to the problems arising in
the development of mobile tomographic devices. The next section provides a brief
overview of the technical basis and Section 1.4.3 discusses possible challenges in
the acquisition and the reconstruction for in loco tomography.
1.4.2 Technical Developments
The X-ray sources commonly used in CT are based on thermionic electron emis-
sion, i. e. a cathode heated to a high temperature emits electrons that are then
accelerated. A metal anode is bombarded with the accelerated electrons to gen-
erate X-rays. Such sources have high power consumption which hinders their use
in mobile devices. Recent developments of the �eld-emission cathodes based on
the use of silicon [38] or carbon nanotubes [39, 40] allow to signi�cantly decrease
the power demand and the size of X-ray tubes, which can operate at room tem-
perature, thus making them more suitable for the use in in loco tomography.
Triboelectric X-ray sources are the other promising candidates for mobile tomo-
graphic devices [41, 42], opening up the possibility to create small, independently
addressable arrays of X-ray sources without a high-voltage power supply.
Portable X-ray �at-panel detectors are commercially available and increasingly
used in clinical practice for digital radiology [43]. Moreover, the promising results
in developing �exible detectors [44�46] provide additional possibilities for mobile
tomography.
While X-ray sources and X-ray detectors are only making the �rst steps into
the mobile applications, mobile robots are widely used in industry, medicine, for
military, security and household tasks. Extensive research is carried out into ev-
ery related aspect, including robot localisation and mapping [47�49] and accurate
positioning of robot manipulators [50, 51].
To sum up, the current achievements and promising results in the development
of X-ray sources, detectors and mobile robots make it technically feasible to create
a mobile robotized tomographic device for in loco tomography in the years ahead.
However, the development of such devices in practice might be hindered by the
challenging problems described in the following section.
1.4.3 Challenges in Data Acquisition and Reconstruction
The following points are likely to present di�culties for the acquisition and the
reconstruction of the datasets from mobile tomographic devices:
� Trajectory selection: conventionally used nowadays circular or helical source-
detector trajectories might be unavailable or impractical due to the obstacles
in the scanning scene;
23
CHAPTER 1. INTRODUCTION
� Limited angular range: some directions around the object can be unavailable
for acquisition depending on the scanning scene;
� Limited number of projections: since acquisition of every projection image
involves repositioning of the source, the detector or both, it might be time-
consuming, resulting in a possibility to acquire only a limited number of
projections in a reasonable time;
� Projection truncation: depending on the size of the object and the detector
as well as the limitations of the scanning scene it might be impractical or
impossible to acquire projections containing the complete object, resulting
in the need to acquire and handle truncated projections;
� Projection alignment: it might be impossible to determine the position and
the orientation of the detector and the source with the accuracy required for
conventionally used reconstruction algorithms, thus demanding projection
alignment.
In this thesis, the �rst three of the above points are addressed. By no means the
complete solutions are provided. Instead, with the techniques described in the
following chapters, we hope to have made the �rst steps towards the thorough
solution to the challenges posed by in loco tomography.
An ad hoc trajectory can be developed for every particular scanning scene,
such as a translation-based acquisition for an object in a corner of a building [33].
In [52], an algorithm for single-photon emission computed tomography (SPECT) is
proposed that calculates optimal detector positions during acquisition with a free-
hand detector in an intra-operative setting. This algorithm uses a surface model
of a patient to describe available detector positions and selects the next detector
position based on the properties of the projection matrix. Therefore, attempts are
made to optimize the acquisition trajectories, but, to the best of our knowledge,
there is no general method for automatic trajectory selection taking into account
the surroundings of an object being scanned and the related limitations. A good
trajectory selection algorithm might also select the most informative projection di-
rections as not every projection direction is equally bene�cial for a reconstruction
(see [53] and Fig. 1.3). Additionally, trajectory variability complicates the use of
the analytical reconstruction methods with such datasets and favours the iterative
reconstruction algorithms (see Section 1.2.2).
The limited angular range and the limited number of projections in the datasets
are both examples of the limited data problem, presenting signi�cant di�culties for
conventional reconstruction algorithms, such as FBP and SIRT (Fig. 1.2). A num-
ber of algorithms have been developed to deal with the small number of projections
or the limited angular range [54�58] and it is still the �eld of active research. One
24
1.4. CURRENT DEVELOPMENTS AND CHALLENGES IN IN LOCO
TOMOGRAPHY
(a) 50 projections (b) FBP reconstruction (c) SIRT reconstruction
(d) Limited angular range (e) FBP reconstruction (f) SIRT reconstruction
Figure 1.2: Examples of the datasets with the limited data. (a) Sinogram from 50 projections,(d) sinogram from a limited angular range and the corresponding FBP (b, e) and SIRT (c,f) reconstructions. Projections from the complete dataset (Fig. 1.1b), not involved in thereconstruction, are shown with lower contrast.
possible way to handle the limited data problem is including extra knowledge
about the object into the reconstruction. For example, if the object is known to
consist of a number of homogeneous materials, than the total variation minimiza-
tion algorithm [59], promoting sparsity of the image derivative magnitude, is a
very suitable reconstruction algorithm. If, in addition, we know the number and
the attenuations of the materials in the object, discrete tomography [60] can come
into play. One particularly important for this thesis algorithm of this kind is Dis-
crete Algebraic Reconstruction Technique (DART) (see Chapter 4 and [61]), which
can e�ectively handle reconstruction problems with the limited data. Exploiting
the prior knowledge about the materials in the object, DART alternates the seg-
mentation steps and the reconstruction only in a set of boundary pixels, allowing
to obtain accurate reconstructions from the datasets with the small number of
projections or acquired from the limited angular range [62, 63].
25
CHAPTER 1. INTRODUCTION
1.5 Proposed steps towards In Loco Tomography
In this thesis, three techniques are presented that contribute towards the develop-
ment of acquisition and reconstruction algorithms for mobile tomographic devices
capable of in loco tomography.
The �rst technique, the variable distance approach, is a step towards an auto-
matic trajectory selection for tomographic data acquisition. It is inspired by the
observation that for an elongated object, such as one presented in Fig. 1.1a, the
detector is not optimally exploited from a circular trajectory, leaving signi�cant
parts of the detector unused (visible in Fig. 1.1b as the black areas adjacent to the
left or right edge of the image in some rows). Using the prior knowledge about
the shape and the size of the object to calculate optimal source positions for the
given projection directions allows to improve the reconstruction quality for such
objects.
(a) Phantom (b) Start at 10◦ (c) Start at 20◦ (d) Start at 30◦
Figure 1.3: In�uence of the projection angles on the reconstruction quality. SIRT reconstruc-tions (b-d) of a phantom (a) using 5 equiangular projections with di�erent starting angles.Projection angles are marked with black arrows.
Just as the variable distance approach, the dynamic angle selection algorithm
aims at maximizing the information contained in an acquired projection, although
from a di�erent perspective. Since some projection angles can be more bene�cial
for the reconstruction of the particular object than the others (Fig. 1.3) and being
able to acquire only a limited number of projections, we might want to acquire
the most informative ones. The algorithm uses already acquired projection data
to dynamically select the next projection angle delivering the biggest amount of
information.
The multiresolution DART algorithm is, in contrast to the above mentioned
techniques, a reconstruction algorithm. Based on DART, this algorithm for dis-
crete tomography reconstructs the given dataset on a coarse reconstruction grid
26
1.6. OUTLINE OF THIS THESIS
and then resamples this reconstruction to use it as an initial point for a new recon-
struction process on a �ner grid. This is repeated iteratively until the target pixel
size is reached allowing to create more accurate reconstructions signi�cantly faster
than DART. The biggest improvement is achieved for the reconstruction problems
with the limited data, making multiresolution DART a good candidate for the use
in in loco tomography.
1.6 Outline of this Thesis
Chapter 1 contains a general introduction.
Chapter 2 presents the variable distance approach, an algorithm that uses the
prior knowledge about the convex hull of an object to modify the circular
trajectory such that the detector is fully exploited for every projection angle.
An experiment using a desktop micro-CT system and a piece of a pencil as
an object demonstrates the ability of the proposed approach to reveal more
details in the object compared to the conventional circular trajectory.
Chapter 3 presents the dynamic angle selection, an algorithm that uses the al-
ready measured projection data to select next projection angles that max-
imize the information gain. A number of simulation experiments show the
ability of the algorithm to select projections leading to more accurate recon-
structions compared to the conventional angle selection strategies.
Chapter 4 presents the multiresolution DART algorithm, which can yield accu-
rate reconstructions in only a fraction of time compared to DART, therefore
making its use for large experimental datasets more feasible. The algorithm
is applied to two micro-CT datasets and to a synchrotron dataset.
Chapter 5 draws general conclusions.
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32
2Variable Distance Approach
Contents
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Noiseless simulations in two dimensions . . . . . . . . . 37
2.3.2 Noiseless simulations in three dimensions . . . . . . . . . 40
2.3.3 Simulations with noise . . . . . . . . . . . . . . . . . . 44
2.3.4 Real experiment . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
This chapter has been published as
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �Adaptive zooming in X-ray
computed tomography,� Journal of X-Ray Science and Technology, vol. 22, no. 1,
pp. 77�89, 2014.
Abstract � In computed tomography (CT), the source-detector system com-
monly rotates around the object in a circular trajectory. Such a trajectory does
not allow to exploit a detector fully when scanning elongated objects. A new ap-
proach is proposed, in which the full width of the detector is exploited for every
projection angle. This approach is based on the use of prior information about the
object's convex hull to move the source as close as possible to the object, while
avoiding truncation of the projections. Experiments show that the proposed ap-
proach can lead to more accurate reconstructions and increased spatial resolution
in the object compared to the conventional circular trajectory.
33
CHAPTER 2. VARIABLE DISTANCE APPROACH
2.1 Introduction
In most X-ray computed tomography (CT) acquisition setups, the source-detector
system rotates around the object in a well-de�ned and geometrically simple man-
ner. In micro-CT imaging, for example, a circular source-detector trajectory is by
far the most popular one. The radius of such a trajectory is often chosen so as to
avoid truncation in the acquired projections. That is, the radius is chosen large
enough so that for each angle the full projection of the object is captured by the
detector. However, for elongated objects, a circular trajectory does not allow to
exploit the detector optimally. In [1], it was shown that non-planar trajectories
yield visually better reconstructions than circular trajectories in applications of to-
mosynthesis to breast imaging. In single-photon emission computed tomography
(SPECT), non-circular orbits have been shown to reduce uniformity artefacts [2],
to improve resolution [3, 4], contrast, edge de�nition, and uniformity [5]. Never-
theless, the use of non-conventional trajectories is still almost unexplored.
To improve reconstruction quality, a new approach is proposed in which the
full width of the detector is exploited for every projection angle. To this end,
projections are taken from the smallest possible distances to the object, while
avoiding truncation. This is achieved by calculating the source position for every
projection angle based on prior knowledge about the convex hull of the object. The
proposed approach is integrated into an algebraic reconstruction framework. Possi-
ble applications of this approach include scanning devices with �exible acquisition
geometries and mobile tomography devices. Objects with substantial di�erences
in their dimensions, such as electronic components, can especially bene�t from
scanning based on the proposed approach.
Prior knowledge about the object can come in various forms. A total variation
(TV) minimization algorithm exploits sparsity of image derivative magnitude to
address the few-view, limited-angle and bad-bin reconstruction problems [6]. In
interior tomography, prior knowledge of the grey values within a small area inside
the object is often readily available and can lead to more clinically feasible imag-
ing [7]. In CT scanning protocols assuming repeated imaging, results of the initial
scan(s) can be involved into the reconstruction of the consecutive scans allowing
to signi�cantly reduce the number of projections required [8]. Information about
the edges of the object is shown to improve the reconstruction quality in the case
of the few-view problem [9]. Finally, prior knowledge about the grey values of each
of a few materials forming the object allows to use Discrete Algebraic Reconstruc-
tion Technique (DART), which can yield accurate reconstructions from a small
number of projections or from a small angular range [10]. In all above-mentioned
cases, prior knowledge is involved during the reconstruction. Our approach, on the
contrary, uses the convex hull of the object as a source of information about the
34
2.2. APPROACH
geometry of the object to optimise the acquisition. In practice, an approximation
of the convex hull of the object can be built from a preparatory scan used to plan
the scanning procedure or from CAD models (for industrial objects) [11].
The structure of this chapter is as follows. In Section 2.2 our approach is
explained. Section 2.3 describes experiment setups and presents the reconstruction
results. The approach is discussed in Section 2.4. Finally, conclusions are drawn
in Section 2.5.
2.2 Approach
The idea of the proposed variable distance approach (VDA) is to acquire a pro-
jection for a particular projection angle by placing the X-ray source as close as
possible to the object, while avoiding truncation. In contrast to the circular tra-
jectory approach (CTA), which keeps the source-object distance constant, VDA
allows to fully use the detector and obtain more information from this angle. To
calculate the smallest possible source-object distance, prior information about the
object must be exploited. In our simulations, we use the convex hull of the object
to calculate this distance.
Figure 2.1: Geometry of trajectory calculation in VDA.
35
CHAPTER 2. VARIABLE DISTANCE APPROACH
Consider a cone-beam CT setup with a circular trajectory and a �at-panel
detector (Fig. 2.1), where the source-detector distance is constant. Let (x, y, z)
be a Cartesian coordinate system in R3 which is �xed with respect to the object
and let O denote the centre of rotation. For a given projection angle, denote the
source position on the circular trajectory with S and the corresponding positions
of the detector corners (in sequential order) with D1, D2, D3, D4. Suppose that
the source-detector system can be shifted along the line l containing O and S.
Consider a point P belonging to the pyramid SD1D2D3D4, which assures that
the point P is projected onto the detector. The source position closest to the
point P while avoiding truncation, say S′, then corresponds to a case when P
belongs to one of the faces of the pyramid S′D′1D′2D′3D′4 except for D′1D
′2D′3D′4,
where S′D′1D′2D′3D′4 is obtained from SD1D2D3D4 by translation along l. Denote
s =−→OS, p =
−−→OP and the normal vectors of the faces SD1D2, SD2D3, . . . , SD4D1
as n1, n2, . . . , n4, respectively. Assume that P belongs to S′D′iD′i+1 (1 ≤ i ≤ 4,
D′5 ≡ D′1 for ease of notation), which has ni as its normal vector. Then, the
position vector r of any point in the plane containing S′D′iD′i+1 (and P ) can be
found from
ni · (r − p) = 0. (2.1)
The intersection of l and the plane de�ned by Eq. (2.1) is si,P = ti,Ps, such that
ni · (si,P − p) = 0, (2.2)
which brings one to
ti,P =ni · pni · s
. (2.3)
S′ can be found as−−→OS′ = tPs, where
tP = max1≤i≤4
ti,P . (2.4)
Consider A1A2 . . . An (n ≥ 4), the convex hull of the object. In our experi-
ments, we suppose that the convex hull is a polyhedron, but the idea can be easily
adapted to other cases. The closest possible source position S′ for this convex hull
can be expressed as−−→OS′ = ts, where
t = maxP∈{Ai,A2,...,An}
tP . (2.5)
From Eq. (2.5), the source position that is closest to the object while truncation
is avoided can be computed. Repeating this procedure for every projection angle
yields the desired trajectory.
36
2.3. EXPERIMENTS
2.3 Experiments
In this section, the proposed variable distance approach is demonstrated on a
number of phantoms in two (Section 2.3.1) and three (Section 2.3.2) dimensions
in noiseless simulations, simulations with noise are presented in Section 2.3.3. The
approach is then applied to a real dataset in Section 2.3.4.
2.3.1 Noiseless simulations in two dimensions
Simulation experiments were run using three phantom images (Fig. 2.2) to demon-
strate the proposed approach. Phantom 1 (Fig. 2.2a) is a Siemens star-like phan-
tom. Phantoms 2 and 3 (Fig. 2.2c and Fig. 2.2e) represent a fragment of foam and
a fragment of pencil CT image, respectively. Reconstructions were performed on a
square reconstruction grid of 1024×1024 pixels while the size of the each phantom
was 2048×2048 pixels to reduce the e�ect of the pixelation on the reconstructions.
A number of m equiangular fan-beam projections were computed from the orig-
inal phantoms using Joseph's projection method [12]. The source trajectory for
VDA was calculated according to Eq. (2.5). It is shown in Fig. 2.2 together with
the source trajectory for CTA and the detector centre trajectories. In CTA the
source was placed at the distance corresponding to the maximum distance used
in VDA. A detector with n = 1024 elements was used. The reconstructions were
built with 300 iterations of the Simultaneous Iterative Reconstruction Technique
(SIRT) [13]. Values outside the convex hull were not involved in the reconstruc-
tion. All experiments presented in this thesis were implemented using the ASTRA
toolbox [14].
The quality of the reconstructions was assessed by calculating the mean squared
errors (MSEs) according to
MSE(I , I)
=1
|C|∑
(i,j)∈C
(I (i, j)− I (i, j)
)2
, (2.6)
where I denotes the reconstruction upsampled by splitting each pixel into 2 × 2
pixels and I is the original phantom with the convex hull C. Table 2.1 shows the
obtained numerical results. Fig. 2.3 shows the examples of the reconstructions of
Phantom 1 using CTA and VDA. These reconstructions suggest that VDA can
yield visually better reconstructions, providing clearer feature borders, e. g. for
vertical ray-like parts of the phantom (marked with white arrows in Fig. 2.3b and
Fig. 2.3d). From Table 2.1, it is clear that VDA is only slightly outperformed
by CTA when reconstructing Phantom 3 from 200 projections, providing notably
better �gures for MSE in the remaining cases.
To further compare the proposed approach with its conventional counterpart,
37
CHAPTER 2. VARIABLE DISTANCE APPROACH
(a) Phantom 1
−5000 0 5000−6000
−4000
−2000
0
2000
4000
6000
Pixels
Pix
els
Source (VDA)Source (CTA)Detector centre (VDA)Detector centre (CTA)
(b) Trajectories for Phantom 1
(c) Phantom 2
−4000 −2000 0 2000 4000
−4000
−3000
−2000
−1000
0
1000
2000
3000
4000
Pixels
Pix
els
Source (VDA)Source (CTA)Detector centre (VDA)Detector centre (CTA)
(d) Trajectories for Phantom 2
(e) Phantom 3
−4000 −2000 0 2000 4000
−4000
−3000
−2000
−1000
0
1000
2000
3000
4000
Pixels
Pix
els
Source (VDA)Source (CTA)Detector centre (VDA)Detector centre (CTA)
(f) Trajectories for Phantom 3
Figure 2.2: Phantoms 1�3, 2048× 2048 pixels, and the corresponding trajectories.
38
2.3. EXPERIMENTS
(a) CTA (b) VDA
(c) CTA error (d) VDA error
0 0.1 0.2 0.3 0.4 0.50.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (lp/pixel)
MT
F
CTAVDA
(e) MTF
Figure 2.3: Reconstructions (a, b), error images (c, d) and the modulation transfer function(e) for Phantom 1 (Fig. 2.2a), m = 30. White arrows in (a, b) point to the edges in theobject, which are better reconstructed using VDA compared to CTA.
39
CHAPTER 2. VARIABLE DISTANCE APPROACH
Table 2.1: MSE of the reconstructions of Phantoms 1�3 (shown in Fig. 2.2).
CTA VDA
Phantom 1m = 30 9.10× 10−2 7.25× 10−2
m = 200 1.46× 10−2 1.35× 10−2
Phantom 2m = 30 7.66× 10−2 7.47× 10−2
m = 200 1.34× 10−2 1.31× 10−2
Phantom 3m = 30 1.58× 10−3 1.48× 10−3
m = 200 3.45× 10−4 3.47× 10−4
the modulation transfer functions (MTFs) of CTA and VDA were calculated as
follows. First, the two-dimensional discrete Fourier transforms of the phantom
and the reconstructions obtained using CTA and VDA were computed. Next,
the magnitudes of the Fourier coe�cients for the reconstructions were divided
by the corresponding magnitudes of phantom's Fourier coe�cients. Finally, the
results were integrated for each frequency. Fig. 2.3e presents the resulting MTFs
for Phantom 1, which con�rm the ability of the proposed approach to produce
reconstructions with improved spatial resolution compared to CTA.
2.3.2 Noiseless simulations in three dimensions
Experiments were performed using two phantoms (Fig. 2.4). Phantom 4 (Fig. 2.4a)
is in fact a low resolution version of Phantom 1 (Fig. 2.2a) stacked 512 times
and intersected with an ellipsoid having axial ratios 0.95 : 0.3 : 0.3. Phantom 5
(Fig. 2.4b) represents the same ellipsoid with a lattice-like structure consisting of
voxel representations of spheres inside. Reconstructions were performed on a cubic
reconstruction grid of 256 × 256 × 256 voxels while the size of each phantom was
512 × 512 × 512 voxels. A number of m equiangular cone-beam projections were
computed from the original phantoms using Joseph's projection method [12]. The
source trajectory for VDA (Fig. 2.4c) was calculated according to Eq. (2.5). In
CTA the source was again placed at the distance corresponding to the maximum
distance used in VDA. The detector had 256× 256 elements. The reconstructions
were built with 300 iterations of SIRT. Values outside the convex hull were not
involved in the reconstruction.
The quality of the reconstructions was assessed by a three-dimensional analogue
of Eq. (2.6). Fig. 2.5 and Fig. 2.6 present the resulting reconstructions. Fig. 2.5d
and Fig. 2.6d present the di�erence
D(ICTA, IV DA, I
)=∣∣∣ICTA − I∣∣∣− ∣∣∣IV DA − I∣∣∣ (2.7)
showing, which approach produces the results closer to the phantom. Table 2.2
40
2.3. EXPERIMENTS
(a) Phantom 4 (b) Phantom 5
−1000 −500 0 500 1000
−1500
−1000
−500
0
500
1000
1500
Pixels
Pix
els
Source (VDA)Source (CTA)Detector centre (VDA)Detector centre (CTA)
(c) Trajectories for Phan-toms 4 and 5
Figure 2.4: Cross sections of Phantoms 4 and 5 (a-b), 512× 512× 512 pixels, by the planez = 0.5, and the trajectories used in reconstruction (c) (in the plane z = 0) ((b) windowedto [0.45, 0.55] for better visual contrast).
represents the obtained numerical results. Visually, the results for VDA seem to
be of better quality compared to the results for CTA. In particular, star rays are
better distinguishable in the central part of the image and seem to have better ver-
tical borders in the VDA reconstruction than in the CTA reconstruction (Fig. 2.5)
of Phantom 4. For Phantom 5, inclusions near the tips of the ellipsoid are bet-
ter distinguishable in the VDA reconstruction (Fig. 2.6). Interior of the phantom
looks more uniform on the VDA reconstruction (Fig. 2.6c) than on the CTA re-
construction (Fig. 2.6b), whose artefacts might be confused with actual object
features. The di�erence image (Fig. 2.6d) con�rms these observations. Numerical
results in Table 2.2 show that in terms of MSE VDA clearly outperforms CTA on
both phantoms. The presented results suggest that VDA can provide an ability
to better handle small features in the objects than CTA.
Table 2.2: MSE of the reconstructions of Phantoms 4�5 (shown in Fig. 2.4).
CTA VDA
Phantom 4m = 200 2.27× 10−2 1.99× 10−2
m = 500 1.97× 10−2 1.66× 10−2
Phantom 5m = 200 2.82× 10−3 2.38× 10−3
m = 500 2.77× 10−3 2.29× 10−3
41
CHAPTER 2. VARIABLE DISTANCE APPROACH
(a) Phantom 4 (b) CTA (c) VDA
(d) Error di�erence (e) CTA error (f) VDA error
(g) Phantom 4, magni�ed (h) CTA, magni�ed (i) VDA, magni�ed
Figure 2.5: Cross sections of Phantom 4 and its CTA and VDA reconstructions by the planez = 0.5 (a-c), corresponding error images (e, f), a di�erence image (d) as de�ned by Eq. (2.7),and magni�ed cross sections of Phantom 4, its CTA and VDA reconstructions by the planex = −6.5 (g-i).
42
2.3. EXPERIMENTS
(a) Phantom 5 (b) CTA (c) VDA
(d) Error di�erence (e) CTA error (f) VDA error
(g) Phantom 5, magni�ed (h) CTA, magni�ed (i) VDA, magni�ed
Figure 2.6: Cross sections of Phantom 5 and its CTA and VDA reconstructions by the planez = 0.5 (a-c), corresponding error images (e, f), a di�erence image (d) as de�ned by Eq. (2.7),and magni�ed cross sections of Phantom 5, its CTA and VDA reconstructions by the planex = −137.5 (g-i) ((a-c) and (g-i) windowed to [0.45, 0.55] for better visual contrast).
43
CHAPTER 2. VARIABLE DISTANCE APPROACH
2.3.3 Simulations with noise
In order to evaluate the proposed approach in more realistic situations the exper-
iments shown in Sections 2.3.1 and 2.3.2 were extended with noise simulations as
follows. Consider a monochromatic X-ray tube which emits Ns photons towards a
detector element with the area of 1 square unit placed perpendicularly to the beam
(to the line connecting the point X-ray source and the centre of the element) at
the distance of ds from the source. Then, the average number of photons reaching
the detector element E at the distance of d is
N = N0e−
∫µ(ξ) dξ = N0e
−A = N0e−k
∫g(ξ) dξ =
Nsd2sS cosα
d2e−k
∫g(ξ) dξ, (2.8)
where A is the ray integral calculated for the detector element E with no noise
introduced, g (ξ) is the grey level of the phantom in the point ξ, k is the scaling
coe�cient which matches g (ξ) with the attenuation coe�cient µ (ξ) of the object
(k is assumed to be 1/100 in our simulations), N0 is the number of photons emitted
towards the considered detector element of the area of S with α being the angle
between the normal of the element and the beam (scattered photons were ignored).
Then, the actual number of photons N ′ counted by E can be selected according to
Poisson statistics [15]. The noisy ray integral for the element E can be calculated
by
A′ = − lnN ′
N0. (2.9)
For each phantom from Sections 2.3.1 and 2.3.2, Eq. (2.8) and Eq. (2.9) were
applied to the noiseless projections to obtain K = 10 noisy sets of projection
data (values Ns = 105 and Ns = 106 were used to represent di�erent noise levels
and ds was equal to the source-object distance used in CTA in all simulations of
this section). For each noisy projection dataset the reconstructions were built as
described earlier and the mean values ofMSE(I , I)over these K reconstructions
were gathered into Table 2.3, from which we see that VDA can yield better results
in the presence of noise than CTA. For none of these cases, the latter outperforms
VDA numerically, yielding reconstructions with visually similar or lower quality
as it was already described in Section 2.3.2 in noiseless simulations.
2.3.4 Real experiment
To mimic a tomographic system with variable source and detector position, the
following experiment was conducted using a desktop micro-CT system SkyScan-
1172 (Bruker-MicroCT, Belgium). A piece of a pencil with a diameter of 7 mm
and a length of 15 mm was used as an elongated object. For this object, seven
full-angle datasets were obtained, each containing 600 images of 880× 666 pixels,
44
2.3. EXPERIMENTS
Table 2.3: MSE of the noisy reconstructions of Phantoms 1�5, as described in Section 2.3.3.
CTA VDA
Phantom 1, Ns = 105 m = 30 1.07× 10−1 8.04× 10−2
m = 200 1.88× 10−2 1.86× 10−2
Phantom 1, Ns = 106 m = 30 9.26× 10−2 7.33× 10−2
m = 200 1.51× 10−2 1.42× 10−2
Phantom 2, Ns = 105 m = 30 8.06× 10−2 8.00× 10−2
m = 200 1.84× 10−2 1.73× 10−2
Phantom 2, Ns = 106 m = 30 7.71× 10−2 7.53× 10−2
m = 200 1.39× 10−2 1.35× 10−2
Phantom 3, Ns = 105 m = 30 6.34× 10−3 6.28× 10−2
m = 200 4.82× 10−3 4.07× 10−3
Phantom 3, Ns = 106 m = 30 2.16× 10−3 1.96× 10−3
m = 200 8.05× 10−4 7.33× 10−4
Phantom 4, Ns = 105 m = 200 2.40× 10−2 2.07× 10−2
m = 500 2.02× 10−2 1.70× 10−2
Phantom 4, Ns = 106 m = 200 2.28× 10−2 2.00× 10−2
m = 500 1.98× 10−2 1.66× 10−2
Phantom 5, Ns = 105 m = 200 4.05× 10−3 3.25× 10−3
m = 500 3.27× 10−3 2.64× 10−3
Phantom 5, Ns = 106 m = 200 2.95× 10−3 2.47× 10−3
m = 500 2.82× 10−3 2.32× 10−3
with the source-object distances ranging from 80.77 to 117.01 mm. The source-
detector distance was 216.392 mm. A dataset obtained from the biggest distance
was used during the reconstruction with CTA. Based on the CTA reconstruction,
an approximate convex hull for VDA was created. In VDA for each projection angle
the closest possible source position was calculated for this convex hull according to
Eq. (2.5) and a projection was chosen from the dataset obtained from the smallest
distance bigger than or equal to the distance from the calculated source position
to the centre of rotation. Resulting trajectories are presented in Fig. 2.7.
Both the CTA and VDA reconstructions were performed on the 880×880×666
voxels reconstruction grid with a voxel size of 19.4 µm using 700 iterations of
SIRT. Fig. 2.8 presents the reconstructions. Visually, both reconstructions seem
to have comparable quality. The area surrounding the biggest dense particle in the
bottom right part of the object contains mild streaks in the VDA reconstruction
whereas in the CTA reconstruction there are two darker areas. Such di�erence in
the artefacts around dense particles can be explained by the anisotropic sampling
in VDA. However, further research of this question is required. Fig. 2.8, and
hence Fig. 2.9, might be in�uenced by the mechanical stability of the system, in
45
CHAPTER 2. VARIABLE DISTANCE APPROACH
−3000 −2000 −1000 0 1000 2000 3000
−3000
−2000
−1000
0
1000
2000
3000
Pixels
Pix
els
Source (VDA)Source (CTA)Detector centre (VDA)Detector centre (CTA)
Figure 2.7: Trajectories used for the reconstruction of the pencil.
(a) CTA (b) VDA
Figure 2.8: Reconstructions of the central (containing optical axis) slice of the pencil withCTA and VDA (voxel size 19.4 µm). White squares mark the regions shown enlarged inFig. 2.9.
46
2.4. DISCUSSION
particular, the vertical alignment of the sample stage compared to the optical axis
of the system.
Figures 2.9a-2.9b present the enlarged portions of the reconstructions shown
in Fig. 2.8, where a border between wood and graphite seems to have better
contrast in the VDA reconstruction and two dense particles in the middle of the
image are easier visually distinguishable. Fig. 2.9c presents the reconstruction of
the same region obtained with FDK [16] implemented in NRecon software [17]
(Bruker-MicroCT) from the dataset with the smallest source-object distance, the
voxel size is 13.4 µm. As the region lies far from the edge of the �eld-of-view,
artefacts caused by the truncated projections are negligible in this part of the
NRecon reconstruction. We therefore consider the NRecon reconstruction as the
ground truth for this region. This reconstruction shows that the above mentioned
di�erences in the CTA and VDA reconstructions are not the artefacts of the latter,
but rather the features truly presented in the object. Hence, experimental studies
agree with the simulations described in Section 2.3.2, showing the ability of VDA
to produce reconstructions which are superior to those produced by CTA in the
realistic setup.
(a) CTA (b) VDA (c) NRecon
Figure 2.9: Comparison of the CTA and VDA reconstructions of the pencil. Enlarged frag-ments of the CTA (a) and VDA (b) reconstructions of the central slice (Fig. 2.8), with whitearrows marking the features visible in the VDA and not visible in the CTA reconstruction. Thesame region reconstructed with FDK implemented in NRecon software (Bruker-MicroCT) (c)(voxel size 13.4 µm), con�rming that the features are truly present in the object.
2.4 Discussion
The proposed approach allows to exploit prior knowledge of the object's shape and
size to optimize the detector usage and to obtain more detailed information when
scanning an elongated object, increasing the reconstruction quality. An approxi-
47
REFERENCES
mate convex hull of the object can be created from a preparatory scan (in clinical
CT), a series of pictures of the object in optical range (in micro-CT) or a CAD
model (in industrial CT). Implementation of the source position selection algorithm
is straightforward and easily adaptable to various setups, e. g. systems with con-
stant object-detector distance (rather than a system with constant source-detector
distance, considered in the chapter). The data collected can be immediately re-
constructed with an algebraic reconstruction procedure, while analytical methods
require rebinning, possibly leading to loss of quality.
Possible applications of the proposed approach include mobile tomographic
devices for use in the �eld and tomography of objects that have substantial dif-
ferences in all three dimensions, such as electronic components. Currently these
objects are imaged in helical or cone beam stacked mode and the source-object
distance is de�ned by the second biggest dimension, no matter how small the
third one is. Use of the variable distance approach will allow to better exploit the
dimension di�erences in this case.
2.5 Conclusions
We proposed the variable distance approach for fan- and cone-beam CT scanning.
This approach is based on the modi�cation of the classic circular trajectory ac-
cording to prior information about the object's convex hull which is used to take
projections from as small as possible distances to the object for every projection
angle providing that the truncation is avoided. Our experiments showed that the
proposed approach can lead to more accurate reconstructions with lower errors.
Reconstruction of the real dataset demonstrated an ability of the approach to
reveal more details in the object compared to the conventional circular trajectory.
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[9] M. Fedrigo, A. Wenger, and C. Hoeschen, �Investigating tomographic reconstructionwith a priori geometrical information,� Journal of X-Ray Science and Technology,vol. 20, no. 1, pp. 1�10, 2012.
[10] K. J. Batenburg and J. Sijbers, �DART: A practical reconstruction algorithm fordiscrete tomography,� IEEE Transactions on Image Processing, vol. 20, no. 9, pp.2542�2553, 2011.
[11] A. Laurentini, �The visual hull concept for silhouette-based image understanding,�IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 2, pp.150�162, 1994.
[12] P. M. Joseph, �An improved algorithm for reprojecting rays through pixel images,�IEEE Transactions on Medical Imaging, vol. 1, no. 3, pp. 192�196, 1982.
[13] J. Gregor and T. Benson, �Computational analysis and improvement of SIRT,� IEEETransactions on Medical Imaging, vol. 27, no. 7, pp. 918�924, 2008.
[14] W. J. Palenstijn, K. J. Batenburg, and J. Sijbers, �The ASTRA tomography tool-box,� in 13th International Conference on Computational and Mathematical Methods
in Science and Engineering (CMMSE), 2013.
[15] H. Q. Guan and R. Gordon, �Computed tomography using algebraic reconstructiontechniques (ARTs) with di�erent projection access schemes: A comparison studyunder practical situations,� Physics in Medicine and Biology, vol. 41, no. 9, pp.1727�1743, 1996.
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[16] L. A. Feldkamp, L. C. Davis, and J. W. Kress, �Practical cone-beam algorithm,�Journal of the Optical Society of America A: Optics, Image Science, and Vision,vol. 1, no. 6, pp. 612�619, 1984.
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tion in Radiology and Nuclear Medicine (Fully 3D), Salt Lake City, USA, 2005, pp.215�218.
50
3Dynamic Angle Selection
Contents
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Information gain . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Upper bound for the diameter of the solution set . . . . 54
3.2.3 Surrogate solutions . . . . . . . . . . . . . . . . . . . . 55
3.2.4 Candidate angles . . . . . . . . . . . . . . . . . . . . . 55
3.2.5 Dynamic angle selection algorithm . . . . . . . . . . . . 55
3.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 Randomly oriented bars . . . . . . . . . . . . . . . . . . 57
3.3.2 Wood phantoms: noiseless simulations . . . . . . . . . . 59
3.3.3 Wood phantom: simulations with noise . . . . . . . . . 61
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
This chapter has been published as
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �Dynamic angle selection in X-
ray computed tomography,� Nuclear Instruments and Methods in Physics Research
Section B: Beam Interactions with Materials and Atoms, vol. 324, pp. 17�24, 2014.
Abstract � In X-ray tomography, a number of radiographs (projections) are
recorded from which a tomogram is then reconstructed. Conventionally, these pro-
jections are acquired equiangularly, resulting in an unbiased sampling of the Radon
space. However, especially in case when only a limited number of projections can
be acquired, the selection of the angles has a large impact on the quality of the
51
CHAPTER 3. DYNAMIC ANGLE SELECTION
reconstructed image. In this chapter, a dynamic algorithm is proposed, in which
new projection angles are selected by maximizing the information gain about the
object, given the set of possible new angles. Experiments show that this approach
can select projection angles for which the accuracy of the reconstructed image is
signi�cantly higher compared to the standard angle selections schemes.
3.1 Introduction
Tomography has applications ranging from 3D imaging of nano-materials by elec-
tron microscopy to the reconstruction of accretion disks from astronomical ob-
servations. In many of these applications, it is highly desirable to reduce the
number of projections taken, or it is even impossible to acquire many projections.
In image-guided radiotherapy, for example, a patient is being imaged for several
times posing a serious radiation safety concern [1]. In astro-tomography, only a few
satellites are capable of imaging the corona of the sun, leading to long acquisition
times. In electron tomography, the electron beam gradually damages the object,
also imposing a restriction on the number of projections that can be acquired [2].
When an image is being reconstructed from a small number of projections, the
angles from which these projections will be acquired will signi�cantly in�uence the
reconstruction quality. In [3], it was shown that the quality of the reconstructions
can be highly dependent on the projection angles in binary tomography. In that
paper, an algorithm was proposed for identifying optimal projection angles based
on a blueprint image known to be similar to the scanned object, which can be
readily applied in the �eld of nondestructive testing. For the more general case of
grey scale tomography, a framework was proposed in [4], which allows to optimize
the set of projection angles based on certain prior knowledge about the object.
In [5], an algorithm was proposed to select new projection angles based on the
quanti�cation of the projection information content using an entropy-like function
of the already acquired projections. For tomography of elliptical objects, a genetic
algorithm was proposed [6], which exploits the preferential direction characteristic
of the objects and uses reconstructions from available projections to select the
next projection directions. In [7], a new strategy was recently proposed for angle
selection in binary tomography, which is based on the concept of information gain
from adding a particular projection angle to the set of projection directions and
does not require specifying prior knowledge about the object.
In this chapter, the dynamic angle selection strategy for binary object scanning
is adapted for use in grey scale tomography. It is a dynamic algorithm, which se-
lects a new angle based on the currently available projection data and incorporates
two major concepts: 1) sampling of the set of images that are consistent with the
already acquired projection data and 2) determining the amount of information
52
3.2. METHOD
that can be gained by acquiring a projection from a particular angle.
The structure of this chapter is as follows. In Section 3.2 our approach is
explained. Section 3.3 describes experiment setups and presents the obtained
results. The approach is discussed in Section 3.4. Finally, conclusions are drawn
in Section 3.5.
3.2 Method
3.2.1 Information gain
The idea of the proposed angle selection algorithm is to select a new projection
direction in such a way that the newly obtained projection will contain as much
information about the object as possible. As a measure of information, a concept
of information gain is used, which is based on the diameter of the set of solutions
that are consistent with already obtained projections [7]. For clarity and easy
reference, the concept of information gain is brie�y explained in this section.
Let Θ = {θ1, . . . , θd} be the current set of d angles, for which projection data
pΘ = WΘv of the unknown image v ∈ [0, 1]nhave already been measured, where n
is the number of pixels in the image, WΘ =(wΘij
)∈ Rm×n is the projection matrix
corresponding to Θ, and m is the total number of measurements in the projection
data pΘ ∈ Rm. Note that if the assumption on the range of the grey values of
the unknown image is not satis�ed in practice, a preprocessing step is needed to
make the assumption valid, which is discussed in Section 3.4. Let SWΘ
(pΘ)
={x ∈ [0, 1]
n: WΘx = pΘ
}be the set of all solutions that are consistent with the
projection data pΘ. Then, the information gain for any image x and set of angles
Θ yielded by taking a projection from angle θ is de�ned by
G (x,Θ, θ) = diam(SWΘ
(WΘx
))− diam
(SWΘ∪{θ}
(WΘ∪{θ}x
)), (3.1)
where diam (V ) = max {‖x− y‖2 |x, y ∈ V } for any V ⊂ [0, 1]n. This de�nes the
information gain as the di�erence of the diameters of the sets of all images having
the same projections as x. Having de�ned the mean information gain of a set of
images V ⊂ [0, 1]nas
G (V,Θ, θ) =
√∫VG (x,Θ, θ)
2dx∫
Vdx
, (3.2)
the next projection angle can be found as
θd+1 = arg maxθ∈[0,π)
G(SWΘ
(pΘ),Θ, θ
). (3.3)
53
CHAPTER 3. DYNAMIC ANGLE SELECTION
In practice, the integrals in Eq. (3.2) have to be approximated. In this chapter,
three approximation steps are proposed. Firstly, the diameter of the solution set in
Eq. (3.1) is substituted with its upper bound which was proposed in [8, 9] for binary
solutions and which also holds for the solutions belonging to [0, 1]n(Theorem 1).
Secondly, integration in Eq. (3.2) is replaced by a summation over a set of surrogate
solutions, which represent the true set SWΘ
(pΘ). Finally, the continuous domain
of the candidate angle θd+1 is substituted with a �nite set of candidate angles,
updated every time a new angle is chosen. These steps are explained in detail in
the following sections.
3.2.2 Upper bound for the diameter of the solution set
The upper bound for the diameter used in Eq. (3.1) is based on the concept
of the central reconstruction x∗, which is the shortest solution in SWΘ
(pΘ), in
the Euclidean sense. This reconstruction can be computed using the Conjugate
Gradient Least Squares (CGLS) method, an iterative Krylov subspace method [10].
De�ne the central radius by R =
√‖pΘ‖1d − ‖x∗‖22. Then, the following theorem
allows to �nd an upper bound for the Euclidean distance between two solutions
from SWΘ
(pΘ).
Theorem 1 Let x, y ∈ SWΘ
(pΘ). Then ‖x− y‖2 ≤ 2R.
To proof this theorem, we start with the following
Lemma 1 Let x ∈ SWΘ
(pΘ). Then ‖x‖22 ≤
‖pΘ‖1
d .
Proof of Lemma 1. For the strip projection model,∑mi=1 w
Θij = d (j = 1, . . . , n),
as the total pixel weight for each projection angle is equal to the area of a pixel,
which is 1, and there are d projection angles. Since pΘi ≥ 0 (i = 1, . . . ,m), we have∥∥pΘ
∥∥1
=∑mi=1 p
Θi and
m∑i=1
pΘi =
m∑i=1
n∑j=1
wΘijxj
=
n∑j=1
(m∑i=1
wΘij
)xj =
n∑j=1
dxj ,
and hence∥∥pΘ
∥∥1
= d∑nj=1 xj .
As x ∈ [0, 1]n, ‖x‖22 ≤ ‖x‖1 =∑nj=1 xj =
‖pΘ‖1
d .
�Proof of Theorem 1. From the de�nition of x∗ we have (x− x∗) ∈ N
(WΘ
)and x∗ ⊥ (x− x∗). Using the Pythagoras' theorem and Lemma 1 yields
‖x− x∗‖22 = ‖x‖22 − ‖x∗‖22 ≤
∥∥pΘ∥∥
1
d− ‖x∗‖22 = R2.
54
3.2. METHOD
Therefore,
‖x− y‖2 ≤ ‖x− x∗‖2 + ‖y − x∗‖2 ≤ 2R.
�
3.2.3 Surrogate solutions
In order to evaluate the mean information gain de�ned by Eq. (3.2), the frac-
tion under the square root is replaced by the mean information gain over the set
of surrogate solutions, which are used as samples representing the true solution
set SWΘ
(pΘ). A surrogate solution is calculated from a template image, a ran-
domly generated member of a given parameterized family of images. This template
image is then used as a starting point for the Simultaneous Iterative Reconstruc-
tion Technique (SIRT) [11] that computes the surrogate solution consistent with
already obtained projection data pΘ. As the system WΘx = pΘ is severely un-
derdetermined, the surrogate solution partly retains the features present in the
template image. Hence, allowing su�cient variation within the set of template im-
ages results in variation of the surrogate solutions obtained and allows to control
the approximation of the true solution set SWΘ
(pΘ).
3.2.4 Candidate angles
An adaptive approach is proposed to modify the set of angles being considered
at the subsequent angle selection steps. Let Ad+1 = {α1, α2, . . . , αl} , 0 ≤ α1 ≤α2 ≤ . . . ≤ αl < π be the set of candidate angles for selecting the next angle
θd+1. Suppose that αi is the best angle and θd+1 := αi. Then, let the can-
didate angle set used for the selection of the angle θd+2 be de�ned as Ad+2 ={α1, . . . , αi−1,
αi−1+αi2 , αi+αi+1
2 , αi+1, . . . , αl
}. This procedure allows to better
sample the candidate angle space near the angles which are likely to reveal more
details in the object and still leaves the possibility for choosing completely new
directions.
3.2.5 Dynamic angle selection algorithm
Combining all the approximation steps, an algorithm for estimating the mean
information gain for a candidate angle can be de�ned (Alg. 1). Based on this
algorithm, the proposed angle selection approach iterates over the set of candidate
angles and chooses the angle yielding the highest mean information gain.
55
CHAPTER 3. DYNAMIC ANGLE SELECTION
Algorithm 1 Computing the mean information gain for a candidate angle θ,based on K surrogate solutions
Input: Θ = {θ1, . . . , θd}, pΘ, θx∗ = CGLS
(WΘ, pΘ
); //compute central reconstruction
D = 2
√‖pΘ‖1d − ‖x∗‖22; //compute the upper bound of the solution set diameter
Φ = Θ ∪ {θ}; //include the candidate angle into the set of projection anglesfor i := 1 to K do //loop over K surrogate solutions
x = generateSurrogate(Θ, pΘ
);
pΦ = WΦx; //calculate new projection data that includes the projection forthe candidate angle
x∗ = CGLS(WΦ, pΦ
); //compute the new shortest solution
Di = 2√‖pΦ‖1d+1 − ‖x∗‖
22; //compute the new upper bound
end
Output: G =
√1K
∑Ki=1
(D − Di
)2
//the mean information gain
3.3 Experiments
Simulation experiments were run to assess the ability of the proposed algorithm to
select favourable projection angles. The size of the phantoms was limited to 128×128 pixels due to the computational complexity of the approach. A quantitative
evaluation of the proposed algorithm is based on the assumption that a good angle
selection scheme will lead to a more accurate reconstruction from fewer angles
compared to a reconstruction from angles chosen by a standard selection scheme.
For all presented experiments, the template images used for the generation of
the surrogate solutions were created as a superposition of 50 2D Gaussian blobs
with randomly chosen orientation and standard deviation along both axes between
3 and 10 pixels. For the selection of each angle, K = 10 surrogate solutions were
generated. Angles with a step of 10◦ were chosen for the initial set of candidate
angles, which was then modi�ed as described in Section 3.2.4.
Three angle selection schemes were chosen as antagonists for the proposed
algorithm:
� Standard. A widely used strategy, in which angles are selected between 0◦
and 180◦, with equiangular spacing. Changing the number of angles actually
changes the entire set of selected angles, which explains the �uctuations of
the numerical results for this strategy.
� Gap-angle. In the gap-angle scheme, a new angle is selected as the midpoint
between the two consecutive angles with the largest angular gap between
56
3.3. EXPERIMENTS
them. If several pairs of angles have equal gaps, one of them is chosen
randomly.
� Entropy-based. In this approach, a new angle is selected based on the �en-
tropy� of the already measured projections. Assuming that∑mi=1 p
θi = 1,
the entropy Eθ for the projection pθ is de�ned as Eθ = −∑mi=1 p
θi log pθi ,
where m is the number of detector elements and pθi log pθi = 0 for pθi = 0.
The next projection is measured between two projections that present the
maximum di�erence in the entropy. For a detailed description and analysis
of this approach, the reader is referred to [5].
Nine angle sets were used as a starting point for all algorithms, containing two
perpendicular angles and having an angular shift of 10◦ with respect to the pre-
vious initial angle set, giving 9 starting con�gurations. For each of the starting
con�gurations, angles were selected with the schemes under consideration and the
selected angles were then used to compute reconstructions using 250 iterations of
SIRT. The mean values of the mean squared errors (MSEs) of the reconstructions
for all starting con�gurations were then calculated and plotted together with the
standard errors (shown as shaded areas in the plots). Two sets of experiments were
run, revealing the ability of the algorithms to deal with phantoms having clear di-
rection preferences and to handle more realistic phantoms, which are described
in Sections 3.3.1 and 3.3.2. For one phantom from Section 3.3.2 simulations with
varying noise levels were also performed, which are described in Section 3.3.3. All
presented experiments were implemented using the ASTRA toolbox [12] where
GPU acceleration was used extensively [13].
3.3.1 Randomly oriented bars
For the �rst series of experiments, a set of phantoms was created. A phantom in
this set consists of six rectangular bars, each oriented along one of the angles from{180◦i
7 , i = 1, . . . , 6}with respect to the vertical edge of the phantom. The number
of di�erent bar-edge angles in the phantom de�nes the orientation of the phantom.
Thus, in the phantom with one orientation, the direction of the bars is the same. In
the phantom with six orientations, the orientation of each bar is randomly chosen
from the set{
180◦i7 , i = 1, . . . , 6
}. For the number of orientations from one to
four, six phantoms were created with randomly chosen bar-edge angles, giving 24
phantoms in total. Examples of the phantoms with one, two and four orientations
are shown in Fig. 3.1a, 3.1b and 3.1c, respectively. Fig. 3.2a-3.2d present average
MSEs for the phantoms with equal number of orientations. These plots suggest
that the proposed approach has a clear advantage over the other three strategies for
the phantoms with one or two orientations and shows comparable or worse results
57
CHAPTER 3. DYNAMIC ANGLE SELECTION
(a) Oneorientation
(b) Twoorientations
(c) Fourorientations
Figure 3.1: Examples of the phantoms with one (a), two (b) and four (c) orientations, usedin the experiments of Section 3.3.1.
for phantoms with a larger number of orientations, showing strong dependency
on the number of orientations in the phantom, whereas the performance of its
antagonists demonstrates little to no such dependency.
To illustrate each step of the algorithm, the selection of one angle is considered
in detail for the phantom shown in Fig. 3.1a. Projections from three angles Θ =
{0◦, 80◦, 90◦} are already available for the phantom with one orientation (Fig. 3.1a)
and the corresponding reconstruction is shown in Fig. 3.3b. Fig. 3.3a presents the
average information gain. Note that zeros in this plot correspond to the already
acquired projections. Fig. 3.3c and 3.3d show the possible reconstructions for cases
when angles 50◦ and 110◦ are chosen as the fourth projection angle. Examples
of the reconstructions from �ve angles yielded by the considered approaches are
presented in Fig. 3.4. Fig. 3.5 depicts the distribution of 20 angles selected with
each algorithm.
The selection of one angle for the phantom shown in Fig. 3.1b, is illustrated
in Figs. 3.6-3.7. Projections from three angles Θ = {0◦, 90◦, 130◦} are already
available for the phantom, the corresponding reconstruction is shown in Fig. 3.6b
and Fig. 3.6a shows the average information gain. Fig. 3.6c and 3.6d present the
possible reconstructions for cases when angles 70◦ and 125◦ are chosen as the next
projection angle. Fig. 3.7 depicts the distribution of 20 angles selected with each
algorithm.
This examples illustrate the correspondence between the average information
gain and the reconstruction quality, con�rming the ability of the proposed approach
to select projection angles according to the directions presented in the object and
to yield more accurate reconstructions compared to the standard angle selection
schemes. Note that while the global maximum in the average information gain
agrees with the bar orientation in the phantoms, local maxima may (Fig. 3.6a) or
may not (Fig. 3.3a) indicate such orientation.
58
3.3. EXPERIMENTS
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
0.06
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(a) One orientation
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
0.06
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(b) Two orientations
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
0.06
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(c) Three orientations
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
0.06
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(d) Four orientations
Figure 3.2: MSE as a function of the number of projection angles for phantoms with one tofour orientations.
3.3.2 Wood phantoms: noiseless simulations
For the second series of the experiments, six phantoms (Fig. 3.8) were created
from micro-CT (Fig. 3.8a-3.8d) and scanning electron microscope (Fig. 3.8e and
3.8f) images of wood samples. Although these phantoms have clear preferential
directions, they are far more complex compared to the bar phantoms described in
Section 3.3.1. The phantoms include �ne structures with (Fig. 3.8a-3.8d) or with-
out (Fig. 3.8e and 3.8f) preferential directions and have two di�erent shapes which
either have or do not have preferential directions. Fig. 3.8b and 3.8d were obtained
by rotating Fig. 3.8a and 3.8c, respectively. However, as the phantoms were de-
�ned on a pixel grid, some interpolation occurred during the rotation, resulting in
minor di�erences in the results for the proposed algorithm compared to the results
for the non-rotated phantoms. Fig. 3.9 presents examples of the reconstructions of
59
CHAPTER 3. DYNAMIC ANGLE SELECTION
0 50 100 1500
0.1
0.2
0.3
0.4
0.5
Angle (degrees)
Info
rmat
ion
gain
(a) Information gain
(b) Currentreconstruction
(c) Red angleadded
(d) Greenangle added
Figure 3.3: Selection of the fourth angle for the phantom shown in Fig. 3.1a. Informationgain (a) and reconstruction from three already obtained projections (b). The reconstructionsfrom four projections are shown in (c) and (d), where the fourth projection was selected asindicated in the information plot by the red and green dot.
(a) Standard (b) Gap-angle (c)Entropy-based
(d) Dynamic
Figure 3.4: Reconstructions of the phantom shown in Fig. 3.1a from �ve projection directionsselected with the standard, gap-angle, entropy-based and dynamic algorithms.
60
3.3. EXPERIMENTS
0
2
4
30°
60°
90°
120°
150°180°
0°
(a) Standard
0
2
4
30°
60°
90°
120°
150°180°
0°
(b) Gap-angle
0
2
4
30°
60°
90°
120°
150°180°
0°
(c) Entropy-based
0
2
4
30°
60°
90°
120°
150°180°
0°
(d) Dynamic
Figure 3.5: Distribution of the �rst 20 angles selected with the standard, gap-angle, entropy-based and dynamic algorithms for the phantom shown in Fig. 3.1a. Bin size is 5◦. Blackarrows indicate feature orientation in the object.
the phantom shown in Fig. 3.8c and Fig. 3.10 shows the MSEs for Fig. 3.8a-3.8f,
which demonstrate the ability of the proposed approach to yield projection angle
sets resulting in more accurate reconstructions from fewer projections. The results
for Fig. 3.8b and 3.8d suggest that the dynamic algorithm accurately reconstructs
the rotated objects as well as the unaltered objects. In general, the dynamic angle
selection algorithm outperforms the other methods with respect to reconstruction
quality. Note that the shape of the phantoms also plays a signi�cant role in the
performance of the proposed algorithm, allowing it to reduce the MSE faster as a
function of the number of projections for the phantoms that have a square shape
compared to circular phantoms.
3.3.3 Wood phantom: simulations with noise
To evaluate the proposed approach in more realistic situations, the experiments
shown in Section 3.3.2 were extended with noise simulations as follows. Poisson
noise was simulated in the projection data based on a number of counts Ns for the
phantom shown in Fig. 3.8d from Section 3.3.2. The values Ns = 105, Ns = 106
and Ns = 107 were used to simulate the di�erent noise levels [14]. The number of
SIRT iterations was 250. The mean values of the mean squared errors (MSEs) of
the reconstructions as a function of the number of projections were then calculated
and plotted together with the standard errors (shown as shaded areas in the plots)
in Fig. 3.11. Fig. 3.12 shows the examples of the reconstructions for Ns = 105.
These results suggest that the proposed algorithm is robust against noise and
can provide projection angle sets leading to more accurate reconstructions in the
61
CHAPTER 3. DYNAMIC ANGLE SELECTION
0 50 100 1500
0.5
1
1.5
2
2.5
3
3.5
Angle (degrees)
Info
rmat
ion
gain
(a) Information gain
(b) Currentreconstruction
(c) Red angleadded
(d) Greenangle added
Figure 3.6: Selection of the fourth angle for the phantom shown in Fig. 3.1b. Informationgain (a) and reconstruction from three already obtained projections (b). The reconstructionsfrom four projections are shown in (c) and (d), where the fourth projection was selected asindicated in the information plot by the red and green dot.
presence of noise. Improvement in the reconstruction quality for the proposed
algorithm decreases compared to the standard angle selection schemes as the level
of noise increases, yet allowing the dynamic approach to outperform other angle
selection strategies for the phantom shown in Fig. 3.8d even in the case of the
highest noise level considered.
3.4 Discussion
The proposed dynamic angle selection algorithm selects projection angles based on
the already measured projection data by maximizing the information gain. The
algorithm does not use any prior knowledge about an object. The only assumption
made is that pixel values fall into the range of [0, 1]. However, even if this assump-
tion is not satis�ed, an upper bound for a pixel value can be calculated based on a
62
3.4. DISCUSSION
0
1
2
3
30°
60°
90°
120°
150°180°
0°
(a) Standard
0
1
2
3
30°
60°
90°
120°
150°180°
0°
(b) Gap-angle
0
1
2
3
30°
60°
90°
120°
150°180°
0°
(c) Entropy-based
0
1
2
3
30°
60°
90°
120°
150°180°
0°
(d) Dynamic
Figure 3.7: Distribution of the �rst 20 angles selected with the standard, gap-angle, entropy-based and dynamic algorithms for the phantom shown in Fig. 3.1b. Bin size is 5◦. Blackarrows indicate feature orientation in the object.
(a) Woodphantom 1
(b) Woodphantom 1,rotated
(c) Woodphantom 2
(d) Woodphantom 2,rotated
(e) Woodphantom 3
(f) Woodphantom 4
Figure 3.8: Phantoms used in the experiments of Section 3.3.2: with (a-d) or without (e-f)preferential directions in �ne structures. Shapes of the phantoms either have or not havepreferential directions, (b) and (d) are rotated versions of (a) and (c).
pixel size and attenuation of the densest material possible (providing that X-rays
are not entirely absorbed by the object). The projection data can then be divided
by this bound, resulting in the corresponding scaling of the reconstructed image
63
CHAPTER 3. DYNAMIC ANGLE SELECTION
(a) Standard (b) Gap-angle (c) Entropy-based (d) Dynamic
Figure 3.9: Examples of the reconstructions of the phantom shown in Fig. 3.8c from tenprojection directions selected with the standard, gap-angle, entropy-based and dynamic algo-rithms.
and the above mentioned assumption being satis�ed.
Experiments show that the proposed approach can produce angle sets that pro-
vide more accurate reconstructions compared to the conventional angle selection
strategies or reconstructions of comparable quality from smaller number of projec-
tions. The biggest improvement in reconstruction quality is achieved for objects
with a few preferential directions in the interior and shape. Intuitively, imaging
quality for this type of objects strongly depends on a small set of selected an-
gles, whereas a relatively large number of projections assures high reconstruction
quality for any reasonable angle set. In contrast, objects with a high number of
preferential directions (or without any preferences) can hardly be reconstructed
with acceptable quality from a small number of projections, and the in�uence of
the angle choice for a large number of projection angles decreases (again, for rea-
sonable choices). Experiments show also that the proposed algorithm reveals the
preferential directions in the object one by one, adding a few projections per each
preferential direction. This behaviour might contribute to the moderate improve-
ments in reconstruction quality achieved by the proposed algorithm compared to
the conventional angle selection schemes for objects with a high number of prefer-
ential directions.
The current algorithm is not feasible for large image sizes because of the high
computational complexity of the method. Indeed, to select one projection angle,
algorithm performs K SIRT reconstructions to generate surrogate solutions and
Kl + 1 CGLS reconstructions to calculate upper bounds for the diameter of the
solution set, where K is the number of surrogate solutions and l is the number of
candidate angles. The number of surrogate solutions K should be kept relatively
high in order to provide adequate sampling of the solution space. Possible ways to
reduce the computational requirements include preselection of the candidate angles
(to lower the value of l) and the use of the computation results on the subsequent
steps. Other important questions include the integrability of the information gain
64
3.4. DISCUSSION
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(a) Wood phantom 1
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(b) Wood phantom 1, rotated
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(c) Wood phantom 2
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(d) Wood phantom 2, rotated
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(e) Wood phantom 3
0 10 20 30 400
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Number of projections
MS
E
StandardGap−angleEntropy−basedDynamic
(f) Wood phantom 4
Figure 3.10: MSE as a function of the number of projection angles for Fig. 3.8.
65
CHAPTER 3. DYNAMIC ANGLE SELECTION
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
0.03
Number of projections
MS
E
Dynamic (Ns=105)
Dynamic (Ns=106)
Dynamic (Ns=107)
Dynamic (noiseless)
(a) Dynamic
0 10 20 30 400.005
0.01
0.015
0.02
0.025
0.03
Number of projections
MS
E
Standard (Ns=105)
Gap−angle (Ns=105)
Entropy−based (Ns=105)
Dynamic (Ns=105)
(b) 105 Photons
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
0.03
Number of projections
MS
E
Standard (Ns=106)
Gap−angle (Ns=106)
Entropy−based (Ns=106)
Dynamic (Ns=106)
(c) 106 Photons
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
0.03
Number of projections
MS
E
Standard (Ns=107)
Gap−angle (Ns=107)
Entropy−based (Ns=107)
Dynamic (Ns=107)
(d) 107 Photons
Figure 3.11: MSE as a function of the number of projection angles for the simulations withdi�erent noise levels for the proposed algorithm (a) and for all considered algorithms (b-d)for Fig. 3.8d.
over the set of solutions (Eq. (3.2)), and the in�uence of the class of template
images and the candidate angle sets on the performance of the proposed approach.
These questions will be addressed in future work.
In this chapter, the dynamic angle selection algorithm was applied to two-
dimensional objects. However, the concept of the information gain is directly
applicable to the three-dimensional case, as no assumptions were made about the
dimensionality of the object or projection data. A three-dimensional analogue
of Theorem 1 can be proven giving the corresponding upper bound. Surrogate
solutions should be calculated from three-dimensional template images and the
selection of candidate angles should conform with the desired resulting trajec-
tory, e. g. circular or helical. In such a way, a three-dimensional analogue of the
66
3.5. CONCLUSIONS
(a) Standard (b) Gap-angle (c) Entropy-based (d) Dynamic
Figure 3.12: Examples of the reconstructions of the phantom shown in Fig. 3.8d from tenprojection directions selected with the standard, gap-angle, entropy-based and dynamic algo-rithms, with noise added to the projection data, Ns = 105.
proposed algorithm can be developed, raising the importance of decreasing the
computational complexity of the algorithm to a new level.
3.5 Conclusions
In this chapter, an acquisition algorithm for dynamic angle selection in grey scale
computed tomography was proposed. In this approach, the angle from which a
new projection needs to be acquired to gain the most information about the object,
is dynamically computed. Prior knowledge about the object itself is not required.
Simulation experiments showed that this approach can forecast projection angles
that lead to more accurate reconstructions from fewer projections compared to
the widely used angle selection approaches. The proposed method is well suited in
X-ray imaging scenarios where the acquisition of a single projection is expensive
(in terms of acquisition time, dose, or object-source positioning). The highest gain
is expected for objects with a small number of preferential directions in the shape
and interior. Currently, the computation time of the proposed algorithm is very
high for large experimental datasets. Future work will focus on improving the
e�ciency of the algorithm.
References
[1] M. J. Murphy, J. Balter, S. Balter, J. A. BenComo, Jr., I. J. Das, S. B. Jiang, C.-M.Ma, G. H. Olivera, R. F. Rodebaugh, K. J. Ruchala, H. Shirato, and F.-F. Yin,�The management of imaging dose during image-guided radiotherapy: Report of theAAPM Task Group 75,� Medical Physics, vol. 34, no. 10, pp. 4041�4063, 2007.
[2] P. A. Midgley and R. E. Dunin-Borkowski, �Electron tomography and holographyin materials science,� Nature Materials, vol. 8, no. 4, pp. 271�280, 2009.
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[3] L. Varga, P. Balazs, and A. Nagy, �Direction-dependency of binary tomographicreconstruction algorithms,� Graphical Models, vol. 73, pp. 365�375, 2011.
[4] Z. Zheng and K. Mueller, �Identifying sets of favorable projections for few-viewlow-dose cone-beam CT scanning,� in 11th International Meeting on Fully Three-
Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Fully 3D),Potsdam, Germany, 2011, pp. 314�317.
[5] G. Placidi, M. Alecci, and A. Sotgiu, �Theory of adaptive acquisition method forimage reconstruction from projections and application to EPR imaging,� Journal ofMagnetic Resonance Series B, vol. 108, no. 1, pp. 50�57, 1995.
[6] M. Venere, H. Liao, and A. Clausse, �A genetic algorithm for adaptive tomographyof elliptical objects,� IEEE Signal Processing Letters, vol. 7, no. 7, pp. 176�178,2000.
[7] K. J. Batenburg, W. J. Palenstijn, P. Balazs, and J. Sijbers, �Dynamic angle selectionin binary tomography,� Computer Vision and Image Understanding, vol. 117, no. 4,pp. 306�318, 2013.
[8] K. J. Batenburg, W. Fortes, L. Hajdu, and R. Tijdeman, �Bounds on the qualityof reconstructed images in binary tomography,� Discrete Applied Mathematics, vol.161, no. 15, pp. 2236 � 2251, 2013.
[9] W. Fortes, K. J. Batenburg, and J. Sijbers, �Practical error bounds for binary to-mography,� in Proceedings of the 1st International Conference on Tomography of
Materials and Structures (ICTMS), Ghent, Belgium, 2013, pp. 97�100.
[10] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. Society for Industrialand Applied Mathematics, 2003.
[11] J. Gregor and T. Benson, �Computational analysis and improvement of SIRT,� IEEETransactions on Medical Imaging, vol. 27, no. 7, pp. 918�924, 2008.
[12] W. J. Palenstijn, K. J. Batenburg, and J. Sijbers, �The ASTRA tomography tool-box,� in 13th International Conference on Computational and Mathematical Methods
in Science and Engineering (CMMSE), 2013.
[13] ��, �Performance improvements for iterative electron tomography reconstructionusing graphics processing units (GPUs),� Journal of Structural Biology, vol. 176,no. 2, pp. 250�253, 2011.
[14] H. Q. Guan and R. Gordon, �Computed tomography using algebraic reconstructiontechniques (ARTs) with di�erent projection access schemes: A comparison studyunder practical situations,� Physics in Medicine and Biology, vol. 41, no. 9, pp.1727�1743, 1996.
68
4Multiresolution DART
Contents
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Motivation and approach . . . . . . . . . . . . . . . . . . . . . 71
4.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.1 Noiseless simulations . . . . . . . . . . . . . . . . . . . 76
4.3.2 Simulations with noise . . . . . . . . . . . . . . . . . . 81
4.3.3 Real experiments . . . . . . . . . . . . . . . . . . . . . 82
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
This chapter has been published as
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �A multiresolution approach to
discrete tomography using DART,� PLoS ONE, vol. 9, no. 9, e106090, 2014.
Abstract � In discrete tomography, a scanned object is assumed to consist
of only a few di�erent materials. This prior knowledge can be e�ectively exploited
by a specialized discrete reconstruction algorithm such as the Discrete Algebraic
Reconstruction Technique (DART), which is capable of providing more accurate
reconstructions from limited data compared to conventional reconstruction algo-
rithms. However, like most iterative reconstruction algorithms, DART su�ers from
long computation times. To increase the computational e�ciency as well as the
reconstruction quality of DART, a multiresolution version of DART (MDART) is
proposed, in which the reconstruction starts on a coarse grid with big pixel (voxel)
size. The resulting reconstruction is then resampled on a �ner grid and used as an
initial point for a subsequent DART reconstruction. This process continues until
69
CHAPTER 4. MULTIRESOLUTION DART
the target pixel size is reached. Experiments show that MDART can provide a
signi�cant speed-up, reduce missing wedge artefacts and improve feature recon-
struction in the object compared with DART within the same time, making its
use with large datasets more feasible.
4.1 Introduction
Computed tomography (CT) is a non-invasive imaging technique which is based on
reconstruction of an object from a series of projection images. CT has applications
on all scales, ranging from 3D imaging of nanomaterials by electron microscopy to
the reconstruction of electron-density maps of the solar corona [1, 2]. In many of
these applications, it is highly desirable to reduce the number of projections taken.
In materials science, for example, reducing the number of acquired projections
leads to faster imaging which allows to increase the time resolution to study the
evolution of structural changes in materials induced by stress or temperature [3].
In electron tomography, the number of projections is kept low either to limit the
acquisition time or because the electron beam may damage the sample [4].
Unfortunately, a low number of acquired projections leads to artefacts in the
image reconstruction. Indeed, analytical reconstruction algorithms, such as Fil-
tered Backprojection (FBP) [5], require a large number of projections acquired
from a full angular range to obtain reconstructions of acceptable quality. Iterative
reconstruction algorithms, such as the Simultaneous Iterative Reconstruction Tech-
nique (SIRT) [6], allow to incorporate prior knowledge about the object into the
reconstruction such that high quality reconstructions can be obtained from even a
low number of projections. Various forms of prior knowledge about the object can
be employed. Sparsity of image derivative magnitude is used in a total-variation
(TV) minimization algorithm to address few-view, limited-angle and bad-bin re-
construction problems [7]. Alternatively, information about the edges of the object
is shown to improve the reconstruction quality in case of limited data problems
[8]. Finally, prior knowledge about the number of materials has also been shown
to yield accurate reconstructions from a small number of projections, which is the
domain of discrete tomography [9].
Recently, a practical algorithm for discrete tomography, the Discrete Algebraic
Reconstruction Technique (DART), was introduced, which is able to produce high
quality reconstructions, even for large datasets [10]. Meanwhile, DART or vari-
ations of DART [11�14] have been successfully applied in electron tomography
[1, 15], micro-CT [16, 17] and magnetic resonance imaging (MRI) [18]. However,
being an iterative reconstruction algorithm, DART su�ers from long computation
times, which limits its use for in applications where computation time is important.
To decrease computation time or, alternatively, improve reconstruction quality
70
4.2. MOTIVATION AND APPROACH
achieved in a certain computation time, a new approach is proposed in which the
available projection data is �rst reconstructed using DART on a coarse grid. The
obtained reconstruction is then resampled on a grid with smaller pixels and used as
a starting point for a subsequent DART reconstruction. This process is iteratively
repeated until the target pixel size is reached. The proposed approach can extend
the area of applicability of DART, allowing its application to large experimental
datasets.
The multiresolution and multigrid approaches were previously applied to to-
mographic reconstruction [19�21], with focus on improvement of computational
e�ciency and memory usage. The approach proposed in this chapter allows to
improve feature reconstruction and reduce missing wedge artefacts in addition to
improved computational e�ciency.
The structure of this chapter is as follows. In Section 4.2 our approach is
explained. Section 4.3 describes experiment setups and presents the obtained
results. The approach is discussed in Section 4.4. Finally, conclusions are drawn
in Section 4.5.
4.2 Motivation and approach
We will now brie�y outline the basic concepts of the DART algorithm [10], after
which the extension to MDART is described.
A �ow chart of DART is shown in Fig. 4.1. The algorithm starts by calculating
an initial reconstruction using an algebraic reconstruction method (ARM). This
reconstruction is then segmented. Usually, only the pixels close to the object
boundary can be misclassi�ed whereas the con�dence in the classi�cation of the
interior of the object and background pixels located far from the object boundary
is high. Therefore all pixels are assigned to either �xed (F ) or non-�xed (U) pixel
sets. The non-�xed pixel set U contains all boundary pixels, i. e. pixels having at
least one adjacent pixel with a di�erent grey level. A randomly chosen fraction
of non-boundary pixels is also added to the set of non-�xed pixels to allow the
formation of new boundaries. The remaining pixels form the �xed pixel set F .
Next, several ARM iterations are performed for the non-�xed pixels while keeping
the values in the �xed pixels unchanged. After that, a termination criterion is
checked (examples of termination criteria are given later in this Section). If the
criterion is not met, the entire reconstruction is smoothed, �nishing one DART
iteration. The process is iteratively repeated until a speci�ed convergence criterion
is met.
Any iterative reconstruction algorithm can be used as the ARM. Throughout
the chapter, SIRT [6] is used as the ARM, which is formulated as follows. Let W ∈Rm×n be a projection matrix and let p ∈ Rm denote a measured projection data.
71
CHAPTER 4. MULTIRESOLUTION DART
Input: projection data
Output: current reconstruction
Compute an initial ARMreconstruction
Segment thereconstruction
Identify fixedpixels F
Identify non-fixedpixels U
Apply new ARM iterationsto U, keep F fixed
Stopcriterion
met?
Smooth thereconstruction
No
Yes
Figure 4.1: Flow chart of DART [10].
Denoting an unknown image with x ∈ Rn, we can formulate the reconstruction
problem as
Wx = p. (4.1)
The update expression for SIRT is given by [6]
xt+1 = xt + CWTR(p−Wxt
), (4.2)
where C ∈ Rn×n and R ∈ Rm×m are diagonal matrices with cjj = 1/∑i wij and
rii = 1/∑j wij .
While DART has shown its e�cacy in reconstruction of micro-CT [17] and
electron tomography [1, 15] datasets, in some cases DART can su�er from slow
convergence, leading to long computation times required to �nd a practically ac-
ceptable reconstruction. Figure 4.2b illustrates one of such cases, where DART
is capable of providing an accurate reconstruction only after a long iteration pro-
cess. For the same phantom, Segmented SIRT (SSIRT) converges rapidly, though
yielding a reconstruction of a poor quality (Fig. 4.2) (the de�nition of the relative
72
4.2. MOTIVATION AND APPROACH
(a) Phantom
0 100 200 300 400 5000
0.05
0.1
0.15
0.2
Computation time, s
RN
MP
SSIRTDART
(b) RNMP
(c) SSIRT,RNMP = 0.206
(d) DART,RNMP = 0.016
Figure 4.2: Example illustrating slow convergence of DART for some datasets. Phantom,4096 × 4096 pixels size, with holes of radius 100 pixels (a) and RNMP as a function ofthe computation time for the reconstruction of this phantom using SSIRT and DART fromm = 20 projections (b). Error images for SSIRT (c) and DART (d) reconstructions after500 s iteration time. Red and green in the error images correspond to misclassi�ed backgroundand object pixels, respectively, black and yellow represent correctly classi�ed background andobject pixels, respectively.
number of misclassi�ed pixels (RNMP) and a detailed description of the experi-
mental conditions are given in the following section). Such behaviour of DART
is explained by a highly inaccurate initial ARM reconstruction. Being calculated
from only a few projections, the initial reconstruction often contains strong arte-
facts which then require many DART iterations in order to reduce these artefacts.
Note that although the initial reconstruction has a certain in�uence on the con-
vergence of DART, it does not determine the resulting reconstruction completely.
Therefore, improving the initial reconstruction will lead to faster convergence and
smaller computation time or to more accurate reconstructions after a �xed com-
putation time.
73
CHAPTER 4. MULTIRESOLUTION DART
In [15], applying masking during the computation of the initial SIRT recon-
struction signi�cantly reduced the missing wedge artefacts in the initial reconstruc-
tion and allowed to improve the resulting DART reconstruction. This improve-
ment was attributed to a better estimation of grey values used in DART as those
grey values were calculated from the initial reconstruction. While inaccurate grey
values may indeed result in inferior quality of the DART reconstructions, even cor-
rect grey values do not guarantee fast and accurate reconstructions simultaneously
(Fig. 4.2).
The idea of the proposed multiresolution approach (MDART) is to �rst start a
DART reconstruction on a coarse reconstruction grid and then use the resampled
resulting reconstruction as a starting point for a subsequent reconstruction on a
�ner grid (Fig. 4.3). The use of coarser grids makes the reconstruction problem
less ill-posed as the number of unknowns decreases and the number of equations
remains the same. This allows to compute a good estimation of the object and then
improve it on �ner grids to reveal �ner structures which cannot be reconstructed
on the initial coarse grid.
Since DART, and hence MDART, is a heuristic algorithm, there is no formal
de�nition of the conditions which guarantee the convergence of the reconstruction
process. The following termination criteria can be used in practice:
� a certain number of iterations are performed;
� the relative number of modi�ed pixels is smaller than a given threshold. If
only a few pixels change their values during the iteration, the object is mainly
reconstructed;
� the di�erence in the projection distance (Eq. (4.3)) between the reconstruc-
tions after two consecutive iterations is smaller than a given threshold. This
means that the reconstruction stops improving.
The projection distance for a reconstruction x∗ ∈ Rn is de�ned as
D (x∗) = ‖Wx∗ − p‖2 . (4.3)
In our experiments, the modi�ed projection distance criterion was used: iter-
ations were stopped if the criterion held for three consecutive iterations. Such a
choice was made to illustrate the applicability of an adaptive switching in the gen-
eral case. For a particular dataset it might be bene�cial to choose another criterion
or use a mixture of them, e. g. a �xed number of iterations for reconstructions on
the coarser grid(s) and the projection distance criterion while reconstructing on
the target grid. Experience with similar datasets might provide some insight here.
Let MDART q denote the multiresolution DART algorithm which operates
on q reconstruction grids or, alternatively, performs q − 1 switchings to a �ner
74
4.2. MOTIVATION AND APPROACH
Input: projection data, currentpixel size, target pixel size
Targetpixel sizereached?
Halve current pixel size,resample the reconstruction
using new pixel size
Output: current reconstruction
No
Yes
Compute an initial ARMreconstruction using
current pixel size
Segment thereconstruction
Identify fixedpixels F
Identify non-fixedpixels U
Apply new ARM iterationsto U, keep F fixed
Stopcriterion
met?
Smooth thereconstruction
No
Yes
Figure 4.3: Flow chart of the MDART algorithm.
reconstruction grid, in which the pixel size is halved. This algorithm starts from the
pixel size which is 2q−1 times bigger than the target pixel size. Note that MDART 1
is identical to the conventional DART. Figure 4.4 illustrates these concepts showing
the reconstruction grids and the projection geometry for MDART 2.
75
CHAPTER 4. MULTIRESOLUTION DART
(a) Coarse grid (b) Target grid
Figure 4.4: Projection geometry, coarse and target reconstruction grids used by MDART 2.
4.3 Experiments
In this section, the proposed multiresolution DART algorithm is demonstrated on
a number of phantoms in noiseless simulations (Section 4.3.1) and simulations with
noise (Section 4.3.2), as well as on real datasets (Section 4.3.3).
4.3.1 Noiseless simulations
A number of simulation experiments were run using phantom images to demon-
strate the proposed approach. In all simulation experiments, the size of the phan-
toms was 4096×4096 pixels while reconstructions were performed on a 1024×1024
reconstruction grid to reduce the e�ect of the pixelation on the reconstructions. A
number of m equiangular fan-beam projections were computed from the original
phantoms using Joseph's projection method [22]. A detector with n = 1024 ele-
ments was used. All experiments presented in this thesis were implemented using
the ASTRA toolbox [23] where GPU acceleration was used extensively [24]. A
desktop PC equipped with an Intel Core i7 930 processor, 12 GiB of RAM and
NVIDIA GeForce GTX 285 graphics card was used for computations.
Four reconstruction algorithms were compared:
� Segmented SIRT (SSIRT). The well known SIRT reconstruction algorithm
[6] was used to calculate the reconstructions which were then segmented
using a global threshold for a fair comparison.
76
4.3. EXPERIMENTS
(a) Phantom 1 (b) Phantom 2
(c) Phantom 3 (d) Phantom 4
Figure 4.5: Phantoms 1�4, 4096× 4096.
� DART [10]. An initial reconstruction was calculated using 50 SIRT itera-
tions; 10 SIRT iterations were applied to the non-�xed pixels during each
DART iteration.
� MDART 2 andMDART 4. All parameters of the underlying DART algo-
rithm were identical to the ones described above. Reconstruction resampling
was performed using the bilinear interpolation.
Correct grey values and a global threshold were used in the simulation exper-
iments. All participating algorithms were stopped after a certain iteration time.
The quality of the reconstructions was assessed by calculating the relative number
of misclassi�ed pixels (RNMP) according to
RNMP(I, I)
=
∣∣∣{(i, j) | I (i, j) 6= I (i, j)}∣∣∣
|{(i, j) | I (i, j) > 0}|, (4.4)
where I is the original phantom and I denotes the reconstruction resampled on
77
CHAPTER 4. MULTIRESOLUTION DART
the same grid as I using the nearest-neighbour interpolation.
In the �rst series of experiments, four phantom images (Fig. 4.5) were used.
Phantom 1 (Fig. 4.5a) is a disk with a number of holes of radius 100 pixels. It
is identical to the phantom used in the previous section (Fig. 4.2a). Phantom 2
(Fig. 4.5b) represents a cylinder head of an internal combustion engine, Phan-
tom 3 (Fig. 4.5c) is a Siemens star-like phantom, Phantom 4 (Fig. 4.5d) consists
of a number of intersecting ellipses and has three grey values, whereas the for-
mer three phantoms are binary. From these phantoms, a number m equiangular
projections were computed. These projections were then reconstructed using the
SSIRT, DART, and MDART.
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(a) Phantom 1, m = 20
0 10 20 30 40 500
0.02
0.04
0.06
0.08
0.1
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(b) Phantom 1, m = 50
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(c) Phantom 2, m = 20
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(d) Phantom 2, m = 50
Figure 4.6: Noiseless simulation results. RNMP as a function of the computation time forthe reconstructions of Phantoms 1�2 (Figs. 4.5a and 4.5b) from m projections. Black andgrey points on the MDART curves mark the moments of switching to a �ner reconstructiongrid.
The obtained results are shown in Figs. 4.6 and 4.7, which suggest that MDART
can provide signi�cantly better reconstruction quality in only a fraction of compu-
tation time compared to SSIRT and DART, especially when there are only a few
78
4.3. EXPERIMENTS
projections available.
0 10 20 30 40 50 600
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(a) Phantom 3, m = 20
0 10 20 30 40 50 600
0.05
0.1
0.15
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(b) Phantom 3, m = 30
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(c) Phantom 4, m = 10
0 5 10 15 20 250
0.05
0.1
0.15
0.2
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(d) Phantom 4, m = 20
Figure 4.7: Noiseless simulation results. RNMP as a function of the computation time forthe reconstructions of Phantoms 3�4 (Figs. 4.5c and 4.5d) from m projections. Black andgrey points on the MDART curves mark the moments of switching to a �ner reconstructiongrid.
For the second series of experiments, a number of phantoms were used, each
consisting of a disk with randomly placed circular holes of a particular size (Fig. 4.8).
Three phantoms were created for each hole size. For these phantoms, projections
from complete and from the limited angular ranges were computed in order to
evaluate the applicability of the proposed approach for objects with features of
various size and for the datasets with the missing wedge.
Figure 4.9 presents the obtained results after 30 s iteration time, demonstrating
the average RNMP over the phantoms with the holes of the particular size together
with the standard errors (shown as shaded areas in the plots). Figure 4.10 shows
the corresponding reconstructions of one of the phantoms with holes of radius 50
pixels calculated from 20 projections with 90◦ missing wedge. These plots demon-
strate the ability of MDART to provide reconstructions of signi�cantly higher
79
CHAPTER 4. MULTIRESOLUTION DART
(a) Hole radius 50 (b) Hole radius 80
Figure 4.8: Examples of the phantoms, 4096× 4096 pixels size, with holes of radius 50 and80 pixels.
30 40 50 60 70 800
0.05
0.1
0.15
Hole radius, pixels
RN
MP
SSIRTDARTMDART 2MDART 4
(a) Varying hole radius
90 60 30 00
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Missing wedge, °
RN
MP
SSIRTDARTMDART 2MDART 4
(b) Varying missing wedge
Figure 4.9: Noiseless simulation results for the phantoms with holes. RNMP for the recon-structions of the phantoms with various hole sizes fromm = 20 projections after 30 s iterationtime: (a) as a function of the hole radius for the 90◦ missing wedge and (b) as a function ofthe missing wedge for the phantoms with the hole radius of 50 pixels.
quality compared to SSIRT and DART and to reduce missing wedge artefacts.
The biggest gain compared to DART is achieved in the experiments with big-
ger missing wedge and smaller number of projections. The poor performance of
MDART 4 on the phantoms with the hole radii of 30 pixels is explained by the
fact that on the coarsest reconstruction grid used by MDART 4 such holes have
a radius of less than one pixel which complicates their detection with a discrete
reconstruction algorithm. Note that for the holes of radius 60 pixels or bigger
MDART 4 shows the best results among all considered algorithms gaining from
the use of coarser grids.
80
4.3. EXPERIMENTS
(a) SSIRT (b) DART (c) MDART 2
(d) RNMP = 0.147 (e) RNMP = 0.064 (f) RNMP = 0.007
Figure 4.10: Reconstructions of the phantom with holes of radius 50 pixels. The reconstruc-tions obtained after iterating for 30 s with SSIRT, DART and MDART 2 using m = 20projections with 90◦ missing wedge together with the corresponding error images (d-f). Redand green in the error images correspond to misclassi�ed background and object pixels, re-spectively, black and yellow represent correctly classi�ed background and object pixels, re-spectively.
4.3.2 Simulations with noise
In order to evaluate the proposed multiresolution approach in a more realistic
situation, Poisson noise was added to one of the noiseless experiments. For the
cylinder head phantom (Fig. 4.5b), K = 5 noisy sets of projection data were
obtained for each noise level. For each noisy projection dataset the reconstructions
were built. The mean values of RNMP(I, I)over these K reconstructions after
25 s iteration time are shown in Fig. 4.11, from which we see that the proposed
method can outperform SSIRT and DART even in the presence of noise. This plot
also demonstrates a slightly higher MDART 4 robustness against noise compared
to MDART 2.
81
CHAPTER 4. MULTIRESOLUTION DART
103
104
1050
0.05
0.1
0.15
0.2
0.25
Photon count, Ns
RN
MP
SSIRTDARTMDART 2MDART 4
Figure 4.11: Results of the simulations with noise. RNMP as a function of the photon countfor the reconstructions of the cylinder head phantom (Fig. 4.5b) from m = 20 projectionswith noise. The iteration process was stopped after 25 s.
4.3.3 Real experiments
The following experiments were conducted in order to demonstrate the perfor-
mance of the proposed multiresolution approach on real data.
For the �rst experiment, a hardware phantom with a diameter of 70 mm was
scanned using the HECTOR micro-CT system developed by UGCT (the Ghent
University Centre for X-ray Tomography, Belgium) in collaboration with X-Ray
Engineering (XRE bvba, Ghent, Belgium) [25]. For this object, a full-angle cone-
beam dataset was acquired containing 2401 projections of 2000× 2000 pixels, the
X-ray tube voltage was 120 kV and the tube current was 333 µA. The source-
detector distance was 1250 mm and the source-object distance was 275 mm. One
slice from this dataset was reconstructed with 1000 iterations of SIRT (Fig. 4.12a)
on a 2000× 2000 reconstruction grid with a pixel size of 44 µm.
In the second experiment, a gypsum jaw model was scanned using a desktop
micro-CT system SkyScan-1172 (Bruker-MicroCT, Belgium). A full-angle cone-
beam dataset consisting of 400 projections of 1984× 524 pixels was acquired, the
X-ray tube voltage was 100 kV and the tube current was 100 µA. One slice from
this dataset was reconstructed on a 1984× 1984 grid with a pixel size of 34.7 µm
using 500 SIRT iterations (Fig. 4.12b).
Finally, a coral was scanned on the TOMCAT beamline [26] at the Swiss Light
Source, Paul Scherrer Institut (Villigen, Switzerland). A full-angle parallel-beam
dataset consisting of 1001 projections of 1022× 378 pixels was acquired, the beam
energy was 28 keV and the ring current was 401 mA. One slice from this dataset
was reconstructed on a 1022×1022 grid with a pixel size of 3.25 µm using 500 SIRT
iterations (Fig. 4.12c).
82
4.3. EXPERIMENTS
(a) Hardware phantom (b) Jaw model (c) Coral
Figure 4.12: SIRT reconstructions of slices of the real datasets using all available projections.(a) The hardware phantom, 2401 projections, (b) the jaw model, 400 projections, (c) thecoral, 1001 projections.
The reconstructions using all available projections (Fig. 4.12) were segmented
using the Otsu segmentation algorithm [27] and used as a ground truth in the
following experiments. A number of m projections of the same slice were chosen
from the corresponding original datasets to form datasets with limited angular
ranges. These datasets were then reconstructed using the algorithms described
above. Since true grey values to be used in DART and MDART were not known,
these values were estimated as mean values in each segmentation class of the Otsu
segmentation of the SIRT reconstructions shown in Fig. 4.12. If the high quality
reconstruction of the object is not available, as is often the case, reconstructions
of another objects containing same materials or table values for the known ma-
terials might be used to estimate grey values to be used in DART and MDART.
Alternatively, the automatic estimation procedure [13] can be adopted.
The obtained results are presented in Fig. 4.13. Figures 4.13a, 4.13c and 4.13e
demonstrate the ability of MDART to signi�cantly speed up the reconstruction
process and to yield more accurate results compared to SSIRT and DART. Fig-
ures 4.13b, 4.13d and 4.13f con�rm that MDART su�ers less from the missing
wedge in the projection data than SSIRT and DART. The decreased performance
of all methods on the jaw model dataset without the missing wedge compared
to the dataset with the 30◦ missing wedge (Fig. 4.13d) may be explained by the
dependency of the reconstruction quality on the actual projection directions for
some objects, especially if there are only a small number of projections used [28].
Moderate performance of MDART 4 on the coral dataset (Figs. 4.13e and 4.13f)
compared to the performance of DART and MDART 2 is caused by the presence
of very �ne details in the object, which cannot be reconstructed on the coarsest
reconstruction grid used by this algorithm. Examples of the reconstructions of the
83
CHAPTER 4. MULTIRESOLUTION DART
0 10 20 30 40 500
0.05
0.1
0.15
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(a) Hardware phantom, m = 20
90 60 30 00
0.02
0.04
0.06
0.08
Missing wedge, °
RN
MP
SSIRTDARTMDART 2MDART 4
(b) Hardware phantom, m = 20, 50 s
0 10 20 30 40 500
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(c) Jaw model, m = 15
90 60 30 00
0.02
0.04
0.06
0.08
Missing wedge, °
RN
MP
SSIRTDARTMDART 2MDART 4
(d) Jaw model, m = 15, 50 s
0 10 20 30 40 500
0.05
0.1
0.15
0.2
Computation time, s
RN
MP
SSIRTDARTMDART 2MDART 4
(e) Coral, m = 20
90 60 30 00
0.1
0.2
0.3
0.4
0.5
Missing wedge, °
RN
MP
SSIRTDARTMDART 2MDART 4
(f) Coral, m = 20, 50 s
Figure 4.13: Results of the real data experiments. RNMP for the reconstructions of the realdatasets (Fig. 4.12) as a function of the computation time from the data with the missingwedge (a, c, e) and as a function of the missing wedge after 50 s iteration time (b, d, f).Missing wedge is 90◦ in (a) and (c) and 30◦ in (e). Black and grey points on the MDARTcurves (a, c, e) mark the moments of switching to a �ner reconstruction grid.
84
4.3. EXPERIMENTS
(a) SSIRT (b) RNMP = 0.084
(c) DART (d) RNMP = 0.042
(e) MDART 2 (f) RNMP = 0.024
(g) MDART 4 (h) RNMP = 0.013
Figure 4.14: Reconstructions of the hardware phantom (Fig. 4.12a) obtained after iteratingfor 50 s using m = 20 projections with 90◦ missing wedge together with the correspondingerror images (b, d, f, h). Red and green in the error images correspond to misclassi�edbackground and object pixels, respectively, black and yellow represent correctly classi�edbackground and object pixels, respectively.
85
CHAPTER 4. MULTIRESOLUTION DART
(a) SSIRT (b) RNMP = 0.083
(c) DART (d) RNMP = 0.074
(e) MDART 2 (f) RNMP = 0.058
(g) MDART 4 (h) RNMP = 0.047
Figure 4.15: Reconstructions of the jaw model (Fig. 4.12b) obtained after iterating for 50 susing m = 15 projections with 90◦ missing wedge together with the corresponding errorimages (b, d, f, h). Red and green in the error images correspond to misclassi�ed backgroundand object pixels, respectively, black and yellow represent correctly classi�ed background andobject pixels, respectively.
86
4.3. EXPERIMENTS
(a) SSIRT (b) RNMP = 0.182
(c) DART (d) RNMP = 0.064
(e) MDART 2 (f) RNMP = 0.062
(g) MDART 4 (h) RNMP = 0.063
Figure 4.16: Reconstructions of the coral (Fig. 4.12c) obtained after iterating for 50 s usingm = 20 projections with 30◦ missing wedge together with the corresponding error images(b, d, f, h). Red and green in the error images correspond to misclassi�ed backgroundand object pixels, respectively, black and yellow represent correctly classi�ed background andobject pixels, respectively.
87
CHAPTER 4. MULTIRESOLUTION DART
hardware phantom (Fig. 4.14), the jaw model (Fig. 4.15) and the coral (Fig. 4.16)
suggest that the proposed approach, and MDART 4 in particular, can signi�cantly
reduce missing wedge artefacts and improve feature reconstruction for real objects.
Therefore, experimental studies conform to the simulation experiments, showing
the ability of the proposed approach to faster yield reconstructions of superior
quality compared to those produced by SSIRT and DART for real datasets.
4.4 Discussion
The proposed multiresolution DART algorithm starts a reconstruction on a coarse
reconstruction grid and then uses the resampled resulting reconstruction as an
initial point for a new reconstruction process on a �ner grid, iteratively switching
to the new grid until the target pixel size is reached. In our experiments, the
next pixel size was always two times smaller than the current one. A certain
variation in the pixel size changing strategy can have additional bene�ts in terms
of computation time.
Experiments show that the proposed approach allows to create accurate recon-
structions signi�cantly faster than DART. Speed-up comes from the following two
facts: iteration time decreases together with the number of pixels in the reconstruc-
tion and DART converges faster when starting from a better initial reconstruction.
More accurate initial reconstruction results from the fact that use of the coarse
grids makes the reconstruction problem less ill-posed decreasing the number of un-
knowns while preserving the number of equations. This is especially important in
case when the limited number of projections is available or the projections were ac-
quired from a limited angular range since the initial reconstruction calculated from
such data can su�er from strong artefacts which sometimes slow down the con-
vergence of conventional DART. With the increasing number of projections, ARM
reconstruction becomes better providing DART with the better starting point and
at some projection number it will be faster to have an ARM reconstruction and a
few extra DART iterations compared to MDART iterations.
The choice of the starting pixel size has a signi�cant in�uence on the perfor-
mance of the proposed approach. On the one hand, the smaller the features present
in the object, the smaller should be the starting pixel size. On the other hand, the
bigger the starting pixel, the higher the potential for a speed-up and for robustness
against noise. This trade-o� should be made having a particular reconstruction
problem in mind.
The proposed multiresolution approach can broaden the use of DART for large
experimental datasets. It also allows to further decrease the number of projections
required to obtain accurate reconstructions in a reasonable time.
88
4.5. CONCLUSIONS
4.5 Conclusions
We proposed a multiresolution DART (MDART) algorithm for discrete tomogra-
phy. This approach is based on the iterative use of a resampled reconstruction
created on a coarse grid as a starting point for a subsequent reconstruction on
a �ner grid. Our experiments showed that MDART can lead to accurate recon-
structions calculated in only a fraction of time compared to DART. The biggest
improvement is reached for the datasets with a very small number of projections
and acquired from a limited angular range. Reconstructions of the real datasets
demonstrated an ability of MDART to signi�cantly decrease the missing wedge
artefacts and improve feature reconstruction in the object compared to the con-
ventional DART algorithm being iterated for the same time.
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91
5Conclusions
X-Ray computed tomography is a well-established, non-invasive imaging technique
with a wide variety of applications in medicine, preclinical research, materials sci-
ence and nondestructive testing. However, one common feature of the acquisition
setups used in these applications is the requirement to put the object into a scan-
ner for imaging, thus implying the limitations on the size of an object that can be
scanned and requiring its displacement.
The advent of in loco tomography based on mobile tomographic devices will
allow to overcome the limitations of the conventional tomographic systems and will
open up numerous applications in nondestructive testing, security, archaeology,
medicine and veterinary. Mounting an X-ray source and a detector on separate
mobile robotized platforms grants a tomographic system the �exibility with respect
to the acquisition geometry and the size of the object and allows tomography of
the objects that cannot be moved. Current advancements in X-ray sources and
detectors as well as in robotics make the development of a mobile robotized device
for in loco tomography technically feasible in the years ahead.
The acquisition and reconstruction of the datasets from mobile tomographic
devices are likely to present challenges for the techniques used in CT nowadays.
Firstly, in the in loco scanning setup, the conventional circular or helical source-
detector trajectories might be unavailable or impractical due to the restrictions in
the scanning scene, giving rise to the trajectory selection problem. Secondly, the
restrictions in the scanning scene might make some directions around the object
unavailable for the acquisition, resulting in a limited angular range. Furthermore,
repositioning of the source and the detector might be time-consuming, resulting
in the possibility to acquire only a limited number of projections in a reasonable
time. Next, accurate positions and orientations of the source and the detector
during the acquisition might be unavailable, resulting in the need for projection
alignment. Finally, it might be impossible to image the complete object from
certain directions, requiting to deal with projection truncation.
93
CHAPTER 5. CONCLUSIONS
In this thesis, we have proposed three techniques adding to the development
of in loco tomography by addressing the �rst three of the above-mentioned chal-
lenges and thus improving the acquisition and the reconstruction with the mobile
tomographic devices.
The �rst technique, the variable distance approach (Chapter 2), uses the prior
knowledge about the convex hull of the object to modify the conventional circular
trajectory. For elongated objects, it allows to optimize the detector usage and to
acquire more detailed information thus increasing the reconstruction quality. The
experiment mimicking the required setup on a desktop micro-CT system con�rmed
the ability of the proposed approach to provide more accurate reconstructions and
to reveal more details in the object compared to the circular trajectory. This
technique is the �rst step towards the optimal trajectory selection.
The dynamic angle selection algorithm (Chapter 3) also aims at improving the
acquisition. In contrast to the variable distance approach, the dynamic angle se-
lection requires no prior knowledge about the object. It is an online algorithm
that uses the already acquired projection data to determine the next projection
direction providing the maximum information gain. The proposed algorithm al-
lows to produce more accurate reconstructions from fewer projections compared
to the conventional angle selection approaches. This algorithm is relevant in the
context of both the trajectory selection and the problem of the limited number of
projections, allowing to select the most informative projections.
The multiresolution DART (Chapter 4) is an improvement of DART, the re-
construction algorithm for discrete tomography that exploits the prior knowledge
about the materials forming the object. DART had been successfully applied to
the datasets from electron tomography and micro-CT, e�ectively dealing with the
limited angular range and the limited number of projections. For some datasets,
however, the accurate reconstructions required long computation times. The mul-
tiresolution DART, proposed in this thesis, allows to signi�cantly decrease compu-
tation time and improve the reconstruction quality compared to DART, decreas-
ing the reconstruction artefacts and making its application to large experimental
datasets more feasible.
There are several possibilities for future research based on the techniques devel-
oped in this thesis. The variable distance approach (Chapter 2) might be extended
to take into account not only the size and the shape of the object, but also the prior
knowledge about the scanning scene and the related restrictions as well as the char-
acteristics of the mobile platforms. Reduction of the computational complexity of
the dynamic angle selection presented in Chapter 3 and studying the in�uence of
the classes of the template images on its performance are also interesting research
directions. Varying the strategy of changing the pixel size in the multiresolution
DART (Chapter 4) might deliver a further improvement in the performance of the
94
algorithm. Additionally, the approaches presented in Chapters 2 and 3 might be
combined into an algorithm that exploits the prior knowledge about the object
and the scene to acquire the most informative projections.
95
Acknowledgements
We would like to thank A. Sasov from Bruker-MicroCT and Elke Van de Casteele
for their help in the data acquisition for the real experiment presented in Chapter 2.
Regarding the datasets in Chapter 4, we are grateful to Jelle Dhaene and
Luc Van Hoorebeke from UGCT, Ghent University (Belgium) for the hardware
phantom dataset; A. Sasov from Bruker-MicroCT and Elke Van de Casteele for
their help in the data acquisition for the jaw model; and Federica Marone from
Paul Scherrer Institut (Switzerland) and Marlene Wall from GEOMAR (Germany)
for providing the coral dataset.
This work was �nancially supported by the University of Antwerp, Antwerp,
Belgium (BOF LP project 25778); iMinds (Interdisciplinary Institute for Technol-
ogy, a research institute founded by the Flemish Government), Flanders, Belgium;
the Agency for Innovation by Science and Technology (IWT), Flanders, Belgium
(SBO project TomFood); and the Netherlands Organisation for Scienti�c Research
(NWO), The Netherlands (Research programme 639.072.005). Networking support
was provided by the EXTREMA COST Action MP1207.
97
Appendix
AScienti�c Contributions
A.1 Journal Articles
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �Adaptive zooming in X-ray com-
puted tomography,� Journal of X-Ray Science and Technology, vol. 22, no. 1, pp.
77�89, 2014.
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �Dynamic angle selection in X-
ray computed tomography,� Nuclear Instruments and Methods in Physics Research
Section B: Beam Interactions with Materials and Atoms, vol. 324, pp. 17�24, 2014.
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �A multiresolution approach to
discrete tomography using DART,� PLoS ONE, vol. 9, no. 9, e106090, 2014.
A.2 Conference Proceedings (full paper)
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �Adaptive zooming in X-ray com-
puted tomography,� in Proc. 1st International Conference on Tomography of Ma-
terials and Structures (ICTMS), vol. Book of Abstracts: Posters, Ghent, Belgium,
2013, pp. 5�8.
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �Dynamic angle selection in X-
ray computed tomography,� in Proc. 1st International Conference on Tomography
of Materials and Structures (ICTMS), vol. Book of Abstracts: Talks, Ghent,
Belgium, 2013, pp. 27�30.
L. F. Alves Pereira, A. Dabravolski, I. R. Tsang, G. D. C. Cavalcanti, and J. Sij-
bers, �Conveyor belt X-ray CT using domain constrained discrete tomography,�
101
APPENDIX A. SCIENTIFIC CONTRIBUTIONS
in Proc. 27th SIBGRAPI Conference on Graphics, Patterns and Images, Rio de
Janeiro, Brazil, 2014, pp. 290�297.
A. Dabravolski, J. De Beenhouwer, and J. Sijbers, �Projection-based polygon es-
timation in X-ray computed tomography,� in Proc. 6th International Conference
on Optical Measurement Techniques for Structures and Systems (OPTIMESS),
Antwerp, Belgium, 2015, in press.
A.3 Conference Proceedings (abstract)
A. Dabravolski, K. J. Batenburg, and J. Sijbers, �A multiresolution approach to the
Discrete Algebraic Reconstruction Technique (DART),� in Proc. 2nd International
Congress on 3D Materials Science (3DMS), Annecy, France, 2014.
102
