INTRINSIC ARTEFACTS OF CIRCULAR CONE-BEAM COMPUTED
TOMOGRAPHY
By
Steven Bartolac
A thesis submitted in conformity with the requirements
for the degree of Master of Science
Graduate Department of Medical Biophysics
University of Toronto
© Copyright by Steven Bartolac (2009)
ii
INTRINSIC ARTEFACTS OF CIRCULAR CONE-BEAM COMPUTED
TOMOGRAPHY
Steven Bartolac
Master of Science, 2009
Department of Medical Biophysics, University of Toronto
Abstract: Circular source and detector trajectories in cone-beam computed tomography
(CT) are known to collect insufficient data for accurate object reconstruction. One model
predicts that the lacking information corresponds to a shift-variant cone of missing spatial
frequency components in the local Fourier domain. These predictions were
experimentally verified by imaging small, localized objects and observing their Fourier
transforms. Measurements indicated that the internal angle of the ‘missing cone’ varies as
the angle of locally intersecting x rays with respect to the horizontal plane, as expected.
Object recovery was also found to depend greatly on the distribution of the object’s
frequency spectrum relative to the missing cone, as predicted. Findings agreed with more
anatomically relevant phantoms, which showed preferential intensity discrepancies at
gradients oriented within or near the missing cone. Methods for artefact correction are in
general limited to approximation unless a priori information is incorporated.
iii
For my wife.
iv
Acknowledgements
This work has only been made possible with the support and contributions of others,
whom I would like to sincerely thank here.
Firstly, I would like to thank my supervisor, Dr. David Jaffray. Aside from abounding
expertise in the field of cone-beam CT, his confidence and encouragement has been an
unwavering support for me. I am ever grateful for the opportunity to work as his student,
and for his invaluable supervision and counsel.
I would also like to thank my committee members, Drs. Lothar Lilge and Jeff
Siewerdsen. Their sincere interest in my research and their guidance in improving all
aspects of my work, from experiment to presentation, has been a great aid to me in
succeeding as a graduate student.
The fundamental theoretical developments in this work were largely based on private
lecture notes and private communications with Dr. Rolf Clackdoyle. I thank both him
and Dr. Frederic Noo for their lengthy discussions that provided invaluable direction and
background for this work, as well as for the beer and wings that accompanied them.
Of course, this work could not have been possible without the support of my loving wife.
I am greatly thankful for her support, encouragement, and the genuine pride and joy she
shares with me in each of my accomplishments.
Lastly, I would also like to thank the support and contributions of all those in the Image-
Guided Therapy Group at Princess Margaret Hospital. (In particular, Noor, Doug, Sun-
mo, Nick, Sami, Thao, Greg, Mike, Saj and Sam).
v
Table of Contents
ABSTRACT: II
ACKNOWLEDGEMENTS IV
TABLE OF CONTENTS V
LIST OF ILLUSTRATIONS VII
LIST OF TABLES VIII
LIST OF ABBREVIATIONS IX
CHAPTER 1: INTRODUCTION 1
1.1 CONE-BEAM COMPUTED TOMOGRAPHY 1
1.2 CONE-BEAM CT IN CLINICAL USE 2
1.2.1 IMAGE-GUIDED THERAPY 2
1.2.2 DIAGNOSTIC IMAGING 4
1.3 CONE-BEAM CT ARTEFACTS DUE TO CIRCULAR TRAJECTORIES 5
1.3.1 INFLUENCE ON CLINICAL IMPLEMENTATION 6
1.3.2 DISTINCTION FROM OTHER ARTEFACTS 7
1.4 MOTIVATION AND CONTENT OF THESIS 8
CHAPTER 2: CONE-BEAM CT ARTEFACTS AND THE LOCAL FOURIER DOMAIN 11
2.1 THEORY 12
2.1.1 FOURIER SLICE THEOREM AND THE RADON TRANSFORM 12
2.1.2 MISSING PLANES IN CONE-BEAM CT 13
2.1.3 A LOCAL FOURIER DESCRIPTION OF CONE-BEAM CT ARTEFACTS 15
vi
2.2 EFFECT OF MISSING SPATIAL FREQUENCY COMPONENTS 19
2.3 SUMMARY 26
CHAPTER 3: CONE-BEAM ARTEFACTS IN REAL IMAGES 27
3.1 EXPERIMENTAL VALIDATION OF A LOCAL FOURIER DESCRIPTION OF CONE-BEAM
CT ARTEFACTS 27
3.1.1 METHODS 27
3.1.2 RESULTS AND ANALYSIS 38
3.2 CONE-BEAM CT ARTEFACT AND IMAGE NOISE 51
CHAPTER 4: CORRECTION SCHEMES 53
4.1 ARTEFACT REDUCTION ALGORITHMS 53
4.2 CONSTRAINTS AND A PRIORI KNOWLEDGE 55
4.3 A FOURIER BASED CORRECTION METHOD 56
4.3.1 IMPLEMENTATION 59
4.3.2 LIMITATIONS 60
4.4 SUMMARY 62
CHAPTER 5: DISCUSSION AND CONCLUSIONS 63
5.1 DISCUSSION 63
5.2 CONCLUSIONS 68
REFERENCES 70
APPENDIX A 76
vii
List of Illustrations
Fig. 1-1. Schematic diagram of fan and cone-beam models used in CT .......................... 3
Fig. 1-2. Cross sectional image of Defrise phantom......................................................... 7
Fig. 2-1. Vector representation of planes used in Radon inversion formula. ................. 13
Fig. 2-2. Illustration of measured and non-measured planes. ......................................... 15
Fig. 2-3. Schematic diagram of missing planes and the missing cone ........................... 17
Fig. 2-4. Schematic diagram of 2D spatial frequency filtering....................................... 20
Fig. 2-5. Close up of image before and after frequency domain filtering ...................... 21
Fig. 2-6. Illustration of frequency filtering of a square................................................... 22
Fig. 2-7. Single square and checkerboard patterns with discrete Fourier transforms..... 24
Fig. 2-8. Filtered square and checkerboard patterns with discrete Fourier transforms... 25
Fig. 2-9. Difference images between filtered and pre-filtered squares........................... 26
Fig. 3-1. Photograph of experimental cone-beam CT bench. ......................................... 28
Fig. 3-2. Surface plots of window function used in Fourier analysis ............................. 33
Fig. 3-3. Schematic representation of mini disk experiment .......................................... 34
Fig. 3-4. Anthropomorphic leg/knee phantom used in experiment. ............................... 37
Fig. 3-5. Sagittal view through acrylic sphere and FFT ................................................. 39
Fig. 3-6. Missing cone surface integrals and derivative plot .......................................... 40
Fig. 3-7. Missing cone angle versus z displacement....................................................... 40
Fig. 3-8. Acrylic spheres imaged on the rotation axis. ................................................... 41
Fig. 3-9. Sagittal views of mini disk phantom at varying orientation and z position ..... 43
Fig. 3-10. Central disk at varying orientation with corresponding FFTs.......................... 44
Fig. 3-11. Artefact simulation by convolution via multiplication in Fourier domain....... 46
Fig. 3-12. Cellular and solid disk phantoms ..................................................................... 47
Fig. 3-13. Sub-volume of rabbit data and FFT. ................................................................ 48
Fig. 3-14. Femur and contour showing intensity changes with displacement . ............... 50
Fig. 3-15. Mean intensity discrepancies along knee contour ........................................... 50
Fig. 3-16. Mean intensity discrepancies along contour of simulated sphere. ................... 51
Fig. 3-17. Spectrum of subvolume of air only volume at differing displacements. ......... 52
Fig. 4-1. Simulatd Shepp-Logan phantom before and after artefact correction ............. 61
viii
List of Tables
Table I: Imaging and Reconstruction Parameters for Small Phantoms………………….29
Table II: Imaging and Reconstruction Parameters for Large Phantoms…………………29
ix
List of Abbreviations
CT – computed tomography
BCT – breast computed tomography
FBP – filtered backprojection
FDK – Feldkamp, Davis and Kress
FPDs – flat panel detectors
FFT – fast Fourier transform
FOV – field of view
IFFT – inverse fast Fourier transform
IGRT – image-guided radiation therapy
MTF – modulation transfer function
NTCP – normal tissue complication probability
NPS – noise power spectrum
PSF – point spread function
TCP – tumour control probability
3D – three-dimensional
1
Chapter 1: Introduction
1.1 CONE-BEAM COMPUTED TOMOGRAPHY*
The term computed tomography (CT) refers to the process of obtaining a three-dimensional
(3D) image from a series of x-ray images (also called projections or radiographs). Typically,
an x-ray source rotates about the object while a digital detector records the projection data.
The first viable CT scanners were developed independently by researchers Sir Godfrey
Newbold Hounsfield at EMI Central Research Laboratories, Hayes, United Kingdom and
Allan McLeod Cormack of Tufts University, Massachusetts, USA, for which they shared the
Nobel Prize in Physics in 1979. These preliminary designs required very long acquisition
times and resulted in poor resolution images due to the immature technology (i.e., limited
detector size, slow data capture, low efficiency, etc.). Second, third and fourth generation CT
designs that spanned over the next three decades have greatly increased the speed and
resolution capabilities of the CT scanner (<0.5 mm voxel sizes, scanning time <0.5
s/rotation). These scanners are found in varied applications, including industrial uses and
medical research. Most notably, CT scanners have had widespread utility in providing high
quality 3D images of patients as clinical aids1.
Before the first multidetector-row CT scanners were introduced in the nineties, CT designs
utilized one-dimensional (1D) detectors with longitudinal resolution and field of view set by
collimation and limited by the width of the linear detector array. This approach required
multiple rotations about the patient in order to cover an extended region of interest, where
each rotation can be thought of as collecting information over a ‘slice’ of the patient. The
* Portions of this work have been published previously as a manuscript in Medical Physics, February 2009, under the title of “A local Fourier description and experimental validation of circular cone-beam computed tomography artifacts.”
2
final 3D image is assembled as a stack of two-dimensional (2D) sections. In contrast to
traditional CT methods, cone-beam CT reconstructs fully 3D images from projections
utilizing a large field of view 2D detector array as illustrated in Fig. 1-1. The term ‘cone-
beam’ was allotted since the x rays are modeled as emanating from a point source that form a
cone rather than a ‘fan-beam’ as with 1D detectors. One advantage of this approach is that it
allows a larger extent of the body to be scanned in a single rotation about the object.
Disadvantages include a number of artefacts that arise from using a wider detector. For
example, a narrow detector has the benefit of rejecting scattered radiation, thereby preventing
the associated loss of contrast in the final images. Any CT scanner with more than a single
detector row utilizes a cone-beam geometry. For relatively few detector rows, however, the
divergence of the beam may be ignored and the projections are often treated as a stack of fan-
beams instead of a cone-beam.
1.2 CONE-BEAM CT IN CLINICAL USE
Although the fields of view and number of detector rows vary broadly, all modern CT
scanners now utilize a cone-beam approach for image acquisition (i.e. they utilize 2D
detector arrays and account for the divergence of the beam in the longitudinal direction).
Modern CT scanner types can typically be divided into two broad clinical streams: image-
guided therapy and diagnostic imaging.
1.2.1IMAGE-GUIDED THERAPY
The onset of digital flat-panel detectors (FPDs), which boasted very large fields of view, first
spurred rapid development of cone-beam CT2,3 in what is now known as image-guided
radiation therapy (IGRT). In this application, a large FPD and a kilovoltage (kV) x-ray tube
3
Fig. 1-1. Schematic diagram showing the progression of CT imaging geometry from a fan-beam (light gray) with a 1D detector array, to a cone-beam (light blue) geometry, with a 2D detector array. Depending on the method of image acquisition, the object of interest may remain stationary during the scan (e.g. circular scan) or be translated longitudinally along the rotation axis (eg. helical scan).
are typically adapted directly to the gantry of the radiation treatment unit. The large field of
view allows the full region of interest to be imaged using a single circular rotation. The
advantage of imaging the patient in the treatment room is that it allows the target lesion to be
determined with very high accuracy with respect to the treatment beam. This accurate
localization translates into increased precision in targeting and reduced probability that
healthy tissue is harmed4-6. Furthermore, higher accuracy means that higher radiation doses
can be effectively delivered with greater tumour control probability (TCP) and without
tradeoff in normal tissue complication probability (NTCP)7,8. Since its conception in IGRT,
interest has grown in applying cone-beam CT to other areas, including its use as an intra-
4
operative image-guidance tool. Extensive research in this field has explored the use of cone-
beam CT in head, neck and spinal surgery9-15, with promising applications to many other
procedures such as guiding liver resection16, or re-alignment of bone fractures17,18.
Typically, these applications involve an FPD adapted to a mobile C-arm, for easy transport to
and from the operating room. The scanner allows the therapist or surgeon to make use of
quick intra-operative scans, which provide 3D information for guidance of surgical tools.
More stationary units have also been designed for specific use in the planning of a wide array
of dental procedures19, with particular interest in the field of implantology20,21.
1.2.2DIAGNOSTIC IMAGING
While image-guided applications aid in directing treatments and operations, diagnostic cone-
beam CT scans are concerned with the detection of abnormalities or changes in the body.
For example, the scan may be used to identify a suspicious lesion (e.g. tumour in the lung),
heart problems (e.g. mitral valve prolapse) or details of a spinal injury (e.g. herniated disk).
The development of cone-beam CT for diagnostic scanners was more gradual than its
development for image-guidance applications. The move from a limited number of rows to
larger multi-row detectors proved to be a technically challenging task. Moreover, FPD
technology was not well suited to meet the speed demands of diagnostic CT designs22,23.
Although these difficulties hindered development, steady progress was nevertheless made,
and diagnostic CT scanners have been produced with double the number of detector rows
approximately every two and a half years23 since 2000, moving the diagnostic CT scanner
away from a fan-beam model and towards a cone-beam model that is now standard. The
latest CT scanner produced by Toshiba (for dynamic volume CT) has 320 detector rows and
a field of view (FOV) comparable to cone-beam systems in IGRT. While their detector
5
technologies are different, the theoretical distinction has essentially been eliminated between
modern image-guidance and diagnostic units. However, it should be noted that in common
usage, “CT” is often meant to refer to diagnostic scanners, while “cone-beam CT” is often
understood as referring specifically to image-guidance units. Moreover, their practical
designs have distinct advantages and disadvantages. For example, coupled with a larger field
of view, the rapid speeds of conventional diagnostic units allow for four-dimensional (4D)
CT imaging, where the fourth dimension is time (i.e. dynamic volume CT).
It should be pointed out that despite slower readout capabilities, distinct advantages of FPDs,
including higher spatial resolution, have prompted investigation for some diagnostic
purposes as well. One large advancement in this field is dedicated breast computed
tomography (BCT), a method employing cone-beam CT for full 3D imaging in breast cancer
screening and diagnosis24-27.
1.3 CONE-BEAM CT ARTEFACTS DUE TO CIRCULAR
TRAJECTORIES
It was well known by the 1980s that cone-beam CT scans using a circular source trajectory
could not obtain complete information for accurate recovery of the imaged object28-32. The
missing information has been described in a number of different ways in the literature,
including Fourier and Radon domain interpretations. These theoretical treatments predict that
accurate reconstruction is achieved only within the plane containing the source trajectory.
Elsewhere, the 3D images will exhibit artefacts, which are most commonly referred to as
cone-beam artefacts. The most notorious phantom demonstrating the failure to achieve
accurate reconstruction is a stack of disks, commonly known as the Defrise phantom, after
6
Michel Defrise who originally suggested this phantom as a test of completeness33. As seen in
Fig. 1-2, images of disks at greater heights above the plane of the source trajectory show
highly degraded recovery of the disk boundary. Such results have led to research producing
various artefact reduction algorithms based on theoretical approximations, as will be
discussed in Chapter 4 of this work. Extensive attention has also been drawn to algorithms
for alternative trajectories such as circle plus line34-36, saddle37-39 or helical39-49 source
trajectories, which provide sufficient information for theoretically exact reconstructions.
1.3.1 INFLUENCE ON CLINICAL IMPLEMENTATION
Despite the wide attention that cone-beam CT artefacts had received, preclinical trials using
simple circular orbits revealed images of surprising quality and utility even with no
corrections attempted for the artefacts2. Cone-beam artefacts that were very evident in the
Defrise phantom were not as obvious in the anatomical specimens imaged. Rather, image
quality seemed to be more dependent on other physical factors, including scattered radiation,
x-ray beam hardening, detector efficiency, electronic noise and detector behaviour changes
with exposure (e.g. image lag and/or ghosting effects). Moreover, the primary use of cone-
beam CT as an image guidance tool, to locate anatomical or other markers, could be done
with relatively high accuracy. Although other orbits provide theoretically complete sampling
of the region of interest, circular orbits were also easiest to implement in a radiotherapy or
surgical setting and reconstructions could be made using fast filtered backprojection (FBP)
algorithms. Circular cone-beam CT was therefore successfully implemented as an image
guidance tool.
7
x
z
z=0
6 cm
Fig. 1-2. Cross sectional image in the y-z plane of a stack of disks, also called the Defrise phantom, showing greater cone beam artefacts (i.e. distortion of the disk edges, shading and streaking) with distance from the plane of the source trajectory (z=0).
In practice, modern diagnostic CT scanners provide options for both helical and circular
scans. Since standard diagnostic CT scanners use helical trajectories in the collection of fan-
beam CT data, the practical transition to helical cone-beam CT was fluid. Typically, helical
scans are achieved by translating the patient through the scanner gantry as the source and
detector rotate continuously. While complete data is theoretically obtainable from helical
scans, some tradeoffs may exist, such as the necessity to move the patient during the scan
which may contribute to motion artefacts. More complex algorithms for non-circular scans
and difficulty utilizing endpoints of the scan have also limited implementation of helical
scans to 64 detector rows at the time of this work. In addition, circular trajectories
conveniently offer a consistent field of view which is advantageous in 4D CT applications.
1.3.2 DISTINCTION FROM OTHER ARTEFACTS
Referring to the image artefacts as ‘cone-beam artefacts’ may be misleading since they are
inherent to the source trajectory rather than the use of a conical beam of x rays. Other names
8
have also been given to cone-beam artefacts such as ‘FDK’ artefacts, so named since the
famous algorithm of Feldkamp, Davis and Kress (FDK)50 is the most widely used
reconstruction algorithm for cone-beam projections under a circular trajectory. However,
this name is also misleading as it suggests that the artefacts are a consequence of the
algorithm rather than the source trajectory used. A more appropriate term for the effect may
therefore be “circular trajectory artefacts”; however, the conventional “cone-beam” moniker
is used for consistency with the literature. It should also be noted that other artefacts,
particularly those due to scattered radiation, may be more directly related to the use of a
wider field of view as in cone-beam CT. However, these artefacts are not referred to as
cone-beam artefacts, and should not be confused with them.
1.4 MOTIVATION AND CONTENT OF THESIS
Given that artefacts are inherent to the circular cone-beam CT geometry, and that this
geometry is increasingly utilized in industry, it is desirable to understand their impact on
clinical images. However, intuitive understanding of the manifestation of the artefacts can be
quite limited. A useful description of the missing information relates the artefacts to missing
spatial frequency components in the local Fourier domain. This description implies that
manifestation of cone-beam artefacts will depend not only on the source geometry, but also
on the frequency content of the object itself, which further suggests that an object’s shape,
texture, and orientation are also necessary parameters in quantification of the expected
artefact. This consequence may lend insight into why varying results in image quality have
been reported in the literature. For example, whereas planar disks (i.e. in the Defrise
Phantom) have been shown to degrade rapidly at higher cone-beam angles, images of
9
complex bony and soft-tissue anatomy have been achieved under similar imaging geometries
with less obvious degradation2.
Although measurements have been made on real cone-beam CT systems in terms of the
PSF51 and the modulation transfer function52 (MTF), the lack of frequency information
proposed by theory has not been validated experimentally in a precise manner. The main
objectives of this thesis are to:
• compile and elucidate theory describing the missing frequency components
• measure the missing cone of frequency content in real data, obtained using circular
cone-beam CT, and compare with theoretical predictions
• quantify cone-beam artefacts in anatomically relevant objects
• propose a novel method for correction of the artefact
Chapter 2 provides a derivation of the Fourier description, and indicates via examples how
the absence of local spatial frequency components can preferentially affect a reconstructed
image. The third chapter supports the local Fourier description of missing data by imaging
simple, localized geometrical objects (disks and spheres) and examining their Fourier
transforms. The implied artefact dependency on the object’s frequency spectrum is
investigated by varying the orientation of a mini disk phantom. Manifestation of the artefact
on a larger scale is then explored via comparisons of large disk phantoms with differing
internal structures and discussed with respect to previous results. More clinically relevant
objects (live rabbit, anthropomorphic knee phantom), are also imaged for the purpose of
10
examining the extent of information loss at anatomical boundaries with distance from the
source plane. In Chapter 4, correction schemes will be briefly reviewed and a method to
correct for the missing frequency components in cone-beam CT is illustrated using a priori
knowledge. This method is tested under simplifying assumptions, with a discussion of its
limitations to real situations. Finally, a discussion of the artefact and its consequences is
summarized in the final chapter. This work contains reprinted material of previously
copyrighted work with permission from the American Association of Medical Physicists53.
11
Chapter 2: Cone-beam CT Artefacts and
the Local Fourier Domain
The information obtained using a circular trajectory has been described in a number of
different ways in the literature. Grangeat represented the missing data in the Radon domain,
showing that ideal cone-beam data from a circular trajectory fill a torus instead of a sphere in
the Radon transform31. Others have described cone-beam artefacts using the point spread
function (PSF), generally basing their derivations on FBP algorithms54-57. Others still have
presented Fourier based descriptions relating cone-beam artefacts to missing spatial
frequency components58-61. The frequency description is of particular interest since it has a
direct link to the spatial resolution capabilities of the imaging system62.
In this section, a local Fourier description will be developed using the Fourier Slice Theorem
and examining local ‘neighbourhoods’ of the object space. A conical region of missing
frequency components will be seen to be absent in the local Fourier domain. This cone has
sometimes been referred to as the “empty cone” in papers on ectomography58, while
elsewhere has simply been referred to as the unsampled59, unmeasured63 or missing cone64,65
of frequency components. In this paper the latter term is adopted, and the region will be
referred to as the ‘missing cone’ herein. The meaning of ‘local’ in ‘local spatial frequency
components’ will be made clear later in the text. Artefacts arising due to these missing
spatial frequency components will be illustrated using 2D analogues as examples.
12
2.1 THEORY
2.1.1 FOURIER SLICE THEOREM AND THE RADON TRANSFORM
The link between plane integrals in the object space and the Fourier domain can be made via
the 3D Radon transform. This transform takes an object defined by some density function
f(x) and transforms it into a set of plane integrals
( , )( , ) ( ) ( ) ( )P sr s f x dP f x x s d xγγ∞ ∞ ∞
−∞ −∞ −∞
= = ⋅∫ ∫ ∫ ∫ ∫ δ γ − , (2.1)
where planes P(γ,s) = {x : x⋅γ = s} have unit normal vector γ with distance s from the origin,
and x is an arbitrary position vector as seen in Fig. 2-1. One version of the Fourier slice
theorem66 states that the linear Fourier transform of r(γ,s) with respect to s
2( , ) ( , ) isR r s e dπ σγ σ γ∞
−
−∞
= ∫ s , (2.2)
is equivalent to the line through the 3D Fourier transform
( ) 2 ( )( ) i x kF k f x e d xπ∞ ∞ ∞
− ⋅
−∞ −∞ −∞
= ∫ ∫ ∫ , (2.3)
that intersects the origin of the Fourier domain and has orientation in the direction γ, such
that F(k)=R(γ,σ) when k= σγ. (Note that traditionally the Radon transform is denoted with a
capital R, whereas in the above equations this notation is reserved to denote its Fourier
transform pair as per eq.(2.3)). Complete knowledge of an object’s plane integrals is
therefore equivalent to knowledge of its 3D Fourier transform. This theorem is exploited by
FBP methods to recover the original object30,67,68. If the set of plane integrals is known, one
can also use the inverse Radon transform69 to reconstruct the object,
13
2
2 2
1( ) ( , )8 s xf x r s
s γ dγ γπ = ⋅
∂= −
∂∫ ∫ (2.4)
,sPγγ
s
Vector Plane Notation
,sPγγ
s
Vector Plane Notation
Fig. 2-1. Vector representation of planes used in the Radon inversion formula.
2.1.2 MISSING PLANES IN CONE-BEAM CT
In cone-beam CT, plane integrals are not measured directly. Linear integrals of the object’s
attenuation coefficient are measured along ideal, straight x-ray paths from the source to the
detector67,68. These line integrals can be parametrized as
0
( , ) ( ( ) )g f tτ ν τ∞
α = + α∫ dt (2.5)
where τ parametrizes the cone-beam source position ν(τ), and the unit vector α indicates the
direction of the emanating ray. It can be seen that any plane that intersects the source
trajectory will contain a fan-beam of rays originating at the source. Integrating over the line
integrals in such a plane will result in an approximate plane integral through the object,
( ( , ) (r gτ τ δ, )dγ) = α α⋅ γ α∫∫ (2.6)
14
If the lines were parallel instead of diverging, then (r τ , γ) would be the true plane integral
r(γ,s) instead of an estimate. The relationship between the approximate plane integral and
the true plane integral has been determined independently by several authors and can be
found via derivatives of the appropriate terms28-31,70,71. Defining
'( , ) ( , )r s r ss
γ γ∂=
∂ (2.7)
and
'( ( , ) '( )r gτ τ δ, dγ) = α α⋅ γ α∫∫ , (2.8)
where δ’ is the derivative of the Dirac delta function, the relationship is simply
'( '( , )r rτπ
−1⎛ ⎞, γ) = γ⎜ ⎟2⎝ ⎠s , (2.9)
which is generally referred to as Grangeat’s result. Tuy made the observation that if all
planes passing through the object intersect the source trajectory then all plane integral
derivatives, r′(γ,s), are obtainable and the object can be fully recovered using the Radon
inversion formula (eq.(2.4)). This condition on the source trajectory is generally known as
Tuy’s condition. When this condition is not met there will be incomplete information for
stable solution of the inverse problem29. In the case of a circular trajectory, Tuy’s condition is
satisfied only for the special case where points lie within the plane containing the source,
herein referred to as the ‘source plane’. For points above or below the source plane a subset
of planes will exist that do not intersect the source trajectory and Tuy’s condition is violated.
Examples of measurable and non-measurable planes are shown in Fig. 2-2.
15
Source Trajectory Source Trajectory
(a) (b)z
yx
zy
x
zy
x
zy
x
Fig. 2-2. (a) Illustration of planes that intersect the source trajectory. (b) The most obvious example of a plane that does not intersect the source trajectory is a plane parallel to it.
2.1.3 A LOCAL FOURIER DESCRIPTION OF CONE-BEAM CT ARTEFACTS
The missing plane integrals can be visualized in the Fourier domain as a shift-variant cone of
local, spatial frequency components. The Fourier description is considered local because it is
derived by considering an object within the local neighbourhood of point xo, which is
sufficiently small and distant from the source that the divergence of the rays can be ignored.
Rays intersecting the local neighbourhood of xo can therefore be grouped into parallel planes,
and the corresponding plane integrals can be measured directly. The planes that are not
measurable at point xo (and by assumption in the local neighbourhood of xo) can then be
identified from the cone-beam geometry. Fig. 2-3 illustrates the case for those planes with
normal vectors restricted to the y-z plane for simplification, and with point xo located at (0, R,
zo). It follows from the Fourier slice theorem that the localized object will have
undetermined lines in the Fourier domain corresponding to the non-measured planes. For
example, the sample plane shown in Fig. 2-3(a) will have a line missing along the
corresponding normal direction in the Fourier space as shown in Fig. 2-3(b). The complete
16
set of missing planes corresponds to a conical region of missing frequency components (see
Fig. 2-3(b)). This missing cone is an oblique, circular cone with its boundary and interior
defined by the set of normal vectors to the missing planes. Proof that the cone is a circular,
oblique cone is provided in the Appendix. A unique cone is associated with each point in
space (i.e. the cone is shift-variant), since a unique set of plane integrals will be missing at
any given location, with the exception being on the source plane. The missing cone can be
defined in terms of its internal angle, η(φ), measured from the vertical axis as a function of
transverse angle, φ. Noting that η(φ) is equivalent to the angle, α(φ) (see Fig. 2-3 and
Appendix), it can be defined as
1
2( ) tan
2 cos( )oz
2R Rη
ρ ρ−
⎛ ⎞⎜φ =⎜ − φ +⎝ ⎠
⎟⎟
(2.10)
with minimum and maximum values
11 tan oz
Rη
ρ− ⎛
= ⎜ +⎝ ⎠
⎞⎟ (2.11)
and
12 tan oz
Rη
ρ− ⎛ ⎞
= ⎜ −⎝ ⎠⎟ , (2.12)
respectively, corresponding to the minimum and maximum values of α(φ) at xo.
Alternatively, the missing cone can be defined in the Fourier domain as
{ }2 2 21:( ) ( )
ox z x yC k a k k k a k= > + − 2 z (2.13)
where a1=zoρ /(ρ2-R2), a2=zoR/(ρ2-R2), and ρ is the radius of the circular trajectory. Note that
for points on the rotation axis, oxC is symmetrical about the vertical axis. Although the above
17
descriptions have restricted xo to the y-z plane, arbitrary xo can be considered by
implementing a rotation of coordinates.
v(-90o) v(90o)
zone of missing plane-sums
xo η2η1
R
non-measured plane
α
η2η1
kykx
kz
yz
x
(b)(a) rotation axis
ko
ρv(-90o) v(90o)
zone of missing plane-sums
xo η2η1
R
non-measured plane
α
η2η1
kykx
kz
yz
x
(b)(a) rotation axis
ko
ρ
Fig. 2-3. (a) Schematic representation of missing plane information at point xo. Shaded area indicates the region of missing planes with normal vectors restricted to the y-z plane for simplicity. An example of a non-measurable plane is indicated by the dashed line. (b) Missing plane information results in unmeasured lines of spatial frequency components that fill a cone in the local Fourier domain as illustrated where ko corresponds to the DC (zeroth frequency) component. The minimum, η1, and maximum, η2, internal angles of the missing cone are shown in (a) as they relate to the angle, α, at source position v(φ), where φ is the transverse angle in degrees, measured counter-clockwise from the x axis.
Since no information is known about the missing frequency components, they are usually
either explicitly or implicitly set to zero by the reconstruction algorithm provided no
additional constraints are introduced. Assuming the measurements are otherwise noiseless,
and no estimations of the missing data are made, the model for the reconstructed image of
objects localized near xo is
2 ( )( ) ( ) ( )o o o
i k xx o x x x y zf x x F k T k e dk dk dkπ
∞ ∞ ∞⋅
−∞ −∞ −∞
− ≅ ∫ ∫ ∫ (2.14)
where
2 ( )( ) ( )o
i k xx oF k f x x e dxdydzπ
∞ ∞ ∞− ⋅
−∞ −∞ −∞
= −∫ ∫ ∫ (2.15)
18
and
if 0( )
1 otherwise o
o
xx
k CT k
∈⎧= ⎨
⎩, (2.16)
where the symbol oxT refers to the transfer function which only passes frequency information
outside the oblique, circular cone, oxC . Note that this transfer function can then be thought of
as a zero pass filter affecting all frequency components (low and high) within the cone oxC .
Although the object of interest is localized near xo, the artefacts associated with the zeroed
frequency components may extend to regions far removed from xo. The non-localized case
can be considered by decomposing the object into smaller sub-regions and analyzing the
artefacts that arise independently for each of these sub-regions. In this non-localized case
(and in the limit as the sub-region approaches infinitesimal size), the reconstruction model
becomes
( ) ( )or x o o o of x f x x dx dy dz
∞ ∞ ∞
−∞ −∞ −∞
= −∫ ∫ ∫ . (2.17)
For localized objects, the predicted missing cone, oxC should be observable in the object’s
Fourier transform. Artefacts resulting from the missing frequency components will in
general depend on the frequency content of the object itself, and therefore on factors such as
its shape, texture and orientation. In particular, reconstructions of objects that have a large
proportion of energy distributed over frequency bands within oxC will be most compromised.
Furthermore, the size of oxC increases with distance from the source plane, implying that
artefacts should become more severe with distance above or below this plane, while accurate
reconstructions should be possible on the source plane itself since oxC vanishes on this plane.
19
It should be noted that the missing frequency data is inherent to the acquisition geometry and
is therefore independent of the reconstruction algorithm. It is also important to note that the
set of recovered frequency components have been described assuming a continuous source
along a circular trajectory (i.e. using infinite projections) and an idealized detector. This
situation is not the case in practice, since projections are sampled at a finite number of
intervals along the circular trajectory and the detector has finite resolution. Conditions for
sufficient sampling in terms of projection number and detector pixel sampling have been
published and are assumed to hold in the analyses herein 61,72.
2.2 EFFECT OF MISSING SPATIAL FREQUENCY
COMPONENTS
The frequency description of missing information in cone-beam CT may lend some intuitive
understanding of the manifestation of cone-beam CT artefacts in images. Recall that a line of
spatial frequency components is associated with changes in the real object along the same
direction. Therefore, large intensity gradients in that direction, such as may exist at an
anatomical boundary, are the most likely regions to suffer resolution loss.
To aid in understanding the consequence of missing lines of spatial frequency components, it
is helpful to observe the effects of removing frequency lines from an image. A 2D image can
be evaluated in order to simplify the visualization. In this case, a ‘wedge’ replaces the ‘cone’
of frequency components described earlier, and plane-sums are replaced by line-sums (see
Fourier Slice Theorem in Kak and Slaney’s text68). Fig. 2-4 shows the process used to
simulate the artefacts in an image with well defined horizontal features. In order to remove
frequency components, the image is Fourier transformed via the Fast Fourier Transform
20
(FFT) and then a subsection of frequency components are removed by multiplying the FFT
with a mask function. It should be noted that the discrete Fourier transform assumes the
image is periodically repeated (in a tile-like manner). Therefore, the removal of frequencies
from the discrete transform is an estimate only. To achieve more accurate results, zero-
padding (literally augmenting zeros to the sides of the image) can be added to the image
before taking the discrete transform. In this example, the original image size is 1704×1704
pixels, and was zero padded to size 3200×3200.
FFT
=
IFFT
×
Input Image
Output Image
FFT
=
IFFT
×
Input Image
Output Image
Fig. 2-4. Schematic diagram of filtering spatial frequency components from a 2D image. The discrete Fourier transform of the original image is multiplied with a mask function of zeros and ones resulting in a wedge of removed missing frequency components. The image without those frequency components is estimated via the inverse transform.
The results before and after the filtering process are seen in Fig. 2-5 (a) and (b). For close
inspection, the bottom right corners of the images are magnified in (c) and (d). There are
some obvious discrepancies between the images that have arisen due to the missing
information. In particular, horizontal edges are less distinguishable; more specifically, the
21
base and top of the canoe as well as the shoreline are notably effected. Since the maximum
intensity gradients introduced by these features are in a direction coincident with the missing
region of frequency components, this loss of resolution might be expected. In addition,
shading and streaking is visible throughout the image. It is noted that this streaking will tend
to blur or obscure features that have the same trend. Interestingly, other details in the
(b) (a)
(c) (d)
Fig. 2-5. Image before (a) and after (b) filtering by the wedge mask of Fig. 2-4. A zoomed in portion shows original details of the canoe (c) are missing in the filtered image (d). Features without strong gradient changes directed within the pie slice of missing frequency components notably appear better preserved.
22
image remain discernible (e.g. canoers’ arms, heads, life vests, and the stern and bow of the
canoe). This example illustrates that missing lines of frequency components may be loosely
related to preferential degradation of intensity gradients along the same direction.
x
y
x
y
P90 (x)O
P0
(y)
O 0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
P90 (x)O
P0
(y)
O
FFT IFFT
Filter
x
y
x
y
x
y
x
y
P90 (x)O P90 (x)O
P0
(y)
O
P0
(y)
O 0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
P90 (x)O P90 (x)O
P0
(y)
O
P0
(y)
O
FFT IFFT
Filter
Fig. 2-6. Removal of frequency components by the same process as used in Fig. 2-4 results in streaking and shading artefacts in the image of a square. Line integrals in the horizontal direction (along the dashed arrows) provide the projection P0
o(y). In the case of the filtered square this projection is just a flat line, corresponding to a constant value for each line integral. Projections in the vertical direction (P90
o(x)) are equivalent for both cases. Note: the regions shown are cropped versions of the original for better visualization of the shading effects (i.e. the artefacts and the integral of P0
o(y) extend over a much larger region than shown). Blue lines indicate the boundary of the removed frequency components.
23
Although a general connection can be made between gradient directions and the direction of
frequency loss, it should be made clear that spatial frequency components are not directly
related to local gradients, but rather depend more generally on changes throughout the
volume. More precisely, a central line of frequency components corresponds to the linear
Fourier transform of the function of plane integrals (or line integrals in 2D) through the
object, as per equation (2.2). Moreover, the precise meaning of a line of zero frequency
components (except the zeroth frequency) is that all integrals over planes perpendicular to
that line will be equal.
Consider as another example the image of a square, which contains large intensity gradients
in both the y and x directions. Repeating the process described in Fig. 2-4, a wedge of
frequency components can be removed, with the results shown in Fig. 2-6. It is clear that the
upper and lowermost surfaces of the square have suffered loss in resolution, similar to that of
the previous examples of the canoe and the disk phantoms shown in the introduction. To see
how the above principles relate to this situation, line integrals along two directions in the
image can be examined by looking at parallel ray projections at 0 and 90o angles before and
after frequency filtering, as shown in Fig. 2-6. In the filtered image, the intensity values have
been redistributed such that the 90o projection appears the same as in the original, while the
0o projection has changed from a square function to a constant value, as predicted. Note that
the Fourier transforms of these projections correspond to lines of frequency components in
the 2D Fourier transform. The constant valued projection has a Fourier transform with all
frequency components equal to zero (except the zeroth frequency which corresponds to the
average intensity value), as expected.
24
These examples illustrate that missing lines of frequency components do not translate to a
simple smoothing function through the object but rather result in a complex redistribution of
intensity values in order to satisfy specific conditions on planar integrals through the object.
A final, simple example, will illustrate this point more definitively.
FFT
Image
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
FFT
Image
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Fig. 2-7. Single and repeated squares in checkerboard pattern with discrete Fourier transforms below (after zero padding). Magnitudes in frequency domain have been normalized to the DC component.
Consider the same square above with additional squares added in a checkerboard fashion as
seen in Fig. 2-7. Filtering the frequency spectra as before results in the images of Fig. 2-8.
Difference images between the filtered and non-filtered cases in Fig. 2-9 highlight the
regions of highest discrepancy between the image sets. Based on the intuitive ideas presented
earlier, it might be suspected that each square will be affected similarly. However, this
25
prediction is evidently not the case. In each image, it is the upper and bottom most regions of
the checkerboard that suffer the greatest intensity discrepancies from the original image.
Greater distortion at these regions, rather than at each individual square, appears necessary
to fulfill the integral conditions posed above.
FFT
Image
-0.2
0
0.2
0.4
0.6
0.8
-0.2
0
0.2
0.4
0.6
0.8
-0.2
0
0.2
0.4
0.6
0.8
-0.2
0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Fig. 2-8. Filtered single and repeated squares in checkerboard pattern with discrete Fourier transforms shown after filtering. Section of missing frequency components are outlined in white.
It is also useful to interpret the data from a Fourier domain perspective. Fig. 2-7 and Fig. 2-8
illustrate the discrete Fourier transforms of the images, showing dramatic changes to the
spectrum as more squares are added. In particular, the dominant frequency components are
located at 45 degrees to the vertical and horizontal axes, instead of coincident with them.
This example shows that the frequency content of an image can change substantially with the
addition of other objects; it also shows that the nature of the distortion in the image depends
on the overall frequency spectrum. Consequently, attempting to predict the extent of
26
resolution loss is quite complex. An attempt to develop best possible resolution equations of
planar features perpendicular to zero frequency lines has been published with specific
application to circular cone-beam CT62.
0.30.20.1
0.30.20.1
0.30.20.1
0.30.20.1
---0
0.10.20.3
---0
0.10.20.3
---0
0.10.20.3
---0
0.10.20.3
Fig. 2-9. Difference images show decreased intensity discrepancies of a single square when surrounded by other squares in a checkerboard pattern. Units are normalized to the intensity of the original square.
2.3 SUMMARY
Theoretical models describe cone-beam CT artefact as arising from lines of local unmeasured
spatial frequency components that fill a cone in the 3D Fourier transform. Intuitive
understanding of the effect of subtracting a line of spatial frequency components from an
image can be difficult to obtain. In general, gradient changes along the direction of the
missing spatial frequency components are likely candidates for having degraded resolution,
while streaking artefacts will trend perpendicular to that direction. A more complete
interpretation relates local gradient changes to changes in the plane integrals in the image,
which more directly correspond to the spatial frequency components. The extent of
resolution loss is complex, and in general depends on the object’s overall frequency
distribution.
27
Chapter 3: Cone-Beam Artefacts
in Real Images
In this chapter, the theoretical prediction of a missing cone of local frequency components is
supported by a series of experiments using small localized objects (disks and spheres) and
examining the related frequency loss in cone-beam CT images. The size and shape of the
cone are measured within the local Fourier transforms. Larger disks of varying texture are
also examined in order to compare the effects on cellular details versus planar features.
Finally, data from an animal specimen and an anthropomorphic phantom are analyzed to
examine the cone-beam effects on clinically relevant features using typical cone-beam angles
of IGRT.
3.1 EXPERIMENTAL VALIDATION OF A LOCAL FOURIER
DESCRIPTION OF CONE-BEAM CT ARTEFACTS
3.1.1 METHODS
A. Imaging Bench for Cone-Beam CT
An amorphous silicon flat panel detector (Paxscan 4030A, Varian, Palo Alto) with 194 µm
pixel pitch, and a 600 kHU x-ray tube (Rad-94, Varian, Palo Alto) were used in a cone-beam
CT laboratory design for the disk and acrylic sphere experiments described below. The main
components of the equipment can be seen in Fig. 3-1. In the test arrangement, the source and
detector remained stationary while the object rotated on the rotational stage under computer
28
control. The axis of rotation is coincident with the z axis. Repeat scans involving vertical
object displacements were achieved by moving the source and detector on vertical linear
rails. Details of the experimental equipment and performance capabilities have been reported
elsewhere72. Cone-beam CT images of a live rabbit were also acquired using a clinical
scanner (Elekta Synergy, Elekta, Stockholm). Tables I and II list imaging parameters used for
each study described below. Detailed descriptions of the methodology for each experiment
performed are provided hereafter. Results and analysis are provided in a separate section.
However, the reader may prefer to read the corresponding results for a given experiment
immediately following the description of its methods.
Flat Panel Detector
X-Ray Tube
Rotation Stage
α
zy
xφ
Fig. 3-1. Photograph of test bench used in acquiring projection data. Phantom shown is for illustrative purposes only (not used in this experiment). Arrows indicate directions of motion of object, x-ray tube and detector during the experiments.
29
Table I: Imaging and Reconstruction Parameters for Small Phantoms
Imaging Parameters Acrylic Sphere Mini Disks Imaging Plaftorm Bench Bench
Source to Axis Distance 60 cm 100 cm Source to Detector
Distance 96 cm 155 cm
X-ray Exposure Mode: pulsed radiographic pulsed radiographic kVp 100 120 mA 80 100 ms 4 5
Filter: 4mm Al + 0.1mm Cu 2mm Al + 0.1mm Cu Rotation/projection
(degrees) 1.125 1.125
No. of Projections 320 320 Frame Rate 1 fps 1 fps
Voxel Size (µm3): 121×121×121 125×125×125
Table II: Imaging and Reconstruction Parameters for Large Phantoms
Imaging Parameters Large Disks Rabbit Knee Imaging Plaftorm Bench Elekta Synergy Bench
Source to Axis Distance 100 cm 100 cm 100 cm Source to Detector
Distance 160 cm 154 cm 155 cm
X-ray Exposure Mode: pulsed fluoroscopic pulsed radiographic pulsed radiographic kVp 120 120 120 mA 40 80 80 ms 7.5 10 8
Filter: 2mm Al + 0.1mm Cu
F1 aluminum bow-tie filter
4mm Al + 0.1mm Cu
Rotation/projection (degrees) 1.2 0.55 1.125
No. of Projections 300 650 320 Frame Rate 1 fps 5.5 fps 1 fps
Voxel Size (µm3): 120×120×120 750×750×750 500×500×500
30
B. Acrylic Sphere Phantom
In order to identify Cxo in localized regions of space, a phantom was constructed using a set
of 3.2 mm diameter acrylic spheres. Spheres were chosen because of their 3D symmetry in
the object space and therefore in the Fourier domain. This property greatly simplifies the
identification of missing frequency components in the Fourier transform. The spheres were
housed in polystyrene foam in order to provide a uniform background of near air density, and
were aligned at 1 cm intervals. This phantom was positioned vertically such that the first
sphere lay on the source plane, while the remaining spheres were at increasing z distances.
The spheres were imaged coincident with the rotation axis, as well as at an offset, R, in the y
direction, in order to observe both the symmetrical and oblique, circular cones within the
local Fourier space of these sub-volumes. Relevant parameters involved in object setup are
seen in Fig. 2-3(a). Fig. 2-3(b) shows the relationship of maximum and minimum internal
angles of the missing cone in frequency space to the real space imaging geometry for an
oblique, circular cone.
The theoretical predictions of the size of Cxo were tested using the acrylic sphere data. All
images were reconstructed using the Feldkamp FBP algorithm. Reconstruction sub-volumes
of dimension 256×256×80 voxels were analyzed (see Table I for voxel dimensions), where
each sub-volume is centred about a single sphere. This dimension was chosen to retain most
information in the x and y directions, where the majority of the artefact is expected, while
restricting influence of artefacts from spheres above or below the one examined. A
background subtraction was made to the data by subtracting the average foam value. Each
sub-volume was then multiplied by a window function, W(i,j,n), such that
31
( ) ( )( , , ) , , ( , , ) 1, 1 ( 1) ( , , )win r h v rf i j n W i j n f i j n W i j W n f i j n= ⋅ = + + ⋅ + ⋅ , (2.18)
where fwin(i,j,n) is the value of the reconstruction volume at index (i,j,n), fr(i,j,n) is the value
of the original reconstruction volume, Wh(i,j)) is a circular Hann window degenerate in n
defined as
( )1
2 2 21( 1, 1) 1 cos 22h
h
i jW i j
Nπ
⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟+ + = −⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
, 0,..., hi N= 0,..., hj N= (2.19) 255hN =
and Wv(n) is a linear Hann window degenerate in i and j defined as
1( 1) 1 cos 22v
v
nW nN
π⎛ ⎞⎛ ⎞
+ = −⎜ ⎜⎜ ⎝ ⎠⎝ ⎠⎟⎟⎟ , 0,..., vn N= 79vN = . (2.20)
Surface plots of central vertical and horizontal cross sections of the resulting window
function are seen in Fig. 3-2. This windowing was performed in order to guarantee a smooth
transition to zero mean values at the boundaries of the volume and therefore reduce spectral
leakage in the Fourier domain73. Note that the separated window functions correspond to
separate convolution kernels in the Fourier domain. These convolutions are not expected to
impact the boundary of the missing cone and therefore are not expected to negatively impact
the results of the following analyses. The data was then zero padded to a volume of
256×256×256 voxels and transformed using the Fast Fourier Transform (FFT). All
measurements were made in the Fourier domain, considering only the absolute magnitudes of
the frequency components. Working with the magnitude was adequate for identification of
the missing frequency components and avoided the necessity of accurate registration of the
sub-volumes that would be required if the phase were to be considered. Various methods are
32
possible for verifying the size of Cxo in the experimental data. The chosen method is similar
to evaluating an edge spread function at the cone boundary. Numerical surface integrals
were evaluated over conical surfaces that ranged in size from less than to larger than the
expected size of Cxo. The conical integration surfaces had the same degree of skew and
orientation as that of Cxo such that at least one surface integral was expected to coincide with
its boundary. The result of each integral was normalized with the corresponding result for the
sphere that was centred on the source plane. Surface integrals within Cxo would ideally be
expected to yield a null value, while values outside it would be expected to have a
normalized value of 1 (since frequency components in this region should ideally be the same
for all spheres). A plot of these integral values as a function of maximum internal angle, η2’,
of the integration surface would be expected to have a maximum derivative at precisely the
boundary of the missing cone (i.e when η2’=η2). This method was tested using simulated
oblique, circular cones of zeros of comparable size created within a volume of ones. The
results indicated that the algorithm could accurately return the internal angle of the simulated
cones with negligible error.
An implicit assumption made in the analysis is that the image of the sphere centered on the
source plane will be a ‘true’ reconstruction, while image data of the spheres above or below
the source plane will exhibit a well-defined region of missing frequency components in the
Fourier domain. This assumption is compromised by several factors. Firstly, since the
missing cone is shift-variant, it will vary in size at different positions within a given sphere.
However, the sphere size was chosen to be small enough to allow for the assumption of shift-
invariance to good approximation over the region of the sphere. Another factor is that the
33
cone-beam artefacts introduced may spread to regions well beyond the sub-volume
examined. Although truncation of the artefacts should introduce inaccuracies in the Fourier
transform, the impact is expected to be minimal since the majority of the object’s energy is
contained within the given sub-volume. Furthermore, the effects of spectral leakage that
would be introduced by truncating the artefacts are reduced by the window function
described previously. Finally, the surface integrals performed excluded regions near the DC
component where the boundary of the edge of the cone is not well defined due to the discrete
sampling of the data.
64
128
192
256
4080
0
0.5
1
Window Function, j = 128
i (voxels)k (voxels)
Mag
nitu
de
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
64128
192256
64128
192256
0
0.5
1
i (voxels)
Window Function, k = 41
j (voxels)
Mag
nitu
de
64
128
192
256
4080
0
0.5
1
Window Function, j = 128
i (voxels)k (voxels)
Mag
nitu
de
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
64128
192256
64128
192256
0
0.5
1
i (voxels)
Window Function, k = 41
j (voxels)
Mag
nitu
de
Fig. 3-2. Surface plots of the function applied to reduce spectral broadening effects in the power spectral density estimation by FFT. The function is applied to avoid sharp discontinuities between the zero padding and the volume edge due to statistical fluctuations.
C. Mini Disk Phantom
A mini disk phantom was constructed in order to test the dependence of object recovery on
the distribution of the object’s frequency spectrum with respect to the missing cone. The
phantom was constructed using three mylar disks 10.2 mm in diameter, 0.21 mm thick, and
spaced by approximately 2.0 mm of polystyrene foam. A schematic representation of the
experimental setup illustrating the key parameters involved is seen in Fig. 3-3. The mini
34
disk phantom was inserted into a polystyrene foam housing and then mounted onto a
rotational stage, which was used to vary the degree of inclination of the disks, θ, with respect
to the source plane (see Fig. 3-3). The disks were then imaged along the rotation axis at
varying distances above the source plane. Since the majority of the energy of the disks lies in
frequency bands that have direction perpendicular to the disk edges, changing the parameter
θ changes the distribution of the frequency spectrum of the phantom with respect to the kz
axis in an obvious way, and allows for a method of probing the frequency response in
localized regions of the image space. The imaging geometry used in this experiment was
chosen to agree with typical geometries used in IGRT (see Table I), observing a maximum
angle, α, of approximately 5.5o. Volume size of the reconstruction was 200×200×100
voxels. Background (foam) subtraction and zero padding to equal dimension was performed
prior to calculation of the FFTs. FFTs were examined for recovery of frequency content with
tilt angle.
z
α
θ
Source Trajectory
z
α
θ
Source Trajectory
Fig. 3-3. Schematic representation of mini disk experiment. The z axis corresponds to the axis of rotation. The size of the disks is greatly exaggerated in the illustration for purposes of clarity. (See text for details.)
35
D. Exclusion of Other Possible Physical Effects
In order to support that the artefact seen is due primarily to loss of frequency content and not
due to other physical effects, the artefact should be reproducible by the theoretical removal of
local frequency components. Using the mini disk phantom centred on the source plane as the
reference image, and using the assumption of shift-invariance, the filtering can be carried out
in the frequency domain,
{ }1( ) ( ) ( )filtf x F k T−= ⋅F k (2.21)
where ffilt(x) is the filtered image, F-1 indicates the inverse Fourier transform and T(k) is a
volume of ones with a missing cone of zeros equivalent to that predicted by theory. This
multiplication in the Fourier domain is equivalent to the convolution in the spatial domain of
the object function with the theoretical PSF, F-1 (T(k)).
E. Large Cellular and Solid Disks
This experiment was designed in order to evaluate artefacts introduced when the internal
structure is more complex. Two distinct large disk phantoms were imaged: one made of solid
acrylic and the other made of cellular polyurethane. The latter material has an internal
structure similar to that of trabecular bone. Both disks were 125 mm in diameter, and 1 cm
thick. The disk phantoms were imaged parallel and at a displacement of 5 cm above the
source plane. The data was analyzed to observe the recovery of internal cellular details at z
displacements where planar features with horizontal orientation are expected to be severely
distorted.
36
F. Rabbit
Images for this experiment were made available from a previous experiment investigating
contrast agents and comprises the control data set (no contrast agent employed). The
specimen was a live, anaesthetized rabbit and was imaged using a clinical Elekta Synergy
unit (Elekta, Stockholm). Data was acquired with the rabbit freely breathing throughout the
scan. The 3D reconstruction of the rabbit was analyzed in order to determine if cone-beam
CT artefacts can be identified in anatomically relevant data acquired using a clinical scanner.
With no ‘ground truth’ for comparison, identifying cone-beam CT artefacts in the spatial
domain may be difficult, especially in the presence of other artefacts (such as beam
hardening). However, analysis in the Fourier domain can provide evidence of cone-beam
artefacts if a region of missing frequency components can be identified. The dimension of
the sub-volume chosen for analysis was 64×64×64 voxels. This sub-volume was chosen to
contain soft tissue, bony anatomy and air. Reconstructions were performed using Elekta
XVI software. Additional reconstruction parameters are found in Table I. The FFT of the
sub-volume was inspected for evidence of the missing cone.
G. Knee Phantom
A knee phantom was selected to reflect real anatomy, while providing a rigid body for best
possible registration of the imaged volumes, therefore allowing analysis in the spatial
domain. The first data set was acquired with the phantom positioned such that the space
between the knee joint was within the central plane. Data was then acquired again with the
knee joint shifted vertically in the field of view by 5 cm. The following analysis was
performed to quantify the intensity discrepancies between registered images of the knee at
these different heights.
37
Fig. 3-4 Anthropomorphic leg/knee phantom used in experiment.
A 3D contour of the knee joint was achieved by applying the following steps. The data
corresponding to the knee joint centred on the source plane were first thresholded using an
intermediary value between the average of the joint material and the surrounding material as
the threshold limit. The resulting image was binary such that all values greater than the limit
were set to 1 and all other points were set to 0. This image was then smoothed, using a
3×3×3 box mean filter to provide a smoother surface. A contour of the knee surface was
extracted by selecting an isosurface value between 0 and 1. The initial isosurface value was
chosen arbitrarily, and then refined to find the contour that highlighted maximum
discrepancies between the registered data sets. Differences along the contour should indicate
which surfaces of the knee joint suffered greatest artefact in the reconstruction. The intensity
38
discrepancies along this contour can be further binned as a function of normal vector angle
with respect to the vertical axis. In order to obtain a basis for comparison, the process was
repeated for a small sphere (6.25 mm diameter) with simulated artefactss by removing a cone
of frequency components as in section D.
It should be noted that the precision of the vertical rails allowed for the displacement of the
knee with respect to the source and detector to very high precision and accuracy, with
positioning errors expected to be less than 0.l voxel lengths. Therefore, no additional method
was required to register the images. Difference images revealed good cancellation of
anatomy, supporting that good registration was achieved.
3.1.2RESULTS AND ANALYSIS
A. Acrylic Sphere Phantom
Fig. 3-5 (a) shows a sagittal view of a sphere reconstructed with an 8 cm offset from the
rotation axis, and a height of 7 cm above the source plane. Note that the noise in the image
tends to obscure any noticeable artefact. However, sectional views through the logarithm of
the 3D FFT, as seen in Fig. 3-5 (b) and (c), show the absence of frequency information
within a conical region of space indicating that artefacts are present in the data. Fig. 3-6(a)
shows a sample plot of the normalized surface integrals, CSI, as a function of internal angle,
η2’, for the same data set. The shape of the curve is as expected, and increases steadily with
increasing η2’ coming to a maximum value near 1. The solid line represents the data after
application of an adjacent mean filter. This filter provided a smoother first derivative while
not expected to shift the location of the peak. Note that the values CSI never approach zero
39
for small η2’; this characteristic may be partly explained by the presence of noise in the data,
partial truncation of the artefacts and spectral leakage not completely eliminated by the
window function. A nearest neighbour approximation to the first derivative of the smoothed
curve is shown in Fig. 3-6(b). A Gaussian peak function was found to fit this smoothed data
adequately with a near unity adjusted R-squared value as well as a small reduced-χ square
value (displayed on the given plots). A third order polynomial was also fit to the peak over a
reduced range for additional validation. The location of the peak provides an estimate of η2
of the missing cone of frequency components. Angles η2 and η1 determined using this
method are drawn in Fig. 3-5 (b). Experimentally determined values of η2 are also plotted as
a function of z for R = 0 cm and R = 8 cm in Fig. 3-7 (a) and (b) respectively. Theoretical
values are shown as solid lines, and indicate that very good agreement exists between
experiment and theory. Fig. 3-8 shows the mean value of 20 slices through the centre of the
reconstruction on the rotation axis in order to demonstrate the increasing artefact at increased
distances z by reducing the influence of noise.
η1 η
z
y
kx=0 ky=0
Fig. 3-5. (a) Sagittacm (windowed). (bshows a measurablelogarithm of FFT at
0 0.05 0.1 0.10 0.05 0.1 0.1
1
(a)
l view through the centre of acry) Central slice of logarithm of F skew in the missing cone as η1 i ky=0. Constant of 1 added prior t
5 0.25 0.2 1 2 3 4 51 2 3 4 5
2(b)
lic sphere imaged at R = 8 cm, FT of acrylic sphere in (a) at s not equal to η2. (c) Central slio log for better visualization.
6 76 7 1 2 3 4 51 2 3 4 5
(c)
cm-z = 7 kx=0 ce of
6 76 7
40
0 2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
CS
I
η2' ( ο )
CSI (normalized) CSI (smoothed)
0 2 4 6 8 10 12-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
dCS
I/dη 2' (
o-1 )
η2' ( ο )
Smoothed Derivative Polynomial Fit Residuals (poly) Gaussian Fit Residuals (Gauss) Baseline
Gaussian Fit:Reduced χ2 = 1.6x10-4
Adjusted R2= 0.99
(a) (b)
Fig. 3-6. (a) Results of surface integrals (normalized) taken over various sized cones as a function of maximum internal angle with respect to the kz axis. Smoothed curve closely follows the original. (b) Derivative of (a) with corresponding Gaussian and polynomial fits to the peak; peak position should be estimate of η2 according to theory.
2 3 4 5 6 7
0
1
2
3
4
5
6
7
8
η 2 ( ο )
Z (cm)
Gaussian Polynomial Theory Residuals (Gauss) Residuals (Poly)
2 3 4 5 6 7
0
1
2
3
4
5
6
7
8
η ( ο
)
Z (cm)
Gaussian Polynomial Theory Residuals (Gauss) Residuals (Poly)
(a) (b)
Fig. 3-7. (a) Experimentally determined missing cone angle plotted for spheres on the rotation axis (internal angle = η2 = η1) using Gaussian and polynomial peak fits to derivative data. (b) Experimentally determined η2 values for spheres at R=8cm. Results show very good agreement with expected values.
41
0 cm
2 cm
4 cm
6 cm
z
Fig. 3-8. Sagittal slice through acrylic spheres imaged on the rotation axis. 20 slices were averaged to better illustrate the cone-beam artefact relative to background noise. Increased artefacts (shading, streaking) are manifested at increasing z distances.
B. Mini Disk Phantom
Sagittal reconstruction slices of the mini disk phantom are provided for varying
displacements and angular orientations in Fig. 3-9. Coronal views are also shown for the
largest z displacement. Note that in the case of the horizontal disk, the artefacts are
symmetric about the rotation axis, and the sagittal view represents any sectional view through
the center of the phantom (i.e. the artefacts extend throughout the axial view). In all cases,
image recovery of the phantom near the source plane appears well defined, while off the
source plane the level of artefact evident is varied. Increasing θ resulted in higher fidelity of
the disk lamina for regions that are far removed from the source plane. In particular, tilting
42
the disks resulted in less edge distortion, greater fidelity of intensity values and a general
reduction of streaking artefacts. Fig. 3-10 demonstrates this result in terms of a frequency
domain representation. The missing cone of frequency components predicted by theory is
evident for the disk parallel to and above the source plane. Conversely, with greater θ, the
disk maintains more of its frequency content since the majority of its frequency spectrum lies
outside the missing cone. Note that artefacts are not completely eliminated by tilting the
disk, because the missing cone still affects some portion of its frequency components. This
effect is expected since all real finite objects have some frequency content in all directions,
and explains the persistence of cone-beam artefacts.
43
Coronal Views
8.6 cm
Sagittal Views
8.6 cm
7.4 cm
6.1 cm
4.9 cm
3.6 cm
2.5 cm
1.2 cm
0 cm
0o 2.4o 3.5o 5.1o 6.4o
Fig. 3-9. Sagittal views of mini disk phantom imaged at varying heights and degrees of orientation. Increase in angular displacement maintains better resolution of disk edge at increased distance from the source plane. Coronal slices for the case where z=8.6 cm are also displayed above the sagittal images.
44
(a)
(b)
0 0.0 0.1 0.150 0.0 0.1 0.15 50 00 20001
Fig. 3-10. Sagittal views of central disk central plane with (a) 0o tilt, and (b) 6.4o oare seen to the right. Missing energy is evi0o and is 8.6 cm off the source plane. By of the range of the missing cone, resulting
0 1000 15magnitude
55cm-
in mini disk phantom imaged on and above f tilt. Sagittal slices of the Fourier transforms
dent in the case where the disk is positioned at tilting the disk most of its energy now lies out in a more well defined image.
45
C.Exclusion of Other Possible Physical Effects
Fig. 3-11 (a) illustrates the process used to reproduce the artefact as indicated by eq. (2.21).
The frequency components removed were equivalent to that of a cone with a uniform internal
angle of 4.9o corresponding to the situation of the mini disk phantom imaged on the rotation
axis and 8.6 cm above the source plane. The same characteristics (i.e. loss of edge resolution,
decreased intensity, streaking) are evident between the simulated and experimental data as
seen in Fig. 3-11 (b). The similarity was verified via the 2D correlation coefficient of
central slices, which increased from 0.73 before the convolution step to 0.95 afterwards
(where 1 indicates the same image). Five central slices were averaged to reduce the
influence of noise in the images before calculation of the correlation coefficient. It should be
observed that a small but non negligible disagreement can be seen in the intensity values
between the simulated and experimental data, as seen in the difference image in Fig. 3-11 (b)
and the vertical profile of the images in Fig. 3-11 (c); this discrepancy may be attributed to
the use of a binary discrete filter in the filtering process, which may have introduced a slight
over filtration of frequency components.
46
Fig. 3-11. (a) Process of artefact simulation by convolution via multiplication in the Fourier domain. The disks imaged on source plane were fast Fourier transformed (FFT) and then multiplied by a function of ones with conical section of zeros resulting in a set of missing frequencies predicted by theory for z = 8.6cm. The filtered Fourier transform was then inverse Fourier transformed (IFFT) showing the simulated artefact. The resulting disks are compared to the disks imaged experimentally at the same z location in (b) showing clear similarity. The difference image and central profile in (c) show small but non-negligible intensity differences that may be due in part to slight over filtration of frequency components by use of a binary filter. Images are shown at the same scale.
(a)
3D FFT
×M(k)
z
3D IFFT3D FFT
×M(k)
z
3D IFFT
Measured Results z=8.6 cm
Simulated Results z=8.6 cm Difference Image
(b)
1
(c)
0 20 0 80 100-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
exp (z=8.6cm)sim (z=8.6cm)exp-sim (z=8.6cmexp (z=0cm)
atte
nuat
ion
coef
ficie
nt (c
m-1
)
cm-
-0.04-0.0200.020.040.060.08
)
z lo ation c (voxels)40 6
47
(a)
5 cm
0 cm0.1
0.15(a)
5 cm
0 cm0.1
0.15
0.1
0.15
(b)5 cm
0 cm
(b)5 cm
0 cm
1
Fig. 3-12. (a) Sagittal reconstruction of cellular disk imaged on and off of the sourcplane. (b) Solid acrylic disk of same dimension as disk in (a) and imaged undequivalent conditions. Difference images between cross sections at different heights ashown below the disk cross sections.
D. Large Disk Phantom
Fig. 3-12 shows the result of imaging disks with differing internal structures on and
source plane. The acrylic disk is obviously distorted when further from the source
characteristic of having high magnitude frequency components on or near the vertical
is worth noting that this effect is very similar to the effect seen in the mini disk phan
on a larger scale. The cellular disk shows similar blurring, contrast differences and st
cm-
0
0.05
0
0.05
0
0.05
0
0.05
0.1
0.15
0
0.05
0.1
0.15
0
0.05
0.1
0.15
cm-1
e er re
off the
plane,
axis; it
tom but
reaking
48
artefacts. While these effects suggest distortion of the planar horizontal features in the
phantom, there is also very good recovery of many internal cellular details under this
moderate cone-beam angle (~2.9o). The difference images confirm these observations,
showing similar characteristics for both types of disks but also showing good cancellation of
internal cellular details in the case of the cellular phantom.
zx
y
kz
ky
zx
yz
xy
zx
y
kz
ky
kz
ky
(a) (b)
Fig. 3-13. (a) Sub-volume of rabbit data showing several slices and isosurface of the rabbit spine. (b) Sagittal of 3D Fourier Transform of volume shown in (a). Volume in (a) is shown prior to use of spherical Hann window.
E. Rabbit Scan
Fig. 3-13 (b) shows a central section of the Fourier transform of the sub-volume of the rabbit
data in (a) after windowing with a spherical Hann window. A clear region of decreased
energy is observable over a conical region within the Fourier transform. The boundaries of
the missing cone that would be expected at the centre of the sub-volume are overlain on the
central section of the FFT for comparison, and indicate fair agreement with observation,
49
supporting that cone-beam CT artefacts are present in the data. Note that non-zero frequency
content within the region of the predicted missing cone is expected for a number of reasons.
Mainly, the approximation of shift-invariance is poor in this case. In addition, artefacts
originating at points near the boundaries of the sub-volume are severely truncated, while
artefacts originating at points outside of the sub-volume extend to regions within it, as per eq.
(2.14).
F. Knee Phantom
Fig. 3-14 shows a contour of the knee joint centred on the source plane. The colourmap
indicates the difference in intensity values along the contour between the data set on the
source plane and the registered data set above the source plane. Observing Fig. 3-14, a bias
can be seen at surfaces with orientation parallel or near parallel to the source trajectory. The
maximum discrepancy was found to be on the order of 10% of the average intensity of the
femur. A bar graph of the average discrepancy as a function of angle of the surface normal
vector is shown in Fig. 3-15. Discrepancies are seen to be greatest at surfaces that are near
horizontal as expected. In addition, the discrepancies are seen to persist for a much broader
range than that of the maximum internal angle of the predicted missing cone. The broad
range may be due in part to streak artefacts affecting nearby surfaces. A small negative bias
also exists for surfaces at high angles. Fig. 3-16 shows the same bar graph for a simulated
sphere for comparison. The same general trends are clearly observable. Differences in
discrepancy magnitudes are expected since the phantoms differ in overall shape, and
surroundings. The negative bias in both cases is an indication that the entire object, and not
just a small region is in general subject to cone-beam artefacts. However, the standard
deviation of the femur intensity was found to be about 1.5% of the its mean intensity. This
50
result indicates that the majority of the discrepancy in the anatomical phantom is on the order
of the noise, which comes in contrast to the case of the disk phantoms where the
discrepancies are clearly visible above the noise level.
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Fig. 3-14. Femur head contour at central source plane, with colourmap corresponding to difference between volumes before and after vertical displacement.
6 12 18 24 30 36 42 48 54 60 66 72 78 84 90-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05Mean Difference
Degrees (o)
Mea
n D
iffer
ence
(Nor
mal
ized
)
Fig. 3-15. Bar graph showing the mean intensity discrepancies along the contour of the knee at different vertical displacements. The difference values are shown as a function of the angle of the normal vector to the surface with respect to the vertical axis. Values are normalized by the mean intensity of the femur head. Error bars correspond to standard deviation of values within a bin.
51
6 12 18 24 30 36 42 48 54 60 66 72 78 84 90-0.01
-0.005
0
0.005
0.01
0.015
0.02Mean Difference
Degrees (o)
Mea
n D
iffer
ence
(Nor
mal
ized
)
Fig. 3-16. Bar graph showing the mean intensity discrepancies along the contour of a simulated sphere. Graph shows very similar trends as for the anatomical phantom. Cone of frequency components removed corresponded to that expected at z=5cm. Error bars correspond to standard deviation of values within a bin.
3.2 CONE-BEAM CT ARTEFACT AND IMAGE NOISE
One of the conclusions of cone-beam CT artefact is that it will be present in the
reconstruction of all objects when using a circular trajectory. This observation should be true
even in the noise power spectrum (NPS) of cone-beam CT images. One way to verify this
prediction is by reconstructing local regions of image noise and evaluating their Fourier
transforms. Fig. 3-17 (a) and (b) show central sections through the Fourier transforms of
reconstructions with no object in the field of view, centred on the source plane and at a
distance of 6 cm above the source plane along the rotation axis. The central section of the
3D Fourier transform of the reconstruction above the source plane shows the conical region
of missing information as expected. Also of interest is that the image centred on the source
52
plane coincidentally has a spectrum that does not appear to indicate frequency information
along the vertical axis. Other factors, such as the choice of pre-processing steps of the
projection data, rather than the diverging beam in the cone-beam CT geometry have been
stated as reasons for this behaviour, as has been explored in depth by Siewerdsen et al74.
) )
(aFig. 3-17. Spectrum of subvolume of air on(b) z = 6 cm.
(b
ly reconstruction centred at (a) z = 0 cm and
53
Chapter 4: Correction Schemes
A short review of the efforts proposed for correction of cone-beam CT artefacts will be
provided in this chapter for completeness. The chapter concludes with an example of how
missing spatial frequency components can be determined using limited a priori knowledge
about the object or its background.
4.1 ARTEFACT REDUCTION ALGORITHMS
Factors such as projection number, projection spacing, noise (e.g. electrical, or Poisson),
scatter, beam hardening, etc. all have bearing on the accuracy and precision of the
measurement of an image function. If noiseless, perfect systems existed, adequate data
sampling would be completely dependent on detector pixel size, projection number, spacing
between projections and the source trajectory. From the Fourier representation of cone-beam
CT artefacts, a straightforward solution is to change the trajectory such that the missing
frequency components are sampled accordingly. Many algorithms have been proposed for
reconstruction using complete orbits (e.g. orthogonal circles, circle plus line, saddle, helical,
etc.), which may be implementable in real world situations. Cone-beam CT adapted to a C-
arm may be well adapted to a trajectory utilizing two orthogonal circles for example. A look
at tradeoffs in terms of image quality when utilizing a different trajectory has been
published72. As mentioned in the introduction, diagnostic CT units now have fields of view
large enough to capture a full organ using a single rotation about the patient. Although these
scanners are well adapted to helical scans, maintaining a circular scan may be desirable in
some circumstances. For example, a circular geometry allows a patient to remain stationary
during the scan decreasing errors that might arise due to motion which may be greater than
54
the introduced cone-beam CT artefacts. Another downside is that the endpoints of the helical
scans are generally not effectively utilized. The complexity of the algorithms for helical
cone-beam CT is also a disadvantage.
Given the utility of circular trajectories, many other methods have been proposed to decrease
cone-beam CT artefacts without changing the scanning trajectory. One approach aims to
correct for the missing information via Radon space interpolation75-77. However, Radon-
based correction methods tend to be less time effective than standard backprojection
algorithms and perform poorly with axially truncated data (although this may not be a factor
in some specific applications [e.g. small animal imaging]). Yang78, more recently, proposed
a shift-variant, FBP method that includes estimated Radon information potentially providing
a more feasible implementation. Analytical forms for the PSF also make deconvolution an
attractive method for artefact correction, as proposed by Peyrin et al55. However, the shift-
variance of the PSF complicates correction by deconvolution without simplifying
assumptions55,56. Other methods suggested for the reduction of cone-beam CT artefacts
include projection weighting schemes79, shift-variant filtering80 and iterative, empirical
methods81,82. A comparison of the benefits of several methods for artefact reduction for large
cone-beam angle geometries has been published83. Despite these varied attempts for artefact
reduction, all such methods are only approximate and accurate reconstruction is not generally
possible in theory, unless strong a priori knowledge is present.
55
4.2 CONSTRAINTS AND A PRIORI KNOWLEDGE
Of course, one can assume that the presence of a priori knowledge can be used to introduce
constraints to the problem of image recovery. The developments of Candes and Romberg84
have shown that surprisingly few sampled frequency components are necessary to recover
the object image given some basic assumptions. From sparse data set analysis they
determined that the number of pre-determined frequency points required to solve the entire
space must be the same or greater than the number of non-zero points in the object. In this
case, the a priori knowledge is some assumption about the number of non-zero points in the
image. Secondly, they hypothesized that the minimum total variation of the image will occur
only when the unknown frequency components are identified correctly. This assumption was
based on the observation that incomplete projection data introduces streaking artefacts which
increases the variation and therefore the total variation of the image. The problem reduces to
solving a minimization problem where total variation is the cost function. Subsequent
developments have applied similar constraints of minimum total variation directly to circular
cone-beam CT85, as well as in MRI86. An alternative and independently derived approach
based on similar principles will be explored below. The approach greatly relies on principles
of frequency domain algebraic reconstruction (ART)87. It will be seen that if the number of
values in the signal that are known a priori are equivalent to the number of unknown
frequency components, a correction for the image can be determined. The specific
contribution to previous methods is the choice of a priori knowledge and solving for the
specific subset of frequency components not recoverable in cone-beam CT.
56
4.3 A FOURIER BASED CORRECTION METHOD
The model for reconstruction that will be utilizied in this section is that each point in the
acquired image represents a (weighted) summation of discrete frequency components. For
simplification, the concept will first be shown in a 1D scenario. The approach will then be
extended to higher dimensions. The discrete linear Fourier transform and its inverse can be
written as:
21
0
i mpMM
p mm
F f eπ−−
=
= ∑ , p=0,1,…,M-1 (3.1)
21
0
1 ipmMM
m pp
f F eM
π−
=
= ∑ . m=0,1,…,M-1 (3.2)
where fm is the signal value at m, and Fp is the Fourier coefficient at p.
Suppose a subset of frequency coefficients is unknown. Then incomplete information is
present for the accurate solution to fm. For simplification, Cmp can be substituted for the
factor of 1/P combined with the exponential term. Equation (3.2) can then be separated into
known and unknown frequency components:
known unknown
m mp p mp P p P
f C F C∈ ∈
= +∑ ∑ p pF . (3.3)
Since the first summation is known it can be carried out and replaced with a constant
complex value Fk. Now suppose that the value of fm at m=a1 is known a priori. In this case,
equation (3.3) can be rewritten in the following way:
1 1
unknown
a p p a kp p
C F f F∈
= +∑ . (3.4)
The right side of equation (3.4) is a known quantity by definition, and can be substituted with
A1:
57
1 1unknown
a p pp p
C F A∈
=∑ (3.5).
At this point it becomes apparent that if the number of known signal values is equal to the
number of unknown Fourier coefficients, a linear set of equations can be formed, as
represented by the augmented matrix below
1 1 1 1U
U U U U U
a w a w
a w a w
C C
C C
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
A
A
(3.6)
where {a1,a2,…,aU} is the subset of values m for which fm is a known value, and
where {w1,w2,…wU) is the subset of values p for which Fp is unknown. If the equations are
linearly independent and assuming the problem is not ill-posed or ill-conditioned, the
unknown frequency components can be found by solving this linear system. Once the
unknown frequency components are derived, all remaining values in the original object
function can be easily corrected.
This method can be extended to higher dimensions. Defining d dimensional vectors, m
=(m1,m2,…,md), and p = (p1,p2,…,pd) with range defined by N-1, where N=(N1,N2,…,Nd), the
discrete Fourier transform can be written in vector notation as
2 ( / )
1
1 im p Nm pd
pl l
f F eN
π ⋅
=
=Π ∑ (3.7)
where the p/N term is defined as (p1/N1,,p2/N2,…,pd/Nd). Similar to the 1D example, this
equation can be rewritten as
2 ( / ) 2 ( / )
1 1
1 1
unknown known
im p N im p Nm p pd d
p P p Pl l l l
f F e F eN N
π ⋅
∈ ∈= =
= +Π Π∑ ∑ π ⋅ (3.8)
58
where sets Pknown and Punknown are now the more generalized sets of vectors p for which Fp is
known and unknown respectively. The second summation is therefore known by definition.
So as before, if the number of values of fm known a priori is equivalent to the number of
unknown values Fp, the missing frequency components can be determined by solving a
straightforward linear set of equations.
Calculation of the sum to the right of equation (3.8) can be avoided given the erroneous
object signal. Define mf as the object function calculated using spatial frequency
components known to be in error, and pF representing its Fourier transform pair. Subtracting
mf from mf gives the following equality:
2 ( / ) 2 ( / )
1 1
1 1
unknown unknown
im k N im k Nm m p pd d
p P p Pl l l l
f f F e F eN N
π ⋅
∈ ∈= =
− = −Π Π∑ ∑ π ⋅ (3.9)
where the summations over known frequency components cancel. In addition, in the special
case where the erroneous frequency components are known to be zero, the second summation
in equation (3.9) can be correspondingly set to zero. Therefore, the linear set of equations can
be solved by considering only the difference between the erroneous real signal values and
their true values known a priori. This aspect eliminates the need to calculate the large
number of frequency coefficients required by equation (3.8). Note that equation (3.9) implies
that the solution to unknown frequency components reduces to a simple, spatially dependant,
additive correction to the erroneous signal.
An alternative approach to solving the linear set of equations is to use a least-square
minimization approach. The difference between the erroneous data points and the a priori
59
values can be minimized by iteratively changing the unknown frequency components (and
using the set of known frequency values as constraints). The least-squared method tends to
be better suited for problems that are ill-posed. Note that this sort of approach is similar to
that used by Candes et al84 except that the minimization is between specific function values
rather than using the total total variation of the image.
Recall that in cone-beam CT the artefacts resulting from a circular trajectory can be modeled
as resulting from the absence of a conical set of local frequency components. Using the
arguments presented above, these absent frequency components can be found if a sufficient
number of values of the object function are known a priori. Fortunately, in almost any scan,
there are values within the reconstruction volume that can be determined with high certainty:
namely, the attenuation coefficient of air surrounding the object. In practice, cone-beam
artefacts manifest as streaking or shading artefacts in the background air values in addition
that can provide measurable differences from the expected value and implemented using
equation (3.9). This aspect suggests that no a priori information about the object itself is
necessary in order to make the correction. Alternatives to using the value of air as a
reference can also be used. For example, an object of known attenuation coefficient could be
placed within the field of view of the object being scanned.
4.3.1 IMPLEMENTATION
Fig. 4-1 (a) shows a 64×64 image of the popular Shepp-Logan phantom which is employed
in the following example as a simplified 2D example. In this contrived example, we
eliminate a single a line of frequency components along the vertical axis from the reference
image such that the filtered spectrum,
60
(3.10)
2 20 if (0, ),
otherwise pp
p p pF
F
= ≠⎧⎪= ⎨⎪⎩
0
where the resulting artefacts are seen in Fig. 4-1 (b), and emphasized in the difference image
in Fig. 4-1 (d). This line of frequency components was empirically found to provide a set of
linearly independent equations. In order to determine the missing frequency components
using equation (3.9), the number of intensity values known a priori must be equal to the
number of missing frequency components. Assuming the reference image is unavailable, the
true intensity values of the phantom itself would be uncertain. However, the intensity values
of the space around the object are known a priori and are clearly expected to be zero in this
case. Using a column of points to the left of the object as the reference pixels, the missing
frequency components can be calculated using equation (3.9). The results seen in Fig. 4-1 (c)
show excellent agreement between the known values and corrected images of the phantom.
4.3.2LIMITATIONS
The method proposed above is promising but very sensitive to a number of limitations. One
of the main issues is that the underlying presumption for real applications is known to be
false. The algorithm depends highly on knowing the ‘true’ values of the Fourier coefficients
outside of the missing cone. However, due to the discrete nature of the Fourier Transform,
noise and other factors such as spectral leakage which are introduced from truncating the
artefact (or object) in any given image, the frequency components in the FFT (even those
outside the missing cone) are estimates only. Furthermore, even in the idealized simulated
scenario, the problem is empirically found to be ill-posed when extended to include missing
Fourier components away from the vertical axis. In this case, the alternative iterative
forcejustification
61
(a) (b)
(c) (d)
Fig. 4-1. (a) Simulated ‘Shepp-Logan’ phantom. (b) Image of phantom after all frequency components along the vertical frequency line have been removed. (c) Image in (b) with restored frequency content (blue line in (a) corresponds to location of points used as a priori knowledge). (d) Difference image between (a) and (b). Difference image between (a) and (c) would yield only null values.
approach is likely to be much more beneficial. An iterative approach would allow
Incorporation of a greater number of a priori values surrounding the object to further
constrain the solution. A second dilemma is that the cone of missing frequency information
is shift-variant, while the example posed above is invariant. This second problem is resolved
without difficulty by accepting the unity of all shift-variant cones as the set of unknown
62
frequency components. In this case, the values in the second summation of equation (3.9)
must be calculated and not set to zero.
4.4 SUMMARY
A natural method to avoid cone-beam artefacts is to use an alternative trajectory. However, in
many applications a circular trajectory continues to be the most practical option. In lieu of
altering the trajectory, attempts to decrease cone-beam CT artefacts under a circular
trajectory are in general limited to approximations, unless sufficient a priori knowledge is
present. The algorithm developed in the above section shows that a priori information of the
surroundings of the object, rather than of the object itself, may be sufficient to correct for
cone-beam CT artefacts.
63
Chapter 5: Discussion and Conclusions
5.1 DISCUSSION
While theoretical developments describing the origin of cone-beam artefacts have been
available for decades, the manifestation of the artefacts in image reconstructions is still
generally non-intuitive. Furthermore, rather than being phased out by alternative trajectories,
applications for circular cone-beam CT continue to grow in industry and research, from
dedicated BCT to dynamic volume CT to intra-operative therapies. Therefore, it is desirable
to gain better understanding of the link between theory and practice. Towards this end, this
work has outlined a Fourier description that lends further insight into the nature of artefact
manifestation. In particular, the relationship between artefacts and the object of interest was
explored via experiments utilizing a variety of phantoms, from simple to anatomically
complex.
Results from the acrylic sphere experiments agreed well with theoretical predictions, giving
strong evidence that cone-beam artefacts can be well described by a shift-variant cone of
missing frequency components in the local Fourier domain. Since the missing cone increases
in size with distance from the source plane, increased artefact is generally observed in all
reconstructions with displacement from the source plane.
However, the extent and nature of the artefacts observed depend on several factors,
including object shape and orientation in space. For example, imaged disks showed better
recovery as the angle of inclination with respect to the source plane is increased. This effect
64
can be explained in terms of the placement of signal energy of the disks with respect to the
missing cone of frequency components predicted by theory. The results support that the
removal of a subset of frequency components will have various effects that depend not only
on the imaging geometry, but also on the object being imaged, and in particular the
frequency content presented to the imaging system. From another point of view, since the
lines of absent frequency components represent changes in the real object along those
directions, it is expected that the resolution of surfaces normal to these directions will be the
most degraded.
Of particular interest is that the horizontal disk phantoms appeared to suffer more distortion
than the acrylic spheres at equivalent positions in space. This differing effect between planar
and cellular features was further illustrated in the images of solid and cellular disks. Whereas
the boundary of the solid acrylic disk was lost even at modest cone-angles, internal cellular
details of the polyurethane disks were evident at the same imaging location. Using the above
rationalization, spherical features may seem to be less degraded in general since the most
likely affected surfaces are more limited in extent (namely the upper and bottom-most
surfaces).
For a more detailed explanation, one can recall that the artefacts really amount to a
redistribution of the intensity values in order to fulfill conditions on the plane integrals
through the object (Chapter 2). In particular, all plane integrals over parallel (infinite) planes
must be equal to a constant value when their normal vectors are aligned within the missing
cone. Moreover, plane integrals must remain unchanged for other orientations. In the case of
65
the disk, large local intensity changes (i.e. bowing out of the disks into nearby planes) appear
to be necessary to achieve these strict conditions, while more subtle shading and streaking
artefacts seem to be sufficient to satisfy these conditions in the case of the sphere. In
comparing these results, it is important to note that subtle changes in the background can sum
to very large changes in the overall planar integral. Therefore, both sets of reconstructions
can achieve the same conditions on planar integrals with radically different intensity changes
observed locally.
The knee phantom presented a more complex, anatomically relevant case for analysis.
Similar to previous observations, maximum distortion in the knee phantom under vertical
displacement was found to preferentially occur at surfaces that were parallel or near parallel
to the source plane, as expected. However, the intensity discrepancies were still much less
than that observed for the horizontal disk phantoms under similar imaging geometry. The
maximum discrepancy in intensity values along the contour was on the order of several times
the noise field, while at other regions was on the order of the noise field or less.
Although spherical, cellular or curved features may appear to maintain overall higher fidelity
in reconstructions than planar features (that are near parallel to the source plane), it should be
made clear that cone-beam artefacts will be present in all real cases, unless corrections are
made successfully. The cumulative effects of the cone-beam artefacts may be of
consequence in terms of introducing contrast reduction, blurring or CT number inaccuracies.
In general, these effects are likely of more importance in diagnostic CT than image-guidance
applications, since without prior knowledge there is greater risk of failing to detect desired
66
features. However, although relatively small, the discrepancies in the knee phantom suggest
that identical contouring criteria may yield different contours if the object has been shifted in
the field of view, which may be relevant for many image-guidance applications. A more in
depth study of such impacts and further study of these effects as a function of such factors as
object texture is recommended for future research.
The methods employed for Fourier analysis also showed utility for identifying the presence
of cone-beam artefacts in clinical data. In analysis of the rabbit image acquired using a
clinical cone-beam CT scanner, no reference (or ‘ground truth’) data was available for
comparison, making it difficult to determine whether cone-beam artefacts were present.
Furthermore, even in the presence of accurate reference data, cone-beam artefacts may have
very low contrast to noise, or be dominated by other artefacts (e.g. scatter, beam hardening)
that obscure noticeable cone-beam artefacts. Analysis of the FFT of a small sub-volume,
however, confirmed decreased energy in the region of the expected missing cone, indicating
that cone-beam artefacts were likely present. This result suggests that similar analyses may
be used to test claims of cone-beam artefact reduction in real data. In addition, the
reconstruction fidelity of any localized feature can also be predicted independently of other
artefacts using the convolution method posed in this paper if a reference image (e.g. a CT
prior) is known. The method can be modified to examine a larger object by piecewise
convolution: the larger volume can be divided into sub-volumes, each filtered by a unique
transfer function defined by the presented theory and the results added.
67
The noise analysis performed may have widespread implications for noise and image quality
analysis. Typical metrics, for example, measure the modulation transfer function using either
a steel plate or a wire. However, these methods make assumptions about the symmetry of the
MTF, and are inadequate for true 3D frequency modulation measurements. The most
appropriate phantom for the 3D MTF is likely a localized object, such as a sphere88, as used
in the present experiments. Other interesting benefits of this analysis may be that intrinsic
details of the assumed imaging geometry may be present in the reconstruction data (eg. air
only scan, imaged sphere) since the missing cone is related to the angle of locally incident x
rays.
In addition, cone-beam registration utilities may also benefit from identifying the region of
the missing cone of frequency components. In particular, registration by phase correlation
would benefit by excluding information within the predicted cone in registration of data sets.
Alternatively, real domain comparisons of data may also benefit by pre-filtering the data such
that information within the region of the largest missing cone is forced to zero for both data
sets.
While a natural method to avoid cone-beam artefacts is to use an alternative trajectory, the
use of circular trajectories are expected to persist in the near future. Attempts to decrease
cone-beam CT artefacts under circular trajectory are generally limited to approximations,
unless sufficient a priori knowledge is present. The algorithm developed in chapter three
shows that a priori information of the surroundings of the object, rather than of the object
itself, may be used in an approach to correct for cone-beam CT artefacts. The presented
68
algorithm is advantageous in that it may not be restricted by the shift-variance of the artefact;
however, it faces significant challenges including sensitivity to data truncation and the ill-
posed nature of the problem. Further exploration of the proposed or similar algorithms is
expected to continue to be of great interest to the CT community. It is also worth mentioning
that in lieu of a feasible correction algorithm, the asymmetry of the artefacts and preferential
degradation of certain surfaces and/or objects suggests that there may exist an optimal
scanning orientation given a particular task. For example, it is obvious that if one were
interested in recovering the surface of a disk, one should orient it, or conversely, the
trajectory, in a way that optimally recovers that edge. Conditions on optimizing scanning
geometry may be of interest for future study.
The presented work has shown that the impact of cone-beam artefacts will vary greatly
depending on the application, the information required and the object of interest. In many
cases, the impact of other artefacts (beam hardening, scatter) may greatly outweigh that of
cone-beam artefacts, especially under moderate cone-angles (5-8o) such as are used in IGRT.
However, as image quality improves with new detector designs and algorithms, cone-beam
artefacts may have increased relative importance. Moreover, in alternative applications with
large missing cones, such as optical cone-beam CT for gel dosimetry, the impact of cone-
beam artefacts may be substantially greater than other artefacts.
5.2 CONCLUSIONS
The results of these experiments support the theoretical predictions of a shift-variant cone of
missing frequency components in the Fourier domain when using a circular source and
69
detector geometry in cone-beam CT. This missing cone was successfully identified and
measured in the Fourier transform of an acrylic sphere phantom. Recovery of the mini disk
phantom was seen to be strongly dependent on the relative energy distribution of the imaged
object with respect to the region of missing frequency components predicted by theory.
Image reconstruction of large disk phantoms with varying internal structure illustrated the
complexity of the observed effect when considering its dependence on the total frequency
content of the imaged object. Analysis of the rabbit data indicated that the results are relevant
to clinical scanners. Results in the knee phantom provided quantifiable support that
anatomical phantoms will exhibit greater artefacts at surfaces that are near parallel to the
source plane. Finally, the proposed method for artefact correction showed promising initial
results under ideal conditions.
70
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76
Appendix A: Proof that the missing cone
is an oblique, circular cone
As before, xo is considered in the y-z plane at (0, R, zo) for simplification. Arbitrary xo can be
considered by a rotation of coordinates. Consider first, cone A, formed by connecting the
source trajectory to xo using straight lines, as shown in Fig. A1. For the sample tangent plane
P, a normal line can be constructed containing xo and intersecting the source plane at n, as
shown in Fig. A2. Likewise, for each plane tangent to the surface of cone A, a
corresponding normal line can be defined. The set of all normal lines constructed in this way
forms a distinct cone, B, as shown in Figure A3. This cone is similar, in the strict
mathematical sense, to the missing cone in the Fourier domain. Cone B will be shown below
to have a circular aperture in the source plane below.
Proof
The vector ν(φ′)-po (Fig. A2) defines the aperture of cone A in the source plane, which by
definition is circular. Using the law of cosines, the magnitude of this vector can be
determined to be (Fig. A4)
( ) ( ) ( )2 2 2cos sinsp R Rρ′ ′φ = φ + − φ′ . (A1)
Similarly, the aperture of cone B in the source plane is defined by vector n(φ′)-po, which has
magnitude ns(φ′). From the similar triangles in Fig. A2, ns(φ′+180) is seen to be inversely
proportional to ps(φ′),
2
( 180)( )o
ss
znp
′φ + =′φ
. (A2)
77
Using equations (A1) and (A2) the ratio of ns(φ′) to ps(φ′) can be formulated as follows
2 2( )( ) ( (
s o o
s s s
n z zp p p c
′φ= =
′ ′ ′φ φ +180) φ )( )2 2c Rρ≡ −, >0 (A3)
and is seen to be constant for all φ′. Therefore, the aperture of cone B in the source plane
must be circular. Since similar cones are defined for arbitrary horizontal plane, the aperture
of cone B must be circular in all horizontal planes. The radius of the aperture in the source
plane must be half the sum of the maximum and minimum magnitude of ns(φ′), and can be
defined as
1 2tan( ) tan( )2 od η η+⎛= ⎜
⎝ ⎠z⎞
⎟ (A4)
where η1 and η2 are the minimum and maximum angles of cone B, with respect to the
vertical axis, obtained directly from the CB geometry. The axis of cone B, lies in the y-z
plane and has angle β to the vertical axis, defined as by
1 2tan( ) tan( )tan2
η ηβ − −⎛= ⎜⎝ ⎠
1 ⎞⎟ (A5).
78
z
Sn
P
xo
B
Figure A1: Illustration of the construction of cone A by connecting xo to the circular trajectory using straight lines.
xo
Sp Snανo
po n
Figure A2: Relationship of the normal line to tangent plane P.
79
z P
xo
A
Source Trajectory
Figure A3: Illustration of the construction of cone B as a function of normal lines to tangent planes to cone A.
φ′ ( )sn ′φ
( 180)sp ′φ +
ps
Figure A4: Aperture of cones A and B in the plane of the circular trajectory.