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.

) )

(a

Fig. 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.