Title Page
 Table of Contents
 List of Tables
 List of Figures
 The compton-scattering process...
 Equipment: design and instrume...
 The radiation fields
 Data collection
 Data analysis and results
 Biographical sketch

Title: Compton-scatter axial tomography with x rays
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00089977/00001
 Material Information
Title: Compton-scatter axial tomography with x rays
Series Title: Compton-scatter axial tomography with x rays
Physical Description: Book
Language: English
Creator: Brateman, Libby Frances
Publisher: Libby Frances Brateman
Publication Date: 1983
 Record Information
Bibliographic ID: UF00089977
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000487135
oclc - 11904159

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
        Page v
    Table of Contents
        Page vi
        Page vii
    List of Tables
        Page viii
        Page ix
    List of Figures
        Page x
        Page xi
        Page xii
        Page xiii
        Page 1
    The compton-scattering process as a method of obtaining information from irradiated objects
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
    Equipment: design and instrumentation
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
    The radiation fields
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
    Data collection
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
    Data analysis and results
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
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        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
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        Page 85
        Page 86
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        Page 88
        Page 89
        Page 90
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        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
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        Page 100
        Page 101
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        Page 108
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        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
        Page 145
        Page 146
        Page 147
        Page 148
        Page 149
        Page 150
        Page 151
        Page 152
        Page 153
        Page 154
        Page 155
        Page 156
        Page 157
        Page 158
    Biographical sketch
        Page 159
        Page 160
        Page 161
        Page 162
Full Text







This research is dedicated

with love

to the memory of my grandfather

Philip Block

for opening my eyes to the natural sciences,

and for encouraging me to read

The Book of Knowledge.


This research project is a result of efforts of many persons. The author

wishes to identify the contributions of the many persons who have made this

work feasible.

The main concept of the project, to extract information from a scatter

field produced in the CT scanning process, and the design of the initial

calculational model are attributed to Alan Jacobs, Ph.D., the research

committee chairperson. Dr. Jacobs' advice in surmounting analytic obstacles

permitted this project to have both a beginning and an ending. In particular,

two other members of the committee assisted in many ways and spent many

hours on various aspects of the project. Lawrence Fitzgerald, Ph.D., advised

and assisted in all of the digital aspects of the project, including instrumentation

design, testing and troubleshooting hardware, computer interfacing and

handshaking, and troubleshooting computer programs. Walter Mauderli, D.Sc.,

advised and assisted in design, fabrication, and testing of the radiation detectors.

Thomas Bowman, Ph.D., advised in mathematical filtering of the data and took

extra time to understand the applications of mathematics to this physically-

realizable system. Genevieve Roessler, Ph.D., advised in photographic and

editorial aspects of the research, and provided sustaining emotional support

throughout the project. Dr. Roessler has been especially helpful as a counselor

throughout the author's student career and is responsible to a large degree for

the author's selection of the graduate program.

Almost every employee in the Bioengineering Services of the J. H. Miller

Health Center has contributed in some way to this project. Special recognition

is given to Charles Rabbit and Don Wallace for their special skills in designing

and fabricating the experimental apparatus, phantoms, and a hundred assorted

parts in an expedient manner; and to James Parker and Otis Adams for designing

and fabricating the motor controller interface and helping when needed.

Donation of the x-ray machine by James Moran, M.D., of Bradenton is

acknowledged, and the excellent care in its disassembly and reassembly by

Kenneth Fawcett, Ovid Gano, and Paul C. Hodges, M.D., Ph.D., is gratefully

appreciated. Mr. Fawcett's expertise in electronics has been especially helpful

in several aspects of the project. Thanks are given to Jack Wood of S & H

Medical Systems for sharing his intuition regarding CT artifacts and insights

in stepping motor requirements at just the right times.

Loan of the x-ray film development system by William Collett, D.D.S.,

is appreciated, and thanks are given to Charles Raynor and his associates for

physically bringing the unit to the laboratory site; and to George Fogle for

his help in setting up its operation. David Kepple of Eastman Kodak provided

darkroom lights and photographic supplies as needed, and his technical advice

and assistance are appreciated. Joe Jackson of Barcia X-Ray Co. helped to

set up the darkroom chemistry procedures and provided the hangers for x-ray

film development. Ronald Franklin took the excellent photographs of the

experimental apparatus and provided their negatives.

The special demonstration of the Victoreen/Nuclear Associates NERO

on the x-ray machine by Robert Duerkes is appreciated. A noninvasive kVp

test was required, and the hours spent by Mr. Duerkes were very helpful.

Many faculty members of the Nuclear Engineering Sciences Department

not members of the research committee have spent time and energy on this

project. They include William Ellis, Ph.D., Edward Dugan, Ph.D., Edward

Carroll, Ph.D., Ronald Dalton, Ph.D., and Samim Anghaie, Ph.D. Special thanks

are given to Dr. Anghaie for his assistance in modification of his program for

the Monte Carlo calculations provided, and for his advice in related matters

as needed. The time spent by Ralph Selfridge, Ph.D., in discussions regarding

pixel rotation is appreciated.

Particular thanks are given to Clyde Williams, M.D., Ph.D., for his support

in transporting the x-ray machine. Without his support the experimental setup

would not have been accomplished.

Special thanks are given to Linda Pigott for her expertise in preparation

of the manuscript, her willingness to work long, hard hours, and her desire to

learn mathematical symbols and equations. Her attention to detail made the

final stages of this project flow easily. The author is grateful to Wesley Bolch

for his ability in taking rudimentary scratching and turning them into

professional drawings. His expertise, patience, and generally good nature are

sincerely appreciated.

A grateful acknowledgement is made to the American Association of

University Women for awarding the author a Dissertation Fellowship, endowed

by Rockwell International, for the academic year 1982-1983. This fellowship

permitted full concentration by the author on this research project, allowing

its completion in a timely manner.

Finally, the author wishes to thank her family and friends for their

continued emotional support throughout her career as a student. She particularly

appreciates their understanding of the stresses, both financial and emotional;

and the time constraints of working so many hours each day/week/month/year,

and subsequent neglect of important relationships.



ACKNOWLEDGEMENTS .............................. iii

LIST OF TABLES .............................. viii

LIST OF FIGURES ................................... x

ABSTRACT ..................... ............. xii


I INTRODUCTION .......................... 1



Radiation-Producing Equipment .. . ......... 7
Objects Irradiated ...................... 7
Scanning Machinery ..................... 10
Radiation-Detection Apparatus . . . . . . 13
Microcomputer and Parallel Interface ......... 15

IV THE RADIATION FIELDS .................... 17

The Entrance X-Ray Beam ................ 17
Significant Radiation Interactions ...... .... 22
The Detected Fields ... ....... .......... 25

V DATA COLLECTION ..................... 33

Experimental Setup ......... ......... .... 33
Data Acquisition ...................... 38
Phantoms Studied ... ................... 42


Discrete Calculation of the Scattered Field . . .. 46
Comparison of Calculated and Measured Signals . . 57
Reconstruction of the Transmitted Field . . . .. 72
Application of the Scattered Field Information . .. 74

VII CONCLUSIONS ........ ......... ....... 101




A PARTS LIST ........................... 104

HARDWARE ......................... 108



THE SCATTERING GEOMETRY ................ 120



H MONTE CARLO CALCULATIONS ................ 129

I SIMULATION OF PHANTOM 2................. 133


ROTATION .......................... 142

L ERRORS IN ROTATING PIXELS ............... 148

LIST OF REFERENCES .......................... 156

BIOGRAPHICAL SKETCH ......................... 159


Table Title Page

IV.1 X-Ray Spectral Intensities in 5 keV Energy Intervals . . 21

IV.2 Percentage of Total Interactions in Polymethylmethacrylate
for Three Major Processes .................... 24

VI.1 True Areas of the Pixels Within the Circle for One
Quadrant .................... .. ........ 49

VI.2 Fitted Areas of the Pixels Within the Circle for One
Quadrant ................... .......... 50

VI.3 Correction Factors as a Function of Traverse Increment . . 60

VI.4 Relative Scatter Factors by Fitted Pixel Along a Row
for the Uniform Phantom .................... 78

VI.5 Pixel Values for Reconstruction of Simulated Scatter
Ratios from Phantom 2 ..................... 93

VI.6 Pixel Values for Reconstruction of Measured Scatter
Ratios from Phantom 2 .................... ... 95

VI.7 Pixel Values for Reconstruction of Measured Scatter
Ratios from Phantom 3 ..................... 97

VI.8 Pixel Values for Reconstruction of Measured Scatter
Ratios from Phantom 4 ........................ 99

B.1 Connections to Parallel Port Interface . . . . . ... 113

C.1 Mass Attenuation Coefficients and Physical Densities
for Four Materials ...................... 115

D.1 Relative Intensities and Exposure Rates for the
Calculated Spectrum ....................... 118

E.1 Summary of Definitions Regarding Figures IV.3 and IV.4 . 121

E.2 Coordinates of Relevant Intersections . . . . . .... 123

F.1 Measurement of Peak Tube Potential . . . . . ... 125

H.1 Photon Energies and Fluxes Used to Approximate the Incident
X-Ray Spectrum for Monte Carlo Calculations . . ... 130

LIST OF TABLES-continued

Table Title Page

H.2 Summary of Multiple and Single Scattering Calculations . . 132

J.1 Elements of the Convolution Kernel . . . . . . ... 137


Figure Title Page

III.1 Block diagram of the research apparatus . . . . . . 9

111.2 The compensating filter ...................... 11

111.3 The scanning assembly ....................... 12

IV.1 The calculated bremsstrahlung spectra . . . . . .... 19

IV.2 The total intensity spectrum . . . . . . . .. .. 20

IV.3 The detected field due to Compton scattering . . . ... 28

IV.4 The geometric parameters of the Compton-scattering
interaction ................. .......... 31

V.1 The focal spot intensity distribution . . . . . . .. 34

V.2 The geometric relationships of the experimental setup . . 36

V.3 The experimental setup ....................... 37

V.4 The x-ray beam size and intensity distribution . . . ... 39

V.5 Images of the x-ray field at the center of rotation of
the scanning assembly ...................... 40

V.6 The phantoms studied .. ................ .... 44

VI.1 Phantom 2 superimposed on the 19x19 pixel matrix . . . 47

VI.2 The discrete Compton-scattering geometry . . . . ... 51

VI.3 Calculated and measured scatter signals for the uniform
phantom ........ ............ ....... .. 59

VI.4 Predicted and measured scatter signals: Phantom 2
rotated 900 clockwise ...................... 63

VI.5 Predicted and measured scattered signals: Phantom 2
rotated 1740 clockwise .................. . 64

VI.6 Ratios of signals from phantom 2 to phantom 1: 900
clockwise rotation ...................... .. 67


Figure Title Page

VI.7 Ratios of signals from phantom 2 to phantom 1: 174
clockwise rotation ...................... 68

VI.8 Ratios of signals from phantom 2 to phantom 1: 1800
rotation ...................... ...... .. 69

VI.9 Sinograms of ratios of the phantoms studied . . . ... 71

VI.10 CT-reconstructed images of the phantoms studied . . ... 76

VI.11 Images of simulated data of phantom 2 . . . . ... 84

VI.12 Descriptive flow chart of the algorithm of method 4 . . 85

VI.13 Scatter signals calculated from the density matrix after
density adjustments by method 4 . . . . . ... 89

VI.14 Resultant images of method 4 applied to the measured
scatter data for the phantoms studied . . . . .... 91

B.1 Interface for the stepping motors . . . . . . ... 108

B.2 The detection circuit ..... ... .............. 110

B.3 The parallel interface ....................... 112

J.1 The backprojection geometry . . . . . . . . ... 139

K.1 Rotated pixelettes, showing their relationship to large
pixels .................. ............. 143

L.1 A 19x19 pixel matrix superimposed on a circular object
with an enclosed discontinuity in density, before and
after rotation .................... .. 150

L.2 The circular object rotated 60 clockwise . . . . ... 152

L.3 The circular object rotated 420 clockwise . . . . ... 155

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Libby Frances Brateman

December, 1983

Chairman: Alan M. Jacobs
Major Department: Nuclear Engineering Sciences

A method of extracting information from the backscattered field pro-

duced in parallel-beam x-ray computed tomography (CT) is presented. A

calculational model to predict the backscattered field based on Compton

scattering is described, and the model is verified by measurements of simple


The phantoms tested, cylinders of polymethylmethacrylate (PMM) with

air gaps and aluminum rods placed internally, are irradiated on a scanning

assembly, built to simulate a first-generation CT scanner. The x-ray source is

an orthovoltage therapy x-ray machine, modified to present an x-ray beam

similar to that of a CT scanner, and the two detectors used are thallium-

activated sodium iodide crystals operated in the current mode. Data from

the transmission detector are reconstructed by traditional CT methods to

provide a transmission image. Data from the backscatter detector are analyzed

in the research project.

After verification of the model for the scattered field calculations, a

method of extracting information from the scattered field is demonstrated,

based on ratios of scatter signals from nonuniform to uniform phantoms. This

method is demonstrated for predicted data of a simulated phantom and for

measured data of the same and two additional phantoms. The method is very

sensitive to air gaps in the phantoms because of the relative electron density

of air with respect to PMM; it is not as sensitive to the aluminum rods for

the same reason. Various methods of applying the scattered field information

to produce an image representing a simulated phantom are considered, and a

preferred method is chosen to reconstruct scattered field data into an image

for the three phantoms studied.

Limitations of the model and limitations of the process are described,

as well as possible uses of the methods considered. Although the system

described is probably not medically useful because of patient dose constraints

and the insensitivity of the method to small density changes, it is believed

that the system described has industrial applications.



This research studies Compton-scattered x rays produced in the geometry

of parallel beam computed tomography (CT). A system is built to simulate a

first-generation CT scanner, using a mechanism to translate and rotate an

object through a well-collimated x-ray beam. In addition to the collimated

sodium iodide transmission detector used in CT, a second detector is placed

near the irradiated object to sense x rays which have undergone backscatter,

almost exclusively due to Compton (incoherent) scattering. Information is

extracted from the backscatter signal, and a method of reconstruction of this

information into an image is demonstrated, allowing comparisons between the

transmission CT image and the scatter image.

Chapter II gives a brief history of the techniques of using the Compton

scattering process to interrogate an object by irradiation and compares other

studies with this one. Chapter III details the experimental apparatus and

instrumentation used in this research.

Chapter IV describes the radiation fields: the incident x-ray spectrum,

the expected modes of interaction with the objects irradiated, and the expected

signals at the transmission and scatter detectors. Chapter V explains the

techniques of data acquisition, and the phantoms studied, and Chapter VI details

the methods of data analysis both for the scattered field and the transmitted

field. Also explained are the model for predicting the scattered field, a

verification of the model, and the use of the model in creating an image of

the object irradiated. Finally the scatter images, reconstructed by the method

described, are shown for the phantoms studied.




That Compton scattering could be used to determine the electron density

of a small volume was demonstrated by Odeblad and Norhagen in 1956 (Od56).

By the 1970s several techniques involving the use of Compton scattered photons

for in vivo medical diagnosis had been developed. Lung pathology was

quantitated by the time-dependent modulation of the scattered field (Re72)

and by direct determination of the lung electron density (Do74, Ka76). Similarly,

bone density and mineral content were quantified by Compton scattering

techniques (Ga73, C173, Hu79). In most cases the sources were monoenergetic

gamma-emitters, and the detectors were scintillators.

Attenuation of the emerging scatter signal by overlying tissues was

found to be a significant problem, and Garnett et al. and Clarke and Van Dyk

individually proposed a two-source method for determining electron density

which would be independent of such attenuation (Ga73, C173). In the two-

source method two sources of different energies are used, where the energy

of the second source corresponds to the Compton-scattered energy of the first

source for the scattering angle of interest. Measurements are made of the

scattered and transmitted beams for two patient orientations, at the initial

position and then after a 1800 rotation, and the electron density in a scattering

volume can then be found, independent of overlying tissues.

Compton-scattered photons have also been detected from diagnostic x-

ray beams. Recent works by Duke and Hanson used the two-source method

with two heavily-filtered x-ray tubes for Compton scatter densitometry (Du82,


Ha83). The x-ray tube allowed a high intensity signal in a well-defined volume,

and it was reported that errors due to the polychromatic nature of the source

were small compared with errors due to multiple scattering.

Using the small sensitive volume made possible by an x-ray tube, time-

dependent signals from epicardial surfaces of dogs have been obtained by Jacobs

and co-investigators Weidner and Tilley for determination of surface motion

abnormalities (We75, Ti76). These signals were then analyzed by Fourier

techniques to determine associated normal and pathological patterns, and more

recently by McInerney, to construct the surfaces (Mc82).

Attempts to use Compton-scattered photons to produce medical images

were first reported by Lale, for the purpose of radiation therapy treatment

planning (La59). Forward-scattered photons from iridium-192 were detected,

and resulting images were intended to provide electron density as a function

of position within a transverse cross-section. In subsequent work, Lale used

5.6 MV x rays collimated to a line source with rotating apertures to create

a scanning point source of radiation. The patient was lowered through the

beam, and the resulting signal, detected by a liquid scintillator, was displayed

on a cathode ray tube (La68).

Direct 900 Compton-scatter imaging has been employed by Pistolesi et

al. and by Guzzardi and Mey, using an Anger camera with a mercury-203 line

source placed perpendicular to the field of view desired (Pi78, Gu81). This

procedure, used for lung tomography, chose the plane of interest by judicious

placement of the source.

In 1965 Clarke studied the use of a collimated sodium iodide scanning

detector with a high energy (cobalt-60 or cesium-137) source to image forward-

scattered photons, as an extension of Lale's work (C165). In 1971 Farmer and

Collins proposed a modification of Lale's method of imaging Compton-scattered

photons by using a cesium-137 source and a lithium-drifted germanium detector,

Ge(Li), sensing 900 Compton-scattered photons (Fa71). Because of the known

relationship between the energy change and scattering angle, the signal detected

for a known primary irradiation path could be related to the position of

scattering along the path, by energy discrimination. Thus the intensity at a

given energy could be directly related to the electron density of the scatterer

as a function of position. The patient was lowered in the beam to cover the

entire cross-sectional area. Problems due to attenuation of the signal and

multiple scattering were encountered, and a second study was designed to

overcome these deficiencies. In the second study, two Ge(Li) detectors,

collimated with focused septa to the plane of interest, were employed (Fa74).

Again, there were problems due to multiple scatter and signal attenuation.

These problems were addressed in a subsequent paper by Battista, Santon

and Bronskill, who described the problem of multiple scatter and proposed a

simple method of determining a multiple-scatter to single-scatter ratio (Ba77).

First, by placing air at the scattering volume to eliminate the single-scatter

signal, the multiple-scatter signal could be determined; then the scatterer could

be replaced in the sensitive volume so that the summed single- and multiple-

scatter signal could be obtained; the ratio could then be extracted. With

respect to the problem of signal attenuation, Battista, Santon and Bronskill

stated that the two-source method (described previously) was not possible for

their physical situation, and then proposed an algorithm for numerically

obtaining values for the attenuation of interest. In the method proposed, the

density was calculated for a pixel with no external attenuation, and, once that

value was obtained, it could be used in the determination of the next pixel

value. It was suggested that transmission data be used as an adjunct to

minimize propagation of errors in the determinations.

Kondic and Hahn extended the two-source method of determining electron

density to a multiple-source and multiple-detection angle method, in which the

various source energies corresponded to the various Compton scattering angles

as described above (Ko70). This work was applied to a two-phase system (that

is, liquid and gas or solid and gas).

Jacobs proposed a method of uncollimated detection in such a system,

using energy discrimination to separate single from multiple Compton scattering

(Ja79b). This proposal, Compton profile densitometry, allowed greater detection

sensitivity than in collimated systems, and it was effected experimentally by

Baker (Ba80), with subsequent analysis by Anghaie in his doctoral research

(An82b). Anghaie showed that spatial resolution in a two-phase system could

be improved on the basis of removal of multiple scattering effects based on

prediction of the multiple scattering from the acquired total signal. That is,

spatial information in the multiple-scattered field is degraded to a blurred

"background" signal, which can be removed by appropriate data analysis.

Techniques employed by Jacobs and Anghaie were applied by Clayton

in her master's research, in which she studied a two-source method of

determining chemical composition in addition to electron density (C183). This

method included photoelectric considerations in addition to the Compton-

scattering process, permitted by the use of two low energy sources (60 and

100 keV).

A different approach to Compton-scatter imaging was taken by Towe

in his doctoral research with Jacobs (To78, Ja79a). Towe used a scanning x-

ray source to irradiate an object and sensed the backscattered radiation with

two detectors; images were reconstructed by algebraic techniques. The peak

energy of the beam was found to correlate with the depth of penetration of

the backscatter signal, so that a form of tomography could be obtained with

image subtraction techniques. Because a small source scanned the object, the

image for an entire scan was produced with a small exposure (approximately

10 mR at the object surface). It was proposed that imaging of the backscattered

field was very sensitive to variations of density at interfaces (such as, air/bone

for the facial sinuses, and air/tissue for the lungs and the larynx) with a small

radiation dose.

Tomographic imaging of Compton-scattered photons has thus been studied

previously in terms of 900 scatter of gamma rays and x rays and either well-

collimated detectors or large field-of-view detectors for direct imaging. A

patent by Danos describes the use of a large focused grid to transmit 900

scattered radiation over a 600 field-of-view to a detector for the purpose of

tomographic tissue characterization (Da80). That a polychromatic source of

x rays could be used for backscatter imaging has been previously demonstrated.

In the present study, a method of extracting information contained in

the backscattered x-ray field is proposed and demonstrated by measurements.

This information is then used to reconstruct a computed axial tomographic

image by a suggested method of reconstruction. Backscattered photons in the

geometry of parallel beam computed axial tomography are studied, using an

uncollimated detector for the scattered field and a collimated detector for

the transmitted field. As in transmission computed tomography the scatter

detector integrates all energy deposited in the detector for a fixed time, rather

than counting photons and using energy discrimination. Thus the instrument

and the methodology employed in this study are designed to simulate a parallel

beam CT scanner with a second detector to sense backscattered x rays from

simple phantoms.



The system, built to simulate a parallel-beam computed tomography

scanner with an additional detector for scattered x rays, consists of several

subsections: the radiation-producing equipment, the objects irradiated, the

scanning machinery which transports the objects through the radiation beam,

the radiation-detection apparatus, and the microcomputer which controls the

scanner and the detectors. These five subsystems are described below. A

block diagram is given in Figure III.1, and a parts list is given in Appendix A.

Radiation-Producing Equipment

A General Electric Maxitron 300 X-Ray Therapy Unit is the radiation

source. The radiation beam used is produced at 140 kVp and 10 mA and is

heavily filtered by 0.41 mm copper. The voltage waveform is pulsatile at

1200 Hz, and the tube is self-rectified. The x-ray beam parameters are

described more completely in Chapter IV.

Objects Irradiated

Each of the phantoms studied is mostly polymethylmethacrylate (PMM).

This material is similar to tissue in elemental composition and density and is

readily obtained. According to Hubbell (Hu69) the elemental composition is

(C5H802)n. The physical density of a sample, calculated by physical measure-

ments of size and mass, is 1.181 g-cm-3. Based on atomic weights the mass

electron density is calculated to be 3.25x1023 electrons per gram (We77).

The phantoms studied are 10 cm diameter PMM and may contain voids

(i.e., holes) and aluminum pins. Descriptions of specific phantoms are given

in Chapter V.

Figure III.1. Block diagram of the research apparatus.


I RESET I_ ___ OPTICAL _______
___ I J_ __PARALLEL--
NaI (TI)
(ROTATION) L_ _ ------- MOTOR


Behind the phantom in the x-ray beam is a compensating filter, also of

PMM. The purpose of the compensator is to provide a uniform, or approximately

uniform, intensity and x-ray spectrum to the transmission detector. The

equation of the curved surface, half of an ellipse, is (with x in cm)

y = 11.13 2(25 x2)2 cm,

and a 2 cm flat surface is provided at each end for calibration. Figure III.2

shows the design for the compensating filter.

Scanning Machinery

A magnesium scanning table supports the phantoms undergoing irradation.

The device, machined and constructed in the Bioengineering Services Machine

Shop, J. H. Miller Health Center, University of Florida, consists of a rotational

platform which moves transverse to the x-ray beam. The scanning mechanism

is observed during a scan by a video camera connected to a display monitor

at the x-ray console and is pictured in Figure III.3.

The turntable is attached by a precision worm and worm gear to a

stepper motor for control of phantom rotation. The platform containing the

turntable is attached to a miniature ball-bearing screw and preloaded nut and

is supported by pillow blocks on two stainless steel shafts. The ball-bearing

screw is attached to a stepper motor, and motion is arrested by limit switches.

Both stepper motors are pulsed by factory-manufactured motor control-

lers which in turn are controlled by the microcomputer described below. Each

motor requires two large power dropping resistors and a 10 A, 24 V power

supply. One 23 A power supply is used for both.

The motor controllers are electrically-isolated by optical coupling from

the microcomputer and interface. Signals from the computer include selection

of motor direction and pulses for stepping. An additional interface separates

the microcomputer interface and the motor controllers: herein is contained

the logic for switching the inputs to the translator modules (i.e., motor direction,

---- 10 cm ---

11.13 cm

1.13 cm

(l--- 14 cm >

Figure III.2. The compensating filter, constructed of polymethylmethacrylate.

Figure III.3. The scanning assembly.

Also shown is the scatter detector.

pulses, and information from the limit switches). This interface inverts and

stretches the pulses from the optical isolators and also provides sufficient

current to operate the motor controllers. A circuit diagram of the interface,

designed and constructed in the Bioengineering Services Electronics Shop of

the University's J. H. Miller Health Center, is shown in Appendix B.

The precision of rotation is 0.040 per step and of translation is

approximately 6.35x10-4 cm per step. The manufacturer's specification on the

preloaded nut and ball-bearing screw is zero backlash, and none is measured.

Rotation is in the clockwise direction only, so backlash is not of consideration;

however, it is important that a complete rotation be reproducibly 3600 exactly,

and such is the case.

Radiation-Detection Apparatus

One detector, collimated to the radiation beam size, detects radiation

transmitted through the phantom, and a second detector, uncollimated, detects

radiation backscattered by the phantom. Each detector consists of a 1.27 cm

diam by 1 cm thick thallium-activated sodium iodide crystal encased in a 0.03

cm aluminum housing, a 1.27 cm diam by 1.27 cm thick quartz light pipe, a

1.3 cm phototube, and a voltage divider network. The detector signal is a

current proportional to the radiation field intensity.

The photomultiplier tubes operate at an indicated -975 V, provided by

a high voltage power supply. This voltage is selected on the basis of the

optimum signal-to-noise ratio (SNR=46) for the transmission detector in the

radiation field attenuated by the reference phantom. Additional high voltage

adjustment is provided by two potentiometers, used to compensate for the

different sensitivities of the two detectors. With the high voltage power supply

set at -975 V, the actual voltage is measured to be -955 V. The transmission

detector operates at a measured -870 V while the scatter detector operates

at -910 V.

The current output from each detector is input to an ultra low bias

current (75 fA) FET operational amplifier, which serves as a current integrator

in conjunction with a low-leakage 2 IF capacitor. The amplifier input is

overload-protected by a picoamp diode, and offset null is controlled by a

potentiometer. A reed relay, operated by the computer, is used to control

measurement (i.e., integration) time and reset, and is protected by a 1 ko

resistor. Across the relay coil is a Zener diode to limit voltage transients.

A printed circuit board contains the components described above for

each detector. A diagram of the detection circuit is given in Appendix B.

Each board is physically mounted very close to its photomultiplier tube, and

a grounded aluminum housing surrounds all electronic components. The high

voltage cable is electrically shielded from the circuit-board components.

Surrounding each aluminum detector housing is a lead shield to protect

the components from stray radiation. A lead cylinder surrounds the

photomultiplier tube and is connected to a larger lead-over-copper cylinder

which encases the aluminum housing. The transmitted beam detector has an

end cap to define the edges of the primary radiation beam, while the scatter

detector is left uncollimated. The end cap of the transmission detector consists

of 0.87 cm Pb, inside of which are 0.20 cm Cu and 0.16 cm Al filters. The

copper and aluminum filters are added to reduce the detected intensities of

characteristic x rays produced in the end cap. The transmission detector

aperture is 0.64x0.95 cm2. The detector leakage current, without high voltage

applied, is found to be 35 fA. At an applied voltage of -900 to -1000 V, the

dark current is measured to be less than 30 nA.

The output signal from each radiation detector is buffered by a wideband

operational amplifier for impedance matching with its associated analog-to-

digital converter. The analog-to-digital converter provides 12-bit data for a

maximum signal of 10 V, allowing output of 8 data bits and then 4 data bits

into the microcomputer. Both the analog-to-digital converters and the buffer

amplifiers are zeroed by nulling potentiometers.

The detector reset relays are isolated electrically from the logic circuitry

by optical couplers using a separate power supply and ground. Logic buffers

are used between the computer interface and the optical couplers to provide

sufficient current for reliable operation.

The sensitivity of the detection system to the x-ray beam of interest

is such that 370 pR-s-1 causes a 10 V-s-1 signal, corresponding to 0.089 jR

per bit. Error in the reproducibility of detection at a signal level of 5 V-s-1

is less than 0.6%.

Microcomputer and Parallel Interace

The microcomputer used for motor control and data acquisition is a

Radio Shack TRS-80 Model III Computer with two disk drives and 48 K

programmable memory. An interface of four 8-bit input and output parallel

ports decodes and encodes information from the 50-pin computer TRS-Bus.

The interface decodes the port address and the data and performs necessary

electronic communications with the computer itself. The interface design is

a modification of one published for the Radio Shack TRS-80 Model I Computer

(Ci80). The circuit diagram as constructed is given in Appendix B, as well as

the input/output pin diagram for the three ports used.

Three Z-80 microprocessor input/output ports are used in the system.

Assembly language programs are used to control the stepping motor rates,

ramp speeds, number of pulses, and motor directions. Control of the reset

relays and analog-to-digital converters is done in BASIC language. A relay is

opened to allow signal integration for a specified interval, controlled by delay

loops. After the desired time, the computer signals an analog-to-digital

converter (ADC) to begin digital conversion. After conversion, the relay is

reset, and the computer signals the ADC to allow data to be read. A logic

switch controls the display of the 8 upper bits versus the 4 lower bits. The

computer inputs the data and converts the two bytes to an integer. Thus a

radiation measurement is obtained. The detectors are activated one at a time,

and the motors are stationary during data collection. After data collection

at one position, the translation motor is moved a specified increment, and the

data collection procedure is repeated. After a specified number of translation

increments, the rotation motor is moved a specified increment. The motor

motions are controlled by assembly language subroutines called by a BASIC

program. This entire procedure is repeated until a complete rotation is

obtained. The experimental use of the equipment is fully described in Chapter V.

The Radio Shack computer is used to reconstruct the transmission

measurements into a computed tomographic numeric image. These data are

transferred via acoustic coupler, along with the scatter measurements, to the

State of Florida's Northeast Regional Data Center for further data processing.

The resultant image data are transferred via acoustic coupler to a Cromemco

Z-2D Computer for display on its associated SDI Graphics Interface display

(189x121 pixels, 16 gray levels). Photographs are made of the display with a

Radx Video Imaging Camera.



The radiation fields used in this experiment are designed to simulate the

x-ray beam in a clinically-operational computed tomography scanner. The x-

ray beam parameters and the model employed are described below.

The Entrance X-Ray Beam

The x-ray beam chosen for this research is generated at a peak tube

potential of 140 kVp (self-rectified, 1200 Hz), using a tube current of 10 mA.

The tungsten-on-copper 450 anode is stationary. In addition to the inherent

filtration of the x-ray tube and housing (4.75 mm Be tube window and 0.25

mm Al transmission ionization chamber), a copper filter 0.41 mm thick is

placed inside the modified cone used for initial primary beam definition. These

parameters are similar to those used in clinical computed tomography scanners

(Bi79). The x-ray beam is then attenuated by 1.94 m of air before reaching

the entrance surface of the phantom under investigation. This distance, longer

than usual in clinical scanners, is chosen mainly for geometric reasons, explained

in detail in Chapter V. The resultant beam intensity is measured to be 0.2

mR-s-1 at the transmission detector.

The Bremsstrahlung Spectrum

The bremsstrahlung component of the x-ray spectrum at the entrance

surface of the phantom is modeled by a modified Kramers method (Kr23).

Since the Kramers method models spectral intensity for a 450 anode at constant

potential, a time-weighted spectrum is calculated for the x-ray beam, based

on the assumption that the high voltage waveform is sinusoidal. A detailed

description of the method used is given in Appendix C. Figure IV.1 gives the

resultant attenuated bremsstrahlung spectra.

The Total Intensity Spectrum

The bremsstrahlung spectra shown in Figure IV.1 are different from their

respective total x-ray spectra by the characteristic radiation from the tungsten

anode. Intensities of the characteristic K-radiations (58, 59, 67 and 69 keV)

are calculated, based on a published spectrum of a beam generated at 140 kV

constant potential and filtered by 0.25 mm Cu, as described below (Bi79). The

bremsstrahlung component of the total intensity at these four energies is

interpolated linearly between the energies above and below the K-radiation

energies. The total intensity-to-bremsstrahlung ratio is calculated for each

energy, and the bremsstrahlung spectrum is adjusted by these ratios at each

of the four energies. The total intensity spectrum is then normalized to 1.0

at 59 keV. This total intensity spectrum at the location of the entrance

surface of the phantom (without backscatter) is presented in Figure IV.2.

For the purposes of attenuation and scatter calculations involving

phantoms, the entrance spectrum is subdivided into 5 keV intervals centered

about energies which are multiple of 5 (e.g., the 50 keV bin contains energies

48-52 keV). These values of relative intensity are given in Table IV.1.

Verification of the Total Spectrum

As direct measurement of the total intensity primary beam spectrum is

not possible with the present instrumentation available, the spectrum is verified

by comparison of measured and calculated half-value layers.

The half-value layer is calculated by first calculating the relative

exposure rate of the x-ray beam and then calculating attenuated exposure

rates by adding attenuating filtration to the beam, as described in Appendix

C, until the proper filtration is chosen. The detailed procedure followed in


4.75mm Be
1600 ------- 4.75 mm Be and 0.25 mm Al
S-* 4.75mm Be and 0.25 mm Al and 0.41 mm Cu

Z 1200 -

w I


600 -

400 / .


0 I I I I I I I I
0 20 40 60 80 100 120 140
Figure IV.1. The calculated bremsstrahlung spectra, attenuated by various filters.



- 0.6


J 0.4


0.0 1 I4I I
0 20 40 60 80 100 120 140
Figure IV.2. The total intensity spectrum at the phantom entrance surface (without backscatter).


X-Ray Spectral Intensities in 5 keV Energy Intervals

Photon Energy Relative Intensity Photon Energy Relative Intensity
(keV) (erg-cm-2-s1) (keV) (erg-cm-2-s-l)

0 0 75 886

5 0 80 831

10 0 85 755

15 0 90 680

20 0 95 583

25 2.2 100 502

30 34.5 105 420

35 177 100 336

40 386 115 257

45 600 120 182

50 769 125 118

55 881 130 70

60 2200 135 24

65 1297 140 1.5

70 970

this determination is given in Appendix D. The calculated first half-value

layer is found to be 8.99 mm Al and second half-value layer, 10.30 mm Al.

Exposure rates are measured with an ionization-chamber instrument

(MDH 1015 X-Ray Monitor with 10X5-6 probe) for various attenuators (Type

1100 alloy aluminum filters) in narrow-beam geometry (Jo71). The first and

second half-value layers, determined graphically, are found to be 8.90 mm Al

and 9.78 mm Al, respectively. The accuracy of the operating tube potential

is described in Chapter V.

It is seen that first half-value layers are in agreement to within 0.1

mm Al (1%), and the ratios between first and second half-value layers (0.87

calculated and 0.91 measured) differ by 0.04 (4.5%). Thus it seems that the

calculated spectrum is a reasonable model of the actual spectrum.

Significant Radiation Interactions

It is assumed that radiation entering the phantom under investigation

consists solely of primary x-ray radiation in the form of a uniform, parallel,

rectangular beam. The beam consists of a spectrum of energies ranging from

25 to 140 keV with relative intensities described in Table IV.1. As this beam

enters the phantom various types of interaction occur, the most significant of

which are photoelectric effect, coherent scattering, and Compton scattering.

Photoelectric Effect

It is the photoelectric effect that controls, to a large extent, subject

contrast in conventional (transmission) diagnostic imaging. It is also mainly

the photoelectric effect that governs the signal detection of photons. The

probability of photoelectric interaction is strongly affected by atomic number

and photon energy, and is proportional to Z3E-3, where Z is the atomic number

of the absorber and E is the photon energy (Jo71). Large variations in this

probability occur at photon energies corresponding to the electron binding

energies of the absorber. After photoelectric interaction, secondary x rays

characteristic of the medium are emitted. Characteristic radiations resulting

from photoelectric interaction in the phantoms irradiated in this study are


Coherent Scattering

The probability of coherent scattering decreases rapidly with increasing

photon energy, and for materials of low atomic number the probability of

coherent scattering is small. As the probability of coherent scattering is

strongly peaked for small scattering angles, such scattered photons do not

contribute significantly to a signal sensed by a detector measuring backscattered

x rays (Jo83). The contribution of coherent-scattered x rays to attenuation in

a phantom can be significant, however, and attenuation coefficients excluding

coherent scatter are used for phantom materials in this research.

Compton Scattering

The probability of Compton scattering is angular dependent, related to

the photon energy and the electron density of the medium. The angular

probability of Compton scattering per electron do(E)/dE is described by the

Klein-Nishina equation (Hu69)

do() 0.5re2 [1 + k(l cose)]-2{1 + cos2 + k2(1 cos0)2
de 1 + k(1 cosG)

where k, the photon energy expressed in units of the electron rest mass, is

(E/511) with E in keV; re, the classical electron radius, is 2.82x10-13 cm, and 0

is the scattering angle. A photon undergoing Compton scattering is shifted

in energy, and the amount of this energy shift is related to the scattering

angle. The scattered photon energy E' in keV is related to the incident energy

E in keV and scattering angle @ by the relationship (Ev68)

E' = E
1 + k(l cosO)

Table IV.2 gives the percentage of interactions in polymethylmethacrylate

by the above three modes, based on tables from Hubbell, for the range of

Table IV.2

Percentage of Total Interactions in Polymethylmethacrylate
for Three Major Processes

Relative Numbers of Processes

energy Coherent Scatter Photoelectric Effect Compto
V) (%) (%) (

8.5 55.0 3

6.4 13.3 8

3.6 4.6 9

2.3 2.0 9

1.8 1.0 9

1.4 0.3 9

n Scatter







Photon E







photon energies studied (Hu69). Here the total Compton scattering probability

is considered (i.e., do(6)/dQ is integrated over all scattering angles). It is

seen that in the range of energies covered by the incident spectrum, Compton

scattering is the dominant interaction process in PMM.

The model used in this research project considers all radiation sensed by

the backscatter detector to result from photons singly-scattered by Compton

interaction. Not considered in the model are secondary characteristic emissions

from the phantom, coherent scatter, and multiple Compton scatter. Judicious

placement of the detector minimizes the proportion of coherent- and to some

degree the multiple Compton-scattered photons reaching the detector, and the

contribution from secondary characteristic radiation is small for the PMM

phantoms compared with the contribution from Compton scatter (Jo83, An83).

The effects due to multiple Compton scatter are studied and described in

Chapter VI.

The Detected Fields

The two radiation detectors sense two different radiation fields: the

transmitted field and the backscattered field. These fields differ in energy

distribution and in intensity.

The Transmitted Field

The transmission detector senses x rays which have passed through the

phantom under investigation and the compensating filter described in Chapter

III. This detector is collimated to the size of the primary x-ray beam.

The primary x-ray beam is attenuated by photoelectric and Compton

interactions. It is assumed that all coherent-scattered photons, but no Compton-

scattered photons or secondary characteristic emissions, are sensed by the

transmission detector. Thus the detected signal rate dS/dt is related to the

entrance spectrum by the relationship

dS/dt = Ii exp [-fx (P/P)iPdx']
1 0

where (P/p)i is the mass attenuation coefficient of the phantom material at

energy of index i, p is the phantom density, x is the total path through the

phantom, and Ii is the incident intensity of energy of index i.

The Scattered Field

The scatter detector senses x rays which enter the phantom, undergo Compton

scattering and are scattered into the field of view of the detector, and then

are transmitted by the phantom. The scatter detector is uncollimated and

placed at an angle 3 with respect to the primary x-ray beam at the phantom

center; the detector face is normal to the line defining $. Figure IV.3 depicts

the scattering geometry. In the case of a pencil beam, the scattered intensity

Isj at detector location D due to Compton scattering by angle 0 at point P2

is related to the entrance intensity at I by the relationship

Isj = Ii exp [_fP2 (/p)ipdx] Pe [do(0)/d ]i exp [-_fP3(S/p)ds]
P1 2 '
where Ii is the incident photon intensity of energy of index i, x is the variable

path length between the phantom entrance location P1 and the site of Compton

scattering P2, Pe is the electron density of the scattering medium at P2,

[da(e)/dQ]i is the Klein-Nishina cross-section for photons per solid angle from

the scattering site P2 to the object exit surface P3, and P is the phantom

density. It should be noted that (V/p)i is the mass attenuation coefficient for

incident photon energy of index i, and (i/P)j is the mass attenuation coefficient

for the scattered photon energy of index j. The Klein-Nishina cross-section

for energy of index i per solid angle [do(0)/dW2]i is calculated by the following


{ -do( 0.5re2 [1 + ki(l-cose)]-2{1 + cos2+ ki2(1 cosE)2
dO 1 + ki(1 cos6)'

Figure IV.3. The detected field due to Compton scattering. The radiation beam enters the phantom at P1,
is attenuated by the phantom from P1 to P2, and is scattered by a volume surrounding P2. The detected
scatter signal is attenuated by the phantom along the exit path. The photons scattered at P2 and entering
the center of the detector at D exit the phantom at P3. The detector is of radius r and thickness t.


and ki = Ei/(mc2),

where re is the classical electron radius, 0 is the scattering angle, m is the

electron rest mass, c is the speed of light, and Ei is the incident photon

energy. The scattered photon energy Ej may be expressed
S 1 + ki(l cose)

It should be noted that the scattering coefficient refers to relative numbers

of photons. When the intensity is used (i.e., the product of the number of

photons and their energy), the scattered intensity must be corrected for the

scattered photon energy, that is, by the ratio Ej/Ei.

For a detector of finite dimensions, the signal detection rate is related

to the solid angle A2 from the scattering site P2 to the detector face centered

at D and to the detector efficiency :j in addition to the scattered intensity.

For a spectrum of incident photons of energies Ei, resulting in a spectrum of

scattered photons of energies Ej, the signal detection rate dS(0)/dt is

dS(O)/dt = E IsjA2 j.
Of course, for a parallel beam of finite size, Compton scattering occurs

within a volume rather than at a point, and Figure IV.3 shows the geometric

relationships among the incident beam, the scattering volume centered at P2,

and the detector of radius r and thickness t located at D. If the scattering

volume is chosen sufficiently small, the solid angle intercepted by the detector

may be considered to originate at point P2.

Given a circular phantom of radius R that is allowed to move along the

y-axis, the scattered intensity at D and the signal detection rate can be

calculated as a function of phantom location. Figure IV.4 shows the necessary

geometric parameters for these calculations. The origin of the x,y-coordinate

system is located at the phantom center. Here, yo is the position from the

center of the phantom to the center of the incident beam, xi is the distance

Figure IV.4. The geometric parameters of the Compton-scattering interaction.
Table E.1 (Appendix E) explains the symbols in detail.

e_ Is

to the scattering at P2 from the y-axis, and 0 is the scattering angle from

the incident beam (+ is in a clockwise direction from the +x-axis). The

dimension ko is the attenuation depth of the incident beam in the phantom,

i.e., the distance between P1 and P2; s is the attenuation depth of the

scattered beam central ray in the phantom, i.e., the distance between P2 and

P3; and rs is the distance between P2 and D. The detector centered on point D

is located distance d from the intersection of the x-ray beam with the y-axis,

and the normal to the detector surface is at angle @ with respect to that

intersection. It should be noted that yo is opposite in direction to the coordinate

y, while xi and o, are in the same direction as the coordinate x.

The scattering angle 0 is related to the scattering depth within the

phantom by the relationship

0 = tan-1 -d sin8
d cos + Po (R2 yo2)

The distance rs from P2 to D can be shown to be

rs = [(R2 yo2) d cos 0o]secO,

and the projection of the scattered beam solid angle onto the detector at D is

irr2 cos( S + 0 T)/rs2.

Additional geometric relationships relevant to Figures IV.3 and IV.4 are given

in Appendix E.

These equations are applied in a discrete manner to specific phantoms,

allowing prediction of the scattered signal detection rate as a function of

phantom position yo. As described previously, it is assumed that the scatter

signal detection rate is a result of only primary, and no secondary or higher-

order, Compton-scattered photons. Furthermore, it is assumed that no other

secondary emissions contribute to the scatter signal detection rate. The

discrete calculations, and their associated algorithms, are described in detail

in Chapter VI.



The specific methods of data collection and analysis employed are

described in this chapter. Following a description of the physical setup of

the experimental apparatus, data-acquisition procedures and the phantoms

employed are discussed.

Experimental Setup

The experimental setup can be described in terms of the x-ray beam

physical characteristics and the physical geometry of the experimental apparatus

with respect to the x-ray beam.

The x-ray spectrum, generated at 140 kVp and 10 mA and filtered by

0.41 mm Cu, is described in detail in Chapter IV. The exposure rate at the

transmission detector is measured to be approximately 0.2 mR-s-1. Because

of the physical design of the high voltage system of the x-ray tube, it is not

possible to measure the peak operating voltage of the x-ray tube by a direct

method. The two indirect methods of estimating the operating tube voltage

employed, and their results, are described in Appendix F. It is found that the

x-ray beam peak tube potential is 140+3 keV.

The x-ray tube focal spot, because of the design of the tube for use

in radiation therapy, is larger than that normally used in clinical computed

tomography scanners. With the use of a small tapered aperture (0.084 cm, 150)

surrounded by four larger apertures (0.14 cm) in a 0.65 cm lead disk machined

to fit the x-ray tube diaphragm assembly, both the focal spot location and

the focal spot size can be determined. Figure V.1 shows the focal spot

intensity distribution at a magnification of 1.75 in the projection of interest,


Figure V.1. The focal spot intensity distribution at a magnification of 1.75.

and the focal spot is calculated to be located 24.4 cm behind the diaphragm

assembly. The focal spot, exceptionally non-uniform in intensity distribution,

is seen to be doughnut-shaped, of approximately 5.6 mm o.d. and 1.4 mm i.d.

in the projection of interest. The beam therefore would exhibit significant

spatial nonuniformity in a phantom placed near the tube housing, using only

one small aperture for collimation; furthermore, the beam would diverge

significantly in passing through a 10 cm phantom located near the aperture.

Therefore a series of apertures in 0.65 cm lead sheet is used to define the x-

ray beam to a uniform intensity and the desired rectangular shape with minimal

penumbra. A long distance is used between the source and the object to

minimize beam divergence (less than 0.1 radian in the horizontal dimension).

Figure V.2 shows the geometric relationships among the focal spot, the

apertures, the phantom, and the transmission and scatter detectors, and Figure

V.3 pictures the equipment and its setup. The central two aperture plates

are supported on an adjustable sawhorse, constructed to simulate an optical


The x-ray beam center is situated at a height of 6.7 cm above the

scanning assembly at the center of rotation, to minimize detection of multiple

scatter from the magnesium platform. The beam center is carefully aligned

with the center of rotation of the scanning assembly such that the beam center

is horizontal with respect to the scanning assembly, and the linear traverse is

perpendicular with respect to the beam.

The scanning assembly sits upon a table, constructed for that purpose,

which holds all of the electronics (except for the detectors, the parallel

interface, and the computer) on lower shelves. After alignment of the x-ray

beam with the scanning mechanism, the two detectors are located in the beam.

The transmission detector aperture is adjusted to accept the primary beam

with minimal penumbra, and the aperture is 0.64 cm in the y-dimension by

36.4 cm cm


47.3 cm


7.62 cm

( 1.27 cm diam )

The geometric relationships of the experimental setup.

24.4 cm

38.4 cm

105.4 cm


0.55 cm


L- -


0.45 cm
0.63 cm

1.00 cm

Figure V.2.

Figure V.3. The experimental setup.

0.95 cm in the z-dimension. The transmission detector is supported by a three-

finger clamp on a frame constructed of large rods. The scatter detector,

shielded only along the detector side, is positioned along a line at 300 with

respect to the intersection of the x-ray beam with the y-axis. The center of

the detector face, placed perpendicular to this 300 line, is 7.62 cm from this

intersection and is centered with respect to the vertical dimension of the x-

ray beam. The scatter detector is supported by a three-finger clamp attached

to a shelf surrounding the table supporting the scanning assembly. Specific

details of the alignment procedures are given in Appendix G.

The x-ray field size at the center of rotation is adjusted to allow 19

juxtaposed measurements within a 10 cm diameter phantom; that is, the

sampling aperture is approximately 0.526 cm at the center of rotation. Figure

V.4 shows the x-ray beam size at the entrance surface of a 10 cm phantom

(0.50x0.74 cm2), at the exit surface of the phantom (0.56x0.80 cm2), and at

the transmission detector aperture (0.64x0.95 cm2). Also shown is an image

of the x-ray field with a shadow of a pin located at the center of the rotation

of the platform at the central traverse increment.

Data Acquisition

Data are acquired for phantoms placed on the scanning apparatus and

irradiated in a discrete manner. The 10 cm diameter objects are irradiated

in linear increments (equal to the x-ray beam width) of 0.526 cm, so that 19

measurements interrogate the entire diameter. Measurements are made with

the transmission detector for an integration time of 1 s and with the scatter

detector for an integration time of 2 s. In order to correct the data for drift in

x-ray intensity and detector sensitivity, two additional data sets are obtained

at each end of the traverse. These two measurement locations are separated

from the phantom by an additional increment. Figure V.5 shows the x-ray



Figure V.4. The x-ray beam size and intensity distribution at (clockwise from upper left): the phantom
entrance, the phantom exit, the transmission detector, and a location just behind the center of rotation.
Note the shadow of a pin at the center of rotation, demonstrating x-ray beam alignment with the scanning

Figure V.5. Images of the x-ray field at the center of rotation of the scanning assembly, showing
the 19 radiation measurement positions and the 2 calibration positions for one complete traverse.

fields incremented for one complete traverse at the center of rotation of the

scanning assembly.

After each traverse the object is rotated clockwise through a 60 angular

increment, and the traverse increments are repeated in the reverse direction.

This six degree increment is selected so that the scatter detector senses the

complete circumference of the object. (The circumference 31.4 cm divided by

the traverse increment 0.526 cm gives 59.69 angular increments, or 6.030 per

angular increment.) The system acquires data for 21 positions in each traverse,

for 60 angular increments, for both the transmission and the scatter detectors.

Each measurement is made as follows. The relay associated with each

transmission detector is activated by a logic one from the computer, thereby

opening the relay and allowing integration by the capacitor. The computer

counts a loop for one second, after which time a logic zero is sent to the

analog-to-digital converter (ADC) to begin conversion (10 V 4095 bits). After

a short interval, chosen experimentally to ensure adequate conversion time,

the relay is closed and the capacitor shorted. A logic one is then sent to

the ADC to allow the data bits to be read into the computer. The Ao logic

switch controls input of the upper eight data bits (one) or the lower four bits

(zero). These two data bytes are input separately into the computer and then

summed to create one integer measurement of the transmission detector. This

procedure is repeated for the scatter detector, except that the integration

time in this case is two seconds.

Each traverse consists of 21 sets of measurements for each detector.

Between each of the central 19 measurements, the computer sends 829 pulses

to the linear motor to effect an increment of 0.5264 cm, so that the total

movement for the central 19 measurements is 10.00 cm, the phantom diameter.

The two sets of measurements obtained at the ends of each traverse are

separated from the next position by 1658 pulses (1.0503 cm). The motor

direction is selected by a logic switch in conjunction with the pulses: a logic

one causes the motor to turn counterclockwise.

At the completion of each traverse, the computer sends 150 pulses to

the rotation motor to effect a 60 clockwise rotation. Measurements continue

for the next traverse in each of the 21 locations, with the measurements

proceeding in the direction opposite to the previous traverse. The

traverse/rotation procedure is repeated for a total of 60 angular increments.

Phantoms Studied

All phantoms evaluated are 10 cm diameter by 5.1 cm thick polymethyl-

methacrylate disks with holes and/or aluminum cylinders of various sizes. The

phantoms are supported on a uniform 10 cm diameter by 5.1 cm high PPM

disk which is attached to the scanning platform at the center of rotation by

a threaded stud.

Phantom 1 is a uniform PMM disk with neither holes nor aluminum pins.

This phantom is used as a reference for calibration and normalization in both

the transmission and the scatter measurements. Details of these procedures

are given in Chapter VI.

Phantom 2 is a PMM disk with a 2.11 cm diameter hole, located at

3.421 cm from the center of the disk. Phantoms 3 and 4 are PMM disks with

both one hole and one aluminum pin. The hole is 1.053 cm in diameter situated

3.947 cm from the disk center for all three phantoms. The aluminum pin is

1.053 cm in diameter at the center of phantom 3, and is 2.632 cm in diameter

in phantoms 4. The aluminum pin is located 1.053 cm from the center of

phantom 4. Figure V.6 shows phantoms 2-4.

All phantoms are measured with the PMM compensating filter described

in Chapter III, designed to provide a constant path length in the transmitted

beam with the reference phantom. In the ideal case the transmitted x-ray

field is uniform in both intensity and spectral distribution. Thus in the ideal

Figure V.6. The phantoms studied, all made of polymethylmethacrylate 5.1 cm thick.
Phantom 1 (not pictured) is a solid disk 10 cm in diameter also 5.1 cm thick.

E E A = 3.947 cm
B = 3.421 cm
D 10cm C = 2.632 cm
I D = 2.105 cm
S B E= 1.053 cm




CM 10 cm


case, variations in the phantom from the uniform are seen as perturbations to

a uniform transmission signal. Practical considerations in both fabrication and

alignment of the compensating filter preclude a perfectly uniform transmission

signal as a function of traverse increment; however, the filter does serve the

function of limiting the expected variations in signal intensity and spectral

distribution, thereby easing the limitation of the linearity of the detector

response. The normalization measurements made at each end of the linear

traverse are made in the flat ends of the compensating filter, with a constant

path length of 11.13 cm.

Compensating filters are used routinely in clinical computed tomography

scanners (Za81). However, the filter is placed in a clinical machine close to

the source and between the x-ray tube and the object (patient), rather than

between the object and the detector, because of patient dose considerations.

Because of the limitations of the scanning apparatus and the required source-

object distance in this experiment, it is not feasible to set up the clinical

situation with regard to this parameter. Furthermore, such placement of the

compensating filter would complicate the scattered beam analysis unnecessarily.

It is believed, however, that this restriction can be removed, if appropriate

considerations are made in future experiments.

The effect of the compensating filter on the transmission measurements

must be removed before image processing. The technique used in removing

the effects of the compensating filter is described in Chapter VI.



Application of the continuous scattered field model described in Chapter

IV to the discrete system of measurements described in Chapter V is described

in this chapter. Relationships are derived between the predicted fields and

the measured fields, and a method is described which uses the scattered field

data to produce images of the phantoms irradiated. These images are compared

with those resulting from the transmitted field measurements, where the

transmitted data are reconstructed by traditional computed tomography means.

Discrete Calculation of the Scattered Field

As measurements are made in linear increments such that 19

measurements cover a 10 cm object, calculations are performed for this spatial

increment (10/19 cm) as well. The 10 cm diameter object is superimposed on

a 19x19 grid, representing a cross-section of the volume element. Thus the

phantoms studied are represented as 19x19 picture elements (pixels), with each

pixel corresponding to (10/19)x(10/19) cm2 in object space. Figure VI.1 shows

phantom 2 superimposed on such a grid.

Pixel Area Fitting

Since the grid is square and the objects are circular, there are many

pixels which lie partially in and partially out of the phantoms. A representation

of the square matrix corresponding to the area of the circular phantom within

each pixel, called the "true" area, is obtained. A second representation, called

the "fitted" area, keeps the area within each row constant but changes the

shape of the edge pixels somewhat. Scatter calculations for a phantom are

performed in increments of pixels, in which the fitted pixel areas are used.


Figure VI.1. Phantom 2 superimposed on the 19x19 pixel matrix.
The shaded area corresponds to a hole 4 pixels in diameter.

The fitted areas are obtained according to the following rules, where

the normalization is such that each pixel has an area of 1. First, adjustments

to pixel areas are performed along a particular row, and the area of that row

of pixels is maintained constant. Second, all pixels contained completely within

the circle are not adjusted. Third, if the outermost pixel intersecting the

circle along a row has its center within the circle, it is not adjusted. Fourth,

if the outermost pixel intersecting the circle along a row has its center outside

the circle, the area of that pixel is set to zero, and its area is summed into

the adjacent pixel in that row. Finally, each pixel, for the purposes of the

scatter calculations, is considered to have its area distributed in a slab of

uniform dimension (equal to 1) across the row, with varying dimensions along

the row, centered upon the location of the true pixel. The true pixel areas

are also considered to be of this shape. All calculations are performed in

units of pixels (or pixel', where 1 pixel" is the linear dimension of a pixel

side). Thus an area of 0.3 pixel implies a pixel of dimensions 0.3 pixell by

1 pixel2. Tables VI.1 and VI.2 show the relationships for one quadrant between

the true pixel areas and the fitted pixel areas.

Prediction of the Signal Detection Rate

Prediction of the signal detection rate dS(G)/dt as described in Chapter

IV is performed in discrete increments for a row of pixels, corresponding to

a volume of a phantom actually irradiated and a measurement actually made.

The calculation is performed for each fitted pixel along a row and the results

summed for all pixels on the row. Figure VI.2 demonstrates the discrete

geometry. For a particular pixel in the row, of coordinates (n,m), 1
consideration of the attenuation path within the object to the scattering voxel,

or volume element, the Compton scattering coefficient per solid angle within

Table VI.1

True Areas of the Pixels Within the Circle for One Quadrant

(rounded to 2 decimal places)

Row 1 2 3 4 5 6 7 8 9 10

1 0 0 0 0 0 0.13 0.51 0.78 0.94 1

2 0 0 0 0.04 0.57 0.98 1 1 1 1

3 0 0 0.09 0.81 1 1 1 1 1 1

4 0 0.04 0.81 1 1 1 1 1 1 1

5 0 0.57 1 1 1 1 1 1 1 1

6 0.13 0.98 1 1 1 1 1 1 1 1

7 0.51 1 1 1 1 1 1 1 1 1

8 0.78 1 1 1 1 1 1 1 1 1

9 0.94 1 1 1 1 1 1 1 1 1

10 1 1 1 1 1 1 1 1 1 1

Table VI.2
Fitted Areas of the Pixels Within the Circle for One Quadrant

(rounded to























2 decimal places)

5 6 7

0 0 0.64

0.61 0.98 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1



























































m = 1,19

Figure VI.2. The discrete Compton-scattering geometry for scattering occurring in pixel (n,m).
Note the fractions of pixels within the exit path of the scattered radiation.

the scattering pixel, the attenuation path of the scattered photon (of lower

energy), and the solid angle subtended by the detector.

Attenuation of the Incident Photons

For each photon energy Ei in the incident spectrum (in increments of

5 keV, where 25
along the path corresponding to the true pixel areas from the entrance to the

center of pixel (n,m), as shown in Figure VI.2. For each pixel of index (l,m),

(2,m), ... (n,m), the mass attenuation coefficient [p/p(n',m)]i for each energy

Ei and the density p(n',m) averaged over the pixel are known, and the integral

exp [-2 (p/p)ipdx]

is replaced by the discrete sum

exp{- { [ (p/p)(n',m)]ip(n',m) o9(n',m)},

where incident photons are attenuated along the true pixel lengths ko(n',m) to

the center of the scattering pixel (n,m).

Compton Scattering

For each photon energy Ei in the attenuated spectrum, again in 5 keV

increments, the Klein-Nishina equation is used to calculate the intensity

scattered by pixel (n,m) in the solid angle A2 intercepted by the detector.

Because the scattering angle 0 is defined by the fixed pixel grid, 0 reduces to a

function of n only: e(n), ln<19. The scattered intensity is shifted in energy

from Ei to Ej, where Ej is also in 5 keV energy increments. For the particular

geometry described (19 pixels across a 10 cm object), as 0 varies from a

minimum angle of 116.00 at pixel (1,m) to a maximum angle of 161.40 at pixel

(19,m), the 140 keV incident photons decrease in energy to 109.9 and 91.3 keV,

respectively; thereby resulting in placement in a new energy bin corresponding

to 110 and 90 keV, respectively.

Of course, scattering occurs within a volume rather than in a plane,

and the volume element in the phantom is considered to be of uniform height,

equal to the size of the radiation beam in that dimension (0.75 cm), across

the entire phantom. The scattering process is assumed to occur within the

voxel at the center of each true pixel, as located in the grid. The Compton

cross-section per electron per steradian is calculated for a point, and the

Compton scattering cross-section for the voxel can be determined, since the

voxel electron density is known.

The solid angle for each pixel (n,m) along the row m can be calculated

from the pixel center to the projection of the detector area normal to the

ray from the pixel center to the center of the detector front face, as in the

continuous case. Because only the 19 pixel centers along row m are of interest,

and because the distance to the detector is fixed for each row, the distance

rs reduces to a function of column: rs(n), 1
the detector area is a function of column n. Figure VI.2 shows the geometry

for irradiation along row m and scattering by e(n) at pixel (n,m).

Attenuation of the scattered photons

Calculations of the scattered photon intensity, performed for each

scattered energy Ej, are more difficult to perform in the grid coordinate

system, because of the conical shape of the attenuation path. Because the

phantoms considered are uniform in construction along the dimension of the

beam height (the z-dimension), a simplifying assumption, consistent with that

made for the incident photon intensity attenuation, is that the calculation can

be considered in the plane of the grid, but weighted as though the plane

represented the volume of interest. Figure VI.2 demonstrates such a plane,

where the solid angle is reduced to a triangle.

The desired attenuation for a particular photon path is expressed in the

continuous case as

exp [-fP3 (p/p)jpds],

where for each point along the path ks, for photons of energy Ej, the mass

attenuation coefficient (p/p)j and the density p are known. This integral is

performed, in the continuous case, over all path lengths within the solid angle

between P2 and the phantom surface.

To perform this calculation exactly is an exceptionally time-consuming

task, involving complicated geometric intersections of the Cartesian voxels

with the conical radiation emanating from the point of scattering. As an

approximation to the exact integrals of the lengths through the pixels with

their respective pixel densities (and mass attenuation coefficients), the following

approximation is made to the discrete calculation: it is considered as an

attenuation over a fixed path length ks(n,m) for a weighted average linear

attenuation coefficient [(v/P)jP] since the mass attenuation coefficient at

energy Ej and the density p are known for all pixels (n,m). This weighted

average is obtained as follows. The area of each pixel within the attenuation

path A(a,b) is determined analytically. This area is then weighted by the

inverse square of the distance between it and the scattering pixel r(n,m)(a,b),

where the distance is obtained between the pixel centers. The product of

each pixel area with its weight is then summed for all pixels lying inside the

triangle of interest. The fraction of this sum that each pixel contributes is

the factor assigned to it. That is, a factor f(n,m)(a,b) for each pixel (a,b)

within the attenuation path due to scattering at pixel (n,m) is obtained:

S[A(a,b)] [r(n,m)(a,b)] 2
Eb [A(a,b)] [r(n,m)(a,b)]-2

This factor, which considers the three-dimensional effect of the solid angle,

is multiplied by the density p and mass attenuation coefficient (p/p)j to calcu-

late the weighted average linear attenuation coefficient. Thus the attenuation

of the scattered intensity is calculated for each scattering pixel (n,m) for each

scattered energy j:

exp { -s(n,m)j Z z [p(a,b)] [(I/P)j(a,b)] [f(n,m)(a,b .

Self-attenuation by the scattering pixel is not considered.

This approximation includes the fact that the contribution of each pixel

to the attenuation through it is related to the amount of the pixel in the

beam and also the fact that, as the beam diverges, the contribution of a

specific amount of a pixel within the beam to the beam attenuation decreases

in an amount related to the beam divergence. For example, if the beam

originating in a pixel crosses totally into the adjacent pixel, the area (or

volume) of intersection is quite small. However, the effect of that pixel on

the scattered radiation beam is significantly larger because the entire solid

angle of the beam crosses through it. At distances farther from the scattering

pixel, the same volume of intersection contributes much less to the beam

attenuation because only a fraction of the beam solid angle passes through it.

While it is recognized that this approximation is not rigorously obtained, it is

felt to be sufficient for the specific task at hand.

The calculated signal

Photons incident on the detector are attenuated by the projected

thickness of the aluminum detector housing. Although in actuality this projected

thickness depends on the scattering site and angle, this calculation is performed

for the path corresponding to photons scattered at the pixel center which are

incident on the center of the detector front surface.

The attenuated spectrum which interacts with the scintillation crystal

is related to the detector response by the detector efficiency Ej, a complicated

function of photon energy Ej. The sodium iodide (thallium-activated) detector

exhibits discontinuities in attenuation and energy absorption for energies

corresponding to the electronic transitions of its constituents, particularly at

the K-edge of iodine at 33 keV (Ev68, Ne65). At other photon energies the

response is not uniform and is governed mainly by the probability of

photoelectric interaction with the crystal. Although it is assumed that all

photons incident on the crystal are absorbed in it, the photon path length

through the crystal varies widely as a function of scattering angle.

The actual detector response, that is, the current which leaves the

photomultiplier tube, is dependent upon the efficiencies of crystal scintillation,

crystal transparency, and photocathode photoemission, all of which depend, to

a greater or lesser extent, on photon energy (Ne65, Kn79). For these reasons,

the effect of the detector efficiency variation is determined experimentally

for the geometries of interest and for the relevant x-ray spectra, but not as

a function of each energy Ej. Comparisons between calculated and measured

responses are therefore made on a relative, not absolute, basis.

For a given position of the phantom with respect to the incident beam

and scatter detector, the calculated signal is determined as follows. For each

pixel (n,m) along the irradiated row m for each incident energy Ei, the incident

spectral intensity Ii is attenuated within the phantom; the Compton-scattering

coefficient is determined by the Klein-Nishina relationship, and the scattered

photon energy Ej is determined and quantized and the solid angle determined;

the scattered intensity is attenuated over the exit path for the scattered

photon energies, and then further attenuated by the detector housing; all

according to the described equations and methods. The resultant intensities

are summed over all energies and over all pixels, thus producing a quantity

proportional to the energy absorbed in the crystal per unit time, not including

detector efficiency. This quantity is referenced as the calculated scattered

intensity or calculated signal. The total calculated signal is then determined

for each pixel row. When the total calculated signal is corrected by the

appropriate detector efficiency effect or correction factor, determined experi-

mentally as a function of row m, the resultant value is the predicted signal.

The above set of calculations is performed for the phantom studied as

a function of rotation increment k with respect to the x-ray beam. Of course,

for a uniform phantom, there is no rotational dependence of the calculated

signal, so the uniform phantom is selected as the reference phantom for

comparison between calculated and measured signals as a function of traverse

increment, and for determination of the correction factors.

Comparison of Calculated and Measured Signals

In order to relate predicted signals to measured signals, the calculated

signals must be corrected as a function of traverse increments. The method

used in determining the necessary correction factors follows.

Determination of Correction Factors

Scattered intensities as a function of traverse increment and rotation

increment are measured for the uniform phantom (phantom 1). The 19 scatter

measurements in each traverse are made concomitant with transmitted intensity

measurements through the compensating filter at the calibration endpoints (i.e.,

at the extreme positions 0 and 20). These transmission measurements are used

to correct for drift in x-ray beam intensity. The data are processed in the

following manner. For each rotation increment, the transmission values obtained

at the two endpoints of one traverse are averaged. These average values are

divided by that obtained for the first rotation increment, producing calibration

factors about unity. Each scatter measurement for one rotation increment is

then divided by its respective calibration factor, and this procedure is followed

for all angular increments. Finally, an average value of the scatter

measurement over all rotation increments is obtained for each of the traverse

increments. The per cent standard deviation is less than 2% in each case.

Calculated values of the scatter signals from a uniform phantom are

compared with the measured scatter signals from that phantom (phantom 1).

Figure VI.3 shows the calculated and measured scatter signals as a function

of traverse increment, where the traverse increment corresponds to the

irradiated row m as depicted previously in Figure VI.2. It should be noted

that two scales are used for ease in visual comparison.

Figure VI.3 demonstrates similar curve shapes between the calculated

and measured scatter signals. The two curves are asymmetric about their

respective central positions because of differences in attenuation along the

exit paths of the scattered photons. The measured data show less asymmetry

with respect to the central position than the calculated data, however, and

these differences are explained on the basis of the differences in spectral shift

as a function of scattered photon attenuation and the nonlinear energy response

of the detector to low energy photons. Additional possible explanations for

the differences among the two data sets are, one, that they result from

multiple photon scattering, not included in the calculational model, and, two,

that they are related to the physical geometry of the experimental setup, with

respect to the finite size of the scattering voxel and the variation in photon

path length in the crystal as a function of scattering angle and site.

Ratios are made of the measured signal from the uniform phantom to

the calculated signal as a function of traverse increment. This set of ratios

is the set of correction factors as a function of traverse increment which

allows conversion of the calculated signals to their respective predicted signals

by multiplication by their respective elements. These correction factors are

given in Table VI.3.


-1 150

-- 100

- 50

5 F-




Figure VI.3. Calculated and measured scatter signals for the uniform
phantom as a function of traverse increment. Measured values shown
are the average of each of the 60 rotation increments, corrected for
drift, and 95% confidence intervals are depicted for five positions.

Table VI.3

Correction Factors as a Function of Traverse Increment

Position Correction Factor

1 12.41

2 11.23

3 11.14

4 11.03

5 10.78

6 11.18

7 10.70

8 10.84

9 10.76

10 10.62

11 10.41

12 10.05

13 9.93

14 9.84

15 9.51

16 9.27

17 9.09

18 8.76

19 8.40

Effects of Multiple Scattering

In order to determine the contribution of multiple Compton scattering

to the total scatter signal, Monte Carlo calculations are performed for the

uniform phantom as a function of traverse increment for three positions. The

calculations are made in terms of the ratio of multiple scattering (including

second-, third-, and fourth-order scattering) to single scattering for the sum

of five energies in the x-ray spectrum, weighted according to their relative

intensities within the spectrum. Appendix H describes the details of the Monte

Carlo calculations, which allow comparisons of the scattered energy shifts as

a function of traverse increment in addition to the effects of multiple

scattering. The results are summarized below.

1) The effect of multiple scattering is on the order of 20% for the 10

cm diam uniform phantom, with variations of approximately 5% due

to traverse increment.

2) The spectral shift in intensities due to attenuation differences

between the two phantom locations is of the order of 40%, with a

small difference between singly- and multiply-scattered photons.

3) The spectral shift in intensities due to single versus multiple

scattering is of the order of 25%, with a small difference due to

the phantom location.

4) Thus it seems that the variation in correction factors shown in Table

VI.3 (25% difference between positions 2 and 18) are due more to

the nonlinearity of the detector response as a function of energy

than to variations in multiple scattering.

Verification of the Algorithm

Phantom 2, a PMM disk with a four-pixel diameter hole near the surface,

described previously in Chapter 5, is used for comparisons of predicted signals

with measured signals among traverse increments and rotation increments.

This phantom is oriented with respect to the x-ray beam such that, for the

first rotation increment, the hole is centered along the positive x-axis when

the incident x-ray beam irradiates along the x-axis in the positive direction.

This orientation is shown in Figure VI.1.

A simulation of phantom 2 is used to generate predicted signals as a

function of traverse increment and a rotation increment. The model consists

of the calculated average density of PMM within each pixel for the 19x19

array for each of the 60 clockwise angular increments k, 1
I describes the details of this procedure. These values of average density

within each pixel are used in the discrete calculations previously described to

determine the predicted signal as a function of traverse increment and rotation


For each traverse increment and rotation increment, measurements of

the scatter signal are obtained and then corrected for drift by use of the

normalized transmission measurements at the endpoints as described above.

Figure VI.4 shows the predicted and measured scatter signals as a function of

traverse increment for phantom 2 at rotation increment k=16, corresponding to

a 900 clockwise rotation. The predicted data set is adjusted at traverse

increment 19 to correct for a discrepancy in x-ray beam intensity and/or

detector sensitivity between the occasions of, one, the measurement of phantom

1, at which time the correction factors given in Table VI.3 were determined,

and, two, the measurement of phantom 2. Excellent agreement between the

two data sets is obtained, with a maximum discrepancy of 3% at position 11.

Figure VI.5 shows the comparison between predicted and measured signals

for rotation increment 30, corresponding to 1740. This increment is chosen

because it is in this orientation that the effect of the hole on the scatter

signal is greatest. It should be noted that the decrease in signal due to the

presence of the hole results from a lack of scattering within the hole.


150 -



50 -




Figure VI.4. Predicted and measured scatter signals
as a function of traverse increment:
Phantom 2 rotated 900 clockwise (k=16).

-J 100






100 F-


8 12 16

Figure VI.5. Predicted and measured scatter signals
as a function of traverse increment:
Phantom 2 rotated 1740 clockwise (k=30).

150 -

Furthermore, when the exit paths of the scattered photons cross the hole, an

increase in signal is obtained, due to less attenuation of the scattered photons.

Excellent agreement in shape between the two data sets as a function

of traverse increment is observed, but larger discrepancies between the two

data sets are noted, at positions 8 and 9, of 11% and 13.5%, respectively.

The reason for these discrepancies is not known, but the following observations

are made. First, both curves show the expected increase in signal at position

8 compared with the comparable position on the other side of the hole, position

14; however, this increase is larger in the predicted case than in the measured

case. Because the exit path is not attenuated as greatly in position 8, the

resultant spectrum contains a larger proportion of low energy radiation, and

the nonlinear energy response of the detector may play a part in this effect.

A second possible explanation is related to the calculational approximation in

determining the weighted average density of the exit path. The extreme

discontinuity of the pixel boundaries and the large size of the pixels compared

with a small portion of the hole lying within, but averaged over the entire

pixel, may explain the differences. Perhaps the inverse-square weighting is

not sufficiently descriptive of the actual attenuation paths because of the

large size of the scattering volume compared with the distances involved. In

any case, the percentage difference is not extremely large. It should be

recognized that, within the hole and at all other positions along the traverse,

all other data are in excellent agreement. Thus it seems that the choice of

the model, while not perfect, is reasonable and quite acceptable.

The Information in the Signal

Comparison of Figure VI.4, phantom 2 rotated 900, with the measured

signals of Figure VI.3, (the uniform) phantom 1, shows a flattening of the

scatter signal for phantom 2 at positions 15-18; however, the perturbations in

the phantom 2 data are much more obvious when ratios are obtained. These

ratios are calculated by dividing the phantom 2 data by the phantom 1 data

for comparable traverse increments for both the measured signals and the

predicted signals. Figure VI.6 shows the ratios for phantom 2 rotated 900.

The location of the hole is striking when observed in this format. The increase

in the signal from the lack of attenuation due to the hole in the exit path is

also apparent. Also seen are small fluctuations in the ratios due to the

statistical nature of the measurements.

Figure VI.7 shows the comparable ratios of the data shown in Figures

VI.5 and VI.3. The measured ratios show a smoother variation than the predicted

ratios, and a somewhat different shape for measurements close to the hole.

(Note especially positions 4-8, and compare the differences with the 95%

confidence intervals shown in Figure VI.3.) That this shape is real is verified

by Figure VI.8, the ratios for 180 rotation, which is obtained for the next

rotation increment after that shown in Figure VI.7. Again it is important to

note that discrepancies are of the order of 5% or much less in most cases,

with the exception of the discrepancies noted previously.

It is these ratios (various phantoms compared with the uniform phantom)

that are used in further data processing of the scatter signals. It is these

ratios which effectively show the character of the phantoms irradiated. A

further demonstration of these ratios as a function of angular increment is

given in Figure VI.9 in sinogram format (Ba81). Figure VI.9a shows the predicted

scatter data ratios from phantom 2 (simulated) as a function of traverse

increment along a row, and as a function of rotation increment along a column,

transformed to 16 shades of gray. The position of the hole, and indeed the

increased signal before the hole, is readily apparent over 1800 of the 3600

rotation. Figure VI.9b shows the measured data ratios for phantom 2. The

variations in the measured data ratios due to system noise are also apparent.







Figure VI.6. Ratios of signals from phantom 2 to phantom 1 as a function
of traverse increment for both predicted signals and measured signals:
900 clockwise rotation (k=16).






-o0 0

00 0 02




Figure VI.7. Ratios of signals from phantom 2 to phantom 1 as a function
of traverse increment for both predicted signals and measured signals:
174 o clockwise rotation (k=30).

1.3 r-

1.2 1-







0 0

it 60







I I I I I I I I I 1
4 8 12 16 2(

Figure VI.8. Ratios of signals from phantom 2 to phantom 1 as a function
of traverse increment for both predicted signals and measured signals:
1800 rotation (k=31).






I.I 1

Figure VI.9. Sinograms of ratios: (a) phantom 2 to
phantom 1 (predicted); (b) phantom 2 to phantom 1
(measured); (c) phantom 3 to phantom 1 (measured);
and (d) phantom 4 to phantom 1 (measured).

(a) (b)


Phantom 2 is a large hole, of 4 pixels in diameter. A more critical test

of the system is with the use of a phantom with a smaller hole. Phantom 3

is such a phantom: its hole is only 2 pixels in diameter. Figures VI.9c and

VI.9d show sinograms of the ratio data obtained from measurements of phantoms

3 and 4, and in both cases, the hole is clearly visible over the 1800 range.

The aluminum rod is not visible in the sinogram of phantom 3 and poorly

visible in that of phantom 4.

The electron densities, in electrons-cm-3, of the materials present in

the phantoms are: polymethylmethacrylate, 3.9x1023; air, 3.6x1020; and

aluminum, 7.8x1023 (Jo71). It should be noted that, while the differences in

electron density between PMM and the other two substances are approximately

equal in magnitude, the ratios of electron densities relative to PMM are 2 for

aluminum and 9.2x10-4 for air. Thus the sinograms, showing ratios of scatter

signals with respect to the uniform phantom, show much greater sensitivity to

air than to aluminum contained within the phantoms.

The sinogram format clearly shows the extraction of information from

the scattered field. The remainder of this chapter describes some of the

methods attempted to display this information in an image. As transmission

measurements are made concomitant with the scatter measurements, the

transmitted data are reconstructed by traditional means into a computed

tomography image.

Reconstruction of the Transmitted Field

The transmission data, acquired for each of the 19 traverse increments

for each of the 60 rotation increments for the five phantoms, are processed

by conventional parallel-beam computed tomography reconstruction techniques.

The result of this processing is a matrix of values related to the linear

attenuation coefficient of the phantom voxel corresponding each cell or pixel


The processing occurs in three major steps. First the data are corrected

for the effects of the compensating filter and for drift in intensity for each

traverse, the redundant halves of the data are averaged, and the negative of

the natural logarithm of the data are obtained. Second, the data are convolved

with the Fourier transform of a simple ramp filter, and, third, the convolved

data are backprojected (Ra71, He80). Details of the reconstruction procedure

are given in Appendix J.

The Effective Attenuation Coefficient

The transmission data are obtained over an essentially constant path

(11.13 cm) of PMM, with perturbations due to discontinuities in the phantoms.

The transmission spectra are thus fairly constant as a function of traverse

increment and phantom. An effective attenuation coefficient for the PMM

phantom-plus-compensator transmission spectrum is calculated, the CT

reconstruction procedure is adjusted to yield this number for the uniform


This effective attenuation coefficient is calculated as follows. The

incident x-ray spectrum described in Chapter IV is attenuated by the attenuation

coefficients, excluding coherent scatter, for 11.13 cm of PMM of density 1.181

g-cm-3. The resultant spectrum is summed to yield a total intensity, which

is compared with the total intensity of the incident spectrum. The effective

attenuation coefficient is then calculated according to the equations given in

Appendix C, based on the known path length and density, under the assumption

that the spectrum is monoenergetic. This procedure follows closely the

determination of the effective energy of a spectrum (Jo71). For the

transmission spectrum postulated, the effective attenuation coefficient is found

to be 0.178 cm2-g-1 or 0.210 cm-1 for the measured density of PMM. This

value of linear attenuation coefficient converts to 0.111 pixe-f.

CT Reconstructed Images

Images of phantoms 1-4 reconstructed by the methods described are

shown in Figure VI.10. The image display is adjusted to present the entire

16 gray-levels for the range of the data within the reconstruction circle in

each case. As a measure of the variability in the data, the mean and per

cent standard deviation of all the pixels whose centers are within the circle

of reconstruction are calculated for the uniform phantom, and the resultant

values are 0.110 and 3.9%, respectively. For the central 49 pixels, the

corresponding values are 0.108 and 0.3%.

Application of the Scattered Field Information

Various methods of applying the information extracted from the scattered

field to create an image are described. The general procedure followed is

described in this section, and the various specific methods attempted follow.

These methods are described for the noise-free simulated scatter ratios of

phantom 2 shown in the sinogram in Figure VI.9a, and one method is chosen

for reconstruction of images from the actual scatter measurements of the


The General Procedure

A simulated uniform CT matrix is used as the 19x19 pixel matrix upon

which scatter calculations are based; that is, with the assumption that the

pixel values can be related to the voxel densities in the phantom, these densities

are used to predict the scattered field as a function of traverse increment m

and rotation increment k. As the CT matrix contains values of the linear

attenuation coefficient of the PMM object for the x-ray spectrum (which is

reasonably uniform for a reasonably uniform object because of the compensating

filter), these pixel values are converted to voxel density by dividing by the

effective mass attenuation coefficient determined for PMM, calculated to be

0.643 pixel-g-1.

Figure VI.10. CT-reconstructed images of the phantoms
studied: (upper left) phantom 1, (upper right) phantom
2, (lower left) phantom 3, and (lower right) phantom 4.


Given simulated voxel density values for PMM, then, the predicted

scatter signal is calculated for each traverse increment, beginning with the

bottom row (m=19). This predicted signal is compared with the measured

signal obtained for a phantom in the comparable orientation, and the ratio

R(m) of the measured to the predicted signal is obtained. If this ratio is

significantly different from unity (more precisely, if the absolute value of the

difference between the measured and predicted signals is greater than 0.015),

then adjustments are made to the pixels along the row of the incident x-ray


Because the scatter signal is weighted heavily toward the entrance

surface of the phantom due to increased attenuation of the scattered signal

by the phantom as a function of depth, adjustments of pixel densities are

weighted accordingly. These weighting factors are determined as follows. For

a uniform phantom the scatter signal is calculated for each row on a pixel-

by-pixel basis. The relative contribution of the scatter signal from each fitted

pixel to the total scatter signal for the row is calculated. A set of relative

scatter factors S(n,m) is thus created, with values for each of the fitted pixels.

Table VI.4 gives these values.

Adjustments of the pixels along a row are made to correct pixel densities

so that the predicted scatter signal becomes equal to the measured scatter

signal. Specific details of the adjustment procedure are given in the following

section. Scatter calculations and subsequent pixel density adjustments are

performed from the bottom of the matrix to the top (i.e., from row m=19 to

m=l) so that adjusted pixel densities are used in the exit paths of the scattered

x rays as the calculations proceed for decreasing m. After all rows of pixels

have been adjusted, the matrix is rotated 1 angular rotation increment (i.e.,

6 clockwise) so that similar calculations can be performed in the next orienta-

tion corresponding to the proper phantom position.

Table VI.4

Relative Scatter Factors by Fitted Pixel Along a Row
for the Uniform Phantom

(a) Left Half

(displayed to 3 decimal places)





























































3 4 5 6 7 8 9 10

0 0 0 0 0.260 0.225 0.189 0.141

0 0 0.223 0.240 0.166 0.116 0.082 0.058

0 0.288 0.214 0.149 0.104 0.074 0.052 0.037

0.272 0.213 0.150 0.107 0.075 0.054 0.038 0.028

0.229 0.161 0.116 0.083 0.059 0.043 0.030 0.022

0.195 0.140 0.102 0.074 0.053 0.038 0.027 0.020

0.165 0.120 0.088 0.064 0.046 0.033 0.024 0.017

0.155 0.114 0.084 0.061 0.044 0.032 0.023 0.016

0.149 0.110 0.081 0.059 0.042 0.031 0.022 0.016

0.147 0.109 0.081 0.059 0.043 0.031 0.022 0.016

0.149 0.111 0.083 0.061 0.044 0.032 0.023 0.017

0.154 0.115 0.086 0.063 0.046 0.033 0.024 0.018

0.168 0.126 0.094 0.069 0.050 0.037 0.027 0.020

0.189 0.142 0.107 0.079 0.058 0.042 0.031 0.023

0.216 0.164 0.123 0.092 0.067 0.050 0.036 0.027

0.229 0.199 0.150 0.112 0.083 0.062 0.046 0.034

0 0.240 0.196 0.148 0.110 0.082 0.062 0.046

0 0 0.168 0.225 0.160 0.121 0.092 0.070

0 0 0 0 0.200 0.206 0.185 0.153

Table VI.4-continued

(b) Right Half


Row 10 11 12 13 14 15 16 17 18 19

1 0.141 0.095 0.057 0.034 0 0 0 0 0 0

2 0.058 0.042 0.030 0.021 0.015 0.007 0 0 0 0

3 0.037 0.027 0.019 0.014 0.010 0.007 0.005 0 0 0

4 0.028 0.020 0.014 0.010 0.007 0.005 0.004 0.002 0 0

5 0.022 0.016 0.011 0.008 0.006 0.004 0.003 0.002 0.001 0

6 0.020 0.014 0.010 0.007 0.005 0.004 0.003 0.002 0.002 0

7 0.017 0.012 0.009 0.006 0.005 0.003 0.002 0.002 0.001 0.001

8 0.016 0.012 0.009 0.006 0.004 0.003 0.002 0.002 0.001 0.001

9 0.016 0.012 0.008 0.006 0.004 0.003 0.002 0.002 0.001 0.001

10 0.016 0.012 0.008 0.006 0.004 0.003 0.002 0.002 0.001 0.001

11 0.017 0.012 0.010 0.006 0.005 0.003 0.003 0.002 0.001 0.001

12 0.018 0.013 0.009 0.007 0.005 0.004 0.003 0.002 0.001 0.001

13 0.020 0.014 0.010 0.008 0.006 0.004 0.003 0.002 0.002 0.001

14 0.023 0.017 0.012 0.009 0.007 0.005 0.004 0.003 0.002 0

15 0.027 0.020 0.015 0.011 0.008 0.006 0.004 0.003 0.001 0

16 0.034 0.025 0.019 0.014 0.010 0.008 0.006 0.004 0 0

17 0.046 0.035 0.026 0.020 0.015 0.012 0.008 0 0 0

18 0.070 0.053 0.041 0.032 0.025 0.013 0 0 0 0

19 0.153 0.115 0.079 0.061 0 0 0 0 0 0

For each rotation increment within a quadrant, a table is generated by

which rotated pixel density values can be calculated. The schema considers

each rotated pixel within the circle and identifies, for each rotation increment,

which stationary pixels intersect that pixel, and the fraction of their respective

areas within the intersection. The method describing the creation of these

tables is given in Appendix K. As few as 4 but as many as 6 pixels may

contribute to a particular rotated pixel within the circle, and this number is

highly dependent upon the location of the pixel within the matrix.

The schema is used in the following way. The rotated pixel density is

calculated by weighting the densities of the intersecting stationary pixels by

their respective fractional intersecting areas and then summing over all the

intersecting pixels. To compensate the pixels which intersect the boundary of

the circle for their respective fractions outside the circle, and therefore to

maintain a constant area within the circle, the rotated pixel densities are

divided by their respective sums of fractional areas. That is,
Z [pi(n',m')s] [fk,i(n',m')s]
p (n,m)R = i

where p (n,m)R is the density of rotated pixel (n,m)R, P i(n',m')s is the density

of the ith stationary pixel of index (n',m')S intersecting pixel (n,m)R, and

fk,i(n',m')S is the fractional intersecting area of the ith stationary pixel (n',m')
with rotated pixel (n,m) for the angular rotation increment k. Because each

stationary pixel contributes some of its associated density to several pixels,

this process causes a blurring of the pixel density information over the several

pixels, and the process of this blurring is dependent on pixel location and

angular rotation; that is, the process is nonstationary. It should be noted that

the process is nonlinear; that is, 2 applications of the 60 rotation scheme pro-

duce more blurring than one application of the 120 rotation scheme. Appendix L

demonstrates some of the errors associated with rotation of the discrete pixel


Pixel densities are rotated by a schema developed for each rotation

increment k within one quadrant (1
thereof is handled differently (and exactly) by appropriate index labelling. For

rotation by more than 900 (k>16), the pixel densities are first rotated exactly

in increments of 900 to the appropriate quadrant, so that the remaining rotation

is an acute angle, and then the rotation schema is applied to effect the

remaining rotation.

After all rotated pixel densities are calculated, the new set of rotated

pixel densities is used as the basis for prediction of scatter signals by row,

and this set of predicted signals is compared with the measured scatter signals

for the phantom in the new orientation. Again, calculations and adjustments

proceed from the bottom of the matrix to the top.

In order to avoid propagation of errors caused by the adjustment scheme,

only the original matrix of pixel density values is rotated and adjusted, on a

view-by-view basis. The adjustments to the rotated pixels must then be rotated

back to the initial orientation so that the overall corrections from each rotation

increment (view) can be compared among views. The backward rotation schema

is generated from the forward schema by re-sorting the table.

After calculations, comparisons, and adjustments are made for each of

the 60 rotation increments, the corrected data are compared and integrated

into a final pixel matrix and then multiplied by the effective mass attenuation

coefficient. The methods applied in this final step for processing of simulated

data are described in the next section.

Generation of the Final Results

In order to allow selection of the adjustment procedure and the method

of integrating the adjusted data into a final pixel matrix, the predicted ratios

displayed in the sinogram of phantom 2 (Figure VI.9a) are used as input data

into the general procedure described above. Several methods are tested, and

a choice is made based on both the visual "quality" of the image (i.e., the

shape and definition of the hole and the minimization of apparent structural

noise) and the magnitudes of the elements of the final pixel matrix. Of the

various methods tested, four are described below, and their resultant images

are shown in Figure VI.11. The method selected for use with the measured

scatter signals is shown in a descriptive flow chart in Figure VI.12.

The four methods selected for description are presented in order of

complexity in terms of adjustments, and manipulations of the adjustments, to

the pixel densities. Method 1 adjusts pixel densities by the measured-to-

predicted scatter signal ratios along each row and rotates back the density

changes to the initial orientation. Method 2 again adjusts pixel densities by

the ratios but the density changes are projected along the row, filtered, and

then backprojected with weighting by the scatter factors. In both methods 1

and 2 the densities of all pixels along a row are changed by the same ratio;

in methods 3 and 4 the changes along each row are weighted by the scatter

factors. As changes to a particular pixel occur in more than one orientation,

the maximum change to a pixel density is used to create the final image in

method 3. In methods 1, 2, and 4, all changes to a particular pixel are summed

to create a total change for the density of each pixel. In all four cases the

final image is the sum of the initial uniform density matrix and the density

change for each pixel. Specific details of the four methods follow.

Method 1. The ratios R(m) are used to adjust each pixel along row m

by p'(n,m) = p (n,m)R(m) for each rotation increment k. After completion of

calculations and adjustments for all 19 rows, these new values of p'(n,m) are

used to recalculate the scatter signals, new values of the calculated scatter

ratios, and new values of the simulated-to-calculated ratios R(m). New adjusted

Figure VI.11. Images of simulated data of phantom
2 processed by four methods of reconstruction: (upper
left) method 1, (upper right) method 2, (lower left)
method 3, and (lower right) method 4. Method 4 is
the procedure chosen for processing measured data.
(See text for description of the methods.)


Begin with measured
scatter data SM(m,k) fo:
traverse increment m an,
rotation increment k

calibration data CAL(k)
to correct for drift
SM (m,k)=SM(m,k)/CAL(k)

Subtract p(n,m) from
p'(n,m) to calculate
differences A(n,m)

Divide SM(m,k) by uniform
phantom scatter data UM(m)
to obtain measured ratios
SM(m,k)=SM(m,k) /UM(m)

Rotate density matrix
to proper orientation
for w=(6k)0 clockwise

traverse increment m=19
iteration j=0

Calculate scatter signal
SC(m) for traverse m

Divide SC(m) by scatter signal
for uniform phantom UC(m) to
obtain calculated ratios

Add all k values of
Ap'(n,m,k) for each
n and m to original
p(n,m) to obtain Ps(n,i

Figure VI.12. Descriptive flow chart of the algorithm of method 4.

Convolve (m) with
kernel to obtain C(m)

Backproject C(m) on a zero
matrix, along rows m, weighted
by relative scatter factors S(n,m)

Define Ap'(n,m,k)=Ap'(n,m)

Rotate Ap'(n,m,k) back to ----- k
initial orientation

values of p'(n,m) are calculated, and the process repeated for a total of 10 sets

of adjustments to all 19 rows. The differences Ak(n,m) = p'(n,m) p(n,m) are

rotated back to the initial orientation, and the sum of these differences for

all rotation increments is made for all pixels to obtain the final scatter

densities pg(n,m):
Ps(n,m) = p(n,m) + Z Ak(n,m).

Method 2. As in method 1, the pixel densities are adjusted by the ratios

R(m) for each rotation increment k: p'(n,m) = p(n,m)R(m); however only one

set of adjustments is made. Because the image from method 1 shows streaking

somewhat akin to the "star" artifact from simple backprojection in CT

reconstruction (Bu74), the differences A(n,m), where A(n,m) = p'(n,m) p(n,m),

are summed for each row to produce As(m) values:

As(m) = E A(n,m).

These As(m) are convolved with the ramp filter of Ramachandran and

Lakshminarayanan described in Appendix J to obtain the convolved differences

C(m) (Ra71). These convolved differences are backprojected along row m with

weighting by the scatter factors S(n,m) for each pixel for each rotation

increment k: Ap'k(n,m) = C(m)S(n,m).

The Ap'k(n,m) values are rotated back to the initial orientation. The final

image densities ps(n,m) are then calculated as a sum over all rotation increments:

Ps(n,m) = p(n,m) + Z Ap'k(n,m).

Method 3. In order to adjust the densities p(n,m) along row m in

proportion to the fraction of signal originating in each pixel for the uniform

phantom, the pixel densities are adjusted for each rotation increment k:

p'(n,m) = p(n,m){l + [R(m) 1]S(n,m)}.

Adjustments are made for all 19 rows, and then new calculated ratios R(m)

are created as in method 1. The process of adjusting the densities is repeated

for 10 sets of adjustments to all 19 rows. As in method 2, the differences

A(n,m) are summed along row m to produce the Ag(m) values which are con-

volved with the ramp filter. These convolved differences C(m) are back-

projected with the S(n,m) weighting as described in method 2, and the resultant

Ap'k(n,m) rotated back to the initial orientation. The change to the image

density for each pixel is calculated by determination of the largest magnitude

difference among all orientations k for each pixel (n,m). That is,

p'(n,m) = p(n,m) + Ap'k,(n,m),

where the specific value k' is chosen by the k for which IAp'kt(n,m) is the largest.

Method 4. As in method 3, the densities p(n,m) are adjusted with

weighting by the S(n,m) values for a total of 10 sets of adjustments. As in

methods 2 and 3, the differences A(n,m) are summed along row m to produce

the Ag(m) values which are convolved with the ramp filter and backprojected

with the S(n,m) weighting. The resultant Ap'k(n,m) values are rotated back

to their respective initial orientations, and the final image density data are

calculated by summing over all 60 rotation increments k:

p'(n,m) = p(n,m) + E Ap'k(n,m).

It is this procedure which is outlined in the flow chart given in Figure VI.12

and which is used for creation of the images using measured, rather than

simulated, data.

The choice of the number of adjustments allowed is determined

empirically through the use of various numbers of adjustments for a particular

phantom orientation. The simulated ratios of phantom 2 for 1800 rotation,

shown previously as the predicted ratios of Figure VI.8, are used as the

simulated data to allow calculation of adjustments to the pixel densities and

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