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An Adapted Modulation Transfer Function for X Ray Backscatter Radiography by Selective Detection

University of Florida Institutional Repository
Permanent Link: http://ufdc.ufl.edu/UFE0021299/00001

Material Information

Title: An Adapted Modulation Transfer Function for X Ray Backscatter Radiography by Selective Detection
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Sabri, Nissia
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: contrast, edge, fourier, modulation, monte, radiography, selective
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The Modulation Transfer Function (MTF) is a quantitative function based on frequency resolution that characterizes imaging system performance. In this study, a new MTF methodology is investigated for application to Radiography by Selective Detection (RSD). RSD is an enhanced, single-side x-ray Compton backscatter imaging (CBI) technique which preferentially detects selected scatter components to enhance image contrast through a set of finned and sleeve collimators. Radiography by selective detection imaging has been successfully applied in many non-destructive evaluation (NDE) applications. RSD imaging systems were designed and built at the University of Florida for use on the external tank of the space shuttle for NDE of the spray-on foam insulation (SOFI) inspection. The x-ray backscatter RSD imaging system has been successfully used for cracks and corrosion spot detection in a variety of materials. The conventional transmission x-ray image quality characterization tools do not apply for RSD because of the different physical process involved. Thus, the main objective of this project is to provide an adapted tool for dynamic range evaluation of RSD system image quality. For this purpose, an analytical model of the RSD imaging system response is developed and supported. Using the Fourier transform and Monte Carlo methods, two approaches are taken for the MTF calculations: one using a line spread function and the other one using a sine function pattern. Calibration and test targets are then designed according to this proposed model. A customized Matlab code using image contrast and digital curve recognition is developed to support the experimental data and provide the Modulation Transfer Functions for RSD.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nissia Sabri.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Dugan, Edward T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021299:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021299/00001

Material Information

Title: An Adapted Modulation Transfer Function for X Ray Backscatter Radiography by Selective Detection
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Sabri, Nissia
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: contrast, edge, fourier, modulation, monte, radiography, selective
Nuclear and Radiological Engineering -- Dissertations, Academic -- UF
Genre: Nuclear Engineering Sciences thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The Modulation Transfer Function (MTF) is a quantitative function based on frequency resolution that characterizes imaging system performance. In this study, a new MTF methodology is investigated for application to Radiography by Selective Detection (RSD). RSD is an enhanced, single-side x-ray Compton backscatter imaging (CBI) technique which preferentially detects selected scatter components to enhance image contrast through a set of finned and sleeve collimators. Radiography by selective detection imaging has been successfully applied in many non-destructive evaluation (NDE) applications. RSD imaging systems were designed and built at the University of Florida for use on the external tank of the space shuttle for NDE of the spray-on foam insulation (SOFI) inspection. The x-ray backscatter RSD imaging system has been successfully used for cracks and corrosion spot detection in a variety of materials. The conventional transmission x-ray image quality characterization tools do not apply for RSD because of the different physical process involved. Thus, the main objective of this project is to provide an adapted tool for dynamic range evaluation of RSD system image quality. For this purpose, an analytical model of the RSD imaging system response is developed and supported. Using the Fourier transform and Monte Carlo methods, two approaches are taken for the MTF calculations: one using a line spread function and the other one using a sine function pattern. Calibration and test targets are then designed according to this proposed model. A customized Matlab code using image contrast and digital curve recognition is developed to support the experimental data and provide the Modulation Transfer Functions for RSD.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Nissia Sabri.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Dugan, Edward T.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021299:00001


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8ef01b4d173563bd9b037b5f1c4043dfa6276ce8







AN ADAPTED MODULATION TRANSFER FUNCTION FOR X-RAY BACKSCATTER
RADIOGRAPHY BY SELECTIVE DETECTION




















By

NISSIA SABRI


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2007




































O 2007 Nissia Sabri


































To my mother









ACKNOWLEDGMENTS

I would like to thank Dr. Edward Dugan and Dr. Alan Jacobs for their guidance, constant

enthusiasm and help. I would like also to thank Dr. James Baciack for being on the committee.

I would like to give a special thanks to my family and friends who were a great source of

motivation. I need to especially thank my husband Julien, for his help support, and endless

patience; my sister and mother, for their constant support; and my friends, especially Benoit

Dionne, Anne Charmeau and Colleen Politt, for their encouragement.

I would like to thank Warren Ussery for the financial funding and my research group,

especially Daniel Shedlock for the invaluable learning experience. Thanks to Ines Aviles-

Spadoni for her help.

I would like to thank Dr. Sj oden for accepting me in his research group to pursue my Ph.D.

Finally, I would like to thank Lockheed Martin Space Systems Co, NASA, Langley Research

Center, NASA, Marshall Space Flight Center and The University of Florida, Department of

Nuclear and Radiological Engineering, for the financial support.











TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ............_...... .__ ...............7....

LIST OF FIGURES .............. ...............8.....


AB S TRAC T ........._. ............ ..............._ 12...

CHAPTER


1 INTRODUCTION ................. ...............14.......... ......


Compton Backscattering Imaging (CBI) ........._..._.. ........._.._ ...............14....
Backscatter Radiography by Selective Detection (RSD) .............. ...............16....
Overview of Previous Work ....._._................. ...............16. ....
Proj ect Obj ectives ................. ...............17......... .....
RSD Scanning System............_..._ .. ............. ........_._ .........1
Moving Table: X-Ray Source and Detectors .............. ...............17....
Image Acquisition :Signal Flow and Software ........._._.._......_.. .............. .....1

2 PROBLEM STATEMENT............... ...............2


General Physics of Photon Interaction ...._.._ ................ ........._.._ ....... 2
Compton Effect .............. ...............25....
Kinematics ........._..... ...._... ...............26.....
Cross Section ........._...... .........._... .. ... .... .._. .. ..... ..........2
Theoretical Approach of the Modulation Transfer Function (MTF) ................. ................ .27
The Fourier Transform Applied to Image Processing ................. ............... ......... ...30
MTF Applied to the RSD Scanning System ......... ......._.._.._ ......... ...........3

3 PRELIMINARY EXPERIMENTS: PULSE AND STEP FUNCTIONS SIMULATION.....36


RSD System Experimental Responses .............. ...............36....
Pulse Input Experiment .............. ...............36....
Step Function Experiment ........._...... ... .. .. ... .. .........._.._........ ..........3
Principles of Statistics and Curve Fitting Applied to MTF Calculation............... ...............3
Results and Analysis............... ...............3
Pulse Function Experiment. ........._.._.. ...._... ...............39...
The Step Function Experiment ........._._.._......_.. ...............41....

4 MTF CALCULATION BASED ON A SINUSOIDAL INPUT FUNCTION ................... ....44


MTF Sinusoidal Pattern Design............... .... .. .............4
System Response to the Input Modulation Function............... ...............44











Digital Output Profile ................... .............. ...... .. .......... .. .............4
Comparison of Detection Properties Between Nal and YSO Crystals ................... .........45
A Model of the Sinusoidal Input Function Using MCNP5 and Variance Reduction
Techniques .............. ............. .. .... .... ................4
Input Function from a 2D Model of the MTF Sine Target ................. ............. .......46
Input Function from a 3D Model of the MTF Sine Target ................. ............. .......50
Volumetric Normalization of the MTF .......................__ ...............51.....
Geometric Normalization ......................... ........... .........5
Volume Calculation Based on an MCNP5 Model .............. ...............54....
Actual MTF Curves Based on a Sine Input Pattern ................. ...............56........... .

5 AN IMPROVED TECHNIQUE FOR THE MTF CALCULATION BASED ON A
STEP FUNCTION............... ...............71


Step Function Target Design for MTF Calculation............... .....................7
A Model of the Step Function Target Using MCNP5 and Variance Reduction
Techniques. ............. ...............72.....

6 PROPOSED TECHNIQUES FOR IMAGE QUALITY AS SESMENT ............... .... ........._..79

Multiple Derivatives and Inflection Points as a Mathematical Criterion for Image
Quality A ssessm ent............... ...... .... .. ......................7
Correlation Between the Different Methods of Calculating the MTF ............... ................82
Resolution Assessment from a Step Function Input ................. ...............82........... ..

7 COMPUTATIONAL PROCESS SING WITH MATLAB. ALGORITHM
ARCHITECTURE FOR MTF CALCULATION (MATLAB) ................. ......................95

Modulation Transfer Function Based on the Sine Target ................. ......... ................95
Modulation Transfer Function Based on a Step Function Target ................. ............... ....96

8 CONCLUSION............... ...............10

APPENDIX

A ENERGY FILTERING USING PAPER ................. ...............105........... ...

B MTF FRAME STRUCTURE ................. ...............107...............

LIST OF REFERENCES ................. ...............109................

BIOGRAPHICAL SKETCH ................. ...............111......... ......











LIST OF TABLES


Table page

4-1 Number of counts at the detector surface. ............. ...............60.....

4-2 Comparison between the Analog and Non-Analog MCNP5 ................ ............. .......61

4-3 Summary of the line diameters and the associated number of line position. ................... ..62

4-4 MCNP5 run condition for Analog versus Non-Analog .......... ................ ...............62

4-5 Comparison between Analog and Non-Analog results in MCNP5 .............. ..................64

6-1 Coefficients used in the fitting function formula for each MTF curve ............................88

6-2 Statistical measures of the fitting accuracy ....__ ......_____ .......___ ..........8

6-3 Roots value of the MTF second derivatives curves ....._._._ .......__. ......._........89

6-4 Different methods of the MTF derivation. ...._. ......_._._ .......__. ..........9










LIST OF FIGURES


FiMr page

1-1 Schematic illustrating X-ray production ................. ...............19........... ...

1-2 Typical spectrum obtained from an X-ray tube with a tungsten anode4 ................... .........19

1-3 Compton Backscattering Imaging (CBI) .............. ...............20....

1-4 Lateral Migration Radiography (LMR) .............. ...............20....

1-5 Photograph of RSD System with 4 Nal Detectors ................. ...............21........... .

1-6 Photograph of RSD System showing YSO detectors mounted to Nal Detectors ..............21

1-7 RSD scanning system mounted on a Eixed frame .............. ...............22....

1-8 Flow chart of the image acquisition process20 ................ ...............23..............

2-1 Photoelectric, Compton and Pair Production ............ ...............34.....

2-2 Kinematics of the Compton Effect ................. ......... ...............34. ..

2-3 Transmission model .............. ...............35....

2-4 Backscatter model ................. ...............35........... ....

3-1 Scanning system output two line pairs placed at 450with respect to the vertical axis.......42

3-2 High exposure scanning output, one sweep of a nylon line (Dirac Simulation)................43

3-3 Scan of a cubic plastic sample: 17.5 mm width, 1 mm beam, 0.5 mm pixels ................43

4-1 Scheme for simulating a sinusoidal input ....__ ......_____ .......___ ..........5

4-2 M TF frame plate .............. ...............58....

4-3 MTF frame plate detailed design ...........__......___ ...............58..

4-4 Output profie from the scan of the MTF Sine target (detector 1 Nal) ................... ...........59

4-5 Scattering-to-absorption ratios for Nal and YSSi20 crystals. ............. .....................5

4-6 MCNP5 model for input profie calculation ................ ...............60........... ..

4-7 Energy spectrum distribution used in the MCNP5 model based on Kramers spectrum....60

4-8 The input sine profie obtained from running MCNP5 ................. .......... ...............62











4-9 Sine profie obtained from modeling 10 nylon lines of different diameters in MCNP5 ...63

4-10 The complete input profie from an MCNP5 simulation as recorded at the detector........63

4-11 MCNP5 model for input profie calculation ................ ...............64........... ..

4-12 Average energy and fraction of the detected signal in each of the six collision bins........65

4-13 Intersection volume of two cylinders ................. ...............65......_.__...

4-14 Two cylinder intersection volume .............. ...............65....

4-15 Integrated profile data ................. ...............66................

4-16 Equivalence between peaks and steps profiles. ............. ...............66.....

4-17 Normalization methodology scheme. ............. ...............66.....

4-18 Experimental and normalized data profile............... ...............67

4-19 A representation of the MCNP5 setup for volume intersection calculations. ................... .67

4-20 Line and beam intersection volume values ................. ...............68........... ..

4-21 A plot of the volumetric normalization of half peaks obtained from MCNP model. ........68

4-22 Visual editor view of the new MCNP setup for volume calculations. .............. ..... ..........69

4-23 Normalization of the MTF sine profile over the intersection volume .............. .... ........._..69

4-24 Statistical smoothing of the normalized profile ....__ ......_____ ...... .....__........7

4-25 MTF function from detector 5 .............. ...............70....

5-1 Edge target made from a junction of lead (absorber) and nylon scattererr) ......................74

5-2 Scanning system response to an edge. ............. ...............74.....

5-3 Fourier transform of the line spread function (black curve) and fitting function (red) .....75

5-4 Geometry of the MTF step target in MCNP5 ................ ................. ...............75

5-5 Data profile obtained from the first MTF step target design in MCNP5 ................... ........76

5-6 Geometry of the second design of the MTF step target ................. ................ ...._..76

5-7 Profile data obtained from the second design of the MTF step target ............... .... ...........77

5-8 Data profile obtained from the third target design; nylon block on top of lead.................77











5-9 Final design profie proposed for the MTF step target ......___ ........__. ..............78

6-1 MTF comparison between Nal and YSSi20 detectors at 45 kVp, 0.5 mm aperture ..........86

6-2 MTF comparison for 3 different aperture diameters............... ...............8

6-3 MTF comparison for different pixel sizes and beam apertures at 45 kVp-45 mA ........_...87

6-4 MTF Boltzmann model fitting function comparison ...._._._.. ..... ..__... ........_._......87

6-5 MTF fitting function first derivative, scan at 45 kVp-45 mA............... ...................8

6-6 MTF fitting function second derivative, scan at 45 kVp-45mA............... ................8

6-7 1 MTF 0.1 mm pixel, 0.5 mm aperture............... ...............89

6-8 2 MTF 0.05 mm pixel, 0.5 mm aperture............... ...............90

6-9 3 MTF 0.1 mm pixel, 1.0 mm aperture............... ...............90

6-10 4 MTF 0.05 mm pixel, 1.0 mm aperture............... ...............90

6-11 5 MTF 0.05 mm pixel, 1.5 mm aperture............... ...............91

6-12 YSO image of MTF Target on a tile panel ......___. ..... ... ...............91

6-13 Selection and smoothing steps for the MTF calculation from a step function ................92

6-14 An example of the edge profile and its first derivative ......... ................. ...............93

6-15 Edge function width estimation .............. ...............93....

6-16 Numerical evaluation of the first derivative of the edge function ................ ................. 94

7-1 Matlab user interface............... ...............9

7-2 MTF menu and data profile .............. ...............98....

7-3 Data profile .............. ...............98....

7-4 User interface for information entries............... ...............99

7-5 Maximum search............... ...............99.


7-6 Saving files............... ...............100.

7-7 Data profile from an edge function and its first derivative ................. ......................100

7-8 Selection of a region of interest in the edge function profile. ................ .............. .....101











7-9 The selected region of interest and the first derivative of the edge function ................... 101

7-10 MTF curves with frequencies expressed in line pairs/pixel and line pairs/mm ...............102

A-1 Comparison between two backscatter images .............. ...............105....

A-2 Line profile evaluation of the paper filtering ................. ...............106......._._..

B-1 MTF frame plate top view ................. ...............107..............

B-2 MTF cover plate top view............... ...............108.









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

AN ADAPTED MODULATION TRANSFER FUNCTION FOR X-RAY BACKSCATTER
RADIOGRAPHY BY SELECTIVE DETECTION

By

Nissia Sabri

August 2007

Chair: Edward T. Dugan
Major: Nuclear Engineering Sciences

The Modulation Transfer Function (MTF) is a quantitative function based on frequency

resolution that characterizes imaging system performance. In this study, a new MTF

methodology is investigated for application to Radiography by Selective Detection (RSD). RSD

is an enhanced, single-side x-ray Compton backscatter imaging (CBI) technique which

preferentially detects selected scatter components to enhance image contrast through a set of

finned and sleeve collimators. Radiography by selective detection imaging has been successfully

applied in many non-destructive evaluation (NDE) applications. RSD imaging systems were

designed and built at the University of Florida for use on the external tank of the space shuttle for

NDE of the spray-on foam insulation (SOFI) inspection. The x-ray backscatter RSD imaging

system has been successfully used for cracks and corrosion spot detection in a variety of

materials.

The conventional transmission x-ray image quality characterization tools do not apply for

RSD because of the different physical process involved. Thus, the main obj ective of this proj ect

is to provide an adapted tool for dynamic range evaluation of RSD system image quality. For this

purpose, an analytical model of the RSD imaging system response is developed and supported.










Using the Fourier transform and Monte Carlo methods, two approaches are taken for the

MTF calculations: one using a line spread function and the other one using a sine function

pattern. Calibration and test targets are then designed according to this proposed model. A

customized Matlab code using image contrast and digital curve recognition is developed to

support the experimental data and provide the Modulation Transfer Functions for RSD.









CHAPTER 1
INTTRODUCTION

The purpose of this investigation is to present and explain the different approaches that

have been taken to develop a Modulation Transfer Function adapted to the Radiography by

Selective Detection RSD imaging systeml-3 for the purpose of defining a process to measure

system response by evaluating the image quality.

The first obj ective of the MTF calculations was to give a complete specification of the

RSD scanning system properties. Therefore a frequency characterization of the output/input

linking was desired. However, the backscattered Hield is highly dependent on the scanned obj ect

meaning that a complete description of the imaging process for all applications is not possible

with a unique transfer function.

After an overview of the physical process involved in this type of imaging, the

experimental results are presented. The maj or sections treated are: the preliminary impulse and

step functions responses, the design of an MTF plate to simulate a sinusoidal input function, the

use of MCNP5 and variance reduction techniques to model the input function, the fitting process

to associate mathematical functions to the experimental data, two proposed models for the MTF

measurements (the sinusoidal and the step functions) and finally, the Matlab codes for practical

calculations.

Compton Backscattering Imaging (CBI)

In this section X-ray production is described for imaging applications. The physics of the

photon interactions with matter is treated in detail in Chapter 2 For a standard transmission

process, X-ray images are maps of the x-ray attenuation coefficient. To a large extent the

attenuation depends on the chemical composition and physical state of the attenuating medium.

In Compton Backscattering Imaging (CBI), images are maps of X-ray photon backscattering4










X-rays are produced by focusing a beam of high energy electrons into a small focal spot on

an anode.

The rapid deceleration of the electrons after they enter the metal of the anode produces a

broad continuous spectrum of X- rays called Bremsstrahlung. Figure 1-1 shows the basic

principle of X-ray production.

There is also a probability for electrons to ionize the atoms in the anode, creating vacancies

in the inner electrons shells. These vacancies are rapidly filled by transitions from outer electron

shells, with the emission of characteristic X-rays .

The energies of these discrete line spectra are characteristic of the anode chemical element.

The total spectrum obtained from a typical X-ray tube with a tungsten anode is shown in Figure

1-2.

As the X-rays traverse the obj ect being scanned, they may be scattered, either elastically or

inelastically, or they may be totally absorbed in a photoionzation process. More details on these

physical processes and their dependence on photon energy can be found in Chapter 2.

A transmission imaging system consists of an X-ray source, the obj ect being radiographed,

and a detector.

From an imaging standpoint there is an important distinction between absorption and

scattering. Usual X-ray scanning systems use transmission (i.e., forward scattered) photons

while CBI uses backscattered photons. The reason for employing a CBI system is simple; for

some applications it is impossible to have film or a detector behind the scanned obj ect.

By illuminating a single point on the target and having a set of detectors collecting the

backscattered photons, it is possible to reconstruct the image with a spatial mapping. The image

is thus a two-dimensional proj section of a three-dimensional obj ect; many planes are collapsed









into one. The information is not given by photons which pass throw the sample like in

transmission radiography, but is given by photons which are scattered back on the same side as

the source.

The detector senses photons coming back from the sample. These photons have interacted

with the medium (Compton interaction) and are scattered back with a different energy. The

energies and angles of backscattered photons depend on the energy of the incident photons and

the medium with which they interact. By counting the number of photons coming back,

information about the target can be deduced.

Backscatter Radiography by Selective Detection (RSD)

Overview of Previous Work

The technique developed at the Nuclear Engineering Department at the University of

Florida, called Lateral Migration Radiography6-14 (Figure 1-4) is similar to the CBI technique

(Figure 1-3), but instead of counting only single-collision backscattered photons, the LMR

technique counts both single- and multiple-collision backscattered photons that have laterally

spread out from the illumination beam entry point.

At the detector surface, signals from single- and multiple-collision backscattered photons

overlap. Therefore, they cannot be expected to cast a sharp shadow image. Instead, the

backscattered radiations form a broad, diffuse distribution on the detector, severely impairing the

distinction between deep and shallow obj ects.

This technique, with some modifications, later led to the Backscatter Radiography by

Selective Detection RSD. By adding adjustable collimators to the detectors it was possible to

select the backscattered photons being counted, especially the depth of the counted photons. By

preferentially selecting specific components of a scattered photon field, information relating to

specific locations and properties of an imaged sample can be extracted.










Project Objectives

The components that form the RSD scanning system are different and complex. Four

maj or parts can be identified: X-ray generator, detectors, the electronics and the image

acquisition and processing.

The obj ective of this study is to characterize the system response depending on different

setups and components. Since the development of the first RSD scanning system, there has not

been an experimental methodology to measure system performance. The global response of the

system depends on the individual performance of each component. The purpose of this proj ect is

to define a process to measure the system response by evaluating the image quality. Since the

image is the system output, it gives an indication on how all the components are performing

together.

From a physical system point of view, the characterization of the response must be defined

through the input/output relationship. Then the challenge is to develop an expression for this

relationship which provides a basis for evaluating the performance of the imaging device and

understanding the nature of its evaluated image properties.

From the image processing standpoint, contrast and resolution characterize the image

quality. Therefore, the calculation of the Modulation Transfer Function (MTF) would be a better

characterization parameter if it is related to the contrast and resolution.

RSD Scanning System

Detector response and image acquisition observed throughout this study are generated

using the RSD scanning system developed for Lockheed.

Moving Table: X-Ray Source and Detectors

The system used in this study consists of four sodium iodide [Nal (TI)] scintillation

detectors, one YSO detector and a Yxlon MCG41 X-ray generator mounted onto a scanning









table with X Y scan motion capabilities. The [Nal (TI)] detectors are positioned at the corners

of an eighteen by eighteen centimeter square, centred on the X-ray beam. The YSO detector

orbits on an aluminium ring around Nal detector two.

YSO images are usually comparable to the Nal images in image contrast. Although the

YSO detector has much less detection surface area (5.06 cm2 VS. 20.3 cm2), it has a slightly

higher quantum efficiency compared to the Nal for low energy X-rays (10-55keV). The detector

is also much lighter and smaller than the Nal detector so it can easily be positioned to obtain

better images. Each [Nal (TI)] detector comprises a two inch diameter by two inch thick Nal

scintillation crystal mounted onto a photomultiplier tube (PMT) and a fast preamplifier

specifically designed to handle high count rates.

A schematic of the RSD [Nal (TI)] detectors components and their configurations is

presented below in Figure 1-5. In Figure 1-6, the YSO is mounted on detector 2 using an

aluminium ring. In Figure 1-7 the RSD system is mounted on a fixed frame.

The 230 ns constant decay time of the Nal(T 1) crystal (230ns) allows sufficient light and

charge collection time from the Nal and PMT, while allowing the detectors to measure

backscatter fields up to 800,000 counts per second, without experiencing statistically significant

pu se pile-upl

Image Acquisition :Signal Flow and Software

The signal recorded from the scanning system is processed and displayed through a

Labview code.

The following flow chart (Figure 1-8) presents the entire image acquisition process from

detection to display.




















X-ray
tube


Figure 1-1. Schematic illustrating X-ray production


350

300

250



SO


O 20 40 60 80 100 (20
energy (keV)

Figure 1-2. Typical spectrum obtained from an X-ray tube with a tungsten anode4








19


Electron
gun


"--























































Figure 1-4. Lateral Migration Radiography (LMR)


Noise


Object


Figure 1-3. Compton Backscattering Imaging (CBI)


Uncollimated
detector


X-ray
generator


I


Collimated
detector


Land mine













Nal
detector ~


Sleeve
collimator
extended


X-ray
beam
tu be


Finned
Collimator
Angle at 90
(degrees)


I \

1


Figure 1-5. Photograph of RSD System with 4 Nal Detectors


Aluminium
ring





YSO
detector


Nal
....etector






A set of
YSO
detectors


Figure 1-6. Photograph of RSD System showing YSO detectors mounted to Nal Detectors




























Figure 1-7. RSD scanning system mounted on a fixed frame



















Image


Complete Pulse train

Labview/computer


X-axis


Y-axis


Compleia Pulse train


NI-Motion
PCI 7344


X-axis


Y-axis


NI-Daq
PCl6602


NI-Motion
breakout box


Step Cr

Complete Pulse train



Pulse train Step
BNC 2121or


A ctive

Limit/Home
Switches


Yes

Digital pulse


SCA is the
pulse in the
voltnnp wNindow N


.X-Motor
Dir
X-axis
X-Motor Amps


Y axis


Y-Motor






Figure 1-8. Flow chart of the image acquisition process20


Y-Motor Amns









CHAPTER 2
PROBLEM STATEMENT

General Physics of Photon Interaction

When considering an X-ray based scanning system, it is highly important to understand

how the photons interact with matter. There are five types of interactions with matter by X-ray

photons which must be taken into account.

* Compton effect
* Photoelectric effect
* Pair production
* Rayleigh (coherent) scattering
* Photonuclear interactions

Since the importance of an interaction for the purpose of this study is being measured by

the energy released in the medium, the three first interactions are the most important. The photon

energy is transferred to electrons, which then impart that energy to matter in many Coulomb-

force interactions along their tracks. Rayleigh scattering is elastic (total energy conserved, and

kinetic energy conserved), meaning that the photon is merely redirected within a small solid

angle with nearly no energy loss. Photonuclear interactions are only significant for photon

energies above a few Mev, where they may create radiation-protection problems through the

(y,n) production of neutrons and consequent radioactivation.

The relative importance of the Compton Effect, photoelectric effect, and pair production

depends on both the photon quantum energy ( Er = hu ) and the atomic number Z of the

absorbing medium.

Figure 2-1 indicates the regions of Z and Er in which each interaction predominates.

The photoelectric effect is dominant at the lower photon energies, the Compton effect takes

over at medium energies, and pair production dominates at the higher energies (with a threshold

of at least 1.02 Mev because the photon energy must exceed twice the rest mass of an electron).










For low-Z (e.g., carbon, air, aluminum, Spray-on Foam Insulation) media the region of

Compton-effect dominance is very broad, extending from approximately 20 keV to 20 Mev. This

gradually narrows with increasing Z. However, for Al, the PE effect is dominant up to about 50

keV.

According to the previous description it is easily understandable why the Compton Effect

is the one that characterizes the photon interactions in an RSD scanning system. The following

description deals with some aspects of the Compton Effect that are essential to understanding

how the image is formed in the RSD scanning system.

Compton Effect

A complete description of the Compton Effect must cover two maj or aspects: kinematics

and cross sections. The first one relates to the energies and angles of the participating particles

when a Compton event occurs; the second predicts the probability that a Compton interaction

will occur.

Two maj or assumptions are made in the following theoretical approach: the electron struck

by the incoming photon is initially unbound and stationary. These assumptions are not rigorous

since the electrons occupy different energy levels and, thus, are in motion and bound to the

nucleus. However, for low Z materials the binding effect does not introduce that much

modification in the cross section value.

As presented in Figure 2-2, a photon of quantum energy E incident from the left strikes an

unbound stationary electron, scattering it at angle 6 relative to the incident photon's direction,

with kinetic energy T.

The scattered photon E' departs at angle cp on the opposite side of the electron direction, in

the same scattering plane. Energy and momentum are each conserved. The assumption of an









unbound electron means that the above kinematics relationships are independent of the atomic

number of the medium.

Kinematics

The relationships between angles and energies are given in Equation 2-1



Shu =T u uuh
hv


cos(0)= (1+ )tan( )
m,c 2

(2-1)

Wherem,c the rest energy of the electron, is 0.5 11 Mev, and hu, hv' and T are


expressed in Mev. There is a one-to-one relation between hv and angle cp of the scattered photon

for a given energy of the incident photon.

The photon transfers a portion of its energy to the electron. All scattering angles 6 for the

photon (between 0 to 1800) are possible and the energy transferred can vary from zero to a large

fraction of the photon energy.

Cross Section

The microscopic cross section is the effective target area presented to an incident photon.

The earliest theoretical description of the process was provided by J.J. Thomson. In this theory

the electron that scatters the incident photon is assumed to be free to oscillate under the influence

of the electric vector.

The Thomson differential cross section per electron for a photon scattered at angle 9, per

unit solid angle is based upon classical mechanics/electrodynamics and is expressed as:










=~o o (1 + cos2 23
daY 2
(2-2)

Later on, Klein-Nishina developed (based upon quantum mechanics) a new definition for

the Compton Effect cross section". This treatment was more successful in predicting the correct

experimental value, even though the electron was still assumed unbound and initially at rest.

The Klein-Nishina differential cross section for photon scattering at angle cp, per unit solid

angle and per electron may be written in the form

d~o, 02 hv hv hv
= -( )2 ( + -S1H2V
daZ 2 hv hv hv
(2-3)

Equation 2-3 is the one usually used for standard calculation of the cross sections, r02 is

squared value of the classical electron radius. In the low-energy limit of Compton scatter (ho less

than about 10 keV), ho' = hu regardless of the photon scatter angle and Equation 2-3 reduces to

Equation 2-2.

Theoretical Approach of the Modulation Transfer Function (MTF)

There are several ways to measure the MTF. Some of them are largely applicable to

different recording systems; either the image is recorded on a film or it is processed to be

displayed on a screen. The two maj or techniques are the Sine Wave Method and the Spread

Function Methodl6

The main problem associated with the first method lies in the production of a spatially-

sinusoidal exposure of known modulation.

A relatively straight forward method is to photograph a variable area test chart for an input

exposure that is a one-dimensional sinusoidal distribution defined by:









f (x) = a + b cos(2xm~i x + e) where 0i is the one-dimensional spatial frequency (or line

frequency), and E is a measure of the phase.

The output is also sinusoidal with the same spatial frequency as the input, but with a

change of amplitude, or modulation. The ratio of the output modulation to the input modulation

depends on the spatial frequency, and turns out to be equal to the modulus of the Fourier

transform of the line spread function.

The modulus of the Fourier transform of the line spread function 1(x) is defined by:






(2-4)

Note that the line spread function of an imaging system is defined as the response of the

system to a line input. A line input may be represented by a single delta function, 3(x, ), which

lies along the yl axis. It is the ratio of output to input modulation that is called the Modulation

fm, fm b
Transfer Function, or MTF. The input modulation is defined by: M~in = a
Imax + fmm a

Since the system response is a convolution of the input and the point spread function of the

system, the output can be written as:


g(x) = f(x-,y, hx,,y y)h6x,d,y)dy


= (~a +bcos(2zm (x x)+ e))h(x ,,y )&, dy,
(2-5)

Integration with respect to yl using (2.4) gives:


g(x) = (~a +bcos(2?izxm +))(x ,) + ))1(,)A
(2-6)








where 1(x,) is the line spread function defined earlier. Using the expansion:

cos(A ) = cos(A) cos(B) + sin(A) sin(B) (2-7)


1(x,) is normalized such that its area is unity, i~e. 1l(x,) dr, = 1, then


g(x) = a+ bcos(2x x + E e)1(x,) cos(2xm x,)dx ,
(2-8)
+ bsin(2xm x+ e) i(x,) sin(2xm x,)&,




g(x) = a + b cos(2xm~i x + E) C(mi) + b sin(2xm~i x + E) S(mi) (2-9)

where


C(m)-i S(m)= T(w)= 1l(x, )exp(-2Himx,)&i, (2-10)

The function T(mi) is the optical transfer function, and C(mi) and S(mi) are its real and

imaginary parts. The optical transfer function is the Fourier transform of the line spread function.

Defining M~(mi) and ~(mi) as the modulus and phase of the optical transfer function, they

can be expressed as:


r~~=r lsr. =- c(m)> (2-11)
c(m), = M~(mi)cos #(mi) andSm)=- m)sn()

And by using these, then Equation 2-9 reduces to:

g(x) = a + M(mi)b cos(2xm~i x + e + #(0i)) (2-12)

Equation 2-12 shows that the output is sinusoidal and has the same frequency as the input.

The output modulation is defined as:










Mowr ax mi=M(wm) (2-13)
Emax + min a

Thus, the ratio of the output modulation to the input modulation is simply equal toM~(m) ,

the modulus of the Fourier Transform of the line spread function.

Since the area under the spread function has been defined as unity, the MTF will be

normalized to unity at zero spatial frequency:


M~(0) = 1(x,), = 1 (2-13)



Given a sinusoidal input of constant modulation -, the system frequency response can be


deduced from the output image contrast Kmax Km'" after dividing byb
Emax gKmn a

Due to the general non-linearity of the scanning process and the uncertainty in

characterizing the input function, the MTF deduced from spread function measurements will not

generally be exactly the same as that obtained from the sine-wave method.

The line spread function method could be performed either by simulating an experimental

pulse with a "Dirac function" or by scanning an edge and differentiating. The last step then is

performing a Fourier Transform calculation.

The Fourier Transform Applied to Image Processing

The general definition of the Fourier Transform of a function f(t) in one dimension is


G;(v) = F, (f (t)) = ~exp(-27r i vt) f (t)dt (2-14)









Two conditions are assumed to be satisfied for f(t) : continuity and periodicity The

extension of this definition to two or three dimensions is straightforward with the spatial

exponential function written as exp(-2 xi i(pu x + 77 y + 5 z) ).

The real utility of the Fourier Transform is that it has a simple inverse.


.f (t) =F (G;(v)) = lexp(+2Ri vt)G (v)dv (2-15)


For a linear system a Fourier Transform of the input is defined as follows


W,,(k) = exp(-2x i k u)w2~(,)d,(ud (2-16)


With the linearity condition, the system output is a superposition of individual outputs.


wa,,(t) = p(t) g0 w,,, (t) = p(t )we,, (t )dt~ .This type of integral is known as a convolution


product where p(t) is the spatial system response function.

The main utility of the Fourier Transform is to give an equivalent expression of the

function in frequency space.

In frequency space the convolution product is equivalent to the usual multiplication. Thus,

in frequency space the output is the multiplication of the input function by the system response

function.

The last important property of the convolution product is that the unit function is Dirac' s

function. Thus, the response to an impulse input is the system response function.

MTF Applied to the RSD Scanning System

The Modulation Transfer Function from a scanning system characterization standpoint -

is the spatial frequency response of an imaging system or a component defined by the contrast,

C, at a given spatial frequency relative to low frequencies.










Spatial frequency is typically measured in cycles or line pairs per millimeter. High spatial

frequencies correspond to fine image details. The more extended the response, the finer the

detail.

Two methods were used to perform the MTF calculation. The first one is based on the

response to a sinusoidal input illumination. The second one uses the magnitude of the Fourier

Transform of the point or line spread function which is the response of an imaging system to a

pulse input such as a point or a line.

Due to technical issues the experiments were performed using sine patterns of various

frequencies and various diameters. A more adapted pattern would have been achieved by

keeping the diameters constant to have a constant modulation. However, the drilling process is

technically difficult for holes of large diameters and small separation. The patterns were

produced using nylon lines (cylindrical shape) of different diameters and spacing.

The following definitions were used

Contrast (f )
M~TF(f ) =100% (2-17)
Contrast (0)


where C(f')= x n is the contrast at the spatial frequency f and C(0)= K is
Vmx+ V~ ,+V

the low frequency contrast (the largest line pair). The above contrast values are the immediate

applications of the theory detailed previously.

V,, V,, Vmn, Vmax represent the luminescence for a pattern at the associated frequency.

V,, V, are maximum (white) and minimum (black) luminescences, respectively, at zero

frequency.

Vmin, Vmax are maximum and minimum luminescences, respectively, at any frequency f.









It is important to notice that in the case of X-ray backscattering, an MTF calculation based

on the output image contrast depends on the spectrum, the target material and geometrical set up

of the system if not properly normalized.

In usual transmission imaging the MTF is a proj section on a 2D plane (Figure 2-3.3). The

signal recorded through the target does not interact with the target pattern. The photons counted

are those that have not been absorbed by the pattern. Thus, the actual volume of the target is not

a critical parameter.

When performing X-ray backscatter imaging, the signal measured is formed by the

photons that interacted with the target pattern (Figure 2-4). Thus, the amplitude of the signal

depends on the volume intersection of the pattern and the beam or the reaction rate.

The use of cylindrical lines in the pattern is to minimize the errors when generating a

sinusoidal input. The lines in the pattern are made of nylon, which has the best ratio of scatter-

to-absorption cross section in the energy range of interest: 5.1 at 35 keV and 26 at 60 keV.

The choice of varying the cylinder diameter with the frequencies introduced an additional

challenge when dealing with the volumetric normalization. The intersection volume of two

cylinders at 900 is easily represented by an integral function. However, because the beam sweeps

continuously over the cylindrical line, a summation of integrals is needed. This aspect will be

treated later on.















iaP* od ti"


Momentum_ h


I I rTll 1 r7TTTT .,7..., .111


~111 IlY _


Metric effect


-


01


120

100
s
f 80
o

a 40
N
20

0.


- ~srr~ r r pr IVuc on
dominant dominant





S6 Compton effect
dominant
11 1 1 1 1.1 1III ~l U


0.05 OJ 0.5 I 5 SO
Photon Energy br, in MeV


50 100


Figure 2-1. Photoelectric, Compton and Pair Production'.









Mlomentu m=P


KE =T


E. = hv


SE' = hv


hv

c


Figure 2-2. Kinematics of the Compton Effect










SX-ray
generator


SRectangular
- shape pattern


Figure 2-3. Transmission model


~ X-ray generator


Nylon
lines


Cylindrical
s-hepren


C


o


So .-


Figure 2-4. Backscatter model


1 1
1 1
II'









CHAPTER 3
PRELIMINARY EXPERIMENTS: PULSE AND STEP FUNCTIONS SIMULATION

RSD System Experimental Responses

One of the first obj ectives was to vary one parameter at a time. The spacing was varied

using a limited number of lines due to the lack of precision in the spacing setup in preliminary

experiments. Experimental results presented in Figures 3-1 show a scanning output of two pairs

of nylon lines with the associated Line Spread Function profile. The two sets of line pairs were

of the same diameter 0.3 mm at 45 degrees with respect to the vertical axis with 3 mm and 1 mm

spacing respectively from left to right on the line profile.

The Line Spread Function (Figure 3-1) shows a typical loss of contrast with increasing

spatial frequency of the line pairs. The decrease of the amplitude between maxima and minima is

the indication of the contrast loss. This experiment was only meant to demonstrate the relation

between the frequency increase and the loss of contrast.

Pulse Input Experiment

Relative to the dimensions of the system, a pulse input can be approximated by a single

thin nylon line (0.3 mm diameter) with a 1 mm beam.

Since the system response depends on the intersection volume of the beam and the line, the

use of a small source beam aperture with a thin line simulates a "Finite" Dirac function. Figure

3-2 is a high resolution, single-line scan of a nylon line (0.3 mm diameter) with 0.02 mm pixel

size.

A convolution product shows that in the ideal case, the system output for a Dirac input

gives the Transfer Function.

Output(x) = Input(x) 0 System response(x) (3-1)









Since the Dirac function is the convolution product unit operator, the output is the system

response. By fitting the experimental data, a mathematical expression for the system response to

a line can be derived.

Step Function Experiment

This experiment simulates an edge function. The Fourier transform of the edge function

should give the same Modulation Transfer Function (MTF) as the line spread function. In the

frequency domain the output is defined as follows:

Output( f) = Input( f )* System response( f) (3 -2)

With indicating regular multiplication.

For modeling an edge function the target is a plastic piece of 17.5 mm width as shown at

the bottom part of Figure 3-3.

Principles of Statistics and Curve Fitting Applied to MTF Calculation

Figure 3-2 and Figure 3-3 show experimental data profiles and the fitting functions

associated with them. To be valid the fitting function must be statistically equal to the

experimental profile. Thus, this section covers the basics of statistics applied to data samples and

more precisely applied to fitting functions.

In order to evaluate the fitting efficiency of a given function, some statistical tests are

performed for each data set. One of these tests is the determination of R, the Correlation

Coefficient. The closer the determination coefficient R2 is to 1, the better is the fit. A correlation

measures the strength of the predicted relation between the experimental data and the fitting

function. The stronger the correlation the better the fitted function approaches the experimental

data.









Given n pairs of observations (x,,y y),with x the experimental data and y the fitting

function value, the sample correlation is computed as


R = (3-3)


Where the sums of squared residuals are defined as

Sw = (y, )2 = SS(Total) (3 -4)


The Chi-square test is a different measure of the goodness-of-fit. The X2 test measures

the deviation between the sample and the assumed probability distribution (i.e., hypothesis). The

value of Chi-square is calculated according to the following formula,


x i (N, Np1) )35
Np,

Where py p, pp.. ,, J is set of,,~, hypothetical probabilitie associate wih Nevents


falling into n categories with observed relative frequencies of(N, /N, N2 /N,..., N,, /N). For

large values of N, the random variable X2 approximately follows the X2 -distribution density

function with n-1 degrees of freedom.

The F-test is another statistical tool that can be used, for example, to test if different MTF

curves are statistically equal. Here are some explanations on how the F-test is performed.

First the two data sets (the measured data and the data from the library) are individually

fitted using the fitting function. Then the two data sets are combined (appending one to the

other), and then a fit is performed on the combined data set with the same function. From these

three fits, the values for the SSR (sum of squares of the difference between the data and fit

values) and the DOF (number of degrees of freedom) are obtained.









Then, SSR1, DOFl, SSR2, and DOF2 are obtained from the individual fits, and

SSRcombined and DOFcombined are obtained from the fit of the combined data.

The following values are computed: SSRseparate = SSR1 + SSR2 and DOFseparate =

DOF 1 + DOF2 .

The last step is performed by computing the F value.


F = (SSRcombinzed- SSRseparante) D)OFsepararte (3 -6)
(DOFcombined DOFseparate) SSRseparate

Once the F value is computed, the p-value is computed using the formula:

p = 1 invf (F, (DOFcombined DOFseparate), DOFseparate) (3 -7)

This p-value is then used to make a statistical statement as to whether the data (not the

parameter values) are significantly different or not. If the p-value is greater than 0.05, we can say

that the data sets are not significantly different at the 95% confidence level.

Results and Analysis

Pulse Function Experiment

In order to obtain the MTF from experimental data, it is necessary to obtain a mathematical

function from a data fit. Once the fitting function is obtained, the Fourier Transform of the

profile gives the system response function in the case of a pulse input. To perform the fitting, a

Lorentz's model was used with the following equation:

2*A w
y = yo +-* (3-8)
Fr(4*(x--x,)2 +2)

where

yo = 2135.57969 f 2.60653
R2 = 0.85644
x,=4638 .06 with statistical tests on the data 7
w = 0.37763 f0.0139 = 26666.39739
A = 459.40838 f 12.69682Do









The data profile used in the Pulse function experiment has been obtained from a scan at 45

kVp, 45 mA with a beam aperture size of 0.5 mm and a pixel size 0.02mm x Imm. The line was

0.050 mm width.

Once the mathematical formulation was established, the next step was to calculate the

Fourier Transform of the obtained function (Equation 3-8). Since the exact formula depends on

different constants that change according to the experimental conditions, it is more valuable to

determine the general shape of the Fourier Transform than the precise mathematical expression.

By using Equation 3-9

2a exp(- (~-m)> ) = exp(-a (~) m = 2xi
a 2 + X2 X (3 -9)



f(x xo)> f(m)~ exp(- jei xo)
letting X = 2(x xo ) and using the following formulas FT 1



The Fourier Transform of Equation 3-8 is obtained as

2*A (2)*w, 2*A 4xi 4mc, 2xi
y = yo + -* '~-l *exp(-w )*exp(-j o) +yo*3( )
S* (2) (4*"(x -xc)2 + 2) z*4 x x x


(3-10)

The Fourier Transform modulus gives the Modulation Transfer Function:

2*A 16xi2x 2*A
MTF dirac function= *exp(-w 2) 0 4 Xp(-w *8 *z2) (3-11)
z*4 x2 zi*4


with z = (mm )


The above formula gives the general behavior.









The Step Function Experiment

When the step function is treated, the best fitting function for this shape is provided by the

Bolzmann's model

(A, A2)
y = A2 1 (3-12)
x-x
1+ exp( 0)

Where

A, = 495.42214 f5.35331
R2 = 0.98645
A, = 971.69652 f 2. 17543
2 and 2 for the statistical tests
xo = 3.7751f 0.0304 = 397.87267
Dof
dre = 0.0441f 0.05193

When using a step function to define the MTF an additional step is needed before the

Fourier Transform. A first derivative is performed.

X-XO
dY(X) (A2 -Al)*e dY
dX YXO(3-13)
(1 +e Y ) 2

Due to the complex form of the above function, a straight forward calculation of the

Fourier transform is not possible.

An alternative approach was to perform the derivative and its Fourier Transform

numerically. Then by fitting the function a mathematical formulation was established.

A z-z
MTF_edge~function= yo + exp(-2 ( c 2) (3-14)




with z = (mm )






























































..


yo = -14.65077 f 2.66317
R2 = 0.99794

c- wi78E' 000 7 th statistical test on data ,
w = 0.65586 f 0.00568 = 76.95055

A = 438.57833 & 5.01373Do


The data profile used in the Pulse function experiment has been obtained from a scan at 45


kVp, 45 mA with a beam aperture size of 1 mm and a pixel size 0.5mm x 0.5mm. The line was

0.050 mm width.


Even though the mathematical expressions for the pulse based MTF and the step function


MTF are not exactly the same, the general behavior follows exp(-a z2), With a a constant.


Detector 1















10-


I'j.



''
1


contrast


L~ 1*~directin mm)~ *5 9 r


Figure 3-1. Scanning system output two line pairs placed at 450with respect to the vertical axis








































I(IIJIJIJI(I
0 2 4 6 8 10

Distance x in (mm)


Figure 3-2. High exposure scanning output, one sweep of a nylon line (Dirac Simulation)


LVVV .


Line spread function(1)
- Lorentz fitting function(2)


3000 -



2800 -



S2600 -



a2400 -



a 2200 -
Z


2000 -


-m- B
-Boltzmann fit of Dai -


16000 -


14000 -


12000 -


10000 -


8000 -


6000 -


4000 -


- 4


II~
1 ; I1 i. r.V: ,1
i .r-.!~..~.. .


Line profile


0 5 10 15 20
Distance x in (mm)


25 30 35


Figure 3-3. Scan of a cubic plastic sample: 17.5 mm width, 1 mm beam, 0.5 mm pixels









CHAPTER 4
MTF CALCULATION BASED ON A SINUSOIDAL INPUT FUNCTION

MTF Sinusoidal Pattern Design

The first idea was to generate a sinusoidal input pattern using nylon line of different

diameters and spacing. Figure 4-1, showing fiye nylon lines, an x-ray generator and two

detectors, illustrates the scheme for simulating a sinusoidal input. As the scanning system sweeps

over the lines, a sinusoidal signal is formed at the detector face

The actual MTF target contains 5 lines for each diameter. This is to ensure good statistics

in the results. The actual MTF target consists of an aluminum frame to hold different diameter

nylon lines with varying spatial frequencies. Figures 4-2 and 4-3 show the MTF plate design.

The target frame is 25.4 cm x 12.7 cm (10 x 5 inches) and 0.3 cm (1/8 inch) thick. The

nylon lines are strung across the 7.6 cm (3 inch) air gap in the center of the frame. A cover plate

was designed to be attached to the back of the frame to protect the nylon lines connections and

provide a flat surface on which the target sits. The cover plate is 0.6 cm (1/4 inch) thick.

Twelve sets of holes were initially designed. Two additional levels of holes sets were

included in the design to vary the frequency while the diameters are kept constant.

System Response to the Input Modulation Function

Digital Output Profile

Figure 4-4 shows the output profie obtained from scanning the MTF Sine target at an X-

ray energy of 45 kVp and a current of 45 mA. This profie was obtained from detector 1 (Nal).

For this particular set up, the decrease in contrast started at the sixth set of lines corresponding to

a diameter of 1.28 mm (0.39 line pairs/mm). The loss of contrast is noticeable when there is an

increase in the minimum values of the profie, i.e. a shift in the baseline.









After the eighth set of lines, the five peaks of each new set are not distinguishable. Thus,

the loss of resolution starts at a line diameter of 0.52 mm (0.96 line pairs/mm). The loss of

resolution is defined with respect to the Full Width at Half Max (FWHM). If the separation

between two maxima is smaller than the width of the individual peak at half its maximum value

than the resolution between the two peaks is lost.

Comparison of Detection Properties Between Nal and YSO Crystals

In the previous section, the output profile was treated from a digital imaging point of view

and no special care was taken to evaluate the best detector configurations. However, since the

detectors themselves have limited efficiencies, it is necessary to quantify their responses with

respect to the backscattered spectrum.

Two types of detectors were used in the MTF experiments: Nal and YSO. Figure 4-5

shows the scattering-to-ab sorption ratios for both Nal and YSO. The values obtained are for Nal

and YSSI20 crystalso7

The lower the scattering-to-absorption ratio the better the detection capabilities. In the

energy range of interest (below 50 keV) the YSSi20 crystal has a more favorable scattering-to-

absorption ratio than the Nal from about 16 keV to 33 keV. At about 16.4 keV, the ratio achieves

a maximum value of 0.0796 for the YSSi20. The Nal crystal is a much better detector at energies

higher than 33 keV.

Since the YSSi20 was the most frequently used detector for the MTF experiments, the

following study will concentrate on characterizing the YSSi20 detection performance with

respect to detected energies. First, it is necessary to calculate the average energy of the

backscattered spectrum using a Monte Carlo simulation. The model used is based on MCNP5

analog simulations and the layout is described in detail in the following section.









The average energy of the incident X-ray beam is 22.73 keV and its maximum energy is 50 kVp.

The average energy of the backscattered spectrum given in Table 4-1 is 26.74 keV. This value

was obtained by averaging over the five energy bins with the number of particles used as

weighting functions. A non-analog run gives essentially the same result with an average detected

energy of 26.75 keV and a relative error of 0.021%. A more detailed analysis on the Analog

versus Non-Analog results will be given in the following section.

A Model of the Sinusoidal Input Function Using MCNP5 and Variance Reduction
Techniques

As shown in the previous section, the output profile is easily obtained from scanning the

MTF Sine target. However, there is no experimental way to precisely determine the input profile.

Thus a Monte Carlo model is necessary to correctly determine the input function, to

correlate the output profile to the system response.

Input Function from a 2D Model of the MTF Sine Target

Figure 4-6 shows the MCNP5 model for a 2D input profile calculation. The profile

obtained from the model presented in Figure 4-6 is not strictly 2D. Actually the entire line (3D

volume) is modeled but only the contribution from the mid-plane region is used to generate the

profile. This is to be compared with the profile obtained from the contribution of the entire line.

Only one line per set is modeled up to the 10th set of holes. The last two sets did not give

good experimental results. Then using the problem symmetry only one half of the line is

modeled.

In the actual experimental design, the X-ray generator and the detector move over the

target. For each mesh cell defined by (x+Ax, y+Ay) the number of photons recorded is used to

display one pixel. To simplify the model in MCNP5 the detector and X-ray beam are kept at the

same position while the line position is varied.










The start position is where the beam and the line axis intercept. Then an offset of 0.01 cm

is added between the two axes for each new simulation. The final position of the line axis is such

that it does not intersect with the beam any more.

The detector is a cylinder of 2.54 cm diameter with 0.635 cm thickness centered at (0,

5.08, 4.317).

The plane source is defined at the bottom surface of the detector. Note that it is not

recommended to use a plane that is a physical boundary in a system as a source plane. This can

cause problems. A "source plane" that can be very slightly offset (e.g., by 0.001 cm) from the

physical plane should be used instead. From which the x-ray beam is sampled using a disc of

0.05 cm diameter along the z axis.

The nylon line is centered for the first position at 3.8 cm along the x axis as is the X-ray

beam. The line is represented by a cylinder along the y axis lying on the xy plane.

To model the experimental set up as closely as possible a sheet of paper underneath the

nylon line and a concrete floor are modeled.

There are ten different diameters to simulate. For each diameter the number of line

positions is equal to the ratio of the radius and the modeled pixel size (constant 0.01 cm).

Two Variance Reduction Techniques are used: DXTRAN sphere and forced collisions for

modeling the input profile.

The DXTRAN sphere enables the simulation to obtain many particles in a small region of

interest that would otherwise be difficult to sample. Because the solid angle that sees the detector

surface from the interaction volume in the line is small, a transport of particle to the surface of

interest is necessary.









Upon sampling a collision, DXTRAN estimates the correct weight fraction that should

scatter toward the detector surface, and arrive without collision at the surface of the sphere. The

DXTRAN method then puts this correct weight on the sphere.

The collision event is sampled in the usual manner, except that the particle is killed if it

tries to enter the sphere because all particles entering the sphere have already been accounted for

deterministically. The DXTRAN sphere is centred on the YSO detector.

Forced collisions are used to increase the frequency of random walk collisions within the

small intersection volume of the beam and the entire nylon line.

A particle can be forced to undergo a collision each time it enters a designated cell that is

almost transparent to it. The particle and its weight are appropriately split into two parts, collided

and uncollided. Forced collisions are often used to generate contributions to point detectors, ring

detectors, or DXTRAN spheres.

Here forced collisions are used as a complementary method to the DXTRAN sphere. The

forced collision card is set such that only the particles entering the cell undergo forced collisions.

The run used a 0.5 mm diameter beam, a 0.1 mm pixel and the beam was centered over the

pixel. The number of runs necessary for this input profile calculation is 132.

The energy card uses a distribution of energies with the associated probabilities at 50kVp.

The distribution is based on the Kramers spectrum' modified for tungsten target attenuation and

beryllium window and aluminum filter attenuation.

Figure 4-7 shows the energy distribution used at 50kVp as a maximum energy of the

incident particles in the MCNP5 model based on the Kramers spectrum. The spectrum is

distributed between 0 and 50 kVp with 74 interpolation points.









Two tallies are used; they are based on the current entering the bottom surface of the

detector. The first tally records the partial and total currents and based on the number of particle

collisions from 1 up to 6. The second tally does not distinguish the particles according to the

number of collisions experienced before reaching the detector but it counts particles coming

from a specific cell in the mid-plane of the nylon lines. Table 4-2 summarizes the number of

simulations needed for modeling the input profile, taking into account the number of different

diameters and for each diameter the number of runs.

In addition to the 132 runs necessary for the line profies, there is one simulation for

modeling the air separation between the lines. Figure 4-8 shows the data profie obtained from a

mid-plane contribution only.

The errors associated with the data profie shown in Figure 4-8 are on the order of a tenth

of a percent. Table 4-3 shows a comparison between an Analog MCNP5 run without any

variance reduction technique and a Non-Analog run using the two indicated variance reduction

techniques. The numbers of counts are given for a single source particle and for a positive

current with respect to the detector entrance surface. Table 4-3 shows that up to 40 keV the

errors associated to both Analog and Non-Analog techniques are below 1%. The last energy bin

from 40 to 50 keV corresponds to the incident beam maximum energy; this is why very few

particles are counted. As explained in Chapter 1, the energy of the backscattered particle is a

fraction of the incident energy.

Also according to Figure 4-5 the fraction of scatter/absorption in the YSO detector

increases continuously above 20 keV and reaches a value of 0. 1 between 45 keV and 50 keV.

This means that a fraction of the positive current is scattered back out of the detector and even

less particles are counted in this energy region leading to an increase in the error.









In a Non-analog Monte Carlo method, the physics is biased such that the quantities to be

calculated are estimated in a shorter time or with a smaller variance. To preserve an unbiased

sample mean, each particle is given a statistical weight which is defined based on the unbiased

and biased density functions.

The effectiveness of the Non-Analog techniques is measured by a quantity called "Figure

of Merit", FOM, defined by:


FOM = (4-1)
time(min) error

Where "error" is the relative error. The higher the FOM, the more efficient the calculation.

Table 4-4 presents the number of particles and calculation time for both Analog and Non-

Analog runs. The Non-Analog run is more than 3 times faster and needs less than 16 times the

number of particles to achieve the same order of accuracy on the results.

As discussed previously another aspect of the Non-Analog technique is to introduce a shift

in number of particles with respect to the energy bins. This is mostly due to the DXTRAN

sphere. Some variance reduction techniques do not preserve the energy spectrum information.

Input Function from a 3D Model of the MTF Sine Target

The 3D input profie was obtained using the same layout as the one used in the previous

section for the 2D profie. The only difference is that the entire volume of the nylon line was

sampled instead of sampling only the mid-plane contribution. Figure 4-9 shows the MCNP5

model used for the calculation of the 3D input profie from a nylon line.

The same variance reduction techniques were used and the detector coordinates were (0, 0,

4.3 17). The profie was obtained using 1000000 particles for each of the 132 runs.

Nine of the ten statistical tests were passed in MCNP5. The last test; the pdf slope was not

passed.









However, the relative errors associated with the obtained profie were between0.32%/ and

2.3 5%. Figure 4-10 and Figure 4-11 show the partial and complete profiles obtained from

modeling the MTF sine target using MCNP5.

Figure 4-10 shows the reconstructed input profie with only one line for a given diameter.

Each peak corresponds to one line and was obtained from the MCNP5 simulation. Then knowing

the actual separation distances between the lines, the complete profie has been reconstructed and

is shown in Figure 4-11.

Table 4-5 shows a comparison between the Analog and Non-Analog results for the 3D

model of the input Sine Target.

Figure 4-12 shows the fraction of the contribution of the particles to the detected signal

according to their number of collisions and the average energy of each collision bin. The signal

is dominated by the first scatter signal up to 94.156%. The sixth collisions component is almost

0%. In order for a particle to have undergone multiple collisions and get back to the detector, it

must have come from the higher end of the source spectrum.

Volumetric Normalization of the MTF

The previous section treated the sine function profie at the detector face. Since the MTF

target used nylon lines of different diameters and spacing, the amplitude of the sine profie varies

with the line pair frequency. This variation is due to the variation line diameters and more

specifically, to the variation in the intersection volumes of the X-ray beam with the nylon line.

The volumetric normalization attempts to normalize over the intersection volume to obtain

a profie with constant amplitude. Two methods used are: a geometric normalization based on

integrals and an MCNP5 model to estimate the volume from the particles path.









Geometric Normalization

It is important to notice that the conventional MTF calculation (e.g., as employed with

transmission X-ray imaging) is performed using a multiple step data profile. This model gives a

constant amplitude of the input signal distribution after normalization per unit volume. The

intersection volume of the cylindrical beam and the target (MTF Sine pattern) sample is easily

calculated in this case and remains constant at a given frequency.

In order to introduce equivalence between the step model and the actual Sine MTF, some

definitions are given below:

First, consider the intersection volume of two cylinders of the same radius in Figure 4-13.



One of the cross sections is a square of side half-length ,v the volume is given by



V2 (r.r)= (2 )2 dz r3~ (4-2)


Figure 4-14 shows the intersection volume of two cylinders.



If the two right cylinders are of different radii r,, and reo with rt > Gom then the


volume common to them is :



V2Ln>Ba) Line Ln2Beam2)Ek (te Beam2Kk)(43


Where K(k) is the complete elliptic integral of the first kind, E(k) is the complete elliptic


integral of the second kind, and k = neam is the elliptic modulus.









However, even with a formula to calculate the intersection volume, the complete physical

process is not covered. The beam sweeps over the lines in a continuous mode. For a given beam

size, the actual intersection volume is related to the number of counts through the exposure time

and the pixel size. This means that at each step a fraction of the volume is covered several times.

The resulting overlapping contributes to the signal (counts per peak) in different

proportions depending on the cylinders' radii.

As a preliminary model, only the intersection at the center is considered to give the most

significant response. Although this is a restrictive approach, it gives an idea of the intersection

volume contribution versus the diameter for the large line diameters.

As previously explained, the data profie has to be redistributed for each given diameter.

Thus, using the integral of the data and the line widths as they appear in the image, the number

of counts is redistributed to flatten the maximum of each peak. Figure 4-15 presents the

integrated profie.

The idea is to obtain an equivalent of the step profie from a peak profile as shown in

Figure 4-16.This is to avoid two competing factors of signal amplitude and frequency variations.

The method consists of transforming the peak shape profile to a step shape profile and

normalizing the number of counts per unit volume.

The first step is performed using the integral under each peak shown in Figure 4-15. The

second step requires knowing the value of the intersection volume (X-ray beam and nylon line).

This volume has been calculated using Equation 4-3 assuming an intersection of the X-ray beam

and the line at the center axis only. Figure 4-18 shows the experimental data and the normalized

profile. From right to left each set of lines of a given diameter is shown in a specific color.










Also from right to left the line diameter decreases. At about 2.5 inches the peak data are

not represented because of mismatch between the line diameter and the drilled hole diameter.

This was fixed on the MTF sine plate for later experiments. Note that up to the ninth set of

lines, the normalized profile is decreasing, and the slope is matching the contrast loss. Up to the

ninth line set, the beam diameter is less than the line diameter. In the tenth set the beam and the

lines are of the same diameter. The two last line sets, 11 and 12, have smaller diameters than the

beam diameter.

Figure 4-18 shows that the employed model is well adapted to the first nine line sets. For

line sets 11 and 12, the signals from two different lines overlap. This overlapping gives an

artificially high response. In fact for each line in these sets, the signal includes the response of

several lines. Recall that the equivalence between cylindrical lines and the multiple step target is

performed using the integral and the width. For these sets the normalization is more challenging

since the number of counts recorded and the intersection volume are related in a more complex

manner.

Volume Calculation Based on an MCNP5 Model

In order to perform the volume intersection calculations the input model used to obtain the

input profile is modified. A spherical source is set to enclose the problem with a radius of 12 cm.

Figure 4-19 shows a sketch representing the MCNP5 input model, the two cylinders

intersecting, detector, paper and concrete. However the figure does not show that the source

sphere is centered at the origin.

Note that the input set up described in Figure 4-19 is not the optimum way for doing

volume calculations. However, because of the large number of input files needed, it was a quick

start method since the inputs did not need maj or modifications.










A more efficient configuration would be achieved by suppressing everything in the model

but the two cylinders and centering a much smaller spherical source on them. Then by setting the

material card to void, the volume is obtained by tallying the flux in the intersection region.

The line radii are the ones used in the previous MCNP5 models (for the input profile

calculations). Figure 4-20 shows a plot of the intersection volume values versus the line radii.

The absolute errors are also plotted. The intersection volume increases with the line radius as

expected and the relative errors are higher for small radii. The plot in Figure 4-20 is given as an

indication of the volume trend versus line radius. The values shown are not exact since MCNP5

scales the flux inside the cell of interest to an unknown volume.

Note that for the very small volume intersections there were zero particles in the volume of

interest after running 7,000,000 histories. From these poor statistics, the volume values are

obviously not reliable for small lines radii and small intersection volumes.

Figure 4-21 shows a normalization based on the volume values calculated from the above

model. It is expected to not have constant amplitude since the volume values over which the

normalization is performed are not expected to be correct.

Either a new model is necessary or a higher number of histories. Figure 4-21 is obtained by

taking half of each peak plotted in Figure 4-10 and then normalizing by the intersection volume

from the above results.

Figure 4-22 shows a new input set up that is proposed to enhance the volume calculation.

As previously mentioned, by modeling only the two cylinders and the spherical source, better

statistics are achieved.

This new set up was done using 7,000,000 histories. Figure 4-23 and Figure 4-24 show the

normalization of the input sine profile over the intersection volume of two cylinders.










The two cylinders have the same dimensions as the nylon line and the X-ray beam. The

figures show that a more effective normalization is achieved when it is performed over the

individual pixels.

Note that in Figure 4-23 the extreme values correspond to 10 cr the average value of the

normalized profile for each line. This is due to the small intersection volume on the line's edges.

Thus a statistical smoothing is performed over these values and the resulting normalization is

shown in Figure 4-24.

This last plot shows the feasibility of a volumetric normalization to obtain a profile of

constant amplitude.

Even if the volumetric MTF couples the volumetric distribution of the target to the

scanning system response, it offers a basis for the system relative evaluation.

This integrated 3-D MTF allows comparison between detectors and gives a basis on which

to test a global improvement in the system. By using the same target, the volume and material

parameters are kept constant in the different scans.

Actual MTF Curves Based on a Sine Input Pattern

An example of the MTF curves obtained from the Sine target is given in Figure 4-25, this

profile is obtained from detector 5 (Y5Si20).

The MTF presented here does not include any normalization processing. Thus, the MTF is

sensitive to the change in the volume intersection of the beam and the lines. The drop in the MTF

values is related to a loss of contrast and a volume variation. The relative difference in MTF

values indicates the quality of the images when using the same target.

The Boltzmann fitting model is given by the following formula:

Al A2
MTF_experimental= Y = A2 + x-xo (4-3)
1+e dY









The corresponding coefficients are listed below. An important note is that in this section X

is a frequency since it represents the MTF's variable.

Al= 113.633
R2 = 0.99907
A2 = -2.25743
XO = 0.77575 = 1.02355
dX = 0.32778Do


X-AXIS
Y-AXIS -30


Figure 4-1. Scheme for simulating a sinusoidal input


Sinusoidal
Input

jprofil





'. ** **
------ **


Figure 4-2. MTF frame plate


Figure 4-3. MTF frame plate detailed design



















$4


Deterioration
Ef contrast

2 Loss of
2~ I eSOlution







0 200 400 600 800 1000 1200 1400 1E
Pixel number


Figure 4-4. Output profile from the scan of the MTF Sine target (detector 1 Nal). Scanned at
45kVp, 45mA with a 0.1 mm pixel size and a 1.0 mm source aperture.




0.70

0.00
Se Average
Energy of the
o .so backscattered -a
+ ,field 26.74 keV -S
I10.40
0 22.7 keV
~o Maximum
0.0 average X-ray beam
~ 030 energy of incident
cnthe energy
a .20 icdn




0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Energy (keV)


Figure 4-5. Scattering-to-ab sorption ratios for Nal and YSSi20 crystals.





Table 4-1. Number of counts at the detector surface for each energy bin and the average energy
of the backscattered spectrum.
Energy bins Mev Counts Error %
2.00E-02 1.08466E-02 0.1300%
3.00E-02 8.65223E-03 0.1500%
4.00E-02 3.15683E-03 0.2500%
5.00E-02 1.34348E-04 1.2200%
total 2.27900E-02 0.0900%
Average energy Mev 2.67437E-02 1.2989%


YSO detector,
R=1.27cm


DXTRAN
sphere
Outer radius
4.635 cm
Inner radius
4 cm


X-Ray beam


SPlane source


SPaper


Concrete


Figure 4-6. MCNP5 model for input profile calculation. 2D profile calculated from mid-plane
contribution


Figure 4-7. Energy spectrum distribution used in the MCNP5 model based on Kramers spectrum


z

--










Table 4-2. Comparison between the Analog and Non-Analog MCNP5
Flagged mid-plane surface Error %
Analog Non-Analog Analog vs
positive Energy bins Counts Counts Error% Nn
current Mev Error % Analog
J+ 2.00E-02 4.77980E-03 0.2000% 4.79438E-03 0.3400% -0.3050%
3.00E-02 4.31184E-03 0.2100% 4.34073E-03 0.3600% -0.6700%
4.00E-02 1.60421 E-03 0.3600% 1.60496E-03 0.6700% -0.0468%
5.00E-02 6.81481E-05 1.7500% 6.80647E-05 3.6700% 0.1224%
total 1.07640E-02 0.1300% 1.08081E-02 0.2300% -0.4097%













Table 4-3. Summary of the line diameters and the associated number of line position simulations
Line set number Line Diameter (mm) Pixels needed Number of runs
10 0.5 8.5 9
9 0.52 8.6 9
8 0.75 9.75 10
7 0.85 10.25 11
6 0.95 10.75 11
5 1.28 12.4 13
4 1.4 13 13
3 1.8 15 15
2 2.05 16.25 17
1 3.33 22.65 23


Table 4-4. MCNP5 run condition for Analog versus Non-Analog
Analog Non-Analog
Time ( min) 25.96 7.01
Number of particles 50000000 3100000


0.02



0.015


0.005 1F


1000


15001111 2000 111 111i11


Figure 4-8. The input sine profile obtained from running MCNP5





O
U
e
m
~ 2.00E-02
c

3
o
a
L
c
~ 1.50E-02
rr
d


rr
c 1.00E-02
o
o


B
5.00E-03
n
E
3
z

0.00E+00


3

Distance in cm


Figure 4-9. Sine profile obtained from modeling 10 nylon lines of different diameters in MCNP5







2 50E-02 ,


2 00E-02







1 50E-02







1 00E-02







5 00E-03


nnncnn


1190

Pixel number


1690


2190


2690


Figure 4-10. The complete input profile from an MCNP5 simulation as recorded at the detector

surface, pixel size 0.1mm.












z


Table 4-5. Comparison between Analog and Non-Analog results in MCNP5
Response from the entire line volume Error %
Analog Non-Analog Analog vs
positive Energy bins Mev Counts Error % Counts Error% Nn
current Aao
J+ 2.00E-02 1.08466E-02 0.1300% 1.08436E-02 0.2300% 0.0277%
3.00E-02 8.65223E-03 0.1500% 8.71375E-03 0.2300% -0.7110%
4.00E-02 3.15683E-03 0.2500% 3.14837E-03 0.4000% 0.2680%
5.00E-02 1.34348E-04 1.2200% 1.35106E-04 2.1000% -0.5642%
total 2.27900E-02 0.0900% 2.28408E-02 0.1400% -0.2229%


YSO detector,
R=1.27cm


X-Ray beam


DXTRAN
sphere
Outer radius
4.635 cm
Inner radius
4 cm


I
I
I


SPlane source


SPaper


Concrete


Figure 4-11. MCNP5 model for input profile calculation. 3D profile calculated from a volume
contribution












Energy in keV
26.77 26.26


25.82 30.8i










C 0.03%


lIll rHI Avarage energy
0.%, Fraction of detected signa


1 2


3 4
Number of collisions


Figure 4-12. Average energy and fraction of the detected signal in each of the six collision bins.














Figure 4-13. Intersection volume of two cylinders


Figure 4-14. Two cylinder intersection volume





60000

,a 40000
0 20000


0C
0


-I


500 1000
Pixel number


1500


Figure 4-15. Integrated profile data


Equivalence


Figure 4-16. Equivalence between peaks and steps profiles.


Step 1


Step 2


Redistribute
to have a
constant
number of
counts along
the line


Normalize the
new profile by
the
intersection
volume of the
X-ray beam
and the line


Obtain
the data
profile


Obtain the
number of
counts per
unit volume
of nvion.


EUI


Figure 4-17. Normalization methodology scheme.


EUI










































I~ ,
~x

L ~u~rv
'0 A~l ; ni'iZ.''t~l Tllln i 3


- Data profile from the image Baseline


Normalised data


70000
60000
50000
40000
30000
20000
10000
0


7 6 5 4 3


0 1 2 3 4
X position in inches


Figure 4-18. Experimental and normalized data profile


'C- Source


YSO


Flux
tally
Paper
Concrete


Figure 4-19. A representation of the MCNP5 setup for volume intersection calculations


2 h ~1A A


,~z t













S2.50

S2.00

S1.50

Sloo


S0.50


S0.00
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18
Nylon line radius in cm


Figure 4-20. Line and beam intersection volume values. Beam radius 0.025 cm and line radii
from 0.1665 cm to 0.025 cm






S0.18
S0.16
.5 0.14

0.12

S 0.1


S0.06

0.04




Distance in (cm )



Figure 4-21. A plot of the volumetric normalization of half peaks obtained from MCNP model.





C olr g






I Facets
(-ww Mesh
SRect
Stalmesh


I : Ilr' I I
I I I~ ,.. I :
1
:- -~


I : 111 I~ I I- ~1 I -~ I ~ ~
I I I~ ~.
'1


SRef resh
SSur F
yj Celll;
rj Color
R FaetIs
r ww Mlesh
r Rect
r talmesh


5


Figure 4-22. Visual editor view of the new MCNP setup for volume calculations.


E 8.00E+02
~ 7.00E+-


7 .00E+02





1 .00E+02

b 6.00E+02


0 100 200


300 400 600
Pixel number


Figure 4-23. Normalization of the MTF sine profile over the intersection volume


Sphere


114 1
5



i 3 12 3












8.00E+02

7.00E+02

6.00E+02

5.00E+02

4.00E+02

3.00E+02

2.00E+02

1.00E+02

0.00E+00


0 100


200


300
Pixel number


400


600


Figure 4-24. Statistical smoothing of the normalized profile


120-

100 -
80-

60-

40-
20-


0 0.5 1 1.5 2 2.5

Frequency line pairs per mm


Figure 4-25. MTF function from detector 5, pixel size 0.05mm and beam aperture 0.5mm at 45
kVp-45 mA


~r MMMMHHH nn









CHAPTER 5
AN IMPROVED TECHNIQUE FOR THE MTF CALCULATION BASED ON A STEP
FUNCTION

Step Function Target Design for MTF Calculation

Figure 5-1 is a calibration target which can be used for the Modulation Transfer Function

calculation based on the edge function method. The left side of the target is lead (absorber) and

the right side of the target is nylon scattererr).

Figure 5-2 is the measured experimental response (black line) of the RSD scanning system

to an edge in units of number of counts/pixel as the scanning system moves across an edge; the

fitting function is shown in red.

The relation and the parameters used in the fitting process are:


Step fitting function = yo + Al*" exp( z) (5-1)


yo = 149.89047 & 368.69193 R2 = 0.81671
t, = 0.25157 f 0.07615 X2
= 663638.53125
Al = 2.3019E 6 f 0.00001 Dof

The line spread function is obtained by differentiating the step response function

formulated from Figure 5-2. Figure 5-3 shows the Fourier transform of the differentiated step

function. Both amplitude and phase are given; the resulting data are then fitted to give an MTF

function.

The Modulation Transfer Function is given by:

A z-z
MTF_edge~function = yo + exp(-2*"( c 2) (5 -2)




z= 2r(mm )









with the following parameters

y, = -214.13974 f16.80154
Ze = 9.8466E -16 f 0.00711 R ('= 0.99966

Sw = 5.59145 f 0.03699
= 395.45637
A = 22048.33177 f 236.34524 DofX

There are two important features to notice. First, the Modulation Transfer Function

obtained from this latest experiment is in agreement with the preliminary experiments performed

with the edge function. Second, the MTF based on the edge function includes the effect of a

geometric edge. Although the nylon and lead are at the same height, the X-rays easily penetrate

the nylon compared to lead and as a result the lead/nylon interface appears as an edge to X-rays.

As expected, the MTF obtained using this method is not exactly the one obtained from a

sine input modeling (with the MTF Sine target) due to amplitude variation. However, the

behavior still follows an exponential decrease. For the sine wave modeling with the MTF target,

the MTF follows an asymptotic behavior proportional to exp(-x), and according to this study the

asymptotic behavior is proportional to exp(-x2)

Finally, for calibration purposes and relative comparison of image quality both methods

are valid. However, for simplicity and efficiency in general calibration procedures the edge

response would provide a much faster tool. Obviously the MTF based on a Sine input is more

accurate in predicting the system response versus frequency.

The MTF Sine target is more sensitive to small variations in contrast and resolution than

the step target.

A Model of the Step Function Target Using MCNP5 and Variance Reduction Techniques.

To achieve the optimum design of the MTF step target, the system response is modeled in

MCNP5. Different configurations were tested to obtain a system response as sharp as possible to










approach the ideal step function. In all MCNP5 runs the same detector set up as in Chapter 4 was

used. Forced collisions and DXTRAN sphere were also used as accelerations techniques. The

maximum error achieved on the number of counts was 1.05%.

The first target design was a cubic plastic piece enclosed in a lead frame of the same

height. Figure 5-4 shows the geometry of the target.

The lead frame is 0.5 cm thick and 2 cm height, the cubic nylon piece is 2 cm by 2 cm by 2

cm. According to the MCNP5 run the mean free path of particles in the nylon piece is 1.9806 cm

and about 0.00263 cm in lead.

This configuration gave the data profie shown in Figure 5-5. The beam source scanned the

target from edge to edge; the detector is on the left hand side at a negative x. This first

configuration did not provide a satisfactory profile shape to model an edge function.

A modified design of the MTF step target was tested by setting the nylon piece 1 cm

higher than the lead frame. Figure 5-6 shows the geometry of the second design of the step

target. This design was chosen to reduce the geometric lead shielding on the edges of the nylon

piece.

Figure 5-7 shows the data profile obtained from the second MTF step target design. The

profile is closer to a sharp edge function than the first design in the central top region, however

the drop near the lead frame is more important than in the first design

A third design was tested where the nylon block (2cm by 2 cm by 2cm) was laid down on a

lead sheet (3 cm by 3 cm by 0.5 cm). The data profile (Figure 5-8) shows an increase on the

nylon block edges that is slightly larger on the detector side (left hand side). This is due to a 2 cm

nylon edge that is contributing to the total signal in addition to the flat top surface.










The contribution of the top center part of the target appears as a dip in the center of the

profile due to the relatively high contribution of the edges. This design gives a sharper profie at

the plastic/edge junction but the high contribution of the plastic step induces a distortion of the

center part of the profie. A better target would be achieved using a thinner plastic piece on a

lead sheet.

The Einal design proposed for the step target is given in Figure 5-9, it includes aluminum

and lead base sheets and a junction of lead and plastic pieces of the same height.

The lead and plastic pieces are sitting on the lead sheet enabling to obtain the two

configurations presented in the first and third designs on the same line profie.













Figure 5-1. Edge target made from a junction of lead (absorber) and nylon scattererr)



Number
of counts




4,00 4,25 4,50 4,75 5,00 5,25 5,50 5,75
Distance in cm
Figure 5-2. Scanning system response to an edge.
































0 2 4 6

Frequency (lines/cm)


Figure 5-3. Fourier transform of the line spread function (black curve) and fitting function (red)


Lead


Nylon


Figure 5-4. Geometry of the MTF step target in MCNP5












6.00E-02


Detector S.00E-02-


4.00E-02


2 3.00E-02-


2.00E-02
Led Plastic Lead\

1.00E-02-


0.00E+00
0 0.5 1 1.5 2 2.5 3 3.5 4
Distance (cm)




Figure 5-5. Data profile obtained from the first MTF step target design in MCNP5









Nylon


m2 cm II Lead III2cm
1cm





Figure 5-6. Geometry of the second design of the MTF step target












8.00E-02

7.00E-02

6.00E-02

5.00E-02

4.00E-02

3.00E-02

2.00E-02

1.00E-02

0.00E+00


2.5 3 3.5 4


Plastic


0 0.5 1 1.5 2

Distance (cm)


Figure 5-7. Profile data obtained from the second design of the MTF step target


1.00E-01

9.00E-02

8.00E-02

7.00E-02

6.00E-02

5.00E-02

4.00E-02

3.00E-02

2.00E-02

1.00E-02

0.00E+00


0 0.5 1 1.5 2

Distance (cm)


2.5 3 3.5 4


Figure 5-8. Data profile obtained from the third target design; nylon block on top of lead











SU.S Incn -1 Incn
0.5 inchI ,



0.5 inchNyo66
1/16 inch
1/16 inch
2 inch
First step Second step
design design


Figure 5-9. Final design profile proposed for the MTF step target


Aluminium









CHAPTER 6
PROPOSED TECHNIQUES FOR IMAGE QUALITY ASSESSMENT

Multiple Derivatives and Inflection Points as a Mathematical Criterion for Image Quality
Assessment

Even if the volumetric Sine MTF couples the target specific variations to the scanning

system response, it offers a basis for relative evaluation of system performance. It is an

integrated 3-D MTF over the vertical direction. This MTF allows comparison between detectors

and gives a basis on which to test a global improvement in the system.

Figure 6-1 presents a comparison between the MTF from Detector 1 (Nal) and Detector 5

(YSSi20). These results show that over a frequency range between 0.2 line pairs/mm and 2 line

pairs /mm, the performance of the Yg Si20 detector is superior to that of the Nal detector.

In Figure 6-2 the MTF plots are compared for three different aperture diameters of 0.5 mm,

1.0 mm and 1.5 mm.

Over the whole range of frequencies the MTF curve is higher for the smallest aperture. The

higher the MTF, the better the image with respect to the contrast and resolution.

The MTF presented here does not include any volumetric normalization processing. The

MTF is sensitive to the change in the volume intersection of the beam and the lines. The drop in

the MTF values is related to a loss of contrast and a volume variation. The relative difference in

MTF values indicates the quality of the images when using the same target.

Since the main purpose of the MTF plate is X-ray imaging system calibration, the main

obj ective is to provide a comparison of image quality.

Figure 6-3 shows several MTF plots for different conditions. In addition to the MTF value

at a given frequency, the curvature and the inflection point characterize the contrast and

resolution losses.










In Figure 6-3 the comparison is done over three aperture sizes of 0.5 mm, 1.0 mm, 1.5 mm

and two pixel sizes of 0.05 mm and 0. 1 mm.

For a given aperture, the larger pixel size has a higher MTF and hence a better image

quality.

In order to use mathematical properties as a criterion to sort the MTF curves, a

mathematical model is established. The plots in Figure 6-4 were generated by fitting the MTF

curves using Boltzmann functions.

Al A2
The formula used for the fitting process is MTF_experimental= Y = A2 + x-xo. The
1+e dY

corresponding coefficients are listed in Table 6-1 for each curve. Note that in this section X is a

frequency since it represents the MTF's variable.

In order to evaluate the fitting efficiency some statistical test results are given in Table 6-3.

Chi 2
The R2 ValUeS are close to 1 indicating a very good fitting function, the are the reduced
DoF

Chi 2 V81UeS obtained from the Nonlinear Least squares fitting and are given as an example

As previously explained, the curvatures and inflection points are of great interest when

comparing images from different set ups. Equation 6-1 gives the first derivative with respect to

the frequency (line pairs per mm). The coefficients are given in Table 6-2 and the plots are

shown in Figure 6-5.

X-XO
dY(X) (A2-Al)*e dY
dX YXO(6-1)
(1 +e Y ) 2
The second derivative is given by Equation 6-2:

YX-X X-Xo
d2Y e dY ed -
= (A2 Al)* (6-2)
dX 2 dX x-xo
(e Y + 1) 3









The inflection points are given by the second derivatives' zeros. By sorting the

corresponding frequencies, the images are compared with respect to their contrasts. The plots are

shown in Figure 6-6.

The zero values of the second derivative are presented in Table 6-4. The higher zero values

characterize better image quality according to criteria developed in this study.

The images corresponding to the MTF curves shown in Figure 6-3 are sorted and presented

in Figure 6-7 to Figure 6-11. The images are sorted using a scale from 1 to 5; 1 is the best

relative quality and 5 the relatively poorest quality.

The proposed MTF target is to be used in large scans for calibration purposes. Figure 6-12

is an image from an uncollimated YSO detector.

The number of counts needs to be increased to achieve a lower statistical error.

The image is shown to give an idea of how a calibration scan would be done. The MTF

target was laid on the sample being scanned. The heterogeneity of the sample (Tile Test Panel

VT70-191037-005) offered a good test to evaluate the MTF target response in a real

environment. However the background is of the same order of magnitude as the MTF target

response (approximately one third). This shows the limit of this MTF target design which is

highly affected by the material background.

The obj ective is to design an optimized small MTF target, such as the effect of the

background material is minimized.

The proposed image assessment techniques used the MTF curves obtained from the MTF

Sine target. However the same techniques can be applied to the MTF curves obtained from the

MTF step target.









Correlation Between the Different Methods of Calculating the MTF

The correlation between the Step function and the Sine function for MTF determination

needs to be done under the same experimental conditions. Once a relation is established between

the two methods one can be used knowing its limitations and advantages.

As previously explained, the Sine function based MTF uses more experimental

interpolation points over the frequency domain than the Step function MTF. This makes the Sine

MTF target more adapted for precise measurements of the contrast and resolution for given

frequencies. Also comparison between different MTF curves is finer and extends over a larger

frequency domain. For these reasons the Sine MTF target will be used as a reference for MTF

calculations.

A comparison between the MTF curves obtained experimentally from the Sine target and

the step target are not of high interest, unless the profiles are normalized over the target

interaction volume. This is because the MTF obtained from the Sine target contains information

on the change in volume. Recall from Chapter 4 that the Sine based MTF decreases less rapidly

(exp(-x)) compared to the Step function based MTF (exp(-x2))

The Sine based MTF uses the output modulation of the Sine input function whereas the

Step function MTF is derived through the Line Spread function.

Resolution Assessment from a Step Function Input

To demonstrate the equivalence between the MTF calculations based on the edge function

and the line spread function the definition of the step function is needed.


wf"(x, ) =< (6-3)

Also










11 y (x, y)= 3(x )dx= j(3(x ) 3(y )dy; )dx =jl 11 (x', y)dx'C (6-4)


Since the system is assumed in first approximation as linear, the output must be:


e(x) = lII (x,y) = J11 (x ,y)dx= 1(x')dx' (6-5)


Hence, the edge spread function is the indefinite integral of the line spread function:

de(x)
1(x) = (6-6)


Figure 6-13 shows the 3 steps needed to perform an MTF calculation based on the edge

function. First the data profie is obtained from the experiment then the profie is truncated to

only use one edge, finally the profie is smoothed using the averaged values of the lower and

higher regions of the profie. This smoothing procedure is necessary because the derivation is a

high pass filter; meaning that the high frequency noise will have a high contribution to the signal.

Another possibility is to apply a Gaussian frequency window to the first derivative of the profie

to discriminate against the high frequency noise.

Once this smoothing step is performed the first derivative is obtained numerically as

shown in Figure 6-14.

The width of the rising edge between 10% and 90% corresponds to the width of the first

derivative at 10% of its maximum. This distance x in pixel or mm can be used as a quick criteria

to compare different scan conditions and to perform resolution assessment using a step function.

There are many advantages to using the edge response for measuring resolution. In fact,

the main reason for wanting to know the resolution of a system is to understand how the edges in

an image are blurred.









The first advantage is that the edge response is simple to measure because edges are easy

to generate in images. If needed, the Line Spread Function can easily be found by taking the first

derivative of the edge response.

The second advantage is that all common edges responses have a similar shape, even

though they may originate from different Point Spread FunctionS20. Since the shapes are similar,

the 10%-90% distance is an excellent single parameter measure of resolution. The third

advantage is that the MTF can be directly found by taking the one-dimensional Fourier

Transform of the Line Spread Function (unlike the PSF to MTF calculation that must use a two-

dimensional Fourier transform).

For example the step function presented in Figure 6-13 is used to calculate the resolution

associated to the 10%-90% edge response. Figure 6-15 shows how the width x of the 10%-90%

edge is calculated.

For the particular conditions of the above edge scan the system has a 10%-90% edge

response of 1.94 mm.

The limiting resolution is a vague term indicating the frequency where the MTF amplitude

has a value of 3% to 10%.

In fact the edge width measured between 10% and 90% can be related to a frequency at

which the MTF is 10% of its maximum value.

Assuming the LSF can be fitted by a Gaussian function, which is the case for most imaging

systems. Then the Fourier Transform is also a Gaussian function as shown in Equation 6-7.


LS~x ep( )4FTLS) f e* x(-1(2zfe) (6-7)
2d 2.:p-~)B(L:~)~i;~ep

The width of the LSF at 10% of its maximum is given by

xiou6 = 2o21n(10) = edge 0 idthl (6-8)









This distance can also be measured directly from the edge width between 10% and 90%.

Now considering the MTF given by the Fourier Transform of the LSF, it has a value of

about 10% of its maximum at a frequency

J21n(10)
on = (6-9)
2xcro

Combining Equations 6-8 and 6-9 gives

2*ln(10) 1.46
on (lp/mm or 1p/pixel) (6-10)
a *" edge 1 idthr edge 1 idthr

The 10% contrast level on the corresponding MTF curves will occur at about: 0.75 1p/mm

or 1p/pixel for an edge width of 1.94 mm. This is a very convenient method to asses the system

limiting resolution between 10% and to compare different images using a single number.

Figure 6-16 shows an example of a numerical calculation of the first derivative and the

Fourier Transform of the edge function used in Figure 6-15. The amplitude of the Fourier

Transform gives the MTF. The predicted frequency at which the MTF value is 10% from the

edge width method gives 0.75 1p/mm the measured value from the MTF curve gives 0.665

1p/mm. The error associated to the measured value with respect to the predicted value is about

11.3%.

This is due to the errors associated to the numerical evaluations of the first derivative and

the Fourier Transform but also the initial assumption of the Gaussian fitting.

As a conclusion the edge width between 10 % and 90% is a convenient single number for

relative comparison of different images. The same edge function can be used to generate an MTF

curve. The theoretical relationship between the edge width and the frequency at which the MTF

value is 10% can be used as an indication of the experimental frequency. In the previous

example an error of 11.3% was calculated between the two frequencies.















120


- detectorl -Nal
--detector5-YSO


1 1.5 2
frequency line pairs/mm


Figure 6-1. MTF comparison between Nal and YSSi20 detectors at 45 kVp, 0.5 mm aperture






120-

100 -1 '* mtf-1.5mm
aperture
80 -1 I

S60 -.
N ~mtf-1.0mm
S40 aperture

z 20-

0 mtf-0.5mm
0 0.5 1 1.5 2 2.5 aperture

Frequency line pairs per mm




Figure 6-2. MTF comparison for 3 different aperture diameters at 45kVp-45mA-0.05mm pixel
1 ze.












-* mtf-0.1mm pix-
100 1 mmap

80
u.. -a- mtf-0.1mm pix-
r 0.5mm ap

.0 mtf-0.05mm pix-
m 40-
E 1.5mm ap
z: 20-
mtf-0.05mm pix-
0 7, 1.0mm ap
0 0.5 1 1.5 2 2.5
-m mtf-0.05mm pix-
Frequency line pairs per m m 0.5mm ap


Figure 6-3. MTF comparison for different pixel sizes and beam apertures at 45 kVp-45 mA



120-

S100 fit-mtf-0.1mm pix-
1 mm ap
1 80-
r fit-mtf-0.1mm pix-
0.5mm ap
N 60-
E \ fit- mtf-0. 05mm
40 pix-1.5mm ap

20 -1 fit- mtf-0. 05mm
pix-1.0mm ap

0 0.5 1 1.5 2 2.5 ftmf00m
pix-0.5mm ap
Frequency line pairs per mm



Figure 6-4. MTF Boltzmann model fitting function comparison for different pixel sizes and
beam apertures at 45 kVp-45 mA














Table 6-1. Coefficients used in the fitting function formula for each MTF curve
Al A2 XO dX
MTF 0.1mm pixel / 0.5mm aperture 113.633 -2.25743 0.77575 0.32778
MTF 0.05mm pixel / 0.5mm aperture 117.8884 3.91869 0.61552 0.29361
MTF 0.1mm pixel / 1.0mm aperture 102.3881 2.23524 0.54824 0.12499
MTF 0.05mm pixel / 1.0mm aperture 107.0763 1.48754 0.4929 0.13709
MTF 0.05mm pixel / 1.5mm aperture 107.1106 3.23425 0.37257 0.08768


Table 6-2. Statistical measures of the fitting accuracy
Chi 2 R 2
DoF;
MTF 0.1mm pixel / 0.5mm aperture 1.02355 0.99907
MTF 0.05mm pixel / 0.5mm aperture 1.25625 0.99916
MTF 0.1mm pixel / 1.0mm aperture 5.94114 0.99724
MTF 0.05mm pixel / 1.0mm aperture 1.94303 0.99906
MTF 0.05mm pixel / 1.5mm aperture 3.41298 0.99825


~Ist aerivative mtT-
0.1mm pix-1 mm
ap
3 1stt derivative mtf-
0.1mm pix-0.5mm
ap
1st derivative mtf-
0.05mm pix-
1.5mm ap
1st derivative mtf-
0.05mm pix-
1.0mm ap
1t st derivative mtf-
0.05mm pix-
0.5mm ap


-5 O


@10
E


-20

-25

-30


Frequency line pairs per mm


Figure 6-5. MTF fitting function first derivative, scan at 45 kVp-45 mA







































Table 6-3. Roots value of the MTF second derivatives curves
Curves first root (Freq-line pairs per mm)
MTF 0.1mm pixel / 0.5mm aperture 0.77575
MTF 0.05mm pixel / 0.5mm aperture 0.61552
MTF 0.1mm pixel / 1.0mm aperture 0.54824
MTF 0.05mm pixel / 1.0mm aperture 0.4929
MTF 0.05mm pixel / 1.5mm aperture 0.37257


__


-*-2nd derivative
mtf-0.1mm pix-
1 mm ap
-2nd derivative
mtf-0.1mm pix-
0.5mm ap
2nd derivative
) .5 p 1 1.5 2 2.5 mtf-0.05mm pix-15 a

2nd derivative
mtf-0.05mm pix-
1.0mm ap
-m-2nd derivative
mtf-0.05mm pix-
Frequency line pairs per m m 0.5mm ap


-500


-1000


-1500


Figure 6-6. MTF fitting function second derivative, scan at 45 kVp-45mA


Detector 6


-11561




-6285




-1008


0 10 20 30 40 50 60 70 80 90
X( direction (mm)


100 110 120 130 140


I
155


Figure 6-7. 1 MTF 0.1 mm pixel, 0.5 mm aperture
















155


II I


_I


IIII


Detector 6


0 10 20 30 40 50l 60~ 70 80 90 100n 110 120 130: 140


-7438 "
O




-1035


Figure 6-8. 2 MTF 0.05 mm pixel, 0.5 mm aperture



Detector 6


0 1 1 3 40 50 6 70 80 90 100 110 120 130 140
:< direction (mm)


I I
155


-24051
o


-4187


Figure 6-9. 3 MTF 0.1 mm pixel, 1.0 mm aperture





Detector 6


0 1 20 30 40 50 60 70 80 90 100 110 120 130 140
:< direction (mm)


'I "
155


a
s


-29111 n
~
C
v,


-4363


Figure 6-10. 4 MTF 0.05 mm pixel, 1.0 mm aperture





11I


Detector 6


-110153




o





-9051


0 10 20 30 40 50l 60~ 70 80 90 100n 110 120 130: 140


Figure 6-11. 5 MTF 0.05 mm pixel, 1.5 mm aperture


Detector 5


Nylon
lines
from the
MTF
target


Aluminium
edge of the
MTF frame









Tile Test
Panel


0 10 20L~ 30 40 50b
:: direction (mm)l


Figure 6-12. YSO image of MTF Target on a tile panel


-,3:77


-e
s
rr
-i~cjo ;

n





























































Figure 6-13. Selection and smoothing steps for the MTF calculation from a step function


MTF

MTF
((,rl)
MTF ((,0)

MTF
((,0)
MTF((,0)


I


Table 6-4. Different methods of the MTF
Input function Output function

Point source (x,y) Point spread
function (x,y)
Line source (x)= I Point Line spread
source (x,y) function (x)
Edge function (x) Edge spread
function (x)
Sine input(x) Sine output (x)


derivation
Intermediate steps

2D FT

1D FT

d(Edge(x)) /dx = Line spread function (x)
and 1D FT
Contrat(()/Contrast(0)=a*MTF


30000

25000

20000

S15000

S10000

5000

0










x 104 Scanning systern output: Counts/pixel




1i



0 10 20 30 40 50 BO 70 80
Pixel nurnber


First derivative


Pixel number


Figure 6-14. An example of the edge profile and its first derivative








25000


20000


15000


10000


5000


90%



Distance X
mm or pixel


10%


0 10 20 30 40 50 60 70
Distance (mm)


Figure 6-15. Edge function width estimation




























0. ..l.. .


Di FFT r




y INbvisigrgj

G~i^~DcF =7.01484
F^2 = 098518

-1.71357 .41015
>0: -1.9351E-16 .00232
w 0.66329 n.07
A 5.0611@ n.E22


14000 -


0 5 1 15 20


25 30 35


Data: Derivativel Data33B
Model: Gauss
Equation: y=y0 + (A/(w*sqrt(Pl/2)))*exp(-2*((x-xc)/w)^2)
Weighting:
y No weighting

Chi^2/DoF = 9422.64067
R^\2 = 0.9963


-1.76655
25.76397
1.01068
-9715.58996


116.04368
10.00675
10.01394
1120.72925


X Axis Title


Figure 6-16. Numerical evaluation of the first derivative of the edge function used in the
example and its Fourier Transform









CHAPTER 7
COMPUTATIONAL PROCESS SING WITH MATLAB. ALGORITHM ARCHITECTURE FOR
MTF CALCULATION (MATLAB)

Modulation Transfer Function Based on the Sine Target

The main result of this task was a code that integrates all the calculations for the MTF

process. The code was written in the MATLAB 7.0.4 programming language. The code was to

be implemented in an image processing tool previously used by the Lockheed-Martin Space

Systems Company.

Figure 7-1 shows the Matlab interface for the profie data generation and the MTF

calculation. The interface is analogous to the code used currently to process the output images

from the system and draw the profies.

After scanning the MTF plate, a couple lines are generated. When saving the profie

(Figure 7-2), the MTF menu appears to enable the MTF processing.

Once the profie is saved in a text format, the code generates a *.dat fie using the same

name. This file will be used in Matlab to generate MTF curves.

The conventional profile used for the MTF calculation should have the maximum peaks on

the left, since they are used to generate the low frequencies. The code is essentially written

following this model. There is an option to reverse the profile data to make user entries easier

(Figure 7-3).

Figure 7-4 shows the user interface for entering the Sine MTF plate information. Default

values are already entered for the Sine MTF target.

The first step is to locate the maxima and minima in the image. Based on these values the

contrast and the MTF are calculated. Figure 7-5 shows how the preliminary peak selection is

displayed.









The local maximums are designated using red crosses. Because of the fluctuations in the

data, it is nearly impossible to pick up one maximum per peak, unless using the Full Width at

Half Maximum (FWHM) for each set of holes. This part is performed in the "Automatic" option

available in the code.

Currently, the MTF calculation requires that the user select for each peak a region of

interest. The region of interest (ROI) does not have to be precisely selected. The code extracts

the x-ordinates from the image to recalculate the overall maximum in the ROI.

After all the peaks have been selected, the MTF plot is generated (Figure 7-6). When

saving the plot, the same name is used to create a new folder that contains the data profie and

the MTF plots in PDF format in addition to a text fie that contains the values of the MTF versus

frequencies.

Modulation Transfer Function Based on a Step Function Target

Figure 7-7 shows a step function profie obtained from the preliminary experiment

(Chapter 3) of an edge function. The first derivative is also given since the derivation is the first

step in using the edge function. Note that the data is noisy and a statistical smoothing would

provide a better data profie to start with.

A discrete Fourier transform is then performed on the first derivative and the modulus is

estimated to give the MTF. Figure 7-8 shows the MTF curve and its first and second derivatives.

As expected, the numerical treatment without any smoothing on the data introduces high

fluctuations in the MTF calculation. These large fluctuations made it nearly impossible to use the

zeros of the second derivative as a criterion for image quality assessment.

Either a denoising algorithm or an iterative least squares estimate fitting of the data using

(FittingFunaction(xi) -M\~easuremevt(xi))2

J J is needed.











The more convenient choice for automated use would be using the fitting tools provided

with the Matlab7.0.4 version.


Figures from 7-8 to 7-10 show the different steps in the Matlab code used to generate MTF

curves from an edge function. Figure 7-8 shows how a region of interest can be selected, Figure


7-9 and 7-10 show the selected region of the edge function its first derivative and the MTF


curves with the frequencies expressed in line pairs/pixel and line pairs/mm.








Data Format l Display DAT

SDisplay TIF

Data File Directory Inents and Settings\Nissia~ly DocumentslFishinglinet-6-1 6-2006-MTF-45kVp-45mA-0 .5mmAperture Fle

Image product M~in Image Image name 6-1 6-2006-MTFplate-55Kvdell _Minlmage
FileListImage list
6-16200 -MT .Isedfe I-le -. .11 ^ Display
..i..10. ril.Ise. l Jleg 1 alCompute -616-2006-MT F-45kVp-45mA- 0.5mmApe rturedt
616-2006-MTFplate-56Kvdetector3.dat -616-2006-MTF-45k~p-45mA-0.5mmAperturedc
61 6-2006-MTFplate-55Kvdetector4 .dat 616-2006-MTF f ,:-..j_.i,-- ...r..--pen...4.1
616-2006-MTFplate-55Kvd etector 5.dat -61-06MF .4..-0i...-: dn
-616-2006-MTF-15i r.ira0 r~r .n. '--820-T ar~.9rr.nosd



Format A







Figure 7-1. Matlab user interface











File Edit View Insert Tools Desllop Window Help
D c a El e la | RR9 | W 0-C E O I n


S1.. rTE i ITF-46kVp-46mA-0. 1pix-D.6mm-ap-HorizordalCorrection-YSOring-p iio lO-10-2C


10000-


8000


S6000-





4001245



Inches


_ _____


File Edit Adjust Filter View Utilities Windov MTF X









Line Drofile


Figure 7-2. MTF menu and data profile


Figure 7-3. Data profile













































Enter the number of I~nes of the same diameter used in the MTF frame:
151
Enter the number of dif ferent lines saaleariosused in the MTF frame:

101 1


~YliY~iiC~VR~i~i~


File Edit View Insert Tools Desidop Window Help


Scanning system output: Counts/pixel
12000


10000-


S8000-


a 6000-


Line 1
02624
Line 2
/0.1614
Line 3
01418

Line 4
01102
Line 6
01008
Line00 6

Line 7


Line 6
0" 0590
Line 9

/0.0408
Line 10
0O 0394
Line 11
0O 0290
Line l2
/0 0180


Figure 7-4. User interface for information entries


200 400 600 800
Pixel number


1000 1200 1400 1600


Figure 7-5. Maximum search











File Edit View Ir.>ert Tuol> De ktop riduw Help
D c l a g A 0 00 0 m i~
rlad its .:.n Tlar.-lar Fur.Chir.









II




I | 1 I I II. iII I:I j I i


Figure 7-6. Saving files


First derivative
S20000

0 15000

S10000

~5000




~-5000
0 10 20 30 40 50 60 10 80
: Pixel number


Figure 7-7. Data profile from an edge function and its first derivative














100


x 104 Scanning system output: Counts/p xel









0 10 2 0 4 0 6 0 8




Pixel number





















x14


5.5 ---- -- -- --- --- Da you want select a Ro?

5 --ys





3.5-



2.5 - - -

2 ------- -------': : ': ------- --

1 5 - - -


= ~I_^_I 1I


lInb =


h-le Edit lext tell lools Debug Desktop Window Help


r FX
a Ill 8 5


2 'HTF processing, this
4 i-system.
5 4After averaging the
6 Used to calculate t1
7 %minlmumi for each ser
8 iulopen('FIGURE') ;

10 4*f*f**** Open the fi
11 [filename, path] = ui
12
13 [fld,messagle] = foper
14
15 %******* Initialize a
16 All=[];
17 A11=0;
18 Al= [];
19 A1=0;
20 maxim= [];
21 minim=[ []
22 C= fl :


As-ILCUYI Iracenimage,13);
images (A2) ;
A2=iredon(Image,10);
images (A2) ;
A2=1radon(Image,10) ;
figure~imagesc(A2);
S6/7/07 11:14 AH --%
-- 6/11/07 7:16 PH --
-- 6/13/07 3:40 PH --


No


count point

0


Figure 7-8. Selection of a region of interest in the edge function profile


x 104 Scanning system output: Counts/pixel









61 I-i-L- I










0 5 10 15 20 25 r

Pixel nurnber S


Figure 7-9. The selected region of interest and the first derivative of the edge function













101


First derivative


10

Pixel nurnber





























MTF, First and second derivative


Fourier Transform : Amplitude and Phase


a 0.8 -







0.2
0


0.1 0.2 0.3 0.4
Frequency [Ip/pixel]


0.2 0.4 0.6 0.8
Frequency [Ip/mm]


0.5 E-


Figure 7-10. MTF curves with frequencies expressed in line pairs/pixel and line pairs/mm









CHAPTER 8
CONCLUSION

In order to properly characterize the X-ray backscattering system several definitions of the

Modulation Transfer Function have been introduced. These definitions and the methodology for

calculating the MTF depend on the input function to the system. Several input functions have

been tested: Point Function, Line Function, Step Function and Sine Function. The relationship

between the different functions and the resulting MTF was treated to understand the benefits and

limitations of each input type function for practical use. The preliminary experiments for an

impulse and step functions showed the expected responses from mathematical derivations. The

key step for a complete analysis was the ability to accurately fit the curves according to statistical

tests and obtain mathematical expressions that were used later for curve recognition.

A Sine target pattern was proposed for precise evaluation of the MTF as a function of

frequency. The design was based on nylon lines of different diameters and separation. This MTF

Sine target was used for maj or comparisons and relative image quality assessment. The

experiments were performed mostly with the new compact system using YSSI20 detectors, but

some experiments used Nal detectors. The large dimensions of the MTF Sine target made it less

desirable for practical use on small scans areas. Also this Sine MTF target was highly dependent

on the background material. Instead, an improved Step target design was proposed to meet a size

constraint of approximately a cube of 0.5 inch by 2 inch by 5/8 inch.

The different designs were supported by MCNP5 models using two variance reduction

techniques; forced collisions and DXTRAN sphere. These models enabled to understand the

different contributions to the signal and their relationships with the target own volume.










A geometrical volumetric normalization of the input sine profile was performed using the

complete elliptic integrals of the first and second kind. However this method was not completely

successful in providing a good volumetric normalization.

Monte Carlo simulations helped provide an understanding of the effect of the volume

decrease in the MTF Sine target through two competing factors: the volumetric interaction rate

and the particle mean free path.

For practical image quality assessment and comparison, the evaluation criterion used with

the Sine MTF target was the first zero of the second derivative of the MTF curve. A method for

resolution assessment based on an edge input function was proposed. This method relates the

rising edge width between 10% and 90% to the frequency at which the theoretical MTF value is

2*ln(10) 1.46.
10%, fioto= (lp/mm or 1p/pixel).
tr edge 11 idthIl edge 11 idthIl

The MTF calculations were performed using MATLAB7.0.4. Customized codes were

written with user interfaces for MTF curve generation.

Finally, some MTF applications in image processing and some of the early results on foil

filtering with the RSD scanning system are presented.










APPENDIX A
ENERGY FILTERING USINTG PAPER

While setting up the experiments for the MTF measurements, placing a regular sheet of

paper under the nylon lines, in addition to the lead on the floor, drops the background noise by

400 counts/pixel.

Figure A-1 shows a comparison between two backscatter images, one with and one

without paper. The maximum intensities are about the same order, while the background

contribution drops off by half.

Note that for case b in Figure A-1 the bright line on the image is above the sheet of paper

while the 3 lines on the left of the image are right under the paper. All of the lines are equally

distant from the paper.

Figure A-2 shows a line profile across image b in Figure A-1. The lines under the paper

show with near half intensity of that of the line above the paper.


Detector 1
Detector 1 .. .


a -..~.. I dlri.l......l ..I b

Figure A-1. Comparison between two backscatter images. a) Scan without paper underneath the
nylon line. b) Scan with paper underneath the nylon line
















5-


4.5-


4-






O .
Inche

Fiue -. Ln rfl vlaino h ae itrn




















10





















NOTE:
1- blamates rare Ilsled tear len 10 right
2- Ditne in inctis.
3-Ue Laext Cdrll sie If exact size Is nall avehible
4-Three hole Ilnes, Irtner. rnkdde and outer.
6- 4Ei chamber 10 depth ofi hole diameter

Irmer M lescit 0.2824 1S)
Mktt ddi hole lh Q.2624 (6)
Oder) hale gatel 0.303441)


~7 9 ilS








;------5.3480

.-----~;I


tapped through
h10le F-32 18)


a~t.028
Inner hole pich 0.1418 {5)

,ra.03B


loner hole plich 0.0740 :6:
Middi hole pikh 0.0744 (6)
0.Aer hole d(1s* 0.112214)



lInner holedt ONilCh~ 0 094(

Catear hole Irhen 0 0531(4)~

Ine0. caB0166034 5

dinner kbhol lcn 0.0290 ;5)

lannr heto rtic1 0,01fl015)


- middle


-OUIlf


Figure B-1. MTF frame plate top view


APPENDIX B

MTF FRAME STRUCTURE


Aluminlum plate C 1250 Ihlck.
MTF frame p ate
TOP VIEW


Nissia Sabri
5-12-2006i































02500 -


Nissia Sabri
5-12-2006


Aluminium plate 0.25 thick
MTF cover plate
TOP VIEW


Figure B-2. MTF cover plate top view









LIST OF REFERENCES


1. E. Dugan, A. Jacobs, S. Keshavmurthy, and J. Wehlburg," Lateral Migration Radiography,"
Research in Nondestructive Evaluation, 10(2) p. 75-108 (1998).

2. A. Jacobs, E. Dugan, S. Brygoo, D. Ekdahl, L. Houssay, and Z. Su, "Lateral Migration
Radiography: A New X-ray Backscatter Imaging Technique," Proceeding of SPIE, 4786 p.
1-16 (2002).

3. E. Dugan, A. Jacobs, L. Houssay, and D. Ekdahl, "Detection of Flaws and Defects Using
Lateral Migration X-ray Radiography," Proceeding of SPIE, 5199 p. 47-61 (2004).

4. H. Barrett, and W.Swindell, Radiological Imaging, The Theory of Image Formation,
Detection, and Processing, Academic Press, Inc San Diego, California 1981.

5. F. Attix, Introduction to Radiological Physics and Radiation Dosimetry, John Wiley & sons,
Inc New York, New York (1986).

6. A. Jacobs, and J. Campbell, "Landmine Detection by Scatter Radiation Radiography,"
Scientific and Technical Final Report, Contract DAAK 70-86-K-0016, U. S. Army Belvoir
Research, Development and Engineering Center, (1987).

7. J. Campbell, and A. Jacobs, "Detection of Buried Land Mines by Compton Backscatter
Imaging," Nuclear Science and Engineering, 110 p. 417-424 (1992).

8. Y. Watanabe, J. Monroe., S. Keshavmurthy, A. Jacobs, and E. Dugan, "Computational
Methods for Shape Restoration of Buried Obj ects in Compton Backscatter Imaging," Nuclear
Science and Engineering, 122 p. 55-67 (1996).

9. J. Wehlburg, S. Keshavmurthy, E. Dugan, and A. Jacobs, "Geometric Considerations
Relating to Lateral Migration Backscatter Radiography (LMBR) as Applied to the Detection
of Landmines," Proceeding of SPIE, 3079 p. 384-393 (1997).

10. Z. Su, J. Howley, J. Jacobs, E. Dugan, and A. Jacobs., "The Discernibility of Landmines
Using Lateral Migration Radiography," Proceeding of SPIE, 3392 p. 878-887 (1998).

11. C. Wells, Z. Su, J. Moore, E. Dugan, and A. Jacobs, "Lateral Migration Radiography
Measured Image Signatures for the Detection and Identification of Buried Landmines",
Proceeding of SPIE, 3710 p. 906-916 (1999).

12. C. Wells, Z. Su, A. Allard, S. Salazar, E. Dugan, and A. Jacobs, "Suitability of Simulated
Landmines for Detection Measurements Using X-ray Lateral Migration Radiography",
Proceeding of SPIE, 4038 p. 578-589 (2000).

13. Z. Su, A. Jacobs, E. Dugan, J. Howley, and J. Jacobs, "Lateral Migration Radiography
Application to Land Mine Detection, Confirmation and Classification," Optical Engineering,
39(9) p. 2472-2479 (2000).









14. E. Dugan, A. Jacobs, Z. Su, L. Houssay, D. Ekdahl, and S. Brygoo, "Development and Field
Testing of a Mobile Backscatter X-ray Lateral Migration Radiography Land Mine Detection
System," Proceeding of SPIE, 4742 p. 120-131 (2002).

15. R. Evans, The Atomic Nucleus, McGraw Hill Boo, Inc. New York, New York (1955).

16. J. Dainty, and R.Shaw, Image Science Principles, Analysis and Evaluation of Photographic-
Type Imaging Processes, Academic Press, London (1974).

17. M.J. Berger, J.H. Hubbell, S.M. Seltzer, J. Chang, J.S. Coursey, R. Sukumar, and D.S.
Zucker, "XCOM: Photon Cross Section Database (version 1.3) http ://physics.nist.gov/xcom",
National Institute of Standards and Technology (May 2007).

18. D. Shedlock, "X-ray Backscatter Imaging for Radiography by Selective Detection and
Snapshot Evolution, Development, and Optimization", Ph.D. Dissertation, University of
Florida (2007).

19. B.T. Addicott, "Characterization and Optimization of Radiography by Selective Detection
Backscatter X-ray Imaging Modality", M. S. Thesis, University of Florida (2006).

20. S. Smith, The Scientist and Engineer's Guide to Digital Signal Processing California
Technical Publishing, San Diego, California (1997).









BIOGRAPHICAL SKETCH

Nissia Sabri is a graduate assistant at the University of Florida. She j oined the Scatter x-ray

laboratory in the Nuclear and Radiological Engineering Department in August of 2005 to

complete a Master of Science in nuclear engineering. She obtained a Master of Science in

applied physics engineering in September 2006 and a Bachelor of Science in physics in May

2005 at The Grenoble National Engineering School for Physics-France.





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1 AN ADAPTED MODULATION TRANSFER F UNCTION FOR X-RA Y BACKSCATTER RADIOGRAPHY BY SELECTIVE DETECTION By NISSIA SABRI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2007

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2 2007 Nissia Sabri

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3 To my mother

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4 ACKNOWLEDGMENTS I would like to thank Dr. Edward Dugan and Dr Alan Jacobs for their guidance, constant enthusiasm and help. I would like also to tha nk Dr. James Baciack for being on the committee. I would like to give a special thanks to my family and frie nds who were a great source of motivation. I need to especially thank my husband Julien, for hi s help support, and endless patience; my sister and mother, for their cons tant support; and my friends, especially Benoit Dionne, Anne Charmeau and Colleen Politt, for their encouragement. I would like to thank Warren Ussery for the financial funding and my research group, especially Daniel Shedlock for the invaluable learning experience. Thanks to Ines AvilesSpadoni for her help. I would like to thank Dr.Sjoden for accepting me in his research group to pursue my Ph.D. Finally, I would like to thank Lockheed Mar tin Space Systems Co, NASA, Langley Research Center, NASA, Marshall Space Flight Center and The University of Florida, Department of Nuclear and Radiological Engineer ing, for the financial support.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............12 CHAPTER 1 INTRODUCTION..................................................................................................................14 Compton Backscattering Imaging (CBI)................................................................................14 Backscatter Radiography by Selective Detection (RSD).......................................................16 Overview of Previous Work............................................................................................16 Project Objectives............................................................................................................17 RSD Scanning System............................................................................................................17 Moving Table: X-Ray Source and Detectors..................................................................17 Image Acquisition :Signal Flow and Software................................................................18 2 PROBLEM STATEMENT.....................................................................................................24 General Physics of Photon Interaction...................................................................................24 Compton Effect...............................................................................................................25 Kinematics..................................................................................................................... ..26 Cross Section.................................................................................................................. .26 Theoretical Approach of the Modul ation Transfer Function (MTF)......................................27 The Fourier Transform App lied to Image Processing............................................................30 MTF Applied to the RSD Scanning System...........................................................................31 3 PRELIMINARY EXPERIMENTS: PULSE AN D STEP FUNCTIONS SIMULATION.....36 RSD System Experimental Responses...................................................................................36 Pulse Input Experiment...................................................................................................36 Step Function Experiment...............................................................................................37 Principles of Statistics and Curve Fitting Applied to MTF Calculation.................................37 Results and Analysis........................................................................................................... ....39 Pulse Function Experiment..............................................................................................39 The Step Function Experiment........................................................................................41 4 MTF CALCULATION BASED ON A SINUSOIDAL INPUT FUNCTION.......................44 MTF Sinusoidal Pattern Design..............................................................................................44 System Response to the I nput Modulation Function..............................................................44

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6 Digital Output Profile......................................................................................................44 Comparison of Detection Propertie s Between NaI and YSO Crystals............................45 A Model of the Sinusoidal Input Func tion Using MCNP5 and Variance Reduction Techniques..................................................................................................................... .....46 Input Function from a 2D Model of the MTF Sine Target..............................................46 Input Function from a 3D Model of the MTF Sine Target..............................................50 Volumetric Normalization of the MTF...................................................................................51 Geometric Normalization................................................................................................52 Volume Calculation Based on an MCNP5 Model..........................................................54 Actual MTF Curves Based on a Sine Input Pattern................................................................56 5 AN IMPROVED TECHNIQUE FOR THE MTF CALCULATION BASED ON A STEP FUNCTION..................................................................................................................71 Step Function Target Design for MTF Calculation................................................................71 A Model of the Step Function Target Using MCNP5 and Variance Reduction Techniques..................................................................................................................... .....72 6 PROPOSED TECHNIQUES FOR IMAGE QUALITY ASSESMENT................................79 Multiple Derivatives and Inflection Points as a Mathematical Criterion for Image Quality Assessment.............................................................................................................79 Correlation Between the Different Me thods of Calculating the MTF....................................82 Resolution Assessment from a Step Function Input...............................................................82 7 COMPUTATIONAL PROCESSING WITH MATLAB. ALGORITHM ARCHITECTURE FOR MTF CALCULATION (MATLAB)..............................................95 Modulation Transfer Function Based on the Sine Target.......................................................95 Modulation Transfer Function Ba sed on a Step Function Target...........................................96 8 CONCLUSION.....................................................................................................................103 APPENDIX A ENERGY FILTERING USING PAPER..............................................................................105 B MTF FRAME STRUCTURE...............................................................................................107 LIST OF REFERENCES.............................................................................................................109 BIOGRAPHICAL SKETCH.......................................................................................................111

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7 LIST OF TABLES Table page 4-1 Number of counts at the detector surface..........................................................................60 4-2 Comparison between the Analog and Non-Analog MCNP5.............................................61 4-3 Summary of the line diameters and th e associated number of line position......................62 4-4 MCNP5 run condition for Analog versus Non-Analog.....................................................62 4-5 Comparison between Analog and Non-Analog results in MCNP5...................................64 6-1 Coefficients used in the fitting function formula for each MTF curve..............................88 6-2 Statistical measures of the fitting accuracy........................................................................88 6-3 Roots value of the MTF second derivatives curves...........................................................89 6-4 Different methods of the MTF derivation..........................................................................92

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8 LIST OF FIGURES Figure page 1-1 Schematic illustrating X-ray production............................................................................19 1-2 Typical spectrum obtained from an X-ray tube with a tungsten anode4............................19 1-3 Compton Backscatte ring Imaging (CBI)...........................................................................20 1-4 Lateral Migration Radiography (LMR).............................................................................20 1-5 Photograph of RSD System with 4 NaI Detectors.............................................................21 1-6 Photograph of RSD System showing YS O detectors mounted to NaI Detectors..............21 1-7 RSD scanning system m ounted on a fixed frame..............................................................22 1-8 Flow chart of the image acquisition process20...................................................................23 2-1 Photoelectric, Compton and Pair Production5...................................................................34 2-2 Kinematics of the Compton Effect....................................................................................34 2-3 Transmission model......................................................................................................... ..35 2-4 Backscatter model.......................................................................................................... ....35 3-1 Scanning system output two line pairs placed at 45with respect to the vertical axis.......42 3-2 High exposure scanning output, one sw eep of a nylon line (Dirac Simulation)................43 3-3 Scan of a cubic plastic sample: 17.5 mm width, 1 mm beam, 0.5 mm pixels...................43 4-1 Scheme for simulating a sinusoidal input..........................................................................57 4-2 MTF frame plate............................................................................................................ ....58 4-3 MTF frame plate detailed design.......................................................................................58 4-4 Output profile from the scan of the MTF Sine target (detector 1 NaI)..............................59 4-5 Scattering to absorption ratios for NaI and Y5Si2O crystals.............................................59 4-6 MCNP5 model for input profile calculation......................................................................60 4-7 Energy spectrum distribution used in the MCNP5 model based on Kramers spectrum....60 4-8 The input sine profile obtained from running MCNP5......................................................62

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9 4-9 Sine profile obtained from modeling 10 nylon lines of diffe rent diameters in MCNP5...63 4-10 The complete input profile from an MCNP 5 simulation as recorded at the detector........63 4-11 MCNP5 model for input profile calculation......................................................................64 4-12 Average energy and fraction of the detected signal in each of the six collision bins........65 4-13 Intersection volume of two cylinders.................................................................................65 4-14 Two cylinder intersection volume.....................................................................................65 4-15 Integrated profile data................................................................................................... .....66 4-16 Equivalence between peaks and steps profiles..................................................................66 4-17 Normalization methodology scheme.................................................................................66 4-18 Experimental and normalized data profile.........................................................................67 4-19 A representation of the MCNP5 set up for volume intersection calculations.....................67 4-20 Line and beam intersection volume values........................................................................68 4-21 A plot of the volumetric normalization of half peaks obtained from MCNP model.........68 4-22 Visual editor view of the new MC NP setup for volume calculations................................69 4-23 Normalization of the MTF sine profile over the intersection volume...............................69 4-24 Statistical smoothing of the normalized profile.................................................................70 4-25 MTF function from detector 5...........................................................................................70 5-1 Edge target made from a junction of lead (absorber) and nylon (scatterer)......................74 5-2 Scanning system response to an edge................................................................................74 5-3 Fourier transform of the line spread function (black cu rve) and fitting function (red).....75 5-4 Geometry of the MTF step target in MCNP5....................................................................75 5-5 Data profile obtained from the firs t MTF step target design in MCNP5...........................76 5-6 Geometry of the second de sign of the MTF step target.....................................................76 5-7 Profile data obtained from the s econd design of the MTF step target...............................77 5-8 Data profile obtained from the third target design; nylon block on top of lead.................77

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10 5-9 Final design profile propos ed for the MTF step target......................................................78 6-1 MTF comparison between NaI and Y5Si2O detectors at 45 kVp, 0.5 mm aperture..........86 6-2 MTF comparison for 3 diffe rent aperture diameters..........................................................86 6-3 MTF comparison for differe nt pixel sizes and beam ap ertures at 45 kVp-45 mA............87 6-4 MTF Boltzmann model fitting function comparison.........................................................87 6-5 MTF fitting function first deri vative, scan at 45 kVp-45 mA............................................88 6-6 MTF fitting function second deri vative, scan at 45 kVp-45mA........................................89 6-7 1 MTF 0.1 mm pixel, 0.5 mm aperture..............................................................................89 6-8 2 MTF 0.05 mm pixel, 0.5 mm aperture............................................................................90 6-9 3 MTF 0.1 mm pixel, 1.0 mm aperture..............................................................................90 6-10 4 MTF 0.05 mm pixel, 1.0 mm aperture............................................................................90 6-11 5 MTF 0.05 mm pixel, 1.5 mm aperture............................................................................91 6-12 YSO image of MTF Target on a tile panel........................................................................91 6-13 Selection and smoothing steps for the MTF calculation from a step function..................92 6-14 An example of the edge pr ofile and its first derivative......................................................93 6-15 Edge function width estimation.........................................................................................93 6-16 Numerical evaluation of the firs t derivative of the edge function.....................................94 7-1 Matlab user interface...................................................................................................... ....97 7-2 MTF menu and data profile...............................................................................................98 7-3 Data profile............................................................................................................... .........98 7-4 User interface for information entries................................................................................99 7-5 Maximum search............................................................................................................. ...99 7-6 Saving files............................................................................................................... ........100 7-7 Data profile from an edge function and its first derivative..............................................100 7-8 Selection of a region of intere st in the edge function profile...........................................101

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11 7-9 The selected region of interest and th e first derivative of the edge function...................101 7-10 MTF curves with frequencies expresse d in line pairs/pixel and line pairs/mm...............102 A-1 Comparison between two backscatter images.................................................................105 A-2 Line profile evaluation of the paper filtering...................................................................106 B-1 MTF frame plate top view...............................................................................................107 B-2 MTF cover plate top view................................................................................................108

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12 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science AN ADAPTED MODULATION TRANSFER F UNCTION FOR X-RA Y BACKSCATTER RADIOGRAPHY BY SELECTIVE DETECTION By Nissia Sabri August 2007 Chair: Edward T. Dugan Major: Nuclear Engineering Sciences The Modulation Transfer Function (MTF) is a quantitative function based on frequency resolution that charact erizes imaging system performance. In this study, a new MTF methodology is investigated for application to Radiography by Selective Detection (RSD). RSD is an enhanced, single-side x-ray Compton backscatter imaging (CBI) technique which preferentially detects selected scatter compone nts to enhance image contrast through a set of finned and sleeve collimators. Radiography by sel ective detection imaging has been successfully applied in many non-destructive evaluation (NDE ) applications. RSD imaging systems were designed and built at the University of Florida for use on the external tank of the space shuttle for NDE of the spray-on foam insulation (SOFI) inspection. The x-ray ba ckscatter RSD imaging system has been successfully used for cracks and corrosion spot detec tion in a variety of materials. The conventional transmission x-ray image qua lity characterization t ools do not apply for RSD because of the different physical process invol ved. Thus, the main objective of this project is to provide an adapted tool for dynamic range evaluation of RSD system image quality. For this purpose, an analytical model of the RSD imag ing system response is developed and supported.

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13 Using the Fourier transform and Monte Carlo methods, two approaches are taken for the MTF calculations: one using a line spread func tion and the other one using a sine function pattern. Calibration and test targets are then designed according to this proposed model. A customized Matlab code using image contrast and digital curve recognition is developed to support the experimental data and provide th e Modulation Transfer Functions for RSD.

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14 CHAPTER 1 INTRODUCTION The purpose of this investigation is to presen t and explain the differe nt approaches that have been taken to develop a Modulation Tr ansfer Function adapte d to the Radiography by Selective Detection RSD imaging system1-3 for the purpose of defining a process to measure system response by evaluating the image quality. The first objective of the MTF calculations was to give a complete specification of the RSD scanning system properties. Therefore a frequency characteriza tion of the output/input linking was desired. However, the backscattered field is highly dependent on the scanned object meaning that a complete descrip tion of the imaging process for al l applications is not possible with a unique transfer function. After an overview of the physic al process involved in this type of imaging, the experimental results are presented. The major sec tions treated are: the preliminary impulse and step functions responses, the desi gn of an MTF plate to simulate a sinusoidal input function, the use of MCNP5 and variance reductio n techniques to model the input function, the fitting process to associate mathematical functions to the expe rimental data, two proposed models for the MTF measurements (the sinusoidal and the step functions) and finally, the Matlab codes for practical calculations. Compton Backscattering Imaging (CBI) In this section X-ray production is described for imagi ng applications. The physics of the photon interactions with matter is treated in de tail in Chapter 2 For a standard transmission process, X-ray images are maps of the x-ray at tenuation coefficient. To a large extent the attenuation depends on the chemical composition and physical state of the attenuating medium. In Compton Backscattering Imaging (CBI), im ages are maps of X-ray photon backscattering4.

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15 X-rays are produced by focusing a beam of high energy electrons into a small focal spot on an anode. The rapid deceleration of the electrons after they enter the metal of the anode produces a broad continuous spectrum of Xrays called Bremsstrahlung. Figure 1-1 shows the basic principle of X-ray production. There is also a probability fo r electrons to ionize the atoms in the anode, creating vacancies in the inner electrons shells. These vacancies ar e rapidly filled by transitions from outer electron shells, with the emission of characteristic X-rays5. The energies of these discrete line spectra are characteristic of the a node chemical element. The total spectrum obtained from a typical X-ray tube with a tungsten anode is shown in Figure 1-2. As the X-rays traverse the object being scanned, they may be scattered, either elastically or inelastically, or they may be totally absorbed in a photoionzation process. More details on these physical processes and their dependence on photon energy can be found in Chapter 2. A transmission imaging system consists of an X-ray source, the object being radiographed, and a detector. From an imaging standpoint there is an important distinction between absorption and scattering. Usual X-ray scanning systems use tr ansmission (i.e., forward scattered ) photons while CBI uses backscattered photons. The reason for employing a CBI system is simple; for some applications it is impossible to have f ilm or a detector behind the scanned object. By illuminating a single point on the target and having a set of detectors collecting the backscattered photons, it is possible to reconstr uct the image with a sp atial mapping. The image is thus a two-dimensional proj ection of a three-dimensional obj ect; many planes are collapsed

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16 into one. The information is not given by photons which pass throw the sample like in transmission radiography, but is given by photons which are scatte red back on the same side as the source. The detector senses photons coming back from the sample. These photons have interacted with the medium (Compton interaction) and are scattered back with a different energy. The energies and angles of backscattered photons de pend on the energy of the incident photons and the medium with which they interact. By counting the number of photons coming back, information about the target can be deduced. Backscatter Radiography by Selective Detection (RSD) Overview of Previous Work The technique developed at the Nuclear Engi neering Department at the University of Florida, called Latera l Migration Radiography6-14 (Figure 1-4) is similar to the CBI technique (Figure 1-3), but instead of counting only single-collision b ackscattered photons, the LMR technique counts both singleand multiple-collisi on backscattered photons that have laterally spread out from the illumination beam entry point. At the detector surface, signals from singleand multiple-collision backscattered photons overlap. Therefore, they cannot be expected to cast a sharp shadow image. Instead, the backscattered radiations form a broad, diffuse di stribution on the detector severely impairing the distinction between deep and shallow objects. This technique, with some modifications, later led to the Backscatter Radiography by Selective Detection RSD. By adding adjustable collimators to the detectors it was possible to select the backscattered photons being counted, especially the de pth of the counted photons. By preferentially selecting specific components of a scattered photon field, information relating to specific locations and properties of an imaged sample can be extracted.

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17 Project Objectives The components that form the RSD scanni ng system are different and complex. Four major parts can be identified: X-ray generator, detectors, the electronics and the image acquisition and processing. The objective of this study is to characterize the system response depending on different setups and components. Since the development of the first RSD scanning system, there has not been an experimental methodology to measure syst em performance. The global response of the system depends on the individual performance of each component. The purpose of this project is to define a process to measure the system re sponse by evaluating the image quality. Since the image is the system output, it gives an indi cation on how all the components are performing together. From a physical system point of view, the char acterization of the response must be defined through the input/output re lationship. Then the challenge is to develop an expression for this relationship which provides a basis for evaluatin g the performance of the imaging device and understanding the nature of its evaluated image properties. From the image processing standpoint, contra st and resolution ch aracterize the image quality. Therefore, the calculation of the Modulat ion Transfer Function (MTF) would be a better characterization parameter if it is related to the contrast and resolution. RSD Scanning System Detector response and image acquisition obser ved throughout this study are generated using the RSD scanning system developed for Lockheed. Moving Table: X-Ray Source and Detectors The system used in this study consists of four sodium iodide [NaI (Tl)] scintillation detectors, one YSO detector and a Yxlon MC G41 X-ray generator mounted onto a scanning

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18 table with X Y scan motion capabilities. The [N aI (Tl)] detectors are positioned at the corners of an eighteen by eighteen centimeter square, centred on the X-ray beam. The YSO detector orbits on an aluminium ri ng around NaI detector two. YSO images are usually comparable to the NaI images in image contrast. Although the YSO detector has much less detection surface area (5.06 cm2 vs. 20.3 cm2), it has a slightly higher quantum efficiency compared to the NaI for low energy X-rays (10-55keV). The detector is also much lighter and smaller than the NaI detector so it can easily be positioned to obtain better images. Each [NaI (Tl)] detector compri ses a two inch diameter by two inch thick NaI scintillation crystal mounted onto a photomu ltiplier tube (PMT) and a fast preamplifier specifically designed to ha ndle high count rates. A schematic of the RSD [NaI (Tl)] detector s components and their configurations is presented below in Figure 1-5. In Figure 1-6, the YSO is mounted on detector 2 using an aluminium ring. In Figure 1-7 the RSD system is mounted on a fixed frame. The 230 ns constant decay time of the NaI(T1) crystal (230ns) allows sufficient light and charge collection time from the NaI and PMT, while allowing the detectors to measure backscatter fields up to 800,000 c ounts per second, without experi encing statistically significant pulse pile-up19. Image Acquisition :Signal Flow and Software The signal recorded from the scanning syst em is processed and displayed through a Labview code. The following flow chart (Figure 1-8) presents the entire image acquisition process from detection to display.

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19 Figure 1-1. Schematic il lustrating X-ray production Figure 1-2. Typical spectrum obtained from an X-ray tube with a tungsten anode4 High voltage + X-ray tube Anode Electron gun X-rays Electron beam Sample

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20 Figure 1-3. Compton Back scattering Imaging (CBI) Figure 1-4. Lateral Migr ation Radiography (LMR) Land mine Collimated detector Noise Signal X-ray generator Earth Uncollimated detector X-Ray Generator Object Detector Signal Noise

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21 Figure 1-5. Photograph of RSD System with 4 NaI Detectors Figure 1-6. Photograph of RSD System showing YSO detectors mounted to NaI Detectors Aluminium ring YSO detector NaI detector 2 A set of YSO detectors NaI detector Sleeve collimator extended X-ray beam tube Finned Collimator Angle at 90 (degrees)

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22 Figure 1-7. RSD scanning syst em mounted on a fixed frame

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23 Figure 1-8. Flow chart of the image acquisition process20 Dir Active Step Dir Step Step X-axis Complete Pulse train Complete Pulse train Complete Pulse train Y-axis Y-axis X-axis X-axis Visible light X-Ray Current Analog pulse Analog pulse Analog pulse Digital pulse Analog pulse Yes X rayscatteredtowarddetector Na I Photo-Cathode PMT PreAmp FastAmp oscilloscope MCA SCA is the pulse in the voltagewindow Counter /Timer Pulse train BNC 2121 NI-Daq PCI 6602 Labview/computer Y axis Y Motor Y MotorAmps or image NI-Motion PCI 7344 NI-Motion breakout box Limit/Home Switches X-Motor X-Motor Amps

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24 CHAPTER 2 PROBLEM STATEMENT General Physics of Photon Interaction When considering an X-ray based scanning sy stem, it is highly important to understand how the photons interact with matter4. There are five types of inte ractions with matter by X-ray photons which must be taken into account. Compton effect Photoelectric effect Pair production Rayleigh (coherent) scattering Photonuclear interactions Since the importance of an in teraction for the purpose of th is study is being measured by the energy released in the medium, the three firs t interactions are the most important. The photon energy is transferred to electrons, which then impart that energy to matter in many Coulombforce interactions along their tr acks. Rayleigh scatteri ng is elastic (total energy conserved, and kinetic energy conserved), meaning that the photon is merely redirected within a small solid angle with nearly no energy lo ss. Photonuclear interactions are only significant for photon energies above a few Mev, where they may cr eate radiation-protection problems through the ( ,n) production of neutrons and consequent radioactivation. The relative importance of the Compton Effect photoelectric effect, and pair production depends on both the photon quantum energy ( h E ) and the atomic number Z of the absorbing medium. Figure 2-1 indicates the regions of Z and Ein which each interaction predominates. The photoelectric effect is dominant at the lo wer photon energies, the Compton effect takes over at medium energies, and pair production domin ates at the higher energies (with a threshold of at least 1.02 Mev because the photon energy mu st exceed twice the rest mass of an electron).

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25 For low-Z (e.g., carbon, air, aluminum, Sprayon Foam Insulation) media the region of Compton-effect dominance is very broad, extend ing from approximately 20 keV to 20 Mev. This gradually narrows with increasing Z. However, fo r Al, the PE effect is dominant up to about 50 keV. According to the previous de scription it is easily understand able why the Compton Effect is the one that characterizes the photon inter actions in an RSD scanning system. The following description deals with some as pects of the Compton Effect th at are essential to understanding how the image is formed in the RSD scanning system. Compton Effect A complete description of th e Compton Effect must cover two major aspects: kinematics and cross sections. The first one re lates to the energies and angles of the participating particles when a Compton event occurs; the second predicts the probability that a Compton interaction will occur. Two major assumptions are made in the followi ng theoretical approach: the electron struck by the incoming photon is initially unbound and st ationary. These assumptions are not rigorous since the electrons occupy different energy le vels and, thus, are in motion and bound to the nucleus. However, for low Z materials the bi nding effect does not introduce that much modification in the cross section value. As presented in Figure 2-2, a photon of quantum energy E incident from the left strikes an unbound stationary electron, scattering it at angle relative to the incident photons direction, with kinetic energy T. The scattered photon E departs at angle on the opposite side of th e electron direction, in the same scattering plane. Energy and momentum are each conserved. The assumption of an

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26 unbound electron means that the above kinematics re lationships are indepe ndent of the atomic number of the medium. Kinematics The relationships between angles and energies are given in Equation 2-1 ) 2 tan( ) 1 ( ) cos( )) cos( 1 )( ( 12 0 2 0 c m h h h T c m h h h (2-1) Where2 0c m the rest energy of the electron, is 0.511 Mev, and ', h h and T are expressed in Mev. There is a one-to-one relation between 'hand angle of the scattered photon for a given energy of the incident photon. The photon transfers a portion of its energy to the electron. All scattering angles for the photon (between 0 to 180) are possible and the ener gy transferred can vary from zero to a large fraction of the photon energy. Cross Section The microscopic cross section is the effective target area pres ented to an incident photon. The earliest theoretical description of the process was provided by J.J. Thomson In this theory the electron that scatters the incident photon is assumed to be free to oscillate under the influence of the electric vector. The Thomson differential cross section pe r electron for a photon scattered at angle per unit solid angle is based upon classical mechan ics/electrodynamics and is expressed as:

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27 ) cos 1 ( 22 2 0 0 r d de (2-2) Later on, Klein-Nishina developed (based upon quantum mechanics) a new definition for the Compton Effect cross section15. This treatment was more succe ssful in predicting the correct experimental value, even t hough the electron was still assume d unbound and initially at rest. The Klein-Nishina differential cross section for photon scattering at angle per unit solid angle and per electron may be written in the form ) sin ( ) ( 22 2 2 0 0 h h h h h h r d de (2-3) Equation 2-3 is the one usually used for standard calculation of the cross sections, 2 0r is squared value of the classical electron radius. In the low-energy limit of Compton scatter (h less than about 10 keV), h h regardless of the phot on scatter angle and Equation 2-3 reduces to Equation 2-2. Theoretical Approach of the Modu lation Transfer Function (MTF) There are several ways to measure the MTF. Some of them are largely applicable to different recording systems; either the image is recorded on a film or it is processed to be displayed on a screen. The two major techniqu es are the Sine Wave Method and the Spread Function Method16. The main problem associated with the firs t method lies in the production of a spatiallysinusoidal exposure of known modulation. A relatively straight forward method is to pho tograph a variable area test chart for an input exposure that is a one-dimensional sinusoidal distribution defined by:

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28 ) 2 cos( ) ( x b a x f where is the one-dimensional spatial frequency (or line frequency), and is a measure of the phase. The output is also sinusoidal with the same spatial frequency as the input, but with a change of amplitude, or modulation. The ratio of the output modulation to the input modulation depends on the spatial frequency, and turns out to be equal to the modulus of the Fourier transform of the line spread function. The modulus of the Fourier tr ansform of the line spread f unction l(x) is defined by: 1 1 1 1 1 1 1 2) ( ) ( ) ( ) ( ) ( ) ( dy y x h dy dx y x h x x x l with dx e x l Mx i (2-4) Note that the line spr ead function of an imaging system is defined as the response of the system to a line input. A line input may be represented by a single delta function, ) (1x which lies along the y1 axis. It is the ratio of output to input modulation that is called the Modulation Transfer Function, or MTF. The input modulation is defined by: a b f f f f Min min max min max. Since the system response is a convolution of the input and th e point spread function of the system, the output can be written as: 1 1 1 1 1 1 1 1 1 1 1) ( )) ) ( 2 cos( ( ) ( ) ( ) (dy dx y x h x x b a dy dx y x h y y x x f x g (2-5) Integration with respect to y1 using (2.4) gives: 1 1 1) ( )) ) ( 2 cos( ( ) (dx x l x x b a x g (2-6)

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29 where ) (1x lis the line spread function define d earlier. Using the expansion: ) sin( ) sin( ) cos( ) cos( ) cos(B A B A B A (2-7) ) (1x lis normalized such that its area is unity, i.e. 1 ) (1 1dx x l, then 1 1 1 1 1 1) 2 sin( ) ( ) 2 sin( ) 2 cos( ) ( ) 2 cos( ) ( dx x x l x b dx x x l x b a x g (2-8) or ) ( ) 2 sin( ) ( ) 2 cos( ) ( S x b C x b a x g (2-9) where 1 1 1) 2 exp( ) ( ) ( ) ( ) ( dx x i x l T S i C (2-10) The function ) ( Tis the optical transfer function, and ) ( Cand ) ( S are its real and imaginary parts. The optical transfer function is the Fourier transform of the line spread function. Defining ) ( ) ( and Mas the modulus and phase of the optical transfer function, they can be expressed as: ) ( sin ) ( ) ( ) ( cos ) ( ) ( ) ( ) ( tan ) ( ) ( ) (1 2 2 M S and M C C S and S C M (2-11) And by using these, then Equation 2-9 reduces to: )) ( 2 cos( ) ( ) ( x b M a x g. (2-12) Equation 2-12 shows that the output is sinusoi dal and has the same frequency as the input. The output modulation is defined as:

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30 a b M g g g g MOUT) (min max min max (2-13) Thus, the ratio of the output modulation to the input modulation is simply equal to) ( M, the modulus of the Fourier Transform of the line spread function. Since the area under the spread function has been defined as unity, the MTF will be normalized to unity at zero spatial frequency: 1 ) ( ) 0 (1 1 dx x l M (2-13) Given a sinusoidal input of constant modulation a b the system frequency response can be deduced from the output image contrast min max min maxg g g g after dividing by a b. Due to the general non-linearity of the s canning process and the uncertainty in characterizing the input function, the MTF deduced from spread function measurements will not generally be exactly the same as that obtained from the sine-wave method. The line spread function method could be perf ormed either by simulating an experimental pulse with a Dirac function or by scanning an edge and differentia ting. The last step then is performing a Fourier Transform calculation. The Fourier Transform Applied to Image Processing The general definition of the Fourier Transfor m of a function f(t) in one dimension is dt t f t i t f F G) ( ) 2 exp( )) ( ( ) (1 (2-14)

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31 Two conditions are assumed to be satisfied fo r f(t) : continuity and periodicity The extension of this definition to two or three dimensions is straightforward with the spatial exponential function written as) ( 2 exp(z y x i ). The real utility of the Fourier Transform is that it has a simple inverse. d G t i G F t f) ( ) 2 exp( )) ( ( ) (1 1 (2-15) For a linear system a Fourier Transform of the input is defined as follows du u w u k i k Win in) ( ) 2 exp( ) ( (2-16) With the linearity condition, the system output is a superposition of individual outputs. ') ( ) ( ) ( ) ( ) (dt t w t t p t w t p t win in out This type of integral is known as a convolution product where p(t) is the spat ial system response function. The main utility of the Fourier Transform is to give an equivalent expression of the function in frequency space. In frequency space the convolution product is eq uivalent to the usual multiplication. Thus, in frequency space the output is the multiplicatio n of the input function by the system response function. The last important property of the convolution product is that the unit function is Diracs function. Thus, the response to an impulse input is the system response function. MTF Applied to the RSD Scanning System The Modulation Transfer Functi on from a scanning system characterization standpoint is the spatial frequency response of an imaging system or a component defined by the contrast, C, at a given spatial frequenc y relative to low frequencies.

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32 Spatial frequency is typically measured in cy cles or line pairs per millimeter. High spatial frequencies correspond to fine image details. Th e more extended the response, the finer the detail. Two methods were used to perform the MTF calculation. The first one is based on the response to a sinusoidal input illumination. The second one uses the magnitude of the Fourier Transform of the point or line sp read function which is the respons e of an imaging system to a pulse input such as a point or a line. Due to technical issues the experiments were performed using sine patterns of various frequencies and various diameters. A more adapted pattern would have been achieved by keeping the diameters constant to have a consta nt modulation. However, the drilling process is technically difficult for holes of large diamet ers and small separation. The patterns were produced using nylon lines (cylindrical shap e) of different diameters and spacing. The following definitions were used ) 0 ( ) ( % 100 ) (Contrast f Contrast f MTF (2-17) where min max min maxV V V V f C is the contrast at the spatial frequency f and B w B wV V V V C 0 is the low frequency contrast (the largest line pair ). The above contrast va lues are the immediate applications of the theory detailed previously. max min, ,V V V VB wrepresent the luminescence for a patt ern at the associated frequency. wV, BV are maximum (white) and minimum (black ) luminescences, respectively, at zero frequency. max min,V V are maximum and minimum luminescences, respectively, at any frequency f.

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33 It is important to notice that in the case of X-ray backscattering, an MTF calculation based on the output image contrast depends on the spectr um, the target material and geometrical set up of the system if not properly normalized. In usual transmission imaging the MT F is a projection on a 2D plane (Figure 2-3.3). The signal recorded through the target does not interact w ith the target pattern. The photons counted are those that have not been absorbed by the patte rn. Thus, the actual volume of the target is not a critical parameter. When performing X-ray backscatter imaging, the signal measured is formed by the photons that interacted with the target pattern (Figure 2-4). Thus, the amplitude of the signal depends on the volume intersection of the pa ttern and the beam or the reaction rate. The use of cylindrical lines in the pattern is to minimize the errors when generating a sinusoidal input. The lines in the pattern are made of nylon, which has the best ratio of scatterto-absorption cross section in the energy range of interest: 5. 1 at 35 keV and 26 at 60 keV. The choice of varying the cylinder diameter wi th the frequencies intr oduced an additional challenge when dealing with the volumetric normalization. The inte rsection volume of two cylinders at 90 is easily represented by an inte gral function. However, because the beam sweeps continuously over the cylindrical line, a summation of integrals is needed. This aspect will be treated later on.

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34 Figure 2-1. Photoelectric, Compton and Pair Production5. Figure 2-2. Kinematics of the Compton Effect h E Momentum= c h e 0 'h E Momentum= c h' KE =T Momentum=P

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35 Figure 2-3. Transmission model Figure 2-4. Backscatter model X-ray generator Detector 1 Detector 2 Backscattered photons Cylindrical shape pattern Nylon lines X-ray generator Rectangular shape pattern Detector 1 Detector 2 Transmitted photons

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36 CHAPTER 3 PRELIMINARY EXPERIMENTS: PULSE AND STEP FUNCTI ONS SIMULATION RSD System Experimental Responses One of the first objectives was to vary one parameter at a time. The spacing was varied using a limited number of lines due to the lack of precision in the spaci ng setup in preliminary experiments. Experimental results presented in Figures 3-1 show a scanning output of two pairs of nylon lines with the associated Line Spread Function profile. The two sets of line pairs were of the same diameter 0.3 mm at 45 degrees with respect to the ve rtical axis with 3 mm and 1 mm spacing respectively from left to right on the line profile. The Line Spread Function (Figure 3-1) show s a typical loss of contrast with increasing spatial frequency of the line pairs. The decrease of the amplitude between maxima and minima is the indication of the contrast loss. This experi ment was only meant to demonstrate the relation between the frequency increase and the loss of contrast. Pulse Input Experiment Relative to the dimensions of the system, a pulse input can be approximated by a single thin nylon line (0.3 mm diameter) with a 1 mm beam. Since the system response depends on the inters ection volume of the beam and the line, the use of a small source beam apertu re with a thin line simulates a Finite Dirac function. Figure 3-2 is a high resolution, singl e-line scan of a nylon line (0.3 mm diameter) with 0.02 mm pixel size. A convolution product shows that in the ideal case, the system output for a Dirac input gives the Transfer Function. ) ( ) ( ) ( x response System x Input x Output (3-1)

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37 Since the Dirac function is the convolution produ ct unit operator, the ou tput is the system response. By fitting the experimental data, a math ematical expression for the system response to a line can be derived. Step Function Experiment This experiment simulates an edge function. The Fourier transform of the edge function should give the same Modulation Transfer Functi on (MTF) as the line spread function. In the frequency domain the output is defined as follows: ) ( ) ( ) (f response System f Input f Output (3-2) With indicating regular multiplication. For modeling an edge function the target is a plastic piece of 17.5 mm width as shown at the bottom part of Figure 3-3. Principles of Statistics and Curve Fitting Applied to MTF Calculation Figure 3-2 and Figure 3-3 show experimental data profiles and th e fitting functions associated with them. To be valid the fitting function must be statistically equal to the experimental profile. Thus, this s ection covers the basics of statis tics applied to data samples and more precisely applied to fitting functions. In order to evaluate the fitting efficiency of a given function, some statistical tests are performed for each data set. One of th ese tests is the determination of R, the Correlation Coefficient. The closer the determination coefficient 2 R is to 1, the better is the fit. A correlation measures the strength of the pr edicted relation between the experimental data and the fitting function. The stronger the correlation the better the fitted function approaches the experimental data.

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38 Given n pairs of observations (i iy x,),with x the experimental data and y the fitting function value, t he sample correlation is computed as yy xx xy yy xx i iS S S S S y y x x R ) )( ( (3-3) Where the sums of squared residuals are defined as i i yyy y S2) (=SS(Total) (3-4) The Chi-square test is a different measure of the goodness-of-fit. The test2 measures the deviation between the sample and the assume d probability distribution (i.e., hypothesis). The value of Chi-square is calculated according to the following formula, n i i i iNp Np N2 2) ( (3-5) Where np p p p,..., ,3 2 1 is a set of hypothetical probabilities associ ated with N events falling into n categories with observed relative frequencies of N N N N N Nn/ ,..., / /2 1. For large values of N, the random variable 2approximately follows the 2-distribution density function with n-1 degrees of freedom. The F-test is another statistical tool that can be used, for example to test if different MTF curves are statistically equal. Here are some explanations on how the F-test is performed. First the two data sets (the measured data and the data from the library) are individually fitted using the fitting function. Then the two data sets are combined (appending one to the other), and then a fit is performed on the combin ed data set with the same function. From these three fits, the values for the SSR (sum of squa res of the difference between the data and fit values) and the DOF (number of de grees of freedom) are obtained.

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39 Then, SSR1, DOF1, SSR2, and DOF2 are obt ained from the individual fits, and SSRcombined and DOFcombined are obtained from the fit of the combined data. The following values are computed: SSRse parate = SSR1 + SSR2 and DOFseparate = DOF1 + DOF2 The last step is performed by computing the F value. e SSRseparat e DOFseparat e DOFseparat d DOFcombine e SSRseparat d SSRcombine F* ) ( ) ( (3-6) Once the F value is computed, the p-value is computed using the formula: ) ), ( ( 1e DOFseparat e DOFseparat d DOFcombine F invf p (3-7) This p-value is then used to make a statistical statement as to whether the data (not the parameter values) are significantly di fferent or not. If the p-value is greater than 0.05, we can say that the data sets are not significantly different at the 95% confidence level. Results and Analysis Pulse Function Experiment In order to obtain the MTF from experimental da ta, it is necessary to obtain a mathematical function from a data fit. Once the fitting function is obtained, the Fourier Transform of the profile gives the system response function in th e case of a pulse input. To perform the fitting, a Lorentzs model was used with the following equation: ) ) ( 4 ( 22 2 0w x x w A y yc (3-8) where 69682 12 40838 459 0139 0 37763 0 00462 0 62368 4 60653 2 57969 21350A w x yc with statistical tests on the data 39739 26666 85644 02 2Dof R

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40 The data profile used in the Pulse function ex periment has been obtained from a scan at 45 kVp, 45 mA with a beam aperture size of 0. 5 mm and a pixel size 0.02mm x 1mm. The line was 0.050 mm width. Once the mathematical formulation was established, the next step was to calculate the Fourier Transform of the obtai ned function (Equation 3-8). Since the exact formula depends on different constants that change acco rding to the experimental conditions, it is more valuable to determine the general shape of the Fourier Transf orm than the precise mathematical expression. By using Equation 3-9 X XFT 2 ) exp( ) ) ( exp( 22 2 (3-9) letting ) ( 20x x X and using the following formulas ) ( 1 ) ( ) exp( ) ( ) (0 0 f x f x j f x x fFT FT The Fourier Transform of E quation 3-8 is obtained as ) 2 ( ) 4 exp( ) 4 exp( 4 2 ) ) ( 4 ( ) 2 ( ) 2 ( 20 0 2 2 0x y x x j x w A w x x w A y yFT c (3-10) The Fourier Transform modulus gives the Modulation Transfer Function: MTF_dirac_function ) 8 exp( 4 2 ) 16 exp( 4 22 2 0 2z w A x x w A (3-11) with ) ( 21 mm x z The above formula gives the general behavior.

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41 The Step Function Experiment When the step function is treated, the best fi tting function for this sh ape is provided by the Bolzmanns model ) exp( 1 ) (0 2 1 2dx x x A A A y (3-12) Where 05193 0 0441 0 0304 0 7751 3 17543 2 69652 971 35331 5 42214 4950 2 1dx x A A and 87267 397 98645 02 2Dof R for the statistical tests When using a step function to define the MT F an additional step is needed before the Fourier Transform. A first derivative is performed. 2 0 0) 1 ( ) 1 2 ( ) (dX X X dX X Xe e A A dX X dY (3-13) Due to the complex form of the above function, a straight forward calculation of the Fourier transform is not possible. An alternative approach was to perform the derivative and it s Fourier Transform numerically. Then by fitting the function a ma thematical formulation was established. MTF_edge_function ) ) ( 2 exp( 2 *2 0w z z w A yc (3-14) with ) ( 21 mm x z

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42 01373 5 57833 438 00568 0 65586 0 00177 0 7481 1 66317 2 65077 1416 0A w E z yC with statistical test on data 95055 76 99794 02 2Dof R The data profile used in the Pulse function ex periment has been obtained from a scan at 45 kVp, 45 mA with a beam aperture size of 1 mm and a pixel size 0.5mm x 0.5mm. The line was 0.050 mm width. Even though the mathematical expressions fo r the pulse based MTF and the step function MTF are not exactly the same, the general behavior follows) exp(2z with a constant. Figure 3-1. Scanning system output two line pairs placed at 45with respect to the vertical axis 01020304050 1450 1500 1550 1600 1650 1700 Number of counts/pixelDistance x (cm) Line profile High contrast Low contrast

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43 0246810 2000 2200 2400 2600 2800 3000 line spread function(1) Lorentz fitting function(2)Number of counts/pixelDistance x in ( mm ) Figure 3-2. High exposure s canning output, one sweep of a nylon line (Dirac Simulation) 05101520253035 2000 4000 6000 8000 10000 12000 14000 16000 Number of counts/pixelDistance x in (mm) B Boltzmann fit of Data33_ B Figure 3-3. Scan of a cubic plastic sample : 17.5 mm width, 1 mm beam, 0.5 mm pixels Line profile Line profile

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44 CHAPTER 4 MTF CALCULATION BASED ON A SINUSOIDAL INPUT FUNCTION MTF Sinusoidal Pattern Design The first idea was to generate a sinusoidal input pattern using nylon line of different diameters and spacing. Figure 4-1, showing five nylon lines, an x-ray generator and two detectors, illustrates the scheme for simulating a si nusoidal input. As the scanning system sweeps over the lines, a sinusoidal signal is formed at the detector face The actual MTF target contains 5 lines for each diameter. This is to ensure good statistics in the results. The actual MTF target consists of an aluminum frame to hold different diameter nylon lines with varying spatial frequencies. Figures 4-2 and 43 show the MTF plate design. The target frame is 25.4 cm x 12.7 cm (10 x 5 inches) and 0.3 cm (1/8 inch) thick. The nylon lines are strung across the 7.6 cm (3 inch) ai r gap in the center of the frame. A cover plate was designed to be attached to the back of the frame to protect the nyl on lines connections and provide a flat surface on which the target sits. The cover plate is 0.6 cm (1/4 inch) thick. Twelve sets of holes were in itially designed. Two additional levels of holes sets were included in the design to vary the frequenc y while the diameters are kept constant. System Response to the Input Modulation Function Digital Output Profile Figure 4-4 shows the output profile obtained fr om scanning the MTF Sine target at an Xray energy of 45 kVp and a current of 45 mA. This profile was obtained fr om detector 1 (NaI). For this particular set up, the de crease in contrast started at the sixth set of lines corresponding to a diameter of 1.28 mm (0.39 line pairs/mm). The lo ss of contrast is noticeable when there is an increase in the minimum values of the profile, i.e. a shift in the baseline.

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45 After the eighth set of lines, the five peaks of each new se t are not distinguishable. Thus, the loss of resolution starts at a line diameter of 0.52 mm (0.96 line pairs/mm). The loss of resolution is defined with respect to the Full Width at Half Max (FWHM). If the separation between two maxima is smaller than the width of the individual peak at half its maximum value than the resolution betwee n the two peaks is lost. Comparison of Detection Propertie s Between NaI and YSO Crystals In the previous section, the output profile was treated from a digital imaging point of view and no special care was taken to evaluate the best detector configurations. However, since the detectors themselves have limited efficiencies, it is necessary to quantify their responses with respect to the backscattered spectrum. Two types of detectors were used in the MTF experiments: NaI and YSO. Figure 4-5 shows the scattering-to-absorption ratios for both NaI and YS O. The values obtained are for NaI and Y5SI2O crystals17. The lower the scattering-to-absorption ratio the better the detection capabilities. In the energy range of interest (below 50 keV) the Y5Si2O crystal has a more favorable scattering-toabsorption ratio than the NaI from about 16 keV to 33 keV. At about 16.4 keV, the ratio achieves a maximum value of 0.0796 for the Y5Si2O. The NaI crystal is a much better detector at energies higher than 33 keV. Since the Y5Si2O was the most frequently used detector for the MTF experiments, the following study will concentrate on characterizing the Y5Si2O detection performance with respect to detected energies. First, it is necessary to calculate the average energy of the backscattered spectrum using a Monte Carlo simulation. The m odel used is based on MCNP5 analog simulations and the layout is descri bed in detail in the following section.

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46 The average energy of the incident X-ray beam is 22.73 keV and its maximum energy is 50 kVp. The average energy of the backscattered spectrum given in Table 4-1 is 26.74 keV. This value was obtained by averaging over the five energy bi ns with the number of particles used as weighting functions. A non-analog ru n gives essentially the same result with an average detected energy of 26.75 keV and a relative error of 0. 021%. A more detailed analysis on the Analog versus Non-Analog results will be given in the following section. A Model of the Sinusoidal Input Functi on Using MCNP5 and Variance Reduction Techniques As shown in the previous section, the output profile is easily obtai ned from scanning the MTF Sine target. However, there is no experiment al way to precisely determine the input profile. Thus a Monte Carlo model is necessary to correctly determine the input function, to correlate the output profile to the system response. Input Function from a 2D Model of the MTF Sine Target Figure 4-6 shows the MCNP5 m odel for a 2D input profile calculation. The profile obtained from the model presented in Figure 4-6 is not st rictly 2D. Actually the entire line (3D volume) is modeled but only the contribution from th e mid-plane region is used to generate the profile. This is to be compared with the profile obtained from the contri bution of the entire line. Only one line per set is modeled up to the 10th set of holes. The last two sets did not give good experimental results. Then using the pr oblem symmetry only one half of the line is modeled. In the actual experimental design, the X-ra y generator and the detector move over the target. For each mesh cell defined by (x+ x, y+ y) the number of photons recorded is used to display one pixel. To simplify th e model in MCNP5 the detector and X-ray beam are kept at the same position while the line position is varied.

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47 The start position is where the beam and the line axis intercept. Then an offset of 0.01 cm is added between the two axes for each new simulation. The final position of the line axis is such that it does not intersect with the beam any more. The detector is a cylinder of 2.54 cm diamet er with 0.635 cm thickness centered at (0, 5.08, 4.317). The plane source is defined at the bottom su rface of the detector. Note that it is not recommended to use a plane that is a physical bo undary in a system as a source plane. This can cause problems. A source plane that can be ve ry slightly offset (e.g., by 0.001 cm) from the physical plane should be used instead. From whic h the x-ray beam is sampled using a disc of 0.05 cm diameter along the z axis. The nylon line is centered for th e first position at 3.8 cm alon g the x axis as is the X-ray beam. The line is represented by a cylinder along the y axis lying on the xy plane. To model the experimental set up as closely as possible a sheet of paper underneath the nylon line and a concrete floor are modeled. There are ten different diameters to simulate. For each diameter the number of line positions is equal to the ratio of the radius and the modeled pixel size (constant 0.01 cm). Two Variance Reduction Techniques are used: DXTRAN sphere and forced collisions for modeling the input profile. The DXTRAN sphere enables the simulation to obtain many particles in a small region of interest that would otherwise be difficult to sample. Because the solid angle that sees the detector surface from the interaction volume in the line is small, a transport of pa rticle to the surface of interest is necessary.

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48 Upon sampling a collision, DXTRAN estimates the correct weight fraction that should scatter toward the detector surf ace, and arrive without collision at the surface of the sphere. The DXTRAN method then puts this co rrect weight on the sphere. The collision event is sampled in the usual ma nner, except that the particle is killed if it tries to enter the sphere because all particles en tering the sphere have already been accounted for deterministically. The DXTRAN sphere is centred on the YSO detector. Forced collisions are used to increase the fr equency of random walk collisions within the small intersection volume of th e beam and the entire nylon line. A particle can be forced to undergo a collision each time it enters a designated cell that is almost transparent to it. The particle and its weight are appropriately split into two parts, collided and uncollided. Forced collisions are often used to generate contributions to point detectors, ring detectors, or DXTRAN spheres. Here forced collisions are used as a co mplementary method to the DXTRAN sphere. The forced collision card is set such that only the particles entering the cell undergo forced collisions. The run used a 0.5 mm diameter beam, a 0.1 mm pixel and the beam was centered over the pixel. The number of runs necessary fo r this input profile calculation is 132. The energy card uses a distributi on of energies with the associ ated probabilities at 50kVp. The distribution is based on the Kramers spectrum5 modified for tungsten target attenuation and beryllium window and alum inum filter attenuation. Figure 4-7 shows the energy distribution us ed at 50kVp as a maximum energy of the incident particles in the MCNP5 model base d on the Kramers spectrum. The spectrum is distributed between 0 and 50 kVp with 74 interpolation points.

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49 Two tallies are used; they are based on th e current entering the bottom surface of the detector. The first tally records the partial and to tal currents and based on the number of particle collisions from 1 up to 6. The second tally doe s not distinguish the particles according to the number of collisions experienced before reach ing the detector but it counts particles coming from a specific cell in the mid-plane of the nylon lines. Table 4-2 summarizes the number of simulations needed for modeling the input prof ile, taking into account the number of different diameters and for each diameter the number of runs. In addition to the 132 runs necessary for th e line profiles, there is one simulation for modeling the air separation between the lines. Figu re 4-8 shows the data profile obtained from a mid-plane contribution only. The errors associated with the data profile s hown in Figure 4-8 are on the order of a tenth of a percent. Table 4-3 shows a comparis on between an Analog MCNP5 run without any variance reduction technique and a Non-Analog run using the two indicated variance reduction techniques. The numbers of counts are given fo r a single source partic le and for a positive current with respect to the detector entrance surface. Table 4-3 shows that up to 40 keV the errors associated to both Analog and Non-Anal og techniques are below 1%. The last energy bin from 40 to 50 keV corresponds to the incident beam maximum energy; this is why very few particles are counted. As explained in Chapter 1, the energy of the backscattered particle is a fraction of the incident energy. Also according to Figure 4-5 the fraction of scatter/absorption in the YSO detector increases continuously above 20 keV and reaches a value of 0.1 between 45 keV and 50 keV. This means that a fraction of the positive current is scattered back out of the detector and even less particles are counted in this energy re gion leading to an increase in the error.

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50 In a Non-analog Monte Carlo method, the physics is biased such that the quantities to be calculated are estimated in a shorter time or with a smaller variance. To preserve an unbiased sample mean, each particle is given a statistical weight which is defi ned based on the unbiased and biased density functions. The effectiveness of the Non-Analog techniqu es is measured by a quantity called Figure of Merit, FOM, defined by: 2* (min) 1 error time FOM (4-1) Where error is the relative error. The higher the FOM, the more efficient the calculation. Table 4-4 presents the number of particles and calculation time for both Analog and NonAnalog runs. The Non-Analog run is more than 3 times faster and needs less than 16 times the number of particles to achieve the sa me order of accuracy on the results. As discussed previously anothe r aspect of the Non-Analog tec hnique is to introduce a shift in number of particles with respect to the en ergy bins. This is mostly due to the DXTRAN sphere. Some variance reduction techniques do not preserve the energy spectrum information. Input Function from a 3D Model of the MTF Sine Target The 3D input profile was obtained using the sa me layout as the one used in the previous section for the 2D profile. The only difference is that the entire volume of the nylon line was sampled instead of sampling only the mid-plan e contribution. Figure 4-9 shows the MCNP5 model used for the calculation of the 3D input profile from a nylon line. The same variance reduction techniques were us ed and the detector coordinates were (0, 0, 4.317). The profile was obtained using 1000000 particles for each of the 132 runs. Nine of the ten statistical tests were passed in MCNP5. The last test; the pdf slope was not passed.

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51 However, the relative errors associated with the obtained profile were between0.32% and 2.35%. Figure 4-10 and Figure 411 show the partial and comp lete profiles obtained from modeling the MTF sine target using MCNP5. Figure 4-10 shows the reconstruc ted input profile with only on e line for a given diameter. Each peak corresponds to one line and was obtai ned from the MCNP5 simulation. Then knowing the actual separation distances between the lines, the complete profile has been reconstructed and is shown in Figure 4-11. Table 4-5 shows a comparison between the An alog and Non-Analog results for the 3D model of the input Sine Target. Figure 4-12 shows the fraction of the contributio n of the particles to the detected signal according to their number of collisions and the av erage energy of each collision bin. The signal is dominated by the first scatter signal up to 94. 156%. The sixth collisions component is almost 0%. In order for a particle to have undergone multip le collisions and get back to the detector, it must have come from the highe r end of the source spectrum. Volumetric Normalization of the MTF The previous section treated the sine function profile at the detector face. Since the MTF target used nylon lines of different diameters and spacing, the amplitude of the sine profile varies with the line pair frequency. This variation is due to the variation line diameters and more specifically, to the variation in the intersection volume s of the X-ray beam with the nylon line. The volumetric normalization attempts to nor malize over the intersection volume to obtain a profile with constant amplitude. Two methods used are: a geometric normalization based on integrals and an MCNP5 model to estimat e the volume from the particles path.

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52 Geometric Normalization It is important to notice that the conven tional MTF calculation (e.g., as employed with transmission X-ray imaging) is performed using a multiple step data profile. This model gives a constant amplitude of the input signal distri bution after normalization per unit volume. The intersection volume of the cylindrical beam and th e target (MTF Sine pattern) sample is easily calculated in this case and remain s constant at a given frequency. In order to introduce equivalen ce between the step model and the actual Sine MTF, some definitions are given below: First, consider the intersection volume of two cylinders of the same radius in Figure 4-13. One of the cross sections is a square of side half-length 2 2z r the volume is given by r rr dz z r r r V3 2 2 2 23 16 ) 2 ( ) ( (4-2) Figure 4-14 shows the intersecti on volume of two cylinders. If the two right cylinders are of different radii Beam Liner and r withBeam Liner r then the volume common to them is : )] ( ) ( ) ( ) [( 3 8 ) (2 2 2 2 2k K r r k E r r r r r VBeam Line Beam Line Line Beam Line (4-3) Where K(k) is the complete elliptic integral of the first kind, E(k) is the complete elliptic integral of the second kind, and Line Beamr r k is the elliptic modulus.

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53 However, even with a formula to calculate the intersection volume, the complete physical process is not covered. The beam sweeps over th e lines in a continuous mode. For a given beam size, the actual intersect ion volume is related to the number of counts through the exposure time and the pixel size. This means that at each step a fraction of the volume is covered several times. The resulting overlapping contributes to th e signal (counts per peak) in different proportions depending on th e cylinders radii. As a preliminary model, only the intersection at the center is considered to give the most significant response. Although this is a restrictive approach, it give s an idea of the intersection volume contribution versus the diameter for the large line diameters. As previously explained, the data profile ha s to be redistributed for each given diameter. Thus, using the integral of the da ta and the line widths as they appear in the image, the number of counts is redistributed to flatten the ma ximum of each peak. Figure 4-15 presents the integrated profile. The idea is to obtain an equivalent of the st ep profile from a peak profile as shown in Figure 4-16.This is to avoid two competing factors of signal amp litude and frequency variations. The method consists of transforming the peak shape profile to a st ep shape profile and normalizing the number of counts per unit volume. The first step is performed using the integr al under each peak shown in Figure 4-15. The second step requires knowing the value of the in tersection volume (X-ray beam and nylon line). This volume has been calculated using Equation 4-3 assuming an intersection of the X-ray beam and the line at the center axis only. Figure 4-18 shows the experimental data and the normalized profile. From right to left each set of lines of a given diameter is shown in a specific color.

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54 Also from right to left the line diameter decr eases. At about 2.5 inches the peak data are not represented because of mism atch between the line diameter and the drilled hole diameter. This was fixed on the MTF sine plate for later ex periments. Note that up to the ninth set of lines, the normalized profile is decreasing, and th e slope is matching the co ntrast loss. Up to the ninth line set, the beam diameter is less than th e line diameter. In the tenth set the beam and the lines are of the same diameter. The two last line sets, 11 and 12, have sm aller diameters than the beam diameter. Figure 4-18 shows that the employe d model is well adapted to the first nine line sets. For line sets 11 and 12, the signals from two differe nt lines overlap. This overlapping gives an artificially high response. In fact for each line in these sets, the signal includes the response of several lines. Recall that the equivalence between cylindrical lines and the multiple step target is performed using the integral and the width. For these sets the normalization is more challenging since the number of counts reco rded and the intersect ion volume are related in a more complex manner. Volume Calculation Based on an MCNP5 Model In order to perform the volume intersection cal culations the input model used to obtain the input profile is modified. A spheri cal source is set to enclose the problem with a radius of 12 cm. Figure 4-19 shows a sketch representing th e MCNP5 input model, the two cylinders intersecting, detector, paper a nd concrete. However the figure does not show that the source sphere is centered at the origin. Note that the input set up described in Figure 4-19 is not the optimum way for doing volume calculations. However, because of the larg e number of input files needed, it was a quick start method since the inputs did not need major modifications.

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55 A more efficient configuration would be ach ieved by suppressing everything in the model but the two cylinders an d centering a much smalle r spherical source on them. Then by setting the material card to void, the volume is obtained by tallying the flux in th e intersection region. The line radii are the ones used in the prev ious MCNP5 models (f or the input profile calculations). Figure 4-20 shows a plot of the in tersection volume values versus the line radii. The absolute errors are also plotted. The inters ection volume increases with the line radius as expected and the relative errors are higher for sma ll radii. The plot in Figure 4-20 is given as an indication of the volume trend versus line radius The values shown are not exact since MCNP5 scales the flux inside the cell of interest to an unknown volume. Note that for the very small volume intersections there were zero particles in the volume of interest after running 7,000,000 hi stories. From these poor statistics, the volume values are obviously not reliable for small lines ra dii and small intersection volumes. Figure 4-21 shows a normalization based on the volume values calculated from the above model. It is expected to not have constant amplitude since the volume values over which the normalization is performed are not expected to be correct. Either a new model is necessary or a higher num ber of histories. Figure 4-21 is obtained by taking half of each peak plotted in Figure 410 and then normalizing by the intersection volume from the above results. Figure 4-22 shows a new input set up that is proposed to e nhance the volume calculation. As previously mentioned, by modeling only the two cylinders and the spherical source, better statistics are achieved. This new set up was done using 7,000,000 hist ories. Figure 4-23 and Figure 4-24 show the normalization of the input sine profile over the intersecti on volume of two cylinders.

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56 The two cylinders have the same dimensions as the nylon line and the X-ray beam. The figures show that a more effective normalization is achieved when it is performed over the individual pixels. Note that in Figure 4-23 the extreme values correspond to 10 the average value of the normalized profile for each line. This is due to the small intersection volume on the lines edges. Thus a statistical smoothing is performed over th ese values and the resu lting normalization is shown in Figure 4-24. This last plot shows the feasibility of a volumetric normalization to obtain a profile of constant amplitude. Even if the volumetric MTF couples the volum etric distribution of the target to the scanning system response, it offers a basi s for the system relative evaluation. This integrated 3-D MTF allows comparison between detectors and gives a basis on which to test a global improvement in the system. By using the same target, the volume and material parameters are kept constant in the different scans. Actual MTF Curves Based on a Sine Input Pattern An example of the MTF curves obtained from th e Sine target is given in Figure 4-25, this profile is obtained from detector 5 (Y5Si2O). The MTF presented here does not include any normalization processing. Thus, the MTF is sensitive to the change in the volume intersection of the beam and the lines. The drop in the MTF values is related to a loss of contrast and a volume variation. The relative difference in MTF values indicates the quality of the images when using the same target. The Boltzmann fitting model is given by the following formula: MTF_experimental= dX X Xe A A A Y01 2 1 2 (4-3)

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57 The corresponding coefficients are listed below. An important note is that in this section X is a frequency since it repr esents the MTFs variable. 32778 0 77575 0 0 25743 2 2 633 113 1 dX X A A 02355 1 99907 02 2Dof R Figure 4-1. Scheme for si mulating a sinusoidal input Sinusoidal Input profile X-ray generator Detector Nylon lines

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58 Figure 4-2. MTF frame plate Figure 4-3. MTF fram e plate detailed design

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59 Figure 4-4. Output profile from the scan of th e MTF Sine target (detector 1 NaI). Scanned at 45kVp, 45mA with a 0.1 mm pixel size and a 1.0 mm source aperture. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 05101520253035404550556065707580859095100Energy (keV)Scattering(Coherent+incoherent)/ photoelectric absorption NaI YSO Figure 4-5. Scattering-to-absorption ratios for NaI and Y5Si2O crystals. Deterioration of contrast Loss of resolution Baseline Average Energy of the backscattered field 26.74 keV 22.7 keV average energy of the incident beam Maximum X-ray beam incident energy

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60 Table 4-1. Number of counts at the detector surface for each energy bin and the average energy of the backscattered spectrum. Energy bins Mev Counts Error % 2.00E-02 1.08466E-02 0.1300% 3.00E-02 8.65223E-03 0.1500% 4.00E-02 3.15683E-03 0.2500% 5.00E-02 1.34348E-04 1.2200% total 2.27900E-02 0.0900% Average energy Mev 2.67437E-02 1.2989% Figure 4-6. MCNP5 model for input profile calculation. 2D prof ile calculated from mid-plane contribution Figure 4-7. Energy spectrum di stribution used in the MCNP5 m odel based on Kramers spectrum Pb shieldz x YSO detector, R=1.27cm Plane source X -Ra y beam Paper Concrete Nylon line R=0.1665cm + DXTRAN sphere Outer radius 4.635 cm Inner radius 4 cm Flagged tally from midplane interactions Forced Collisions Photons

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61 Table 4-2. Comparison between the Analog and Non-Analog MCNP5 Flagged mid-plane surface Analog Non-Analog positive current Energy bins Mev Counts Error % Counts Error % Error % Analog vs NonAnalog J+ 2.00E-02 4.77980E-03 0.2000% 4.79438E-03 0.3400% -0.3050% 3.00E-02 4.31184E-03 0.2100% 4.34073E-03 0.3600% -0.6700% 4.00E-02 1.60421E-03 0.3600% 1.60496E-03 0.6700% -0.0468% 5.00E-02 6.81481E-05 1. 7500%6.80647E-05 3.6700% 0.1224% total 1.07640E-02 0.1300%1. 08081E-02 0.2300% -0.4097%

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62 Table 4-3. Summary of the line diameters and the associated number of line position simulations Line set number Line Diameter (mm) Pixels neededNumber of runs 10 0.5 8.5 9 9 0.52 8.6 9 8 0.75 9.75 10 7 0.85 10.25 11 6 0.95 10.75 11 5 1.28 12.4 13 4 1.4 13 13 3 1.8 15 15 2 2.05 16.25 17 1 3.33 22.65 23 Figure 4-8. The input sine prof ile obtained from running MCNP5 Table 4-4. MCNP5 run conditi on for Analog versus Non-Analog Analog Non-Analog Time ( min) 25.96 7.01 Number of particles 50000000 3100000

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63 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02 0123456 Distance in cmNumber of photon detected per source particle Figure 4-9. Sine profile obtai ned from modeling 10 nylon lines of different diameters in MCNP5 0.00E+00 5.00E-03 1.00E-02 1.50E-02 2.00E-02 2.50E-02 -3101906901190169021902690 Pixel numberCount/ pixel (photon detected per source particle) Figure 4-10. The complete input profile from an MCNP5 simulation as recorded at the detector surface, pixel size 0.1mm.

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64 Figure 4-11. MCNP5 model for in put profile calculation. 3D prof ile calculated from a volume contribution Table 4-5. Comparison between Anal og and Non-Analog results in MCNP5 Response from the entire line volume Analog Non-Analog positive current Energy bins Mev Counts Error % Counts Error % Error % Analog vs NonAnalog J+ 2.00E-02 1.08466E-02 0.1300% 1.08436E-02 0.2300% 0.0277% 3.00E-02 8.65223E-03 0.1500% 8.71375E-03 0.2300% -0.7110% 4.00E-02 3.15683E-03 0.2500% 3.14837E-03 0.4000% 0.2680% 5.00E-02 1.34348E-04 1.2200% 1.35106E-04 2.1000% -0.5642% total 2.27900E-02 0.0900% 2.28408E-02 0.1400% -0.2229% Pb shieldz x YSO detector, R=1.27cm Plane source X -Ra y beam Pa p er Concrete Nylon line R=0.1665cm + DXTRAN sphere Outer radius 4.635 cm Inner radius 4 cm Forced Collisions Photons

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65 123456 Fraction of detected signa Avarage energy Number of collisions Figure 4-12. Average energy and fraction of the dete cted signal in each of the six collision bins. Figure 4-13. Intersection volume of two cylinders Figure 4-14. Two cylinder intersection volume Energy in keV 26.77 26.26 25.82 30.87 36.51 25.37 94.145% 5.512% 0.297% 0.03% 0.005% 0.0%

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66 0 20000 40000 60000 050010001500Pixel numbercount/pixel Figure 4-15. Integrated profile data Figure 4-16. Equivalence betw een peaks and steps profiles. Figure 4-17. Normalization methodology scheme. Equivalence Obtain the number of counts per unit volume of n y lon. Step 2 Normalize the new profile by the intersection volume of the X-ray beam and the line Step 1 Redistribute to have a constant number of counts along theline Obtain the data profile

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67 0 10000 20000 30000 40000 50000 60000 70000 0123456 X position in inchesNumber of counts/pixel Data profile from the image Baseline Normalised data Figure 4-18. Experimental a nd normalized data profile Figure 4-19. A representation of the MCNP 5 setup for volume intersection calculations Source z x Paper Concrete Flux tally YSO detector 12 1110 9 8 7 6 5 4 3 2 1

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68 0.00 0.50 1.00 1.50 2.00 2.50 00.020.040.060.080.10.120.140.160.18 Nylon line radius in cmVolume values from flux tallies in cm3 Figure 4-20. Line and beam intersection volume values. Beam radius 0.025 cm and line radii from 0.1665 cm to 0.025 cm 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0123456Distance in (cm ) Volumetric normalization of the input profile Figure 4-21. A plot of the volumetric normalizat ion of half peaks obtained from MCNP model.

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69 Figure 4-22. Visual editor view of th e new MCNP setup for volume calculations. 0.00E+00 1.00E+02 2.00E+02 3.00E+02 4.00E+02 5.00E+02 6.00E+02 7.00E+02 8.00E+02 0100200300400500600 Pixel numberNormalized MTF Sine input over the volume Figure 4-23. Normalization of the MTF sine profile over the intersection volume Sphere source Nylon line X-ray beam

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70 0.00E+00 1.00E+02 2.00E+02 3.00E+02 4.00E+02 5.00E+02 6.00E+02 7.00E+02 8.00E+02 0100200300400500600 Pixel numbe r Normalized MTF Sine input over the volume Figure 4-24. Statistical smoot hing of the normalized profile 0 20 40 60 80 100 120 00.511.522.5 Frequency line pairs per mmNormalized MTF (% ) Figure 4-25. MTF function from detector 5, pi xel size 0.05mm and beam aperture 0.5mm at 45 kVp-45 mA

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71 CHAPTER 5 AN IMPROVED TECHNIQUE FOR THE MTF CALCULATION BASED ON A STEP FUNCTION Step Function Target Design for MTF Calculation Figure 5-1 is a calibratio n target which can be used for the Modulation Transfer Function calculation based on the edge function method. The le ft side of the target is lead (absorber) and the right side of the target is nylon (scatterer). Figure 5-2 is the measured experimental res ponse (black line) of the RSD scanning system to an edge in units of number of counts/pixel as the scanning system moves across an edge; the fitting function is shown in red. The relation and the parameters used in the fitting process are: ) 1 exp( 1 function fitting Step0t z A y (5-1) 0.00001 6 2.3019E 1 0.07615 0.25157 368.69193 149.890471 0A t y 25 663638.531 0.816712 2Dof R The line spread function is obtained by di fferentiating the step response function formulated from Figure 5-2. Figure 5-3 shows th e Fourier transform of the differentiated step function. Both amplitude and phase are given; the resulting data are then fitted to give an MTF function. The Modulation Transfer Function is given by: ) ) ( 2 exp( 2 unction MTF_edge_f2 0w z z w A yc (5-2) ) ( 21 mm x z

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72 with the following parameters 236.34524 7 22048.3317 0.03699 5.59145 0.00711 16 9.8466E 16.80154 -214.139740A w z yC 395.45637 0.999662 2Dof R There are two important features to notice. First, the Modulation Transfer Function obtained from this latest experiment is in agre ement with the preliminary experiments performed with the edge function. Second, the MTF based on the edge function includes the effect of a geometric edge. Although the nylon and lead are at the same height, the X-rays easily penetrate the nylon compared to lead and as a result the lead /nylon interface appears as an edge to X-rays. As expected, the MTF obtained using this met hod is not exactly the one obtained from a sine input modeling (with the MT F Sine target) due to amplitude variation. However, the behavior still follows an exponent ial decrease. For the sine wave modeling with the MTF target, the MTF follows an asymptotic behavior proporti onal to exp(-x), and acc ording to this study the asymptotic behavior is proportional to exp(-x2). Finally, for calibration purposes and relativ e comparison of image quality both methods are valid. However, for simplicity and efficiency in general calibration procedures the edge response would provide a much fa ster tool. Obviously the MTF based on a Sine input is more accurate in predicting the system response versus frequency. The MTF Sine target is more sensitive to sma ll variations in contrast and resolution than the step target. A Model of the Step Function Target Usin g MCNP5 and Variance Reduction Techniques. To achieve the optimum design of the MTF step target, the system response is modeled in MCNP5. Different configurations were tested to obtain a system response as sharp as possible to

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73 approach the ideal step function. In all MCNP5 runs the same detector set up as in Chapter 4 was used. Forced collisions and DXTRAN sphere were also used as accelerations techniques. The maximum error achieved on the number of counts was 1.05%. The first target design was a cubic plastic pi ece enclosed in a lead frame of the same height. Figure 5-4 shows the geometry of the target. The lead frame is 0.5 cm thick and 2 cm hei ght, the cubic nylon piece is 2 cm by 2 cm by 2 cm. According to the MCNP5 run the mean free path of particles in the nylon piece is 1.9806 cm and about 0.00263 cm in lead. This configuration gave the data profile show n in Figure 5-5. The beam source scanned the target from edge to edge; the detector is on the left hand side at a negative x. This first configuration did not provide a satisfactory profile shape to model an edge function. A modified design of the MTF step target was tested by setting the nylon piece 1 cm higher than the lead frame. Figure 5-6 shows th e geometry of the second design of the step target. This design was chosen to reduce the ge ometric lead shielding on the edges of the nylon piece. Figure 5-7 shows the data profile obtained fr om the second MTF step target design. The profile is closer to a sharp edge function than the first design in the cen tral top region, however the drop near the lead frame is more important than in the first design A third design was tested where the nylon bloc k (2cm by 2 cm by 2cm) was laid down on a lead sheet (3 cm by 3 cm by 0.5 cm). The data profile (Figure 5-8) shows an increase on the nylon block edges that is slightly la rger on the detector side (left ha nd side). This is due to a 2 cm nylon edge that is contributing to the total signal in addition to the flat top surface.

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74 The contribution of the top center part of the target appears as a dip in the center of the profile due to the relatively high contribution of the edges. This design gives a sharper profile at the plastic/edge junction but the high contributio n of the plastic step i nduces a distortion of the center part of the profile. A better target woul d be achieved using a thinner plastic piece on a lead sheet. The final design proposed for the step target is given in Figure 5-9, it includes aluminum and lead base sheets and a junction of lead and plastic pieces of the same height. The lead and plastic pieces are sitting on the lead sheet enabling to obtain the two configurations presented in the first a nd third designs on the same line profile. Figure 5-1. Edge target ma de from a junction of lead (absorber) and nylon (scatterer) 4,004,254,504,755,005,255,505,75 0 2000 4000 Y A Figure 5-2. Scanning system response to an edge. Scatterer Absorber Distance in c m Number of counts

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75 02468 0 500 1000 1500 2000 2500 3000 r Gauss fit of FFT1_r ###Frequency (Hz)Amplitude0 1000 2000 3000 4000 5000 02468 Frequency (Hz)Angle(deg) Figure 5-3. Fourier transform of the line spread function (black curve) and fitting function (red) Figure 5-4. Geometry of th e MTF step target in MCNP5 Initial curve Gauss fitting function F r e q uenc y ( lines/cm ) 0.5 cm 2 cm Top view N y lon Lead Front view 2cm

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76 0.00E+00 1.00E-02 2.00E-02 3.00E-02 4.00E-02 5.00E-02 6.00E-02 00.511.522.533.54 Distance (cm)Counts/pixel Figure 5-5. Data profile obt ained from the first MTF step target design in MCNP5 Figure 5-6. Geometry of the sec ond design of the MTF step target Plastic Lead Lead Detector Ai r Top view 2cm Front view Lead N y lon 0.5 cm 2 cm 1cm

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77 0.00E+00 1.00E-02 2.00E-02 3.00E-02 4.00E-02 5.00E-02 6.00E-02 7.00E-02 8.00E-02 00.511.522.533.54 Distance (cm)Counts/pixel Figure 5-7. Profile data obt ained from the second design of the MTF step target 0.00E+00 1.00E-02 2.00E-02 3.00E-02 4.00E-02 5.00E-02 6.00E-02 7.00E-02 8.00E-02 9.00E-02 1.00E-01 00.511.522.533.54 Distance (cm)Counts/pixel Figure 5-8. Data profile obtai ned from the third target desi gn; nylon block on top of lead Plastic Lead Lead

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78 Figure 5-9. Final design profile proposed for the MTF step target 2 inch 1/16 inch 1/16 inch 0.5 inch 0.5 inch 0.5 inch 0.5 inch 1 inch Nylon 6/6 Aluminium Lead First step design Second step design

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79 CHAPTER 6 PROPOSED TECHNIQUES FOR IMAGE QUALITY ASSESMENT Multiple Derivatives and Inflection Points as a Mathematical Criterion for Image Quality Assessment Even if the volumetric Sine MTF couples the target specific varia tions to the scanning system response, it offers a basis for relative evaluation of system performance. It is an integrated 3-D MTF over the vertical direction. This MTF allows comparison between detectors and gives a basis on which to test a global improvement in the system. Figure 6-1 presents a comparison between the MTF from Detector 1 (NaI) and Detector 5 (Y5Si2O). These results show that over a frequenc y range between 0.2 line pairs/mm and 2 line pairs /mm, the performance of the Y5Si2O detector is superior to that of the NaI detector. In Figure 6-2 the MTF plots are compared for th ree different aperture diameters of 0.5 mm, 1.0 mm and 1.5 mm. Over the whole range of frequencies the MTF cu rve is higher for the smallest aperture. The higher the MTF, the better the image with respect to the contrast and resolution. The MTF presented here does not include a ny volumetric normalization processing. The MTF is sensitive to the change in the volume intersection of the beam and the lines. The drop in the MTF values is related to a loss of contrast and a volume variation. The relative difference in MTF values indicates the quality of th e images when using the same target. Since the main purpose of the MTF plat e is X-ray imaging system calibration, the main objective is to provide a comparison of image quality. Figure 6-3 shows several MTF plots for differe nt conditions. In addition to the MTF value at a given frequency, the curvature and the inflection point characterize the contrast and resolution losses.

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80 In Figure 6-3 the comparison is done over thr ee aperture sizes of 0.5 mm, 1.0 mm, 1.5 mm and two pixel sizes of 0.05 mm and 0.1 mm. For a given aperture, the la rger pixel size has a higher MTF and hence a better image quality. In order to use mathematical properties as a criterion to sort the MTF curves, a mathematical model is established. The plots in Figure 6-4 were generated by fitting the MTF curves using Boltzmann functions. The formula used for the fitting process is MTF_experimental= dX X Xe A A A Y01 2 1 2 The corresponding coefficients are listed in Table 6-1 fo r each curve. Note that in this section X is a frequency since it represen ts the MTFs variable. In order to evaluate the fitting efficiency some statistical test results are given in Table 6-3. The 2 R values are close to 1 indicati ng a very good fitting function, the DoF Chi2 are the reduced 2Chi values obtained from the Nonlinear Least squares fitting and are given as an example As previously explained, the curvatures and inflection points are of great interest when comparing images from different set ups. Equation 6-1 gives the first deri vative with respect to the frequency (line pairs per mm). The coeffici ents are given in Tabl e 6-2 and the plots are shown in Figure 6-5. 2 0 0) 1 ( ) 1 2 ( ) (dX X X dX X Xe e A A dX X dY (6-1) The second derivative is given by Equation 6-2: 3 2 2) 1 ( 1 ) 1 2 (0 0 0 dX X X dX X X dX X Xe e dX e A A dX Y d (6-2)

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81 The inflection points are given by the second derivatives zeros. By sorting the corresponding frequencies, the images are compared with respect to their c ontrasts. The plots are shown in Figure 6-6. The zero values of the second derivative are pr esented in Table 6-4. The higher zero values characterize better image quality according to criteria developed in this study. The images corresponding to the MTF curves s hown in Figure 6-3 are sorted and presented in Figure 6-7 to Figure 6-11. The images are sorted using a scale from 1 to 5; 1 is the best relative quality and 5 the relatively poorest quality. The proposed MTF target is to be used in large scans for calibrati on purposes. Figure 6-12 is an image from an uncollimated YSO detector. The number of counts needs to be increased to achieve a lower statistical error. The image is shown to give an idea of how a calibration scan would be done. The MTF target was laid on the sample being scanned. Th e heterogeneity of the sample (Tile Test Panel VT70-191037-005) offered a good test to evaluate the MTF target response in a real environment. However the background is of the same order of magnitude as the MTF target response (approximately one third). This shows the limit of this MTF target design which is highly affected by the material background. The objective is to design an optimized small MTF target, such as the effect of the background material is minimized. The proposed image assessment techniques used the MTF curves obtained from the MTF Sine target. However the same techniques can be applied to the MTF curves obtained from the MTF step target.

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82 Correlation Between the Different Met hods of Calculating the MTF The correlation between the Step function a nd the Sine function fo r MTF determination needs to be done under the same experimental co nditions. Once a relation is established between the two methods one can be used know ing its limitations and advantages. As previously explained, the Sine function based MT F uses more experimental interpolation points over the freque ncy domain than the Step function MTF. This makes the Sine MTF target more adapted for precise measuremen ts of the contrast and resolution for given frequencies. Also comparison be tween different MTF curves is finer and extends over a larger frequency domain. For these reasons the Sine MT F target will be used as a reference for MTF calculations. A comparison between the MTF curves obtained experimentally from the Sine target and the step target are not of hi gh interest, unless the profiles are normalized over the target interaction volume. This is because the MTF obtai ned from the Sine target contains information on the change in volume. Recall from Chapter 4 th at the Sine based MTF decreases less rapidly (exp(-x)) compared to the St ep function based MTF (exp(-x2)). The Sine based MTF uses the output modula tion of the Sine input function whereas the Step function MTF is derived thr ough the Line Spread function. Resolution Assessment from a Step Function Input To demonstrate the equivalence between the MTF calculations based on the edge function and the line spread function the defin ition of the step function is needed. 0 0 0 1 ) ( x x y x wedge in (6-3) Also

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83 ' ') ( ) ) ( ) ( ( ) ( ) ( dx y x w dx dy y x dx x y x wx Line in xx edge in (6-4) Since the system is assumed in first a pproximation as linear, the output must be: x x Line out edge outdx x l dx y x w y x w x e' ') ( ) ( ) ( ) ( (6-5) Hence, the edge spread function is the indefini te integral of the line spread function: dx x de x l ) ( ) ( (6-6) Figure 6-13 shows the 3 steps needed to perform an MTF calculation based on the edge function. First the data profile is obtained from the experiment then the profile is truncated to only use one edge, finally the profile is smoothe d using the averaged values of the lower and higher regions of the profile. This smoothing proc edure is necessary becau se the derivation is a high pass filter; meaning that the high frequency noise will have a high contribution to the signal. Another possibility is to apply a Gaussian frequency window to the first derivative of the profile to discriminate against the high frequency noise. Once this smoothing step is performed the fi rst derivative is obtained numerically as shown in Figure 6-14. The width of the rising edge between 10% and 90% corresponds to the width of the first derivative at 10% of its maximum. This distance x in pixel or mm can be used as a quick criteria to compare different scan conditions and to perf orm resolution assessment using a step function. There are many advantages to using the edge response for measuring resolution. In fact, the main reason for wanting to know the resolution of a system is to understand how the edges in an image are blurred.

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84 The first advantage is that the edge response is simple to measure because edges are easy to generate in images. If needed, the Line Sp read Function can easily be found by taking the first derivative of the edge response. The second advantage is that all common edge s responses have a similar shape, even though they may originate from different Point Spread Functions20. Since the shapes are similar, the 10%-90% distance is an ex cellent single parameter measur e of resolution. The third advantage is that the MTF can be direc tly found by taking the one -dimensional Fourier Transform of the Line Spread Function (unlike the PSF to MTF calculation that must use a twodimensional Fourier transform). For example the step function presented in Figu re 6-13 is used to calculate the resolution associated to the 10%-90% edge response. Fi gure 6-15 shows how the width x of the 10%-90% edge is calculated. For the particular conditions of the above edge scan the system has a 10%-90% edge response of 1.94 mm. The limiting resolution is a vague term indicating the frequency where the MTF amplitude has a value of 3% to 10%. In fact the edge width measured between 10% and 90% can be relate d to a frequency at which the MTF is 10% of its maximum value. Assuming the LSF can be fitted by a Gaussian f unction, which is the case for most imaging systems. Then the Fourier Transform is also a Gaussian function as shown in Equation 6-7. ) ) 2 ( 2 1 exp( 2 ) )( ( ) 2 1 exp( ) (2 2 2 f f LSF FT x x LSF (6-7) The width of the LSF at 10% of its maximum is given by width edge x ) 10 ln( 2 2% 10 (6-8)

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85 This distance can also be measured direc tly from the edge widt h between 10% and 90%. Now considering the MTF given by the Fourier Transform of the LSF, it has a value of about 10% of its maximum at a frequency 2 ) 10 ln( 2% 10 f (6-9) Combining Equations 6-8 and 6-9 gives width edge width edge f 46 1 ) 10 ln( 2% 10 (lp/mm or lp/pixel) (6-10) The 10% contrast level on the corresponding MTF curves will occur at about: 0.75 lp/mm or lp/pixel for an edge width of 1.94 mm. This is a very convenient method to asses the system limiting resolution between 10% and to compar e different images using a single number. Figure 6-16 shows an example of a numerical calculation of the first derivative and the Fourier Transform of the edge function used in Figure 6-15. The amplitude of the Fourier Transform gives the MTF. The predicted freque ncy at which the MTF value is 10% from the edge width method gives 0.75 lp/mm the meas ured value from the MTF curve gives 0.665 lp/mm. The error associated to the measured valu e with respect to the predicted value is about 11.3%. This is due to the errors associated to the num erical evaluations of th e first derivative and the Fourier Transform but also the initi al assumption of the Gaussian fitting. As a conclusion the edge width between 10 % and 90% is a convenient single number for relative comparison of different images. The same edge function can be used to generate an MTF curve. The theoretical relationship between the edge width and the frequency at which the MTF value is 10% can be used as an indication of the experimental frequency. In the previous example an error of 11.3% was calcu lated between the two frequencies.

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86 0 20 40 60 80 100 120 00.511.522.5 frequency line pairs/mmMTF % detector1-NaI detector5-YSO Figure 6-1. MTF comparison between NaI and Y5Si2O detectors at 45 kVp, 0.5 mm aperture 0 20 40 60 80 100 120 00.511.522.5 Frequency line pairs per mmNormalized MTF (% ) mtf-1.5mm aperture mtf-1.0mm aperture mtf-0.5mm aperture Figure 6-2. MTF comparison fo r 3 different aperture diamet ers at 45kVp-45mA-0.05mm pixel size.

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87 0 20 40 60 80 100 00.511.522.5 Frequency line pairs per mmNormalized MTF (%) mtf-0.1mm pix1mm ap mtf-0.1mm pix0.5mm ap mtf-0.05mm pix1.5mm ap mtf-0.05mm pix1.0mm ap mtf-0.05mm pix0.5mm ap Figure 6-3. MTF comparison for different pixel sizes and beam apertures at 45 kVp-45 mA 0 20 40 60 80 100 120 00.511.522.5Frequency line pairs per mmNormalized MTF (%) fit-mtf-0.1mm pix1mm ap fit-mtf-0.1mm pix0.5mm ap fit-mtf-0.05mm pix-1.5mm ap fit-mtf-0.05mm pix-1.0mm ap fit-mtf-0.05mm pix-0.5mm ap Figure 6-4. MTF Boltzmann model fitting function comparison for different pixel sizes and beam apertures at 45 kVp-45 mA

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88 Table 6-1. Coefficients used in the fitting function formula for each MTF curve A1 A2 X0 dX MTF 0.1mm pixel / 0.5mm aperture 113.633 -2.257430.775750.32778 MTF 0.05mm pixel / 0.5mm aperture 117.88843.91869 0.615520.29361 MTF 0.1mm pixel / 1.0mm aperture 102.38812.23524 0.548240.12499 MTF 0.05mm pixel / 1.0mm aperture 107.07631.48754 0.4929 0.13709 MTF 0.05mm pixel / 1.5mm aperture 107.11063.23425 0.372570.08768 Table 6-2. Statistical measures of the fitting accuracy DoF Chi2 2 R MTF 0.1mm pixel / 0.5mm aperture 1.023550.99907 MTF 0.05mm pixel / 0.5mm aperture 1.256250.99916 MTF 0.1mm pixel / 1.0mm aperture 5.941140.99724 MTF 0.05mm pixel / 1.0mm aperture 1.943030.99906 MTF 0.05mm pixel / 1.5mm aperture 3.412980.99825 -30 -25 -20 -15 -10 -5 0 5 0123Frequency line pairs per mmMTF first derivative (%/line pairs per mm) 1 s t d er i va ti ve m tf 0.1mm pix-1mm ap 1st derivative mtf0.1mm pix-0.5mm ap 1st derivative mtf0.05mm pix1.5mm ap 1st derivative mtf0.05mm pix1.0mm ap 1st derivative mtf0.05mm pix0.5mm ap Figure 6-5. MTF fitting function firs t derivative, scan at 45 kVp-45 mA

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89 -1500 -1000 -500 0 500 1000 1500 00.511.522.5Frequency line pairs per mmMTF 2nd derivative (%/ (line pairs pe r mm)^2) 2nd derivative mtf-0.1mm pix1mm ap 2nd derivative mtf-0.1mm pix0.5mm ap 2nd derivative mtf-0.05mm pix1.5mm ap 2nd derivative mtf-0.05mm pix1.0mm ap 2nd derivative mtf-0.05mm pix0.5mm ap Figure 6-6. MTF fitting function second derivative, scan at 45 kVp-45mA Table 6-3. Roots value of th e MTF second derivatives curves Curves first root (Freq-line pairs per mm) MTF 0.1mm pixel / 0.5mm aperture 0.77575 MTF 0.05mm pixel / 0.5mm aperture 0.61552 MTF 0.1mm pixel / 1.0mm aperture 0.54824 MTF 0.05mm pixel / 1.0mm aperture 0.4929 MTF 0.05mm pixel / 1.5mm aperture 0.37257 Figure 6-7. 1 MTF 0.1 mm pixel, 0.5 mm aperture

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90 Figure 6-8. 2 MTF 0.05 mm pixel, 0.5 mm aperture Figure 6-9. 3 MTF 0.1 mm pixel, 1.0 mm aperture Figure 6-10. 4 MTF 0.05 mm pixel, 1.0 mm aperture

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91 Figure 6-11. 5 MTF 0.05 mm pixel, 1.5 mm aperture Figure 6-12. YSO image of MTF Target on a tile panel Nylon lines from the MTF target Aluminium edge of the MTF frame Tile Test Panel

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92 Table 6-4. Different met hods of the MTF derivation Input function Output func tion Intermediate steps MTF Point source (x,y) Point spread function (x,y) 2D FT MTF ( ) Line source (x)= Point source (x,y) Line spread function (x) 1D FT MTF ( ,0) Edge function (x) Edge spread function (x) d(Edge(x)) /dx = Line spread function (x) and 1D FT MTF ( ,0) Sine input(x) Sine output (x) Contrat( )/Contrast(0)= *MTF MTF( ,0) Figure 6-13. Selection and smoothing steps fo r the MTF calculation from a step function 0 5000 10000 15000 20000 25000 30000 01020304050607 0 Distance (mm)Normalized counts /pixel 0 5000 10000 15000 20000 25000 30000 010203040506070 Distance (mm)Normalized counts /pixel 0 5000 10000 15000 20000 25000 30000 01020304050607 Distance (mm)Normalized counts /pixel

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93 Figure 6-14. An example of the ed ge profile and its first derivative Figure 6-15. Edge function width estimation 0 5000 10000 15000 20000 25000 010203040506070 Distance (mm)Normalized counts /pixel10% 90% Distance X mm or pixel 90% 10% 10% Distancex Distancex

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94 Figure 6-16. Numerical evaluati on of the first derivative of th e edge function used in the example and its Fourier Transform 8101214161820222426283032 7 000 6 000 5 000 4 000 3 000 2 000 1 000 0 Data: Derivative1_Data33B Model: Gauss Equation: y=y0 + (A/(w*sqrt(PI/2)))*exp(-2*((x-xc)/w)^2) Weighting: yNo weighting Chi^2/DoF= 9422.64067 R^2= 0.9963 y0-1.76655.04368 xc25.76397.00675 w1.01068.01394 A-9715.589960.72925X Axis Title 05101520253035 0 2000 4000 6000 8000 10000 12000 14000 XAiTil 0.00.20.40.60.81.01.2 0 10 20 30 40 50 60 70 r Gauss fit of FFT7_r Data: FFT7_r M odel: Gauss Equation: y=y0 + (A/(w*sqrt(PI/2)))*exp(-2*((x-xc)/w)^2) W eighting: yNo weighting Chi^2/DoF= 7.01484 R^2= 0.98518 y0-1.713570.41015 xc-1.9351E-160.00232 w0.653290.00754 A50.061940.76622 F(H) Amplitude0 5000 10000 15000 20000 0 0 0 2 0 4 0 6 0 8 1 0 1 2 Angle(deg)

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95 CHAPTER 7 COMPUTATIONAL PROCESSING WITH MATLAB. ALGORITHM ARCHITECTURE FOR MTF CALCULATION (MATLAB) Modulation Transfer Function Based on the Sine Target The main result of this task was a code that integrates all the calculations for the MTF process. The code was written in the MATLA B 7.0.4 programming language. The code was to be implemented in an image processing tool previously used by the Lockheed-Martin Space Systems Company. Figure 7-1 shows the Matlab interface for th e profile data generation and the MTF calculation. The interface is analogous to the code used currently to process the output images from the system and draw the profiles. After scanning the MTF plate, a couple line s are generated. When saving the profile (Figure 7-2), the MTF menu appears to enable the MTF processing. Once the profile is saved in a text format, th e code generates a *.dat file using the same name. This file will be used in Matlab to generate MTF curves. The conventional profile used for the MTF cal culation should have the maximum peaks on the left, since they are used to generate the low frequencies. The code is essentially written following this model. There is an option to revers e the profile data to make user entries easier (Figure 7-3). Figure 7-4 shows the user interface for enteri ng the Sine MTF plate information. Default values are already entered for the Sine MTF target. The first step is to locate the maxima and mi nima in the image. Based on these values the contrast and the MTF are calcula ted. Figure 7-5 shows how the pr eliminary peak selection is displayed.

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96 The local maximums are designated using red cr osses. Because of the fluctuations in the data, it is nearly impossible to pick up one maximum per peak, unless using the Full Width at Half Maximum (FWHM) for each set of holes. This part is performed in the Automatic option available in the code. Currently, the MTF calculation requires that the user select for each peak a region of interest. The region of interest (ROI) does not have to be precise ly selected. The code extracts the x-ordinates from the image to recalc ulate the overall maximum in the ROI. After all the peaks have been selected, the MTF plot is generated (Figure 7-6). When saving the plot, the same name is used to create a new folder that contai ns the data profile and the MTF plots in PDF format in addition to a text file that contains the va lues of the MTF versus frequencies. Modulation Transfer Function Based on a Step Function Target Figure 7-7 shows a step function profile obt ained from the preliminary experiment (Chapter 3) of an edge function. The first derivative is also give n since the derivation is the first step in using the edge function. Note that th e data is noisy and a st atistical smoothing would provide a better data profile to start with. A discrete Fourier transform is then performe d on the first derivativ e and the modulus is estimated to give the MTF. Figure 7-8 shows th e MTF curve and its first and second derivatives. As expected, the numerical treatment without an y smoothing on the data introduces high fluctuations in the MTF calculation. These large fl uctuations made it nearly impossible to use the zeros of the second derivative as a cr iterion for image quality assessment. Either a denoising algorithm or an iterative le ast squares estimate fitting of the data using j jxi t Measuremen xi ction FittingFun min )) ( ) ( (2 2is needed.

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97 The more convenient choice for automated use would be using the f itting tools provided with the Matlab7.0.4 version. Figures from 7-8 to 7-10 show the different st eps in the Matlab code used to generate MTF curves from an edge function. Figure 7-8 shows how a region of interest can be selected, Figure 7-9 and 7-10 show the selected region of th e edge function its firs t derivative and the MTF curves with the frequencies expressed in line pairs/pixel and line pairs/mm. Figure 7-1. Matlab user interface

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98 Figure 7-2. MTF me nu and data profile Figure 7-3. Data profile Lineprofile

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99 Figure 7-4. User interface for information entries Figure 7-5. Maximum search

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100 Figure 7-6. Saving files Figure 7-7. Data profile from an edge function and its first derivative

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101 Figure 7-8. Selection of a region of interest in the edge function profile Figure 7-9. The selected regi on of interest and the first de rivative of the edge function

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102 Figure 7-10. MTF curves with frequencies e xpressed in line pairs/ pixel and line pairs/mm

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103 CHAPTER 8 CONCLUSION In order to properly characterize the X-ray back scattering system several definitions of the Modulation Transfer Function have been intr oduced. These definitions and the methodology for calculating the MTF depend on the input function to the system. Several input functions have been tested: Point Function, Line Function, Step F unction and Sine Function. The relationship between the different functions a nd the resulting MTF was treated to understand the benefits and limitations of each input type function for practic al use. The preliminary experiments for an impulse and step functions showed the expected responses from mathematical derivations. The key step for a complete analysis was the ability to accurately fit the curves according to statistical tests and obtain mathematical expressions th at were used later for curve recognition. A Sine target pattern was proposed for prec ise evaluation of the MTF as a function of frequency. The design was based on nylon lines of different diameters and separation. This MTF Sine target was used for major comparisons and relative image quality assessment. The experiments were performed mostly w ith the new compact system using Y5SI2O detectors, but some experiments used NaI detectors. The large di mensions of the MTF Sine target made it less desirable for practical use on small scans areas. Also this Sine MTF target was highly dependent on the background material. Instead, an improved Step target desi gn was proposed to meet a size constraint of approximately a cube of 0.5 inch by 2 inch by 5/8 inch. The different designs were supported by MC NP5 models using tw o variance reduction techniques; forced collisions and DXTRAN s phere. These models enabled to understand the different contributions to the signal and their relationships with the target own volume.

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104 A geometrical volumetric normalization of the input sine profile was performed using the complete elliptic integrals of the first and s econd kind. However this method was not completely successful in providing a good volumetric normalization. Monte Carlo simulations helped provide an understanding of the effect of the volume decrease in the MTF Sine targ et through two competing factors: the volumetric interaction rate and the particle mean free path. For practical image quality assessment and comparison, the evaluation criterion used with the Sine MTF target was the first zero of the second derivative of the MTF curve. A method for resolution assessment based on an edge input function was proposed. Th is method relates the rising edge width between 10% a nd 90% to the frequency at whic h the theoretical MTF value is 10%, width edge width edge f 46 1 ) 10 ln( 2% 10 (lp/mm or lp/pixel). The MTF calculations were performed using MATLAB7.0.4. Customized codes were written with user interfaces for MTF curve generation. Finally, some MTF applications in image processing and some of the early results on foil filtering with the RSD scanning system are presented.

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105 APPENDIX A ENERGY FILTERING USING PAPER While setting up the experiments for the MTF measurements, placing a regular sheet of paper under the nylon lines, in ad dition to the lead on the floo r, drops the background noise by 400 counts/pixel. Figure A-1 shows a comparison between two backscatter images, one with and one without paper. The maximum intensities are about the same order, while the background contribution drops off by half. Note that for case b in Figure A-1 the bright line on the image is above the sheet of paper while the 3 lines on the left of the image are right under the paper. All of the lines are equally distant from the paper. Figure A-2 shows a line profile across image b in Figure A-1. The lines under the paper show with near half intensity of that of the line above the paper. a b Figure A-1. Comparison between two backscatter images. a) Scan without paper underneath the nylon line. b) Scan with pa per underneath the nylon line

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106 Figure A-2. Line profile evaluation of the paper filtering

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107 APPENDIX B MTF FRAME STRUCTURE Figure B-1. MTF fr ame plate top view

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108 Figure B-2. MTF c over plate top view

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109 LIST OF REFERENCES 1. E. Dugan, A. Jacobs, S. Keshavmurthy, and J. Wehlburg," Lateral Migration Radiography," Research in Nondestructive Evaluation, 10(2) p. 75-108 (1998). 2. A. Jacobs, E. Dugan, S. Brygoo, D. Ekdahl, L. Houssay, and Z. Su, Lateral Migration Radiography: A New X-ray Backscatter Imaging Technique, Proceeding of SPIE, 4786 p. 1-16 (2002). 3. E. Dugan, A. Jacobs, L. Houssay, and D. Ekdahl, Detection of Flaws and Defects Using Lateral Migration X-ray Radiogr aphy, Proceeding of SPIE, 5199 p. 47-61 (2004). 4. H. Barrett, and W.Swindell, Radiological Im aging, The Theory of Image Formation, Detection, and Processing, Academic Pr ess, Inc San Diego, California 1981. 5. F. Attix, Introduction to Radiological Physics and Radiation Dosimetry, John Wiley & sons, Inc New York, New York (1986). 6. A. Jacobs, and J. Campbell, Landmine De tection by Scatter Radiation Radiography, Scientific and Technical Fi nal Report, Contract DAAK 70-86-K-0016, U.S. Army Belvoir Research, Development and Engineering Center, (1987). 7. J. Campbell, and A. Jacobs, Detection of Buried Land Mines by Compton Backscatter Imaging, Nuclear Science and Engineering, 110 p. 417-424 (1992). 8. Y. Watanabe, J. Monroe., S. Keshavmurthy, A. Jacobs, and E. Dugan, Computational Methods for Shape Restoration of Buried Obje cts in Compton Backscatter Imaging, Nuclear Science and Engineering, 122 p. 55-67 (1996). 9. J. Wehlburg, S. Keshavmurthy, E. Dugan, and A. Jacobs, Geometric Considerations Relating to Lateral Migration Backscatter Radiography (LMBR) as Applied to the Detection of Landmines," Proceeding of SPIE, 3079 p. 384-393 (1997). 10. Z. Su, J. Howley, J. Jacobs, E. Dugan, and A. Jacobs., The Discer nibility of Landmines Using Lateral Migration Radiography, Proceeding of SPIE, 3392 p. 878-887 (1998). 11. C. Wells, Z. Su, J. Moore, E. Dugan, a nd A. Jacobs, "Lateral Migration Radiography Measured Image Signatures for the Detection and Identification of Buried Landmines, Proceeding of SPIE, 3710 p. 906-916 (1999). 12. C. Wells, Z. Su, A. Allard, S. Salazar, E. Duga n, and A. Jacobs, Suitability of Simulated Landmines for Detection Measurements Usi ng X-ray Lateral Migration Radiography, Proceeding of SPIE, 4038 p. 578-589 (2000). 13. Z. Su, A. Jacobs, E. Dugan, J. Howley, a nd J. Jacobs, Lateral Migration Radiography Application to Land Mine Detection, Confirma tion and Classification, Optical Engineering, 39(9) p. 2472-2479 (2000).

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110 14. E. Dugan, A. Jacobs, Z. Su, L. Houssay, D. Ekdahl, and S. Brygoo, Development and Field Testing of a Mobile Backscatter X-ray Late ral Migration Radiogra phy Land Mine Detection System, Proceeding of SPIE, 4742 p. 120-131 (2002). 15. R. Evans, The Atomic Nucleus, McGraw Hill Boo, Inc. New York, New York (1955). 16. J. Dainty, and R.Shaw, Image Science Principl es, Analysis and Evaluation of PhotographicType Imaging Processes, Academic Press, London (1974). 17. M.J. Berger, J.H. Hubbell, S.M. Seltzer, J. Chang, J.S. Coursey, R. Sukumar, and D.S. Zucker, XCOM: Photon Cross Section Database (version 1.3) http://physics.nist.gov/xcom National Institute of Standa rds and Technology (May 2007). 18. D. Shedlock, X-ray Backscatter Imaging for Radiography by Selective Detection and Snapshot Evolution, Development, and Optimi zation, Ph.D. Dissertat ion, University of Florida (2007). 19. B.T. Addicott, Characterization and Optimi zation of Radiography by Selective Detection Backscatter X-ray Imaging M odality, M.S. Thesis, University of Florida (2006). 20. S. Smith, The Scientist and Engineers Guid e to Digital Signal Processing California Technical Publishing, San Diego, California (1997).

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111 BIOGRAPHICAL SKETCH Nissia Sabri is a graduate assist ant at the University of Florid a. She joined the Scatter x-ray laboratory in the Nuclear and Radiological En gineering Department in August of 2005 to complete a Master of Science in nuclear engi neering. She obtained a Master of Science in applied physics engineering in September 2006 a nd a Bachelor of Science in physics in May 2005 at The Grenoble National Engineer ing School for PhysicsFrance.