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8f9bbd993c66d391ca687f8e3b20986b 646cd84fa0233b3674219b862114f746957c519c 24250 F20101211_AAATZK sabri_n_Page_085.QC.jpg 9e0f01f5323b718d59aeba4bc9e82383 3fcfd36b4056f006ba260d460a3c602481e94493 55643 F20101211_AAATYV sabri_n_Page_088.jpg cf51cc11a2a8dde4eb53324660499490 1bed41f7cf08cbdb98396e3dc49a927567f2bdf7 15900 F20101211_AAAUFE sabri_n_Page_074.QC.jpg 52753fb4ee37ca9ea674b4f0da4bd9ef 8ef01b4d173563bd9b037b5f1c4043dfa6276ce8 AN ADAPTED MODULATION TRANSFER FUNCTION FOR XRAY 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: XRay 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 41 Number of counts at the detector surface. ............. ...............60..... 42 Comparison between the Analog and NonAnalog MCNP5 ................ ............. .......61 43 Summary of the line diameters and the associated number of line position. ................... ..62 44 MCNP5 run condition for Analog versus NonAnalog .......... ................ ...............62 45 Comparison between Analog and NonAnalog results in MCNP5 .............. ..................64 61 Coefficients used in the fitting function formula for each MTF curve ............................88 62 Statistical measures of the fitting accuracy ....__ ......_____ .......___ ..........8 63 Roots value of the MTF second derivatives curves ....._._._ .......__. ......._........89 64 Different methods of the MTF derivation. ...._. ......_._._ .......__. ..........9 LIST OF FIGURES FiMr page 11 Schematic illustrating Xray production ................. ...............19........... ... 12 Typical spectrum obtained from an Xray tube with a tungsten anode4 ................... .........19 13 Compton Backscattering Imaging (CBI) .............. ...............20.... 14 Lateral Migration Radiography (LMR) .............. ...............20.... 15 Photograph of RSD System with 4 Nal Detectors ................. ...............21........... . 16 Photograph of RSD System showing YSO detectors mounted to Nal Detectors ..............21 17 RSD scanning system mounted on a Eixed frame .............. ...............22.... 18 Flow chart of the image acquisition process20 ................ ...............23.............. 21 Photoelectric, Compton and Pair Production ............ ...............34..... 22 Kinematics of the Compton Effect ................. ......... ...............34. .. 23 Transmission model .............. ...............35.... 24 Backscatter model ................. ...............35........... .... 31 Scanning system output two line pairs placed at 450with respect to the vertical axis.......42 32 High exposure scanning output, one sweep of a nylon line (Dirac Simulation)................43 33 Scan of a cubic plastic sample: 17.5 mm width, 1 mm beam, 0.5 mm pixels ................43 41 Scheme for simulating a sinusoidal input ....__ ......_____ .......___ ..........5 42 M TF frame plate .............. ...............58.... 43 MTF frame plate detailed design ...........__......___ ...............58.. 44 Output profie from the scan of the MTF Sine target (detector 1 Nal) ................... ...........59 45 Scatteringtoabsorption ratios for Nal and YSSi20 crystals. ............. .....................5 46 MCNP5 model for input profie calculation ................ ...............60........... .. 47 Energy spectrum distribution used in the MCNP5 model based on Kramers spectrum....60 48 The input sine profie obtained from running MCNP5 ................. .......... ...............62 49 Sine profie obtained from modeling 10 nylon lines of different diameters in MCNP5 ...63 410 The complete input profie from an MCNP5 simulation as recorded at the detector........63 411 MCNP5 model for input profie calculation ................ ...............64........... .. 412 Average energy and fraction of the detected signal in each of the six collision bins........65 413 Intersection volume of two cylinders ................. ...............65......_.__... 414 Two cylinder intersection volume .............. ...............65.... 415 Integrated profile data ................. ...............66................ 416 Equivalence between peaks and steps profiles. ............. ...............66..... 417 Normalization methodology scheme. ............. ...............66..... 418 Experimental and normalized data profile............... ...............67 419 A representation of the MCNP5 setup for volume intersection calculations. ................... .67 420 Line and beam intersection volume values ................. ...............68........... .. 421 A plot of the volumetric normalization of half peaks obtained from MCNP model. ........68 422 Visual editor view of the new MCNP setup for volume calculations. .............. ..... ..........69 423 Normalization of the MTF sine profile over the intersection volume .............. .... ........._..69 424 Statistical smoothing of the normalized profile ....__ ......_____ ...... .....__........7 425 MTF function from detector 5 .............. ...............70.... 51 Edge target made from a junction of lead (absorber) and nylon scattererr) ......................74 52 Scanning system response to an edge. ............. ...............74..... 53 Fourier transform of the line spread function (black curve) and fitting function (red) .....75 54 Geometry of the MTF step target in MCNP5 ................ ................. ...............75 55 Data profile obtained from the first MTF step target design in MCNP5 ................... ........76 56 Geometry of the second design of the MTF step target ................. ................ ...._..76 57 Profile data obtained from the second design of the MTF step target ............... .... ...........77 58 Data profile obtained from the third target design; nylon block on top of lead.................77 59 Final design profie proposed for the MTF step target ......___ ........__. ..............78 61 MTF comparison between Nal and YSSi20 detectors at 45 kVp, 0.5 mm aperture ..........86 62 MTF comparison for 3 different aperture diameters............... ...............8 63 MTF comparison for different pixel sizes and beam apertures at 45 kVp45 mA ........_...87 64 MTF Boltzmann model fitting function comparison ...._._._.. ..... ..__... ........_._......87 65 MTF fitting function first derivative, scan at 45 kVp45 mA............... ...................8 66 MTF fitting function second derivative, scan at 45 kVp45mA............... ................8 67 1 MTF 0.1 mm pixel, 0.5 mm aperture............... ...............89 68 2 MTF 0.05 mm pixel, 0.5 mm aperture............... ...............90 69 3 MTF 0.1 mm pixel, 1.0 mm aperture............... ...............90 610 4 MTF 0.05 mm pixel, 1.0 mm aperture............... ...............90 611 5 MTF 0.05 mm pixel, 1.5 mm aperture............... ...............91 612 YSO image of MTF Target on a tile panel ......___. ..... ... ...............91 613 Selection and smoothing steps for the MTF calculation from a step function ................92 614 An example of the edge profile and its first derivative ......... ................. ...............93 615 Edge function width estimation .............. ...............93.... 616 Numerical evaluation of the first derivative of the edge function ................ ................. 94 71 Matlab user interface............... ...............9 72 MTF menu and data profile .............. ...............98.... 73 Data profile .............. ...............98.... 74 User interface for information entries............... ...............99 75 Maximum search............... ...............99. 76 Saving files............... ...............100. 77 Data profile from an edge function and its first derivative ................. ......................100 78 Selection of a region of interest in the edge function profile. ................ .............. .....101 79 The selected region of interest and the first derivative of the edge function ................... 101 710 MTF curves with frequencies expressed in line pairs/pixel and line pairs/mm ...............102 A1 Comparison between two backscatter images .............. ...............105.... A2 Line profile evaluation of the paper filtering ................. ...............106......._._.. B1 MTF frame plate top view ................. ...............107.............. B2 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 XRAY 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, singleside xray 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 nondestructive 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 sprayon foam insulation (SOFI) inspection. The xray backscatter RSD imaging system has been successfully used for cracks and corrosion spot detection in a variety of materials. The conventional transmission xray 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 systeml3 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 Xray 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, Xray images are maps of the xray 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 Xray photon backscattering4 Xrays 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 11 shows the basic principle of Xray 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 Xrays . The energies of these discrete line spectra are characteristic of the anode chemical element. The total spectrum obtained from a typical Xray tube with a tungsten anode is shown in Figure 12. As the Xrays 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 Xray source, the obj ect being radiographed, and a detector. From an imaging standpoint there is an important distinction between absorption and scattering. Usual Xray 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 twodimensional proj section of a threedimensional 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 Radiography614 (Figure 14) is similar to the CBI technique (Figure 13), but instead of counting only singlecollision backscattered photons, the LMR technique counts both single and multiplecollision backscattered photons that have laterally spread out from the illumination beam entry point. At the detector surface, signals from single and multiplecollision 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: Xray 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: XRay 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 Xray 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 Xray 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 Xrays (1055keV). 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 15. In Figure 16, the YSO is mounted on detector 2 using an aluminium ring. In Figure 17 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 pileupl 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 18) presents the entire image acquisition process from detection to display. Xray tube Figure 11. Schematic illustrating Xray production 350 300 250 SO O 20 40 60 80 100 (20 energy (keV) Figure 12. Typical spectrum obtained from an Xray tube with a tungsten anode4 19 Electron gun " Figure 14. Lateral Migration Radiography (LMR) Noise Object Figure 13. Compton Backscattering Imaging (CBI) Uncollimated detector Xray generator I Collimated detector Land mine Nal detector ~ Sleeve collimator extended Xray beam tu be Finned Collimator Angle at 90 (degrees) I \ 1 Figure 15. Photograph of RSD System with 4 Nal Detectors Aluminium ring YSO detector Nal ....etector A set of YSO detectors Figure 16. Photograph of RSD System showing YSO detectors mounted to Nal Detectors Figure 17. RSD scanning system mounted on a fixed frame Image Complete Pulse train Labview/computer Xaxis Yaxis Compleia Pulse train NIMotion PCI 7344 Xaxis Yaxis NIDaq PCl6602 NIMotion 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 .XMotor Dir Xaxis XMotor Amps Y axis YMotor Figure 18. Flow chart of the image acquisition process20 YMotor Amns CHAPTER 2 PROBLEM STATEMENT General Physics of Photon Interaction When considering an Xray based scanning system, it is highly important to understand how the photons interact with matter. There are five types of interactions with matter by Xray 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 radiationprotection 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 21 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 lowZ (e.g., carbon, air, aluminum, Sprayon Foam Insulation) media the region of Comptoneffect 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 22, 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 21 Shu =T u uuh hv cos(0)= (1+ )tan( ) m,c 2 (21) 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 onetoone 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 (22) Later on, KleinNishina 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 KleinNishina 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 (23) Equation 23 is the one usually used for standard calculation of the cross sections, r02 is squared value of the classical electron radius. In the lowenergy limit of Compton scatter (ho less than about 10 keV), ho' = hu regardless of the photon scatter angle and Equation 23 reduces to Equation 22. 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 onedimensional sinusoidal distribution defined by: f (x) = a + b cos(2xm~i x + e) where 0i is the onedimensional 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: (24) 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, (25) Integration with respect to yl using (2.4) gives: g(x) = (~a +bcos(2?izxm +))(x ,) + ))1(,)A (26) where 1(x,) is the line spread function defined earlier. Using the expansion: cos(A ) = cos(A) cos(B) + sin(A) sin(B) (27) 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 , (28) + 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) (29) where C(m)i S(m)= T(w)= 1l(x, )exp(2Himx,)&i, (210) 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)> (211) c(m), = M~(mi)cos #(mi) andSm)= m)sn() And by using these, then Equation 29 reduces to: g(x) = a + M(mi)b cos(2xm~i x + e + #(0i)) (212) Equation 212 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) (213) 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 (213) 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 nonlinearity 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 sinewave 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 (214) 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 (215) For a linear system a Fourier Transform of the input is defined as follows W,,(k) = exp(2x i k u)w2~(,)d,(ud (216) 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% (217) 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 Xray 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 23.3). 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 Xray backscatter imaging, the signal measured is formed by the photons that interacted with the target pattern (Figure 24). 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 toabsorption 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 21. Photoelectric, Compton and Pair Production'. Mlomentu m=P KE =T E. = hv SE' = hv hv c Figure 22. Kinematics of the Compton Effect SXray generator SRectangular  shape pattern Figure 23. Transmission model ~ Xray generator Nylon lines Cylindrical shepren C o So . Figure 24. 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 31 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 31) 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 32 is a high resolution, singleline 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) (31) 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 33. Principles of Statistics and Curve Fitting Applied to MTF Calculation Figure 32 and Figure 33 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 = (33) Where the sums of squared residuals are defined as Sw = (y, )2 = SS(Total) (3 4) The Chisquare test is a different measure of the goodnessoffit. The X2 test measures the deviation between the sample and the assumed probability distribution (i.e., hypothesis). The value of Chisquare 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 n1 degrees of freedom. The Ftest 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 Ftest 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 pvalue is computed using the formula: p = 1 invf (F, (DOFcombined DOFseparate), DOFseparate) (3 7) This pvalue is then used to make a statistical statement as to whether the data (not the parameter values) are significantly different or not. If the pvalue 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 +* (38) Fr(4*(xx,)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 38). 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 39 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 38 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 (310) 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) (311) 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 (312) xx 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. XXO dY(X) (A2 Al)*e dY dX YXO(313) (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 zz MTF_edge~function= yo + exp(2 ( c 2) (314) 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 31. 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 32. 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 33. 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 41, showing fiye nylon lines, an xray 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 42 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) 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 44 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 45 shows the scatteringtoab sorption ratios for both Nal and YSO. The values obtained are for Nal and YSSI20 crystalso7 The lower the scatteringtoabsorption ratio the better the detection capabilities. In the energy range of interest (below 50 keV) the YSSi20 crystal has a more favorable scatteringto 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 Xray beam is 22.73 keV and its maximum energy is 50 kVp. The average energy of the backscattered spectrum given in Table 41 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 nonanalog 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 NonAnalog 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 46 shows the MCNP5 model for a 2D input profile calculation. The profile obtained from the model presented in Figure 46 is not strictly 2D. Actually the entire line (3D volume) is modeled but only the contribution from the midplane 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 Xray 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 Xray 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 xray 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 Xray 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 47 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 midplane of the nylon lines. Table 42 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 48 shows the data profie obtained from a midplane contribution only. The errors associated with the data profie shown in Figure 48 are on the order of a tenth of a percent. Table 43 shows a comparison between an Analog MCNP5 run without any variance reduction technique and a NonAnalog 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 43 shows that up to 40 keV the errors associated to both Analog and NonAnalog 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 45 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 Nonanalog 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 NonAnalog techniques is measured by a quantity called "Figure of Merit", FOM, defined by: FOM = (41) time(min) error Where "error" is the relative error. The higher the FOM, the more efficient the calculation. Table 44 presents the number of particles and calculation time for both Analog and Non Analog runs. The NonAnalog 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 NonAnalog 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 midplane contribution. Figure 49 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 410 and Figure 411 show the partial and complete profiles obtained from modeling the MTF sine target using MCNP5. Figure 410 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 411. Table 45 shows a comparison between the Analog and NonAnalog results for the 3D model of the input Sine Target. Figure 412 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 Xray 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 Xray 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 413. One of the cross sections is a square of side halflength ,v the volume is given by V2 (r.r)= (2 )2 dz r3~ (42) Figure 414 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 415 presents the integrated profie. The idea is to obtain an equivalent of the step profie from a peak profile as shown in Figure 416.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 415. The second step requires knowing the value of the intersection volume (Xray beam and nylon line). This volume has been calculated using Equation 43 assuming an intersection of the Xray beam and the line at the center axis only. Figure 418 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 418 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 419 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 419 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 420 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 420 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 421 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 421 is obtained by taking half of each peak plotted in Figure 410 and then normalizing by the intersection volume from the above results. Figure 422 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 423 and Figure 424 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 Xray beam. The figures show that a more effective normalization is achieved when it is performed over the individual pixels. Note that in Figure 423 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 424. 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 3D 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 425, 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 + xxo (43) 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 XAXIS YAXIS 30 Figure 41. Scheme for simulating a sinusoidal input Sinusoidal Input jprofil '. ** **  ** Figure 42. MTF frame plate Figure 43. 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 44. 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 Xray 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 45. Scatteringtoab sorption ratios for Nal and YSSi20 crystals. Table 41. Number of counts at the detector surface for each energy bin and the average energy of the backscattered spectrum. Energy bins Mev Counts Error % 2.00E02 1.08466E02 0.1300% 3.00E02 8.65223E03 0.1500% 4.00E02 3.15683E03 0.2500% 5.00E02 1.34348E04 1.2200% total 2.27900E02 0.0900% Average energy Mev 2.67437E02 1.2989% YSO detector, R=1.27cm DXTRAN sphere Outer radius 4.635 cm Inner radius 4 cm XRay beam SPlane source SPaper Concrete Figure 46. MCNP5 model for input profile calculation. 2D profile calculated from midplane contribution Figure 47. Energy spectrum distribution used in the MCNP5 model based on Kramers spectrum z  Table 42. Comparison between the Analog and NonAnalog MCNP5 Flagged midplane surface Error % Analog NonAnalog Analog vs positive Energy bins Counts Counts Error% Nn current Mev Error % Analog J+ 2.00E02 4.77980E03 0.2000% 4.79438E03 0.3400% 0.3050% 3.00E02 4.31184E03 0.2100% 4.34073E03 0.3600% 0.6700% 4.00E02 1.60421 E03 0.3600% 1.60496E03 0.6700% 0.0468% 5.00E02 6.81481E05 1.7500% 6.80647E05 3.6700% 0.1224% total 1.07640E02 0.1300% 1.08081E02 0.2300% 0.4097% Table 43. 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 44. MCNP5 run condition for Analog versus NonAnalog Analog NonAnalog Time ( min) 25.96 7.01 Number of particles 50000000 3100000 0.02 0.015 0.005 1F 1000 15001111 2000 111 111i11 Figure 48. The input sine profile obtained from running MCNP5 O U e m ~ 2.00E02 c 3 o a L c ~ 1.50E02 rr d rr c 1.00E02 o o B 5.00E03 n E 3 z 0.00E+00 3 Distance in cm Figure 49. Sine profile obtained from modeling 10 nylon lines of different diameters in MCNP5 2 50E02 , 2 00E02 1 50E02 1 00E02 5 00E03 nnncnn 1190 Pixel number 1690 2190 2690 Figure 410. The complete input profile from an MCNP5 simulation as recorded at the detector surface, pixel size 0.1mm. z Table 45. Comparison between Analog and NonAnalog results in MCNP5 Response from the entire line volume Error % Analog NonAnalog Analog vs positive Energy bins Mev Counts Error % Counts Error% Nn current Aao J+ 2.00E02 1.08466E02 0.1300% 1.08436E02 0.2300% 0.0277% 3.00E02 8.65223E03 0.1500% 8.71375E03 0.2300% 0.7110% 4.00E02 3.15683E03 0.2500% 3.14837E03 0.4000% 0.2680% 5.00E02 1.34348E04 1.2200% 1.35106E04 2.1000% 0.5642% total 2.27900E02 0.0900% 2.28408E02 0.1400% 0.2229% YSO detector, R=1.27cm XRay beam DXTRAN sphere Outer radius 4.635 cm Inner radius 4 cm I I I SPlane source SPaper Concrete Figure 411. 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 412. Average energy and fraction of the detected signal in each of the six collision bins. Figure 413. Intersection volume of two cylinders Figure 414. Two cylinder intersection volume 60000 ,a 40000 0 20000 0C 0 I 500 1000 Pixel number 1500 Figure 415. Integrated profile data Equivalence Figure 416. 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 Xray beam and the line Obtain the data profile Obtain the number of counts per unit volume of nvion. EUI Figure 417. 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 418. Experimental and normalized data profile 'C Source YSO Flux tally Paper Concrete Figure 419. 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 420. 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 421. 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 422. 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 423. 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 424. 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 425. MTF function from detector 5, pixel size 0.05mm and beam aperture 0.5mm at 45 kVp45 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 51 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 52 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) (51) 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 52. Figure 53 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 zz 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 Xrays easily penetrate the nylon compared to lead and as a result the lead/nylon interface appears as an edge to Xrays. 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 54 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 55. 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 56 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 57 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 58) 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 59, 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 51. 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 52. Scanning system response to an edge. 0 2 4 6 Frequency (lines/cm) Figure 53. Fourier transform of the line spread function (black curve) and fitting function (red) Lead Nylon Figure 54. Geometry of the MTF step target in MCNP5 6.00E02 Detector S.00E02 4.00E02 2 3.00E02 2.00E02 Led Plastic Lead\ 1.00E02 0.00E+00 0 0.5 1 1.5 2 2.5 3 3.5 4 Distance (cm) Figure 55. Data profile obtained from the first MTF step target design in MCNP5 Nylon m2 cm II Lead III2cm 1cm Figure 56. Geometry of the second design of the MTF step target 8.00E02 7.00E02 6.00E02 5.00E02 4.00E02 3.00E02 2.00E02 1.00E02 0.00E+00 2.5 3 3.5 4 Plastic 0 0.5 1 1.5 2 Distance (cm) Figure 57. Profile data obtained from the second design of the MTF step target 1.00E01 9.00E02 8.00E02 7.00E02 6.00E02 5.00E02 4.00E02 3.00E02 2.00E02 1.00E02 0.00E+00 0 0.5 1 1.5 2 Distance (cm) 2.5 3 3.5 4 Figure 58. 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 59. 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 3D 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 61 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 62 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 Xray imaging system calibration, the main obj ective is to provide a comparison of image quality. Figure 63 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 63 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 64 were generated by fitting the MTF curves using Boltzmann functions. Al A2 The formula used for the fitting process is MTF_experimental= Y = A2 + xxo. The 1+e dY corresponding coefficients are listed in Table 61 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 63. 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 61 gives the first derivative with respect to the frequency (line pairs per mm). The coefficients are given in Table 62 and the plots are shown in Figure 65. XXO dY(X) (A2Al)*e dY dX YXO(61) (1 +e Y ) 2 The second derivative is given by Equation 62: YXX XXo d2Y e dY ed  = (A2 Al)* (62) dX 2 dX xxo (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 66. The zero values of the second derivative are presented in Table 64. 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 63 are sorted and presented in Figure 67 to Figure 611. 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 612 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 VT70191037005) 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, ) =< (63) Also 11 y (x, y)= 3(x )dx= j(3(x ) 3(y )dy; )dx =jl 11 (x', y)dx'C (64) 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' (65) Hence, the edge spread function is the indefinite integral of the line spread function: de(x) 1(x) = (66) Figure 613 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 614. 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 onedimensional 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 613 is used to calculate the resolution associated to the 10%90% edge response. Figure 615 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 67. LS~x ep( )4FTLS) f e* x(1(2zfe) (67) 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 (68) 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 = (69) 2xcro Combining Equations 68 and 69 gives 2*ln(10) 1.46 on (lp/mm or 1p/pixel) (610) 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 616 shows an example of a numerical calculation of the first derivative and the Fourier Transform of the edge function used in Figure 615. 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 detector5YSO 1 1.5 2 frequency line pairs/mm Figure 61. MTF comparison between Nal and YSSi20 detectors at 45 kVp, 0.5 mm aperture 120 100 1 '* mtf1.5mm aperture 80 1 I S60 . N ~mtf1.0mm S40 aperture z 20 0 mtf0.5mm 0 0.5 1 1.5 2 2.5 aperture Frequency line pairs per mm Figure 62. MTF comparison for 3 different aperture diameters at 45kVp45mA0.05mm pixel 1 ze. * mtf0.1mm pix 100 1 mmap 80 u.. a mtf0.1mm pix r 0.5mm ap .0 mtf0.05mm pix m 40 E 1.5mm ap z: 20 mtf0.05mm pix 0 7, 1.0mm ap 0 0.5 1 1.5 2 2.5 m mtf0.05mm pix Frequency line pairs per m m 0.5mm ap Figure 63. MTF comparison for different pixel sizes and beam apertures at 45 kVp45 mA 120 S100 fitmtf0.1mm pix 1 mm ap 1 80 r fitmtf0.1mm pix 0.5mm ap N 60 E \ fit mtf0. 05mm 40 pix1.5mm ap 20 1 fit mtf0. 05mm pix1.0mm ap 0 0.5 1 1.5 2 2.5 ftmf00m pix0.5mm ap Frequency line pairs per mm Figure 64. MTF Boltzmann model fitting function comparison for different pixel sizes and beam apertures at 45 kVp45 mA Table 61. 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 62. 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 pix1 mm ap 3 1stt derivative mtf 0.1mm pix0.5mm 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 65. MTF fitting function first derivative, scan at 45 kVp45 mA Table 63. Roots value of the MTF second derivatives curves Curves first root (Freqline 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 mtf0.1mm pix 1 mm ap 2nd derivative mtf0.1mm pix 0.5mm ap 2nd derivative ) .5 p 1 1.5 2 2.5 mtf0.05mm pix15 a 2nd derivative mtf0.05mm pix 1.0mm ap m2nd derivative mtf0.05mm pix Frequency line pairs per m m 0.5mm ap 500 1000 1500 Figure 66. MTF fitting function second derivative, scan at 45 kVp45mA 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 67. 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 68. 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 69. 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 610. 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 611. 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 612. YSO image of MTF Target on a tile panel ,3:77 e s rr i~cjo ; n Figure 613. Selection and smoothing steps for the MTF calculation from a step function MTF MTF ((,rl) MTF ((,0) MTF ((,0) MTF((,0) I Table 64. 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 614. 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 615. Edge function width estimation 0. ..l.. . Di FFT r y INbvisigrgj G~i^~DcF =7.01484 F^2 = 098518 1.71357 .41015 >0: 1.9351E16 .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*((xxc)/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 616. 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 LockheedMartin Space Systems Company. Figure 71 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 72), 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 73). Figure 74 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 75 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 xordinates from the image to recalculate the overall maximum in the ROI. After all the peaks have been selected, the MTF plot is generated (Figure 76). 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 77 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 78 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 78 to 710 show the different steps in the Matlab code used to generate MTF curves from an edge function. Figure 78 shows how a region of interest can be selected, Figure 79 and 710 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 DocumentslFishinglinet61 62006MTF45kVp45mA0 .5mmAperture Fle Image product M~in Image Image name 61 62006MTFplate55Kvdell _Minlmage FileListImage list 616200 MT .Isedfe Ile . .11 ^ Display ..i..10. ril.Ise. l Jleg 1 alCompute 6162006MT F45kVp45mA 0.5mmApe rturedt 6162006MTFplate56Kvdetector3.dat 6162006MTF45k~p45mA0.5mmAperturedc 61 62006MTFplate55Kvdetector4 .dat 6162006MTF f ,:..j_.i, ...r..pen...4.1 6162006MTFplate55Kvd etector 5.dat 6106MF .4..0i...: dn 6162006MTF15i r.ira0 r~r .n. '820T ar~.9rr.nosd Format A Figure 71. Matlab user interface File Edit View Insert Tools Desllop Window Help D c a El e la  RR9  W 0C E O I n S1.. rTE i ITF46kVp46mA0. 1pixD.6mmapHorizordalCorrectionYSOringp iio lO102C 10000 8000 S6000 4001245 Inches _ _____ File Edit Adjust Filter View Utilities Windov MTF X Line Drofile Figure 72. MTF menu and data profile Figure 73. 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 74. User interface for information entries 200 400 600 800 Pixel number 1000 1200 1400 1600 Figure 75. 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 76. Saving files First derivative S20000 0 15000 S10000 ~5000 ~5000 0 10 20 30 40 50 60 10 80 : Pixel number Figure 77. 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 = hle Edit lext tell lools Debug Desktop Window Help r FX a Ill 8 5 2 'HTF processing, this 4 isystem. 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 : AsILCUYI 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 78. Selection of a region of interest in the edge function profile x 104 Scanning system output: Counts/pixel 61 IiL I 0 5 10 15 20 25 r Pixel nurnber S Figure 79. 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 710. MTF curves with frequencies expressed in line pairs/pixel and line pairs/mm CHAPTER 8 CONCLUSION In order to properly characterize the Xray 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 A1 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 A1 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 A2 shows a line profile across image b in Figure A1. 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 A1. 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. 3Ue Laext Cdrll sie If exact size Is nall avehible 4Three 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 F32 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 B1. MTF frame plate top view APPENDIX B MTF FRAME STRUCTURE Aluminlum plate C 1250 Ihlck. MTF frame p ate TOP VIEW Nissia Sabri 5122006i 02500  Nissia Sabri 5122006 Aluminium plate 0.25 thick MTF cover plate TOP VIEW Figure B2. 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. 75108 (1998). 2. A. Jacobs, E. Dugan, S. Brygoo, D. Ekdahl, L. Houssay, and Z. Su, "Lateral Migration Radiography: A New Xray Backscatter Imaging Technique," Proceeding of SPIE, 4786 p. 116 (2002). 3. E. Dugan, A. Jacobs, L. Houssay, and D. Ekdahl, "Detection of Flaws and Defects Using Lateral Migration Xray Radiography," Proceeding of SPIE, 5199 p. 4761 (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 7086K0016, 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. 417424 (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. 5567 (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. 384393 (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. 878887 (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. 906916 (1999). 12. C. Wells, Z. Su, A. Allard, S. Salazar, E. Dugan, and A. Jacobs, "Suitability of Simulated Landmines for Detection Measurements Using Xray Lateral Migration Radiography", Proceeding of SPIE, 4038 p. 578589 (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. 24722479 (2000). 14. E. Dugan, A. Jacobs, Z. Su, L. Houssay, D. Ekdahl, and S. Brygoo, "Development and Field Testing of a Mobile Backscatter Xray Lateral Migration Radiography Land Mine Detection System," Proceeding of SPIE, 4742 p. 120131 (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, "Xray 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 Xray 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 xray 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 PhysicsFrance. PAGE 1 1 AN ADAPTED MODULATION TRANSFER F UNCTION FOR XRA 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 PAGE 2 2 2007 Nissia Sabri PAGE 3 3 To my mother PAGE 4 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. PAGE 5 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: XRay 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 PAGE 6 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 PAGE 7 7 LIST OF TABLES Table page 41 Number of counts at the detector surface..........................................................................60 42 Comparison between the Analog and NonAnalog MCNP5.............................................61 43 Summary of the line diameters and th e associated number of line position......................62 44 MCNP5 run condition for Analog versus NonAnalog.....................................................62 45 Comparison between Analog and NonAnalog results in MCNP5...................................64 61 Coefficients used in the fitting function formula for each MTF curve..............................88 62 Statistical measures of the fitting accuracy........................................................................88 63 Roots value of the MTF second derivatives curves...........................................................89 64 Different methods of the MTF derivation..........................................................................92 PAGE 8 8 LIST OF FIGURES Figure page 11 Schematic illustrating Xray production............................................................................19 12 Typical spectrum obtained from an Xray tube with a tungsten anode4............................19 13 Compton Backscatte ring Imaging (CBI)...........................................................................20 14 Lateral Migration Radiography (LMR).............................................................................20 15 Photograph of RSD System with 4 NaI Detectors.............................................................21 16 Photograph of RSD System showing YS O detectors mounted to NaI Detectors..............21 17 RSD scanning system m ounted on a fixed frame..............................................................22 18 Flow chart of the image acquisition process20...................................................................23 21 Photoelectric, Compton and Pair Production5...................................................................34 22 Kinematics of the Compton Effect....................................................................................34 23 Transmission model......................................................................................................... ..35 24 Backscatter model.......................................................................................................... ....35 31 Scanning system output two line pairs placed at 45with respect to the vertical axis.......42 32 High exposure scanning output, one sw eep of a nylon line (Dirac Simulation)................43 33 Scan of a cubic plastic sample: 17.5 mm width, 1 mm beam, 0.5 mm pixels...................43 41 Scheme for simulating a sinusoidal input..........................................................................57 42 MTF frame plate............................................................................................................ ....58 43 MTF frame plate detailed design.......................................................................................58 44 Output profile from the scan of the MTF Sine target (detector 1 NaI)..............................59 45 Scattering to absorption ratios for NaI and Y5Si2O crystals.............................................59 46 MCNP5 model for input profile calculation......................................................................60 47 Energy spectrum distribution used in the MCNP5 model based on Kramers spectrum....60 48 The input sine profile obtained from running MCNP5......................................................62 PAGE 9 9 49 Sine profile obtained from modeling 10 nylon lines of diffe rent diameters in MCNP5...63 410 The complete input profile from an MCNP 5 simulation as recorded at the detector........63 411 MCNP5 model for input profile calculation......................................................................64 412 Average energy and fraction of the detected signal in each of the six collision bins........65 413 Intersection volume of two cylinders.................................................................................65 414 Two cylinder intersection volume.....................................................................................65 415 Integrated profile data................................................................................................... .....66 416 Equivalence between peaks and steps profiles..................................................................66 417 Normalization methodology scheme.................................................................................66 418 Experimental and normalized data profile.........................................................................67 419 A representation of the MCNP5 set up for volume intersection calculations.....................67 420 Line and beam intersection volume values........................................................................68 421 A plot of the volumetric normalization of half peaks obtained from MCNP model.........68 422 Visual editor view of the new MC NP setup for volume calculations................................69 423 Normalization of the MTF sine profile over the intersection volume...............................69 424 Statistical smoothing of the normalized profile.................................................................70 425 MTF function from detector 5...........................................................................................70 51 Edge target made from a junction of lead (absorber) and nylon (scatterer)......................74 52 Scanning system response to an edge................................................................................74 53 Fourier transform of the line spread function (black cu rve) and fitting function (red).....75 54 Geometry of the MTF step target in MCNP5....................................................................75 55 Data profile obtained from the firs t MTF step target design in MCNP5...........................76 56 Geometry of the second de sign of the MTF step target.....................................................76 57 Profile data obtained from the s econd design of the MTF step target...............................77 58 Data profile obtained from the third target design; nylon block on top of lead.................77 PAGE 10 10 59 Final design profile propos ed for the MTF step target......................................................78 61 MTF comparison between NaI and Y5Si2O detectors at 45 kVp, 0.5 mm aperture..........86 62 MTF comparison for 3 diffe rent aperture diameters..........................................................86 63 MTF comparison for differe nt pixel sizes and beam ap ertures at 45 kVp45 mA............87 64 MTF Boltzmann model fitting function comparison.........................................................87 65 MTF fitting function first deri vative, scan at 45 kVp45 mA............................................88 66 MTF fitting function second deri vative, scan at 45 kVp45mA........................................89 67 1 MTF 0.1 mm pixel, 0.5 mm aperture..............................................................................89 68 2 MTF 0.05 mm pixel, 0.5 mm aperture............................................................................90 69 3 MTF 0.1 mm pixel, 1.0 mm aperture..............................................................................90 610 4 MTF 0.05 mm pixel, 1.0 mm aperture............................................................................90 611 5 MTF 0.05 mm pixel, 1.5 mm aperture............................................................................91 612 YSO image of MTF Target on a tile panel........................................................................91 613 Selection and smoothing steps for the MTF calculation from a step function..................92 614 An example of the edge pr ofile and its first derivative......................................................93 615 Edge function width estimation.........................................................................................93 616 Numerical evaluation of the firs t derivative of the edge function.....................................94 71 Matlab user interface...................................................................................................... ....97 72 MTF menu and data profile...............................................................................................98 73 Data profile............................................................................................................... .........98 74 User interface for information entries................................................................................99 75 Maximum search............................................................................................................. ...99 76 Saving files............................................................................................................... ........100 77 Data profile from an edge function and its first derivative..............................................100 78 Selection of a region of intere st in the edge function profile...........................................101 PAGE 11 11 79 The selected region of interest and th e first derivative of the edge function...................101 710 MTF curves with frequencies expresse d in line pairs/pixel and line pairs/mm...............102 A1 Comparison between two backscatter images.................................................................105 A2 Line profile evaluation of the paper filtering...................................................................106 B1 MTF frame plate top view...............................................................................................107 B2 MTF cover plate top view................................................................................................108 PAGE 12 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 XRA 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, singleside xray 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 nondestructive 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 sprayon foam insulation (SOFI) inspection. The xray ba ckscatter RSD imaging system has been successfully used for cracks and corrosion spot detec tion in a variety of materials. The conventional transmission xray 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. PAGE 13 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. PAGE 14 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 system13 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 Xray 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, Xray images are maps of the xray 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 Xray photon backscattering4. PAGE 15 15 Xrays 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 11 shows the basic principle of Xray 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 Xrays5. The energies of these discrete line spectra are characteristic of the a node chemical element. The total spectrum obtained from a typical Xray tube with a tungsten anode is shown in Figure 12. As the Xrays 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 Xray source, the object being radiographed, and a detector. From an imaging standpoint there is an important distinction between absorption and scattering. Usual Xray 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 twodimensional proj ection of a threedimensional obj ect; many planes are collapsed PAGE 16 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 Radiography614 (Figure 14) is similar to the CBI technique (Figure 13), but instead of counting only singlecollision b ackscattered photons, the LMR technique counts both singleand multiplecollisi on backscattered photons that have laterally spread out from the illumination beam entry point. At the detector surface, signals from singleand multiplecollision 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. PAGE 17 17 Project Objectives The components that form the RSD scanni ng system are different and complex. Four major parts can be identified: Xray 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: XRay 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 Xray generator mounted onto a scanning PAGE 18 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 Xray 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 Xrays (1055keV). 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 15. In Figure 16, the YSO is mounted on detector 2 using an aluminium ring. In Figure 17 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 pileup19. 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 18) presents the entire image acquisition process from detection to display. PAGE 19 19 Figure 11. Schematic il lustrating Xray production Figure 12. Typical spectrum obtained from an Xray tube with a tungsten anode4 High voltage + Xray tube Anode Electron gun Xrays Electron beam Sample PAGE 20 20 Figure 13. Compton Back scattering Imaging (CBI) Figure 14. Lateral Migr ation Radiography (LMR) Land mine Collimated detector Noise Signal Xray generator Earth Uncollimated detector XRay Generator Object Detector Signal Noise PAGE 21 21 Figure 15. Photograph of RSD System with 4 NaI Detectors Figure 16. 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 Xray beam tube Finned Collimator Angle at 90 (degrees) PAGE 22 22 Figure 17. RSD scanning syst em mounted on a fixed frame PAGE 23 23 Figure 18. Flow chart of the image acquisition process20 Dir Active Step Dir Step Step Xaxis Complete Pulse train Complete Pulse train Complete Pulse train Yaxis Yaxis Xaxis Xaxis Visible light XRay Current Analog pulse Analog pulse Analog pulse Digital pulse Analog pulse Yes X rayscatteredtowarddetector Na I PhotoCathode PMT PreAmp FastAmp oscilloscope MCA SCA is the pulse in the voltagewindow Counter /Timer Pulse train BNC 2121 NIDaq PCI 6602 Labview/computer Y axis Y Motor Y MotorAmps or image NIMotion PCI 7344 NIMotion breakout box Limit/Home Switches XMotor XMotor Amps PAGE 24 24 CHAPTER 2 PROBLEM STATEMENT General Physics of Photon Interaction When considering an Xray 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 Xray 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 radiationprotection 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 21 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). PAGE 25 25 For lowZ (e.g., carbon, air, aluminum, Sprayon Foam Insulation) media the region of Comptoneffect 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 22, 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 PAGE 26 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 21 ) 2 tan( ) 1 ( ) cos( )) cos( 1 )( ( 12 0 2 0 c m h h h T c m h h h (21) 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 onetoone 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: PAGE 27 27 ) cos 1 ( 22 2 0 0 r d de (22) Later on, KleinNishina 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 KleinNishina 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 (23) Equation 23 is the one usually used for standard calculation of the cross sections, 2 0r is squared value of the classical electron radius. In the lowenergy limit of Compton scatter (h less than about 10 keV), h h regardless of the phot on scatter angle and Equation 23 reduces to Equation 22. 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 onedimensional sinusoidal distribution defined by: PAGE 28 28 ) 2 cos( ) ( x b a x f where is the onedimensional 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 (24) 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 (25) Integration with respect to y1 using (2.4) gives: 1 1 1) ( )) ) ( 2 cos( ( ) (dx x l x x b a x g (26) PAGE 29 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 (27) ) (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 (28) or ) ( ) 2 sin( ) ( ) 2 cos( ) ( S x b C x b a x g (29) where 1 1 1) 2 exp( ) ( ) ( ) ( ) ( dx x i x l T S i C (210) 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 (211) And by using these, then Equation 29 reduces to: )) ( 2 cos( ) ( ) ( x b M a x g. (212) Equation 212 shows that the output is sinusoi dal and has the same frequency as the input. The output modulation is defined as: PAGE 30 30 a b M g g g g MOUT) (min max min max (213) 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 (213) 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 nonlinearity 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 sinewave 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 (214) PAGE 31 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 (215) 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( ) ( (216) 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. PAGE 32 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 (217) 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. PAGE 33 33 It is important to notice that in the case of Xray 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 23.3). 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 Xray backscatter imaging, the signal measured is formed by the photons that interacted with the target pattern (Figure 24). 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 scattertoabsorption 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. PAGE 34 34 Figure 21. Photoelectric, Compton and Pair Production5. Figure 22. Kinematics of the Compton Effect h E Momentum= c h e 0 'h E Momentum= c h' KE =T Momentum=P PAGE 35 35 Figure 23. Transmission model Figure 24. Backscatter model Xray generator Detector 1 Detector 2 Backscattered photons Cylindrical shape pattern Nylon lines Xray generator Rectangular shape pattern Detector 1 Detector 2 Transmitted photons PAGE 36 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 31 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 31) 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 32 is a high resolution, singl eline 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 (31) PAGE 37 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 (32) 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 33. Principles of Statistics and Curve Fitting Applied to MTF Calculation Figure 32 and Figure 33 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. PAGE 38 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 ) )( ( (33) Where the sums of squared residuals are defined as i i yyy y S2) (=SS(Total) (34) The Chisquare test is a different measure of the goodnessoffit. The test2 measures the deviation between the sample and the assume d probability distribution (i.e., hypothesis). The value of Chisquare is calculated according to the following formula, n i i i iNp Np N2 2) ( (35) 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 2distribution density function with n1 degrees of freedom. The Ftest 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 Ftest 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. PAGE 39 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* ) ( ) ( (36) Once the F value is computed, the pvalue is computed using the formula: ) ), ( ( 1e DOFseparat e DOFseparat d DOFcombine F invf p (37) This pvalue 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 pvalue 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 (38) 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 PAGE 40 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 38). 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 39 X XFT 2 ) exp( ) ) ( exp( 22 2 (39) 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 38 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 (310) 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 (311) with ) ( 21 mm x z The above formula gives the general behavior. PAGE 41 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 (312) 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 (313) 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 (314) with ) ( 21 mm x z PAGE 42 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 31. 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 PAGE 43 43 0246810 2000 2200 2400 2600 2800 3000 line spread function(1) Lorentz fitting function(2)Number of counts/pixelDistance x in ( mm ) Figure 32. 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 33. Scan of a cubic plastic sample : 17.5 mm width, 1 mm beam, 0.5 mm pixels Line profile Line profile PAGE 44 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 41, showing five nylon lines, an xray 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 42 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 44 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. PAGE 45 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 45 shows the scatteringtoabsorption ratios for both NaI and YS O. The values obtained are for NaI and Y5SI2O crystals17. The lower the scatteringtoabsorption ratio the better the detection capabilities. In the energy range of interest (below 50 keV) the Y5Si2O crystal has a more favorable scatteringtoabsorption 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. PAGE 46 46 The average energy of the incident Xray beam is 22.73 keV and its maximum energy is 50 kVp. The average energy of the backscattered spectrum given in Table 41 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 nonanalog 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 NonAnalog 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 46 shows the MCNP5 m odel for a 2D input profile calculation. The profile obtained from the model presented in Figure 46 is not st rictly 2D. Actually the entire line (3D volume) is modeled but only the contribution from th e midplane 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 Xra 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 Xray beam are kept at the same position while the line position is varied. PAGE 47 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 xray 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 Xray 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. PAGE 48 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 47 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. PAGE 49 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 midplane of the nylon lines. Table 42 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 48 shows the data profile obtained from a midplane contribution only. The errors associated with the data profile s hown in Figure 48 are on the order of a tenth of a percent. Table 43 shows a comparis on between an Analog MCNP5 run without any variance reduction technique and a NonAnalog 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 43 shows that up to 40 keV the errors associated to both Analog and NonAnal 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 45 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. PAGE 50 50 In a Nonanalog 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 NonAnalog techniqu es is measured by a quantity called Figure of Merit, FOM, defined by: 2* (min) 1 error time FOM (41) Where error is the relative error. The higher the FOM, the more efficient the calculation. Table 44 presents the number of particles and calculation time for both Analog and NonAnalog runs. The NonAnalog 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 NonAnalog 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 midplan e contribution. Figure 49 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. PAGE 51 51 However, the relative errors associated with the obtained profile were between0.32% and 2.35%. Figure 410 and Figure 411 show the partial and comp lete profiles obtained from modeling the MTF sine target using MCNP5. Figure 410 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 411. Table 45 shows a comparison between the An alog and NonAnalog results for the 3D model of the input Sine Target. Figure 412 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 Xray 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. PAGE 52 52 Geometric Normalization It is important to notice that the conven tional MTF calculation (e.g., as employed with transmission Xray 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 413. One of the cross sections is a square of side halflength 2 2z r the volume is given by r rr dz z r r r V3 2 2 2 23 16 ) 2 ( ) ( (42) Figure 414 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 (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 Line Beamr r k is the elliptic modulus. PAGE 53 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 415 presents the integrated profile. The idea is to obtain an equivalent of the st ep profile from a peak profile as shown in Figure 416.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 415. The second step requires knowing the value of the in tersection volume (Xray beam and nylon line). This volume has been calculated using Equation 43 assuming an intersection of the Xray beam and the line at the center axis only. Figure 418 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. PAGE 54 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 418 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 419 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 419 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. PAGE 55 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 420 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 420 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 421 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 421 is obtained by taking half of each peak plotted in Figure 410 and then normalizing by the intersection volume from the above results. Figure 422 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 423 and Figure 424 show the normalization of the input sine profile over the intersecti on volume of two cylinders. PAGE 56 56 The two cylinders have the same dimensions as the nylon line and the Xray beam. The figures show that a more effective normalization is achieved when it is performed over the individual pixels. Note that in Figure 423 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 424. 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 3D 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 425, 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 (43) PAGE 57 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 41. Scheme for si mulating a sinusoidal input Sinusoidal Input profile Xray generator Detector Nylon lines PAGE 58 58 Figure 42. MTF frame plate Figure 43. MTF fram e plate detailed design PAGE 59 59 Figure 44. 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 45. Scatteringtoabsorption 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 Xray beam incident energy PAGE 60 60 Table 41. Number of counts at the detector surface for each energy bin and the average energy of the backscattered spectrum. Energy bins Mev Counts Error % 2.00E02 1.08466E02 0.1300% 3.00E02 8.65223E03 0.1500% 4.00E02 3.15683E03 0.2500% 5.00E02 1.34348E04 1.2200% total 2.27900E02 0.0900% Average energy Mev 2.67437E02 1.2989% Figure 46. MCNP5 model for input profile calculation. 2D prof ile calculated from midplane contribution Figure 47. 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 PAGE 61 61 Table 42. Comparison between the Analog and NonAnalog MCNP5 Flagged midplane surface Analog NonAnalog positive current Energy bins Mev Counts Error % Counts Error % Error % Analog vs NonAnalog J+ 2.00E02 4.77980E03 0.2000% 4.79438E03 0.3400% 0.3050% 3.00E02 4.31184E03 0.2100% 4.34073E03 0.3600% 0.6700% 4.00E02 1.60421E03 0.3600% 1.60496E03 0.6700% 0.0468% 5.00E02 6.81481E05 1. 7500%6.80647E05 3.6700% 0.1224% total 1.07640E02 0.1300%1. 08081E02 0.2300% 0.4097% PAGE 62 62 Table 43. 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 48. The input sine prof ile obtained from running MCNP5 Table 44. MCNP5 run conditi on for Analog versus NonAnalog Analog NonAnalog Time ( min) 25.96 7.01 Number of particles 50000000 3100000 PAGE 63 63 0.00E+00 5.00E03 1.00E02 1.50E02 2.00E02 0123456 Distance in cmNumber of photon detected per source particle Figure 49. Sine profile obtai ned from modeling 10 nylon lines of different diameters in MCNP5 0.00E+00 5.00E03 1.00E02 1.50E02 2.00E02 2.50E02 3101906901190169021902690 Pixel numberCount/ pixel (photon detected per source particle) Figure 410. The complete input profile from an MCNP5 simulation as recorded at the detector surface, pixel size 0.1mm. PAGE 64 64 Figure 411. MCNP5 model for in put profile calculation. 3D prof ile calculated from a volume contribution Table 45. Comparison between Anal og and NonAnalog results in MCNP5 Response from the entire line volume Analog NonAnalog positive current Energy bins Mev Counts Error % Counts Error % Error % Analog vs NonAnalog J+ 2.00E02 1.08466E02 0.1300% 1.08436E02 0.2300% 0.0277% 3.00E02 8.65223E03 0.1500% 8.71375E03 0.2300% 0.7110% 4.00E02 3.15683E03 0.2500% 3.14837E03 0.4000% 0.2680% 5.00E02 1.34348E04 1.2200% 1.35106E04 2.1000% 0.5642% total 2.27900E02 0.0900% 2.28408E02 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 PAGE 65 65 123456 Fraction of detected signa Avarage energy Number of collisions Figure 412. Average energy and fraction of the dete cted signal in each of the six collision bins. Figure 413. Intersection volume of two cylinders Figure 414. 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% PAGE 66 66 0 20000 40000 60000 050010001500Pixel numbercount/pixel Figure 415. Integrated profile data Figure 416. Equivalence betw een peaks and steps profiles. Figure 417. 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 Xray beam and the line Step 1 Redistribute to have a constant number of counts along theline Obtain the data profile PAGE 67 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 418. Experimental a nd normalized data profile Figure 419. 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 PAGE 68 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 420. 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 421. A plot of the volumetric normalizat ion of half peaks obtained from MCNP model. PAGE 69 69 Figure 422. 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 423. Normalization of the MTF sine profile over the intersection volume Sphere source Nylon line Xray beam PAGE 70 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 424. 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 425. MTF function from detector 5, pi xel size 0.05mm and beam aperture 0.5mm at 45 kVp45 mA PAGE 71 71 CHAPTER 5 AN IMPROVED TECHNIQUE FOR THE MTF CALCULATION BASED ON A STEP FUNCTION Step Function Target Design for MTF Calculation Figure 51 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 52 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 (51) 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 52. Figure 53 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 (52) ) ( 21 mm x z PAGE 72 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 Xrays easily penetrate the nylon compared to lead and as a result the lead /nylon interface appears as an edge to Xrays. 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 PAGE 73 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 54 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 55. 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 56 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 57 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 58) 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. PAGE 74 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 59, 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 51. 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 52. Scanning system response to an edge. Scatterer Absorber Distance in c m Number of counts PAGE 75 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 53. Fourier transform of the line spread function (black curve) and fitting function (red) Figure 54. 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 PAGE 76 76 0.00E+00 1.00E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 00.511.522.533.54 Distance (cm)Counts/pixel Figure 55. Data profile obt ained from the first MTF step target design in MCNP5 Figure 56. 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 PAGE 77 77 0.00E+00 1.00E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 7.00E02 8.00E02 00.511.522.533.54 Distance (cm)Counts/pixel Figure 57. Profile data obt ained from the second design of the MTF step target 0.00E+00 1.00E02 2.00E02 3.00E02 4.00E02 5.00E02 6.00E02 7.00E02 8.00E02 9.00E02 1.00E01 00.511.522.533.54 Distance (cm)Counts/pixel Figure 58. Data profile obtai ned from the third target desi gn; nylon block on top of lead Plastic Lead Lead PAGE 78 78 Figure 59. 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 PAGE 79 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 3D 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 61 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 62 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 Xray imaging system calibration, the main objective is to provide a comparison of image quality. Figure 63 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. PAGE 80 80 In Figure 63 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 64 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 61 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 63. 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 61 gives the first deri vative with respect to the frequency (line pairs per mm). The coeffici ents are given in Tabl e 62 and the plots are shown in Figure 65. 2 0 0) 1 ( ) 1 2 ( ) (dX X X dX X Xe e A A dX X dY (61) The second derivative is given by Equation 62: 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 (62) PAGE 81 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 66. The zero values of the second derivative are pr esented in Table 64. 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 63 are sorted and presented in Figure 67 to Figure 611. 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 612 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 VT70191037005) 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. PAGE 82 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 (63) Also PAGE 83 83 ' ') ( ) ) ( ) ( ( ) ( ) ( dx y x w dx dy y x dx x y x wx Line in xx edge in (64) 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' ') ( ) ( ) ( ) ( (65) Hence, the edge spread function is the indefini te integral of the line spread function: dx x de x l ) ( ) ( (66) Figure 613 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 614. 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. PAGE 84 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 613 is used to calculate the resolution associated to the 10%90% edge response. Fi gure 615 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 67. ) ) 2 ( 2 1 exp( 2 ) )( ( ) 2 1 exp( ) (2 2 2 f f LSF FT x x LSF (67) The width of the LSF at 10% of its maximum is given by width edge x ) 10 ln( 2 2% 10 (68) PAGE 85 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 (69) Combining Equations 68 and 69 gives width edge width edge f 46 1 ) 10 ln( 2% 10 (lp/mm or lp/pixel) (610) 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 616 shows an example of a numerical calculation of the first derivative and the Fourier Transform of the edge function used in Figure 615. 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. PAGE 86 86 0 20 40 60 80 100 120 00.511.522.5 frequency line pairs/mmMTF % detector1NaI detector5YSO Figure 61. 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 (% ) mtf1.5mm aperture mtf1.0mm aperture mtf0.5mm aperture Figure 62. MTF comparison fo r 3 different aperture diamet ers at 45kVp45mA0.05mm pixel size. PAGE 87 87 0 20 40 60 80 100 00.511.522.5 Frequency line pairs per mmNormalized MTF (%) mtf0.1mm pix1mm ap mtf0.1mm pix0.5mm ap mtf0.05mm pix1.5mm ap mtf0.05mm pix1.0mm ap mtf0.05mm pix0.5mm ap Figure 63. MTF comparison for different pixel sizes and beam apertures at 45 kVp45 mA 0 20 40 60 80 100 120 00.511.522.5Frequency line pairs per mmNormalized MTF (%) fitmtf0.1mm pix1mm ap fitmtf0.1mm pix0.5mm ap fitmtf0.05mm pix1.5mm ap fitmtf0.05mm pix1.0mm ap fitmtf0.05mm pix0.5mm ap Figure 64. MTF Boltzmann model fitting function comparison for different pixel sizes and beam apertures at 45 kVp45 mA PAGE 88 88 Table 61. 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 62. 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 pix1mm ap 1st derivative mtf0.1mm pix0.5mm ap 1st derivative mtf0.05mm pix1.5mm ap 1st derivative mtf0.05mm pix1.0mm ap 1st derivative mtf0.05mm pix0.5mm ap Figure 65. MTF fitting function firs t derivative, scan at 45 kVp45 mA PAGE 89 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 mtf0.1mm pix1mm ap 2nd derivative mtf0.1mm pix0.5mm ap 2nd derivative mtf0.05mm pix1.5mm ap 2nd derivative mtf0.05mm pix1.0mm ap 2nd derivative mtf0.05mm pix0.5mm ap Figure 66. MTF fitting function second derivative, scan at 45 kVp45mA Table 63. Roots value of th e MTF second derivatives curves Curves first root (Freqline 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 67. 1 MTF 0.1 mm pixel, 0.5 mm aperture PAGE 90 90 Figure 68. 2 MTF 0.05 mm pixel, 0.5 mm aperture Figure 69. 3 MTF 0.1 mm pixel, 1.0 mm aperture Figure 610. 4 MTF 0.05 mm pixel, 1.0 mm aperture PAGE 91 91 Figure 611. 5 MTF 0.05 mm pixel, 1.5 mm aperture Figure 612. YSO image of MTF Target on a tile panel Nylon lines from the MTF target Aluminium edge of the MTF frame Tile Test Panel PAGE 92 92 Table 64. 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 613. 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 PAGE 93 93 Figure 614. An example of the ed ge profile and its first derivative Figure 615. 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 PAGE 94 94 Figure 616. 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*((xxc)/w)^2) Weighting: yNo weighting Chi^2/DoF= 9422.64067 R^2= 0.9963 y01.76655.04368 xc25.76397.00675 w1.01068.01394 A9715.589960.72925X 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*((xxc)/w)^2) W eighting: yNo weighting Chi^2/DoF= 7.01484 R^2= 0.98518 y01.713570.41015 xc1.9351E160.00232 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) PAGE 95 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 LockheedMartin Space Systems Company. Figure 71 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 72), 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 73). Figure 74 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 75 shows how the pr eliminary peak selection is displayed. PAGE 96 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 xordinates 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 76). 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 77 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 78 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. PAGE 97 97 The more convenient choice for automated use would be using the f itting tools provided with the Matlab7.0.4 version. Figures from 78 to 710 show the different st eps in the Matlab code used to generate MTF curves from an edge function. Figure 78 shows how a region of interest can be selected, Figure 79 and 710 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 71. Matlab user interface PAGE 98 98 Figure 72. MTF me nu and data profile Figure 73. Data profile Lineprofile PAGE 99 99 Figure 74. User interface for information entries Figure 75. Maximum search PAGE 100 100 Figure 76. Saving files Figure 77. Data profile from an edge function and its first derivative PAGE 101 101 Figure 78. Selection of a region of interest in the edge function profile Figure 79. The selected regi on of interest and the first de rivative of the edge function PAGE 102 102 Figure 710. MTF curves with frequencies e xpressed in line pairs/ pixel and line pairs/mm PAGE 103 103 CHAPTER 8 CONCLUSION In order to properly characterize the Xray 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. PAGE 104 104 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. PAGE 105 105 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 A1 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 A1 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 A2 shows a line profile across image b in Figure A1. The lines under the paper show with near half intensity of that of the line above the paper. a b Figure A1. Comparison between two backscatter images. a) Scan without paper underneath the nylon line. b) Scan with pa per underneath the nylon line PAGE 106 106 Figure A2. Line profile evaluation of the paper filtering PAGE 107 107 APPENDIX B MTF FRAME STRUCTURE Figure B1. MTF fr ame plate top view PAGE 108 108 Figure B2. MTF c over plate top view PAGE 109 109 LIST OF REFERENCES 1. E. Dugan, A. Jacobs, S. Keshavmurthy, and J. Wehlburg," Lateral Migration Radiography," Research in Nondestructive Evaluation, 10(2) p. 75108 (1998). 2. A. Jacobs, E. Dugan, S. Brygoo, D. Ekdahl, L. Houssay, and Z. Su, Lateral Migration Radiography: A New Xray Backscatter Imaging Technique, Proceeding of SPIE, 4786 p. 116 (2002). 3. E. Dugan, A. Jacobs, L. Houssay, and D. Ekdahl, Detection of Flaws and Defects Using Lateral Migration Xray Radiogr aphy, Proceeding of SPIE, 5199 p. 4761 (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 7086K0016, 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. 417424 (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. 5567 (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. 384393 (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. 878887 (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. 906916 (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 Xray Lateral Migration Radiography, Proceeding of SPIE, 4038 p. 578589 (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. 24722479 (2000). PAGE 110 110 14. E. Dugan, A. Jacobs, Z. Su, L. Houssay, D. Ekdahl, and S. Brygoo, Development and Field Testing of a Mobile Backscatter Xray Late ral Migration Radiogra phy Land Mine Detection System, Proceeding of SPIE, 4742 p. 120131 (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, Xray 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 Xray 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). PAGE 111 111 BIOGRAPHICAL SKETCH Nissia Sabri is a graduate assist ant at the University of Florid a. She joined the Scatter xray 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. 