The Pennsylvania State University The Graduate School College of Engineering DEVELOPMENT OF A NEUTRON COMPUTED TOMOGRAPHY SYSTEM AT THE PENNSYLVANIA STATE UNIVERISTY A Thesis in Nuclear Engineering by Liang Shi 2008 Liang Shi Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2008
The thesis of Liang Shi was reviewed and approved* by the following: Jack Brenizer Chair of Nuclear Engineering Professor of Mechanical and Nuclear Engineering Thesis Adviser Kenan ünlü Senior Scientist and Affiliate Professor of Nuclear Engineering Matthew M. Mench Associate Professor of Mechanical and Nuclear Engineering * Signatures are on file in the Graduate School.
ABSTRACT Neutron tomography has been proved to be a powerful tool in the non-destructive testing and many other engineering application fields. Compared to x- rays, neutrons can be attenuated by some light materials, i.e. hydrogen, but can penetrate many heavy materials. Thus, neutron tomography has been extensively used to obtain the sample s interior structure and material properties that other traditional methods can not provide. A neutron computed tomography system was installed at the Pennsylvania State University s Radiation Science and Engineering Center (RSEC) to investigate the water distribution in operating polymer electrolytic fuel cells (PEFCs) under normal and frozen conditions. The neutron tomography experiment and the initial reconstructions were successfully performed. The neutron computed tomography projection data has also been simulated and will be implemented for quantitative tomography research. The Modulation Transfer Function (MTF) technique was used to quantify the true resolution of the imaging system and this technique will continue to be expanded and combined with the quantification technique for the imaging system. For quantitative tomography purposes, major enhancements were made to the previous imaging system to improve the system spatial resolution. A two mirror open imaging system was designed that incorporates substantial radiation shielding to protect the cooled CCD camera. The new system has been successfully tested and is being used for taking radioscopic data and some of the results from these tests will be given in this thesis. iii
TABLE OF CONTENTS LIST OF FIGURES......VI NOMENCLATURE..IX ACKNOWLEDGEMENTS...XII CHAPTER 1: INTRODUCTION.....1 1.1: Neutron Computed Tomography Imaging...1 1.2: Summary of Neutron Imaging at PSU....2 1.3: Thesis Objective...4 CHAPTER 2: THEORY....5 2.1: Basics of Neutron Radioscopic/Radiography Formation...5 2.2: Neutron Source........6 2.3: Image Detector System.....8 2.3.1: Scintillation Screen...8 2.3.2: Charged Coupled Device (CCD) Camera....8 2.4: Neutron computed tomography.....10 2.4.1: Reconstruction algorithm for parallel projections..12 2.5: CT Reconstruction Errors and Beam Hardening Effect...25 2.6: Octopus Tomography Reconstruction Software...30 iv
CHAPTER 3: SIMULATION AND EXPERIMENTAL RESULT OF NCT PROJECTION DATA...33 3.1: Simulation of Neutron Computed Tomography Projection Data..33 3.1.1: Simulation of an Aluminum Cylinder with Copper Core in the Center...34 3.1.2: Simulation of Aluminum Cylinder having a Water Column with Different Densities..36 3.2: Neutron Computed Tomography Experiments..46 3.2.1: Summary of Initial Neutron Computed Tomography System 46 3.2.2: Setup of Neutron Computed Tomography System.48 3.2.3: Measurement and Results......49 3.2.4: Summary and New Data from Upgraded Neutron Computed Tomography System.....53 CHAPTER 4: MODULATION TRANSFER FUNCTION ANALYSIS FOR THE NEUTRON IMAGING SYSTEM.59 4.1: Modulation Transfer Function Technique for Radioscopic System Characterization.....59 4.1.1: Modulation Transfer Function Theory...61 4.2: MTF Experimental Measurement Procedure 73 4.2.1: MTF Measurement for the Original Imaging System.73 4.2.2: MTF Measurement for the Upgraded Imaging System...83 4.3: Comparison of Original and Upgraded Neutron Imaging System Resolution..90 CHAPTER 5: SUMMARY, CONCLUSIONS AND FUTURE WORK.91 5.1: Summary.....91 5.2: Conclusions...92 5.3: Future Work..94 References....95 v
LIST OF FIGURES Figure 2.1: Neutron radioscopic system components 5 Figure 2.2: Illustration showing how parallel projection data are taken by measuring a set of parallel beams for different angles... 11 Figure 2.3: Illustration of how fan beam projection data are taken by measuring a set of fanned beams for different angles... 11 Figure 2.4: Back-projection reconstruction by using filtered projection which is smeared back over the reconstruction plane along lines of constant t 16 Figure 3.1: Aluminum cylinder with copper core radioscopic image with calculated data using simplified exponential attenuation equation and the corresponding cross section reconstruction result using Octopus software 34 Figure 3.2: Visualization aluminum cylinder with copper core 3-D information using VG-Studio Max... 35 Figure 3.3: Gray values from the top to the bottom of the cylinder with a copper core in the center. 36 Figure 3.4: The simulation result of the projection data for the first water density section in the aluminum cylinder described in the text... 39 Figure 3.5: A cross-sectional reconstruction result for the first density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.4.. 39 Figure 3.6: The simulation result of the projection data for the second water density section in the aluminum cylinder described in the text.. 40 Figure 3.7: A cross-sectional reconstruction result for the second density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.6.. 40 Figure 3.8: The simulation result of the projection data for the third water density section in the aluminum cylinder described in the text... 41 Figure 3.9: A cross-sectional reconstruction result for the third density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.8.. 41 Figure 3.10: The simulation result of the projection data for the fourth water density section in the aluminum cylinder described in the text 42 Figure 3.11: A cross-sectional reconstruction result for the fourth density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.10... 42 Figure 3.12: The simulation result of the projection data for the fifth water density section in the aluminum cylinder described in the text. 43 Figure 3.13: A cross-sectional reconstruction result for the fifth density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.12... 43 vi
Figure 3.14: The simulation result of the projection data for the sixth water density section in the aluminum cylinder described in the text. 44 Figure 3.15: A cross-sectional reconstruction result for the sixth density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.14.. 44 Figure 3.16: Gray values of the aluminum from the top to the bottom of the aluminum cylinder with six different water density columns... 45 Figure 3.17: A sketch of the initial Penn State neutron computed tomography system 47 Figure 3.18: A photograph of the test object (left) and the corresponding 2-D radioscopic image (right) from the initial NCT system.. 51 Figure 3.19: A photograph of the test object (left) and the corresponding crosssectional slice reconstruction using 361 projections in 180 at 0.2s/frame from the initial NCT system.... 51 Figure 3.20: The Volume reconstruction results (left) and isolated copper tubing (right) from the initial NCT system.. 52 Figure 3.21: Sketch of the upgraded neutron computed tomography system 54 Figure 3.22: A photograph of the test object (left) and the corresponding 2-D radioscopic image (right) from the upgraded NCT system.. 56 Figure 3.23: A photograph of the test object (left) and the corresponding 2-D radioscopic image (right) after beam shape and white noise correction from the upgraded NCT system 56 Figure 3.24: A photograph of the test object (left) and the corresponding crosssectional slice reconstruction using 361 projections in 180 at 54s/frame from the upgraded NCT system 57 Figure 3.25: A photograph of the test object (left) and the corresponding 3-D volume reconstruction result using 361 projections in 180 at 54s/frame from the upgraded NCT system. 57 Figure 3.26: Isolated copper tubing (right) and aluminum cylinder (left) from the upgraded NCT system... 58 Figure 4.1: Sampling shift for an impulse input to a real time video system (a): input to the system, (b): possible output to the system, (c): possible output to the system... 61 Figure 4.2: Left side of gadolinium foil edge image using white light optical microscopic characterization.. 73 Figure 4.3: Middle of gadolinium foil edge image using white light optical microscopic characterization... 74 Figure 4.4: Right side of gadolinium foil edge image using white light optical microscopic characterization.. 74 Figure 4.5: Blank beam image at Mag0 condition when the reactor power is 1 MW. 75 Figure 4.6: Knife-edge gadolinium foil at Mag0 condition when the reactor power is 1 MW 76 Figure 4.7: Blank beam image at Mag1 condition when the reactor power is 1MW. 76 vii
Figure 4.8: Knife-edge gadolinium foil image at Mag1 condition when the reactor power is 1 MW... 77 Figure 4.9: Sample gadolinium foil knife edge data and the curve fit for that data for the old neutron imaging system at Mag0 condition 78 Figure 4.10: A sample response function calculated from gadolinium foil edge data at Mag0 condition for the old neutron imaging system.. 79 Figure 4.11: A sample MTF calculation result of the old neutron imaging system at Mag0 condition... 79 Figure 4.12: A sample gadolinium foil knife edge data and curve fit for that data for the old neutron imaging system at Mag1 condition 80 Figure 4.13: A sample system response function calculated from gadolinium foil edge data at Mag1 condition for the old neutron imaging system. 80 Figure 4.14: A sample MTF calculation result of the old neutron imaging system at Mag1 condition.. 81 Figure 4.15: Calculated values of β along the edge as a function of edge location at Mag1 condition for the old neutron imaging system... 82 Figure 4.16: MTF family calculation results using the edge resolution parameters at Mag1 condition for the original neutron imaging system... 83 Figure 4.17: Blank beam image for the upgraded neutron imaging system when the reactor power is 1MW... 84 Figure 4.18: Knife-edge gadolinium foil image for the upgraded neutron imaging system when the reactor power is 1MW... 84 Figure 4.19: A sample gadolinium foil knife edge data and curve fit for that data for the upgraded neutron imaging system.. 85 Figure 4.20: A sample system response function calculated from gadolinium foil edge data for the upgraded neutron imaging system 85 Figure 4.21: A sample MTF calculation result of the upgraded neutron imaging system 86 Figure 4.22: Calculated values of β along the edge as a function of edge location for the upgraded neutron imaging system.. 87 Figure 4.23: MTF family calculation results using the edge resolution parameters for the upgraded neutron imaging system. 88 Figure 4.24: Modulation transfer function as a function of increasing distance from neutron converter surface for the upgraded neutron imaging system... 89 viii
NOMENCLATURE Acronyms NCT CCD LUT RSEC TRIGA ASTM MTF IFFT FFT IFT CBP FBP ART SIRT SART MOS PScF ZP Neutron Computed Tomography Charged-Coupled Device Look-up Table Radiation Science and Engineering Center Training Research Isotope production General Atomics American Society of Materials and Testing Modulation Transfer Function Inverse Fast Fourier Transform Fast Fourier Transform Inverse Fourier Transform Convolution Back Projection Filtered Back Projection Algebraic Reconstruction Technique Simultaneous Iterative Reconstructive Technique Simultaneous Algebraic Reconstruction Technique Metal-Oxide-Semiconductor Point Scattering Function Zero-padding Abbreviation GdO 2 S Gadolinium oxi-sulfide scintillation material Symbols Units p θ (t) Projection function unitless f ( x, y) Two-dimensional function of the sample unitless P θ (t) Radon transform function unitless F ( u, v) Two dimensional Fourier transform of the object function unitless S θ (w) Fourier transform function of P θ (t) unitless t Displacement from the beginning of projection zero cm Q θ (w) Filtered projection function unitless W Frequency higher than the highest frequency component In each projection cycles/cm T Intervals between the projections cm N Number of projections unitless K Integer unitless m Integer unitless n Integer unitless τ Sampling interval cm ix
h (t) Band-limited function unitless H (w) The Hilbert transform function unitless w Function in the frequency domain unitless H ( m(2w / N)) Hamming window function unitless M proj Number of projections unitless N rays Number of particles collected for each projection unitless ρ ( x, y) Sample density at position (x,y) g/cm 3 t ( x, y) Sample thickness at position (x,y) cm m atomic Atomic Mass g/mol sample I Measured gray values of the sample image unitless reference I Measured gray values of the reference image unitless f (x) System input function unitless g (x) System response to the input f (x) unitless h B (x) System impulse response to the imaging beam unitless h I (x) System impulse response to the intensifier/ video unitless h i (x) System impulse response to the component i unitless F (u) System input unitless G (u) System output unitless θ Effective divergence of neutron beam degree l Distance between the aperture and the intensifier surface cm H B ( u, l, θ ) Fourier transform of system impulse response function uniltess h (x) Line spread function unitless H B (u) Fourier transform function of h B (x) unitless H I (u) Fourier transform function of h I (x) unitless H i (u) Fourier transform function of h i (x) unitless x Spatial position cm u Spatial frequency cycles/cm β System resolution parameter pixels/cm x 0 x coordinate of knife edge center cm g (x) Knife edge function unitless β I System resolution parameter for intensifier/video pixels/cm β System resolution parameter for component i pixels/cm i β I,i System resolution parameter when component I and i are serially connected pixels/cm α System resolution parameter in Gaussian function pixels/cm u Nyquist sampling frequency cycles/cm N f N Fraction of information passed through the system unitless between zero frequency and the Nyquist frequency x Sampling space or pixel width pixels x
s(x) Noise-free impulse data unitless s c (x) Discrete center sampled data unitless s cβ (x) Center sampled data unitless S cβ (u) Fourier transform of center sampled data unitless s oβ (x) Off center sampled data unitless S oβ (u) Fourier transform of off center sampled data unitless x min Cut off system resolution limits cm β c On center resolution parameter pixels/cm β Off center resolution parameter pixels/cm o xi
ACKNOWLEDGEMENTS First, I would like to thank all the staff in Radiation Science and Engineering Center at Pennsylvania State University, who put a lot of time and patient for scheduling reactor beam time used for neutron tomography research work. I am also especially thankful to Mr. Arthur Kevin Heller, for his help for my research in neutron radiography/tomography. Most of all, I would like to thank Dr. Brenizer and Dr. Mench for getting me involved in this project and give me a lot of inspirations and make it successful. It has been a great honor to working with all of the people involved in the project. Last, special thanks will go to my parents, who not only gave me my life and raised me, but also gave me consistent moral support and love, which is a source of strength for me to study in the U.S. Liang Shi April 2008 xii
DEDICATION This thesis is dedicated to my beloved parents and all the people that helped me to go through difficulties and hardship xiii
CHAPTER 1 INTRODUCTION 1.1: Neutron Computed Tomography Imaging Neutron computed tomography is a powerful nondestructive method that provides 3-D information of the sampling object by recording the sequences of 2-D transmission radioscopy images taken from different angles. Compared to x-rays, neutrons can penetrate deeper into most materials by orders of magnitude. For x-rays, electromagnetic interactions depend on the number of electrons present in the target material. Since neutrons have no charge and are not affected by orbital electrons, there is no such correlation for neutrons [1]. Thin layers of certain light materials, like hydrogenous materials, can be significantly attenuated by neutrons, but are hardly detected by x-rays for the same thickness. Compared to water, aluminum is almost transparent to neutrons, but is attenuating to x-rays due to its higher atomic number. Neutron computed tomography can be used to obtain important 3-D information about the object s internal structure and material properties that other traditional methods can not provide. Typically, a neutron computed tomography system consists of a neutron source with a collimator, a sample rotation device, a 2-D neutron imaging system and a motion control system which synchronizes sample rotation with the imaging system [2]. A computer to capture, store and reconstruct the 3-D images is also needed. The first step in the development of a neutron tomography system is to select and optimize the neutron imaging system. A film and converter assembly is not commonly used in NCT investigation because of inconvenient processing and the film s limited linearity, 1
dynamic range and sensitivity. With film or photo-stimulatable phosphor plates, spatial registration of the projection images is difficult, as even small registration errors results in artifacts in the reconstructed images. The second step is the preparation of the image data and subsequent calculation of the 3-D voxel array using one of many reconstruction techniques, such as a filtered back-projection or an algebraic reconstruction algorithm [3]. Tomography image visualization software that recombines the 2-D vertical images into a 3-D image is commercially available and is useful to analyze the 2-D image projections [4]. 1.2: Summary of Neutron Imaging at PSU A fellow graduate student, Arthur Kevin Heller, began the neutron imaging of fuel cells at Pennsylvania State University in 2003. He utilized the beam port of the 1 MW TRIGA III thermal reactor at Pennsylvania State University experimental facility, where a radioscopic neutron imaging system was installed. Heller s work was primarily focused on using neutron imaging to quantify water mass content in operating polymer electrolytic fuel cells (PEFCs) and to perform the error analysis associated the water mass value due to pixel gray level fluctuations. A novel software package called PSUMagic was programmed in Java for the purpose of aiding in data analysis [5]. Processing of radioscopic images to isolate pixels of water attenuation, generation of the look up table (LUT) from the standard, quantification of water mass within the radioscopic image through use of the LUT, and false colorization of water within the final processed images is all performed by PSUMagic with a few user inputs. The PSUMagic software yielded 2
water mass results with an average error due to pixel gray level fluctuation of 2.7%, the average error of the measured water mass with respect to the theoretical was 6.5% [5]. The author of this thesis and Heller worked together to build a new highresolution computed tomography system for the purpose of quantitative tomography research. New cross-section reconstruction software package, Octopus (version 8), and three-dimensional reconstruction software, VG-Studio Max (version 1.2), were used. This work was divided into two parts: the first part of the work was to convert the existing neutron imaging system to a tomography system. Heller developed software that allowed the CCD camera to be controlled by the image data capture computer and to be synchronized with the motion control system of the rotation device. The initial tomography data was successfully collected and reconstructed, the result was promising and encouraging. The second part of the work was to install a new two mirror neutron imaging system. This system has highly effective gamma ray radiation shielding features that minimizes the radiation to the CCD camera. A mechanical automation will also be included in the system so the camera focal length can be adjustable for certain high resolution requirements. The author utilized the MTF technique to evaluate the true spatial resolution of the imaging system by imaging a gadolinium sharp edge sample. This study also focused on the simulation of tomography projection data to investigate the change in pixel gray value when ignoring the effects of neutron scattering, and to understand the tomography cross-section reconstruction process. 3
1.3: Thesis Objective The initial neutron computed tomography work provided the basic system capable of producing low-resolution neutron tomography for the fuel cell research group at Penn State University. The thesis objective was to develop a high-resolution imaging system to meet the requirement of quantitative tomography research. This thesis summarized the work that has been done since the basic and upgraded neutron imaging system was installed at Penn State Univeristy, corresponding information can be found in Chapters 1, 2 and 3 of the thesis. Chapter 4 discusses the MTF technique that was used in determining the spatial resolution for the basic and upgraded imaging system. Finally, Chapter 5 summarizes the work, makes some conclusions, and suggests some ideas for future work. 4
CHAPTER 2 THEORY The neutron computed tomography method is a non-destructive testing method that provides 3-D information of a sample object by recording a series of 2-D images at different angles and reconstructing the volume data. Therefore, it is important to understand the 2-D neutron imaging method, which provides raw projection data for tomography reconstruction. 2.1: Basics of Neutron Radioscopic/Radiography Formation A neutron radioscopic system consists of a nearly parallel neutron beam and a detector system (Figure 2.1). Neutron source Object Scintillation screen CCD camera Figure 2.1: Neutron radioscopic system components When the beam of thermal neutrons passes through the object, the thickness and composition of the object attenuate the beam by absorbing neutrons from their flight path 5
or scattering them, neutrons that successfully pass through the object then impact the detector. In radioscopy, the detector is a scintillator/image intensifier where the neutron beam is converted to light and amplified and then recorded with a CCD camera. In radiography the detector is a converter screen and radiographic film placed inside a lighttight cassette. Both radiography and radioscopy produce gray scale images, variations in neutron beam attenuation cause the differing shades of gray. Therefore, each shade of gray in the resulting image is indicative of the attenuation properties of the object. 2.2: Neutron Source Thermal and cold neutrons are usually used for neutron imaging because most materials exhibit higher attenuation for low energy neutrons. After the production of neutrons by fission reaction, the neutrons usually have high kinetic energies (approximately peaks at 2 MeV), they must be slowed down by scattering process in a moderator in order to reach the thermal energy range. Heavy water ( D 2 O ) is often used as a moderator because of the relatively large scattering cross section and the small absorption cross section of deuterium atom. The fission neutrons usually reach the thermal energy state after a few scattering interactions with deuterium atoms. At room temperature, the thermal neutron energy spectrum usually yields an average energy of 0.0253eV and with a velocity of 2200m/sec [8]. When the neutron energy reaches the thermal energy range, the neutrons are collimated and directed towards the sample and detector. The neutron source in the Penn State Univeristy Radiation Science and Engineering Center (RSEC) is a 1MW TRIGA research reactor. The neutron beam at the imaging plane is approximately 30 cm in 6
diameter with a L/D ratio of 150 (as measured using ASTM 803) [5], where L is the length of collimator and D is the diameter of aperture, which corresponds to a divergence half angle of approximately 1.4 o [9]. The energy distribution of moderated thermal neutrons is roughly according to the Maxwell Boltzman energy spectrum distribution [10], which is described by the following equation E E Φ, kt kt 0 ( E) de exp( ) de E E th (2-1) Where k is the Boltzman constant, E is the energy of the neutrons, Φ ( ) is the neutron flux at the energy of E, and T is the temperature of the media. The neutron beam consists of moderated thermal neutrons, neutrons still in the moderation process, and neutrons that are in the epithermal energy region of the spectrum and is described as the following equation 0 E de Φ, E 0 ( E) de Eth E E epi (2-2) The direction in which the scattered neutrons are moving is approximately isotropically distributed after the moderation process. Only neutrons within a small solid angle in the desired direction of movement within the isotropic distribution are directed through the collimator, therefore the neutron beam moving out of the collimator and reaching sample object can be seen as a nearly parallel beam for most neutron imaging purposes. 7
2.3: Image Detector System Typically, an image detection system consists of a converter and an image recorder, such as a scintillation screen and Charged Coupled Device(CCD) camera, or a gadolinium foil and film. The scintillation screen and CCD camera are discussed in this thesis since that is currently being used at Penn State for quantitative neutron imaging research. 2.3.1: Scintillation Screen The scintillation screen is used to converted neutrons to visible photons (light) through neutron capture and phosphorescence processes. Phosphorescence is a process in which radiant energy absorbed by certain substances is released relatively slowly compared to the subatomic reactions required to re-emit the light occur less often. In the scintillation screen, the neutrons are absorbed by a high cross section neutron absorption materials, the resulting products interact with the phosphor to produce a visible light that is detected by the CCD camera. A 6 Li based screen is being used at Penn State. 2.3.2: Charged Coupled Device (CCD) Camera The Charged Coupled Device integrated circuit chip is an important electronic part of the CCD camera. Typically, the CCD chip is an array of Metal-Oxide- Semiconductor (MOS) capacitors, each capacitor represents one pixel. An applied external voltage on the top MOS plates results in charges (electrons or holes) being deposited in the resulting potential well. The charges can be shifted from one pixel to the 8
other by the digital pulses applied to the top plates so that the charges can be transferred row by row to a series output. In a neutron imaging system, the electrons are generated by the light photons emitted by scintillation screen, i.e. the electrons are transferred from the valence band to the conduction band by light photons. Some of these electrons can be generated through thermal excitation that constitutes the noise in the CCD and is called dark current [7]. The noise can be reduced by lowering the temperature of the CCD. So in the real-time neutron imaging system, a cooled CCD was chosen for the purpose of reduction of the noise and dark current in the CCD camera. For the preliminary work at Penn State, a Cohu 7700 CCD camera with a sensor array format of 1004 1004 pixels was used. The image system s spatial resolution is dependent on many factors, i.e. the type of lens used, the size of the CCD chip and the distance between scintillator and the test object. Increased spatial resolution can be achieved by using a thinner scintillator at the cost of efficiency [4]. The Cohubased system has a spatial resolution of (148 148) µm 2 with image integration time ranging from 1/30s to 36 minutes. White spots found in the real-time neutron imaging experimental data are caused by the gamma photon and neutron radiation interacting in the CCD camera sensor array. Therefore, neutron and gamma photon shielding of the CCD camera is required. The latest neutron imaging system at Penn State incorporates a specially designed radiation shield and the white spots are reduced by approximately 70% as compared to the initial system. 9
2.4: Neutron Computed Tomography In neutron computed tomography, the 3-D neutron image of the object is reconstructed from a series of 2-D images of the sample taken at different angles by rotating the sample at a constant increment relative to the neutron source. The 2-D image data collected at different angles are called projection data of the sample. The 3-D computed tomography (CT) image is the result of reconstructing the series of 2- D images data. Dependent upon the geometry of beam being used, the project data can be divided into two types: 1): Fan beam Fan beam projection data. 2): Parallel beam Parallel beam projection data. The two types of beam geometry are shown in the following Figure 2.2 and Figure 2.3. The reconstruction technique for parallel beam geometry will be discussed in section 2.4.1 10
y f ( x, y) θ x (t) p θ Figure 2.2: Illustration showing how parallel projection data are taken by measuring a set of parallel beams for different angles [11]. y f ( x, y) x β (t) i p β Figure 2.3: Illustration of how fan beam projection data are taken by measuring a set of fanned beams for different angles [11]. 11
For reconstruction purposes, different algorithms can be implemented. Each has its advantages and disadvantages. The most commonly used algorithms are: Fourier transformation based techniques, i.e. Fourier techniques, Convolution Back Projection (CBP), Filtered Back Projection (FBP), Wavelet based reconstruction. Algebraic reconstruction technique, i.e. Algebraic Reconstruction Technique (ART), Simultaneous Iterative Reconstructive Technique (SIRT), Simultaneous Algebraic Reconstruction Technique (SART), etc. Statistical based reconstruction technique, i.e. Maximum Likelihood and Expectation Maximization Techniques. Unconventional but mixed techniques, i.e. CBP and ART combined reconstruction techniques, etc. In the Fourier-transformation based reconstruction technique, the series projection function p θ (t) which the detector recorded from the object is called the sonogram. In order to calculate the reconstruction of the sonogram, it is necessary to calculate the inverse Radon transform of the projection data p θ (t). 2.4.1: Reconstruction Algorithm for Parallel Projections The discussion in this section was summarized from book C. Kak and Slaney [11]. The Filtered Back Projection (FBP) technique is the most commonly used technique for tomography reconstruction. In this technique, each projection is projected back along the same angle. The sample object is represented by a two-dimensional function f ( x, y) and each line integral by the ( θ,t) parameters as Pθ ( t) = f ( x, y) ds. (2-3) ( θ, t) 12
Introducing a delta function, the above equation can be written as + + P θ ( t) = f ( x, y) δ ( x cosθ + y sinθ t) dxdy, (2-4) where t in above equation can be written as t = x cosθ + y sinθ. (2-5) The function P θ (t) is known as the Radon transform of the function f ( x, y). Each projection consists of a set of combing density line integrals. The simplest projection is a collection of a set of parallel beam integrals given by function P θ (t) with a constant θ, which is illustrated in the Figure 2.2 and is known as a parallel projection. Using the formula for the inverse Fourier transformation, the object function is given f ( x, y) + + j 2π ( ux+ vy) = F( u, v) e dudv, (2-6) where in this equation, F( u, v) is the two-dimensional Fourier transform of the object function, which is F( u, v) + + j 2π ( ux+ vy) = f ( x, y) e dxdy. (2-7) Exchanging the rectangular coordinate system in the frequency domain ( u,v), for a polar coordinate system ( w,θ ), gives u = wcosθ, (2-8) v = wsinθ, (2-9) and the differential relationship between these two types of coordinates is 13
dudv = wdwdθ. (2-10) Knowing function F ( u, v), the inverse Fourier transform of a polar function is written as the following f 2π j 2πw( x cosθ + y sinθ ) ( x, y) = F( w, θ ) e wdwd 0 0 θ, (2-11) this integral can be divided into two separated integrals, one is integrating θ from 0 to 180 and the other one is integrating θ from 180 to 360, which is given f π π j 2πw( x cosθ + y sinθ ) j 2πw[ x cos( θ + 180 ) + y sin( θ + 180 )] ( x, y) = F( w, θ ) e wdwdθ + F( w, θ + 180 ) e wdwd 0 0 0 0 and then using the property of Fourier transform (2-12) F( w, θ + 180 ) = F( w, θ ), (2-13) substituting Equation (2-5) to Equation (2-13), one can have π [ j 2πwt f ( x, y) = F( w, θ ) w e dw] dθ, (2-14) a projection at an angleθ, P θ (t) and its Fourier transform by 0 θ j2πwt = Pθ ( t) e dt S ( w). (2-15) θ The Fourier Slice Theorem can be done in a more solid foundation by considering the (t, s) coordinate system to be a rotated version of the original (x, y) system as expressed by t = s cosθ sinθ x. (2-16) sinθ cosθ y In the (t, s) coordinate system a projection along the lines of constant t is written as 14
+ P θ ( t) = f ( t, s) ds, (2-17) substituting the definition of a projection in the above equation, which results S θ + j 2πwt = e dt [ f ( t, s) ds ] ( w). (2-18) This result can be transformed into the (x, y) coordinate system by using Equation (2-16) j 2πw( x cosθ + y sinθ ) = f ( x, y) e dxdy Sθ ( w), (2-19) the right hand side of the above equation represents a two-dimensional Fourier transform at a spatial frequency of ( u = wcosθ, v = wsinθ ) S ( w) = F( wcosθ, wsinθ ) = F( w, θ ), (2-20) θ so Equation (2-14) can be written as the following π j2πwt f ( x, y) = Sθ ( w) w e dw dθ. (2-21) 0 The above equation can also be expressed as where in the equation, π f ( x, y) = Qθ ( x cosθ + y sinθ ) dθ, (2-22) 0 j2πwt Q ( w) = S ( w) w e dw. (2-23) θ θ The Equation (2-23) is called filtered projection and represents a filtering operation where the frequency response of the filter is dominated by w. The function f ( x, y) is then estimated by calculating the projections for different anglesθ for every point (x, y), in the detector (image plane) corresponding to a t value for a givenθ, which is given in Equation (2-5). Figure 2.4 below illustrates that the filtered projection Q θ 15
contributes to the reconstruction its value at t. y (t) Q θ t t = x cosθ + y sinθ M θ L x Figure 2.4: Back-projection reconstruction by using filtered projection which is smeared back over the reconstruction plane along lines of constant t [11]. It is easily understood that for a given θ value, the value of t is the same for all given (x, y) on the line LM. So the filtered projection Q θ (t) will make the same contribution to the reconstruction at all of these points. In other words, for the reconstruction process, each filtered projection Q θ (t) is smeared back or projected all over the image plane. The integration of Equation (2-23) will be carried out for all spatial frequencies due to the fact that the parameter w has the dimension of spatial frequency. The energy contained in the Fourier transform above certain frequencies can be negligible, so all the projections are considered as band limited. Let W be a frequency that is higher than the highest frequency component in each of the projection, then 16
from the sampling theorem, the sample intervals between the projection is calculated by using the following without any associated errors T = 1 2W. (2-24) If one assumes that the projection data are equal to zero for large values of t, then a projection can be expressed as P θ (mt), m N N =,, 0,, 1 2 2. (2-25) A Fast Fourier Transform (FFT) algorithm can be implemented to approximate the Fourier transform, S θ (w) of a projection, P θ (mt), if N is large enough, which is approximately given as N / 2 1 2W 1 2 ( / ) ( ) ( ) k j π mk N Sθ w S m = Pθ ( ) e. (2-26) N 2W 2W k = N / 2 If the samples of the projection, P θ (mt), are known then the samples of its Fourier transform will be given by Equation (2-26), and the modified projection, Q θ (t), may be digitally evaluated using the following approximation W N / 2 j2πwt 2W 2W 2W j2πm(2w / N ) t Qθ ( t) = S θ ( w) w e dw Sθ ( m ) m e. (2-27) N N N W m= N / 2 If N is large enough, and if we need to determine the projections Q θ (t) for these t at which P θ (t) are being sampled, the following result k W Qθ ( 2 N / 2 2W 2W 2W j2π ( mk / N ) ) ( ) Sθ ( m ) m e, (2-28) N N N m= N / 2 where, k = N / 2,,-1,0,1, N/2. (2-29) 17
From the above equation it is obvious that the digital projection Q θ (t) is approximately given by the inverse Discrete Fourier Transform (DFT) of the product of S θ ( m(2w / N) and m ( 2W / N. Since the reconstructed images have noise in them, better quality images will be obtained if one multiplies the filtered projection ( m(2w / N) S θ m ( 2W / N, by a Hamming window function, which is given as 2W Qθ ( k / 2W ) ( ) N N / 2 S θ n= N / 2 2W 2W ( m ) m N N 2W H ( m ) e N j2π ( mk / N ), (2-30) where in the above equation the H ( m(2w / N)) is the Hamming window function to be implemented. The window function is used to deemphasize the higher frequencies that in many cases contribute most of the observed noise in the image. Using the convolution theorem, the discrete transform of Equation (2-30) can be written as k W Qθ ( 2 2W k k ) Pθ ( ) φ( ), (2-31) N 2W 2W where in the equation, denotes the periodic or circular convolution and φ( k / 2W ) is the 2W 2W inverse DFT of the discrete function of m H ( m ), m=-n/2,,-1,0,1,,n/2. N N By using the Fourier domain Equation (2-27), one can obtain function (t) at the sampling points of a projection, or one can use space domain Equation (2-31) to obtain this function. The reconstructed result of f ( x, y) is calculated by using the discrete approximation K π f ( x, y) = Qθ ( x cosθ i + y sinθ i i ). (2-32) K i= 1 Q θ 18
The value of x cosθi + y sinθ may not necessary correspond to the values of t that determine Q θ in Equation (2-30) or (2-31). However, a linear interpolation is usually i adequate to determineq θ for such values of t, because the projection function (t) can be expressed in t domain i Q ( t) = P ( α) p( t α) dα. (2-33) θ θ In this equation, p(t) is the inverse Fourier transform of w function in the frequency Q θ domain. Its inverse transform does not exist in ordinary form since w function is not a square integration function. The following inverse Fourier transform can be examined to serve this purpose w e w, (2-34) as 0, the inverse Fourier transform of the above equation can be written as p t) = 2 ( 2 2 (2π t) 2 + (2π t) ) ( 2. (2-35) The filtered projection (Equation (2-23)) can be rewritten as where, j j2πwt Qθ ( w) = j2π wsθ ( w) sgn( w), 2 e dw (2-36) π 1, w > 0, sgn( w ) = (2-37) 1, w < 0. Equation (2-36) can be written as the following according to convolution theorem j Qθ ( t) = { IFT( j2πwsθ ( w))} { IFT( sgn( w))}, (2-38) 2π in this equation, the operator * stands for the convolution operation and IFT represents 19
Inverse Fourier Transform. From Inverse Fourier Transform theory, it is known that the inverse Fourier transform calculation result of j2πws ( w) is equal to ( P ( t) / t), and the j inverse Fourier transform calculation result of ( sgn( w) ) is equal to 1/t. Therefore, 2π Equation (2-38) can be written as θ θ Q 1 Pθ ( t) t) =, 2π t t θ ( 2 Equation (2-39) is also called Hilbert Transformation of (2-39) Pθ ( t). The Hilbert transform t can be defined by using the following frequency response function j, w > 0, H ( w) = (2-40) j, w < 0. If the projection data are taken with a sampling interval of τ cm, and assuming there is no aliasing, this implies that in the transform domain the projection data do not contain any energy outside the frequency interval (-W,W), where 1 W = cycles / cm. (2-41) 2τ Let the sampled projection be denoted by P ( kτ ) where k is an integer. For each sampled θ projection P ( kτ ), a filtered projection data Q ( kτ ) is calculated by using Equation (2- θ 27). Equation (2-27) is very attractive since it directly conforms to the definition of the DFT and, if the N is decomposable, possesses a fast FFT implementation. Please note that Equation (2-27) is only valid when the projections are of finite bandwidth and finite order. In practice, the above two conditions can not be satisfied at the same time, and the computer implementation of Equation (2-27) usually leads to interperiod interference artifacts created when the aperiodic convolution (Equation 2-23) is used as a periodic convolution. θ 20
The artifacts that are created can be eliminated by the alternative implementation of Equation (2-23). When the highest frequency in the projections is finite, Equation (2-23) can be expressed as where j2πwt Qθ ( t) = Sθ ( w) H ( w) e dw, (2-42) H ( w) = w b ( w). (2-43) In the equation, function b w (w) can be expressed as w 1, w < W, b w ( w) = (2-44) 0, otherwise. The function H (w) represents the transfer function of a filter with which the projections must be processed and the impulse response h (t), of this filter is given by the inverse Fourier transform of H (w) h( t) = H ( w) e + j 2πwt dw = 1 2 2τ sin 2π t / 2τ 1 2 2π t / 2τ 4τ sinπ t / 2τ ( ) π t / 2τ 2. (2-45) Since the projection data are measured with a sampling interval ofτ, and for the digital processing the impulse response, the sampling interval must be known, then the samples of h ( nτ ) are calculated by the following equation 2 1/ 4τ, n = 0, h( nτ ) = 0, n = even, (2-46) 1, n = odd. 2 2 2 n π τ 21
Both P θ (t) and h(t) are band-limited functions that can be expressed as sin 2πW ( t kτ ) P θ ( t) =, (2-47) kτ ) Pθ ( kτ ) k = 2πW ( t sin 2 ( ) ( ) πw t kτ h t = h( kτ ). (2-48) k = 2πW ( t kτ ) Applying the convolution theorem, the filtered projection Equation (2-42) can be written as ' ' ' Qθ ( t) = Pθ ( t ) h( t t ) dt. (2-49) Substituting Equation (2-47) and Equation (2-48) into Equation (2-49), the following as the filtered projection data at the sampling points can be obtained ( ) Qθ nτ = τ h( nτ kτ ) Pθ ( kτ ). (2-50) k = In practice each projection is of only finite extent. Suppose each P ( kτ ) has valid data within the range of k = 0, N-1 and Equation (2-50) can be rewritten as θ or N 1 h k = ( N 1) Qθ ( nτ ) = τ ( kτ ) Pθ ( nτ kτ ), n=0, 1, 2,, N-1. (2-51) N 1 k = 0 Qθ ( nτ ) = τ h( nτ kτ ) Pθ ( kτ ), n=0, 1, 2, N-1. (2-52) The above equation indicates that in order to determine the value of Q ( nτ ), the value of sequence h( nτ ) should be from k=-(n-1) to k=(n-1). Comparing Equation (2-51) to Equation (2-28), one can see that these two approaches aren t identical because the discrete Fourier transform of h( nτ ) with n is within in a finite range from (N-1) to (N-1) rather than in the range from N/2 to N/2 with a sequence of k [( 2W ) / N]. The discrete θ 22
convolution Equations (2-51) and (2-52) can be implemented directly on the computer and it is much faster to implement it in the frequency domain using FFT algorithms. One must keep in mind that in frequency domain, only periodic (or circular) convolution can be implemented, the convolution used in Equation (2-52) is aperiodic. One way to eliminate the interperiod interference artifacts that exist in the periodic convolution is to pad the projection data with a sufficient number of zeros. It is easy to see if the projection data of P ( kτ ) are padded with zeros to make (2N-1) elements long, θ interperiod interference over N samples will be avoided. For most of the cases, a base-2 FFT algorithm is most frequently implemented. The sequences of P ( kτ ) and h ( kτ ) have to be zero-padded so that each is (2N-1) 2 elements long, where (2N-1) 2 is the smallest integer that is a power of 2 and also greater than 2N-1. The modified frequency domain implementation can be expressed as Q ( nτ ) = τ IFFT{[ FFT P ( nτ ) withzp] [ FFT h( nτ ) withzp]}, (2-53) θ θ where IFFT and FFT denote inverse fast Fourier transform and fast Fourier transform respectively, and ZP denotes zero-padding. If some smoothing is incorporated in Equation (2-53) by multiplying the product of the two FFTs by a Hamming window, the above equation can be rewritten as {[ Q ( nτ ) = τ IFFT FFT P ( nτ ) withzp] [ FFT h( nτ ) withzp] smoothing window θ θ (2-54) After the filtered projection Q ( nτ ) is calculated with the method presented, the θ construction process is the same as that described in the proceeding section. Sometimes in order to reduce computation time required for interpolation, a pre-interpolation of the function Q ( nτ ) is used. Before performing the back-projection calculation, the function θ θ }. 23
Q θ (t) is pre-interpolated onto 10-1000 times the number of points in the projection data. This computation technique can be combined with Equation (2-54). From the resulting data set one can simply find out the nearest neighbor to obtain the value of i i Q θ at i x cos θ + y sinθ. One computation technique of pre-interpolation involves performing the calculation of the IFFT in Equation (2-54) after the frequency domain function is padded with a large number of zeros. The inverse transform of this sequence yields a pre-interpolated functionq θ. The problem with this algorithm is that if the data i sequence contains fractional frequencies, this approach can lead to large errors especially near the beginning and the end of the data sequence. However, with appropriate programming, the back-projection for parallel projection data can be accomplished. 24
2.5: CT Reconstruction Errors and Beam Hardening Effect For x-ray computed tomography projection data, several standard correction methods have been developed and can be applied in the spatial or in the frequency domain. These standard methods and the corresponding techniques for correcting neutron projection data will be discussed in this section. The equations presented in the previous sections are the standard formulas for reconstructing images from the projection data sets assuming that the projection data sets are continuous functions. Implementation of these formulas as a computer algorithm requires that each of the projection data functions, the convolution functions and the smoothing filters have some discrete sampling size. The convolution method can be implemented as direct convolutions in the spatial domain to approximate the equations necessary to implement reconstruction. Less computation time will be required if the Fourier Transform of the data and filters are taken and applied in the frequency domain and subsequently the final reconstructed image data is obtained by using the IFFT. As described in the previous section, the application of the algorithms in the frequency domain can create interperiod interference artifacts in the reconstructed image. These artifacts can be eliminated by padding the projection data sets with sufficient zeros, which forces the interperiod artifacts out of the unimportant zeroed space and the interference over the whole component image space will be avoided. The other common problem encountered with CT reconstruction is beam hardening effect. This usually occurs when the radiation source used in CT reconstruction is a poly-energetic beam of particles. Both x-ray and neutron attenuation properties are energy dependent and a polychromatic beam of particles can result in 25
unwanted artifacts that are displayed as image distortions. This effect is well known as cupping effect. In the non-destructive testing field, thermal neutrons and X-rays used in diagnostics usually display a radiation attenuation decrease property if the particle energy increases. In this case, when the radiation particles pass through the test object, the radiation particle energy spectrum will shift towards higher energies because the lower energy particles are preferentially absorbed and thus, the remaining particle spectrum appears to be shifted to higher energies, which is also called beam hardening effect. Several algorithms were developed as a standard method to correct for this beam hardening effect in x-ray tomography. The techniques use a correction factor, usually it is a higher order polynomial, which is obtained from experimentally collected CT projection data and the known attenuation properties of the materials [13]. Each projection data is multiplied by this correction factor before the reconstruction algorithm is implemented. An example of this algorithm developed for correcting beam hardening effect in fast neutron tomography is by Pfister et al [12], which follows the same general correction procedure by multiplying the projection data by a higher order polynomial and then using the image reconstruction routine [14]. Generally, the pixel resolution of the reconstructed image is correlated to the incremental sampling size present in the detector arrays and the pixel size in the reconstructed image should be less or equal to the detector size. The detector size determines the number of ray paths passing through and collected with each projection angle. Only the rays passing through the test object provide the useful information for the image reconstruction purpose. From this point, determining the number of ray paths 26
is very important for image reconstruction. When the number of rays for each projection has been set, the data collection system must determine the number of projection data sets are to be collected. Setting an appropriate number is important for image reconstruction processing. A higher arbitrarily number can result in large sets of over sampled data which requires additional computer processing time, but at the same time the resolution of the reconstructed image is not improved. However, collecting fewer projection data will result in image artifacts that are called aliasing distortions. These distortions will appear as thin streak (Moirĕ patterns) radiating from the reconstructed components. A. Kak and M. Slaney derived a sampling relationship which relates the number of projections, M proj and the number of rays collected in each projection, N rays, which is: M N proj ray 2 (2-55) Theoretically, the number of tomography projection data needed for image reconstruction purposes is between 1 to 1.5 times the number of rays collected for every projection [11]. In order to avoid Moirĕ patterns in the reconstructed image, the image function of the defining grid should be approximately equal to the number of the rays collected for each projection. Typically, an N times N image requires N rays for each projection and approximately N projection angles for reconstruction. For the image reconstruction process, the relationship between the test object center of rotation and the central particle path is important. The filtered back-projection methods require that each ray path striking the central detector site in the projection data pass through the center of rotation of the test component. Errors of one detector site will result in the circular distortions in the reconstructed image [15]. The center of the 27
rotation for the sample object is determined experimentally by noting the pixel location of an edge, then rotating 180 0 and determining if the new pixel location is along the same edge. If the reconstructed image displays circular artifacts then the projection data must be centered and the reconstruction algorithms repeated in order to eliminate the artifacts. As discussed before, a CT reconstruction calculation requires that only particles from the source passing through the test object along a straight beam path without any interaction be counted at the detector position. The recorded uncollided particles on the detector represent the line integral along the given particle path in addition to object scattered particles. The limits in collimation result in CT system that records the scattered particles as part of the projection line integral data. Detector collimation is used to limit the scattered particles from contributing to the projection data, but eliminating scattering particles from contributing to the projection data completely is not possible. Practically, the scattering effect of the particles to CT projection data will introduce errors. Particles that have absorption or scattering interactions would experience changes in their flight path and will not be recorded if their paths of travel are outside of the solid angle viewed by the detector. For example, pencil beam CT systems experience relatively small particle scattering effects in the projection data. The problem with this type of beam setup is that while the particle scattering contribution to the projection data is decreased, the time required for the data collection is increased by the need for mechanical translation of the object through the pencil beam. The detector for a CT system that is built around arrays or banks of detectors must consider particles that are scattered out of their beam path and are counted by a neighboring detector. x-ray computerized tomography systems experiencing particle 28
scattering have artifacts present in the reconstructed image. The correction method for scattering effect contributions in the CT projection data for x-ray tomography is to assume the additional scattering intensity contribution to each projection data is a constant. Each projection data represents the combination components of the scattered intensity and the primarily particle intensity. While the intensity of x-ray scattering has been determined to be approximately equal for every projection taken for a test object, the intensity of the primarily x-ray is directly dependent on the material through which it passes [16]. If the projected intensity of the primary particle is large then the scattering intensity will have little effect on the projection data. However, the scattering effect can dominate when intensity of the projected primary particle is small. Corrections for the scattering effect in the projection data assume an uniform scattering triangle which is convolved with the projection data [17]. The Monte Carlo simulation method can be used to determine an approximation of the scattering effect created by different materials. A pencil beam, with a diameter less than one pixel width consists of thermal neutrons direct towards a test object with homogenous materials of known geometry. A bank or strip of detectors is placed directly behind the test object and records the neutrons which arrive at the location. The primary particles are directed towards the center of a single detector and the greatest number of particles will be detected at that location. Any particles recorded at the neighboring detector locations can be attributed to scattering effect [12]. This is known as the Point Scattering Function (PScF) model. 29
2.6: Octopus Tomography Reconstruction Software The reconstruction software, Octopus V8, was used for cross-sectional reconstruction for the neutron tomography projection data since the NCT system has been installed and tested at Penn State. For quality cross-sectional reconstruction purposes, the following practices are recommended: Projection image requirements: Parallel beam geometry requires projection images with equi-angular separation between 0 and 180. The last image at 180 is not used for reconstruction but it is used for the calculation of the rotation centre (COR). For fan and cone beam geometry, the projections over 360 opening angle of the fan or cone are often required. In case of large opening angles (large flat panel detectors), a 360 scan is often required. Due to the statistical nature of the measurement, noise is always present in the projection images. Detectors need sufficient grey levels, therefore, if the detector is 12 bit or less, one should acquire a number of images at every angle and use the sum or the average of these images. This approach can improve the reconstruction quality. Open beam images and offset images: Open beam images are often used to correct the beam profile and scintillator /or taper imperfections. Every projection will be divided by the open beam image and every imperfection in this beam profile image will be present in the normalized images. This makes the open beam image the most important one. In theory, one open beam image is enough, but when using the Octopus software, one should acquire many open beam 30
images. Offset images or dark images are needed to correct the dark current, read out noise and ADC offset of the detector. Projections, open beam images and offset images are measured with the same integration time. Also, it is recommended to acquire a large number of offset images since these images will be subtracted from the projection images and the open beam images. An Octopus dataset consists of projections, open beam images (flat fields) and offset images, all stored in the same directory. Integration time: In order to obtain the best reconstruction quality, it is recommended to use a full dynamic range detector. For non-linear detectors, make sure that only the linear range of the detector is used. Usually the CCD cameras have a good linearity. When using Octopus, a full dynamic range detector is preferred. System alignment: For a good tomography acquisition and Octopus reconstruction calculation, the object rotation axis is required to be perfectly vertical. Tilts of the rotation axis are difficult to correct. The beam axis should coincide with the center of the detector. Ring artifact correction: One of the most difficult reconstruction artifacts to correct is the ring artifacts. These rings are usually caused by the abnormal behavior of pixels and some non-linear behavior of the pixels. Post processing algorithms are usually not able to filter out all rings. The rings can be avoided by taking good projection images using the recommended procedure discussed above. 31
Field of View (FOV): At all times, the sample should be in the field of view (the boundaries of the radioscopic image). Artifacts will result if the sample rotates out of the FOV. Before the tomography acquisition starts, it is important for one to check the projections of the object at different rotation angles. Beam hardening correction: Octopus V8 offers two beam hardening correction methods that work when the object is imaged in the air: an advanced beam hardening correction algorithm and a faster polynomial correction. 32
CHAPTER 3 SIMULATION AND EXPERIMENTAL RESULT OF NCT PROJECTION DATA 3.1: Simulation of Neutron Computed Tomography Projection Data A primarily goal of neutron computed tomography research at Penn State University is to develop a technique for water quantification in fuel cells [18]. In order to develop this technique, it is necessary to thoroughly understand and observe the tomographic images generated by ideal neutron computed tomography projection. Ideal neutron computed tomography projection data are images that have no neutron scattering and beam geometrical un-sharpness in the projection images. These projection data sets were simulated by using the single exponential neutron attenuation equation. The source was assumed to be a parallel, monoenergetic neutron beam. The objects modeled in the work are two types of cylinder. The first type was a solid aluminum cylinder with a copper core in the center. The second type was a solid aluminum cylinder with a hole in the center and at a distance from the center, an artifact was introduced. The artifact was divided into six different sections from the top to the bottom, and each section was filled with different density material. The purpose of the simulation work was to create ideal projection data sets to help understand how the voxel gray level of a water column in the reconstructed image was affected by different surrounding materials. 33
3.1.1: Simulation of an Aluminum Cylinder with Copper Core in the Center The first type of modeling test object was a 10-cm diameter and 8.8-cm high aluminum cylinder with a copper core that was 2.5-cm diameter in the center. Figure 3.1 shows the modeling results of the object radioscopic image with calculated data using simplified exponential attenuation equation and the corresponding cross section reconstruction results using Octopus software. Figure 3.1: Aluminum cylinder with copper core radioscopic image with calculated data using simplified exponential attenuation equation and the corresponding cross section reconstruction result using Octopus software [18]. After the reconstruction calculation using Octopus, the cross-sectional slice information was put into VG-Studio Max visualization software to get the 3-D detailed internal information. Figure 3.2 shows the visualization results of the modeled object. Other objects are being similarly modeled in order to investigate the neutron scattering effects on the tomography projection data and develop water quantification technique for 34
Figure 3.2: Visualization aluminum cylinder with copper core 3-D information using VG-Studio Max. neutron computed tomography [18]. Since modeling result of the tomography projection data was without neutron scattering and geometrical un-sharpness, the gray value for the same location in each reconstructed cross-sectional slice should be the same. Figure 3.3 shows a plot of the gray values of the same specific location for each reconstructed slice from the top to the bottom of the simulated aluminum cylinder with copper core in the center. 35
Gray values of aluminum cylinder from top to the bottom 7000 6000 5000 Gray values 4000 3000 2000 1000 0 0 50 100 150 200 250 Slice numbers (pixels) Figure 3.3: Gray values of the aluminum from the top to the bottom of the cylinder with a copper core in the center. From Figure 3.3, one can see that the gray values for the same location from the top to the bottom of the cylinder was a constant, which was expected. 3.1.2: Simulation of Aluminum Cylinder having a Water Column with Different Densities. The second type of object being simulated was a 10-cm diameter and 8.8-cm high aluminum cylinder with a hole that was 2.5-cm diameter in the center, and at 3.8-cm from the center, there were six different sections from the top to the bottom, each section was 0.5-cm diameter and approximately 1.47-cm high and was filled with different density material. The purpose of modeling this object was to find out how the gray value of aluminum changes due to the presence of various material densities. 36
Experimentally, if the sample is inside an enclosing material (e.g. water in a container), this can affect the evaluation of the sample s cross section [19]. This effect is understood with the PScF (Point Scattering Function). In neutron imaging experiment, the pixel value at a given image position is proportional to the measured neutron beam intensity. The corresponding flat field correction consists of a division of the sample image by the blank beam image. This gives a corrected neutron transmission image of the sample material. According to the neutron attenuation law, one can obtain the mass thickness (i.e. the product of the sample density multiplied by the sample thickness) if the density and the cross section are known: sample matomic I ( x, y) ρ( x, y) t( x, y) = ln (3-1) reference σ I ( x, y) total where ρ( x, y) and t ( x, y) are the sample density and thickness at the position (x, y), the atomic mass material properties m atomic and the total microscopic cross-section σ total are intrinsic sample sample I and reference I are the measured gray values of the sample and the reference image. Typically, errors of experimental result of neutron imaging data originally from two main reasons: 1) scattering effects and 2) spectral effects. From the above analysis, it would be interesting to investigate if there was no neutron scattering and geometrical un-sharpness, what would the gray value changes due to the enclosing materials. (e.g. different density material column in the aluminum cylinder). Figures 3.4 through 3.15 show the reconstruction results of the simulated object for each of the different material density section and the corresponding projection data, respectively. The average gray value of each reconstructed cross-sectional slice section was not a constant for the same specific location from the top to the bottom of the 37
aluminum cylinder as was anticipated, but a steady increase step function instead. For each reconstructed cross-sectional slice section (50 slices each), the gray values was a constant as expected. The gray value difference from the top to the bottom was 30000. The cause for this change in gray level was unknown and needs further investigation. 38
Figure 3.4: The simulation result of the projection data for the first water density section in the aluminum cylinder described in the text. Figure 3.5: A cross-sectional reconstruction result for the first density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.4. 39
Figure 3.6: The simulation result of the projection data for the second water density section in the aluminum cylinder described in the text Figure 3.7: A cross-sectional reconstruction result for the second density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.6. 40
Figure 3.8: The simulation result of the projection data for the third water density section in the aluminum cylinder described in the text. Figure 3.9: A cross-sectional reconstruction result for the third density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.8. 41
Figure 3.10: The simulation result of the projection data for the fourth water density section in the aluminum cylinder described in the text. Figure 3.11: A cross-sectional reconstruction result for the fourth density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.10. 42
Figure 3.12: The simulation result of the projection data for the fifth water density section in the aluminum cylinder described in the text. Figure 3.13: A cross-sectional reconstruction result for the fifth density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.12. 43
Figure 3.14: The simulation result of the projection data for the sixth water density section in the aluminum cylinder described in the text. Figure 3.15: A cross-sectional reconstruction result for the sixth density section of aluminum cylinder. The location of this cross-sectional slice is indicated by the horizontal line in Figure 3.14. 44
Figure 3.16 below shows the gray values at the same location within the solid aluminum cylinder versus slice numbers from the top to the bottom with six different water density columns. 70000 60000 Gray values from the top to the bottom of the aluminum cylinder with six different water density columns Gray Level Value 50000 40000 30000 20000 10000 0 0 50 100 150 200 250 300 Slice numbers (pixels) Figure 3.16: Gray values of the aluminum from the top to the bottom of the aluminum cylinder with six different water density columns. 45
3.2: Neutron Computed Tomography Experiments 3.2.1: Summary of Initial Neutron Computed Tomography System The Penn State NCT system consisted of a neutron source, a collimated beam, an object turntable, a Thomson tube scintillation screen and image intensifier, a CCD camera, a mirror, and an image acquisition computer system, which is illustrated in Figure 3.17. The neutron source is a 1 MW TRIGA III research reactor. The neutron beam at the imaging plane is approximately 30 cm in diameter with an L/D ratio of 150 (as measured using ASTM 803) and a maximum divergence half angle of 1.4. The thermal neutron flux at the imaging plane for 1 MW power is 1.7 10 7 n/cm 2 -s. The object turntable was fixed to an aluminum stand and was used to rotate the object to different orientations using a computer program. The active diameter of the Thomson tube is 23 cm (9 inches), and yields a light gain of ~10 3. Image data were collected in a 10 bit, tiff format. Image acquisition and reconstruction was done using two different computer workstations. One computer controlled the CCD camera, the object turntable and image data capture. The other computer provided tomography cross section reconstruction and image visual analysis functions. The most important requirement for a CCD neutron tomography camera is its light sensitivity. At Penn State, a Cohu 7700 CCD camera with a sensor array format of 1004 1004 pixels was used. The imaging system s spatial resolution is highly dependent upon the lens, the size of the chip and the distance between scintillator and test 46
object. The system has a spatial resolution of (148 148) µm 2 with image integration times from 1/30 s up to 36 minutes. A Newport 855C rotary table controller with resolution of 0.001 was used to rotate the sample object for the tomography experiments. Neutron and gamma photon shielding of the CCD was required. Computer for control of the rotary table and camera CCD Camera Reactor Sample on rotary table Neutrons Light Photons Electrons D 2 O Tank Phosphorescent screen GdO 2 S Scintillator Photocathode Figure 3.17: A sketch of the initial Penn State neutron computed tomography system [4]. 47
3.2.2: Setup of Neutron Computed Tomography System The first step in implementing NCT was to choose a suitable imaging device for neutron tomography investigations that fulfills the high demands of system spatial resolution, large dynamic range and good linearity. Detectors fulfilling these requirements are CCD-camera based neutron radiography detectors [20]. The CCD camera chosen for the initial tomography investigations was the Cohu 7700. This CCD was used extensively in Penn State s neutron radiography work and was therefore known to fulfill the requirements for tomography. The CCD camera was controlled by the image data capture computer and was synchronized with the motion control system of a rotary table using a software developed by Heller [4]. After each projection was captured, the computer sent a signal to the rotary table and instructed it to rotate by a defined angle value in preparation for the next projection. The CCD camera produces a digital image that can be immediately used for the 2-D reconstruction. A plug-in for the image acquisition software, Streampix, was written so the computer synchronized rotary table and CCD camera acquires a sequence of digital images from the different view angels automatically [4]. 48
3.2.3: Measurement and Results Before an NCT imaging experiment begins, special care must be taken to ensure the projections are taken with the rotary table s axis of rotation parallel to the vertical pixel lines of the CCD that were aligned perpendicular to the floor. A bubble-level gauge was used to on the rotary table to obtain alignment. If the table was not level, the stand upon which the rotary table rested was adjusted. Table alignment can also be performed by placing a sample test object on the rotary table, taking an image, rotating it by 180 and taking another image. If the second image was flipped and subtracted from the first image, and the resulting image was totally dark, then the rotary table is adjusted well, otherwise, the resulting image shows some white areas indicating the rotary table needs to be adjusted [21]. For NCT imaging, two kinds of images are usually required: at least one blank beam image and a series of 2-D images of the object from different view angles, each image taken at evenly spaced between 0 and 180. After collecting these images, 3-D images reconstruction is possible. The general steps used for image reconstruction are as follows: 1) Acquire images from each projection image, 2) Perform a reactor power fluctuation correction on each projection image, 3) Perform beam shape correction on each projection image, 4) Perform white noise correction on the blank beam and projection images by using the Octopus software, 49
5) Utilize Octopus to perform the cross-sectional reconstructions of the sample object, and 6) Create a 3-D visualization of the object volume using VG-studio using the object cross-sectional reconstruction slices from Octopus to present the detailed internal information of the sample object. The initial reconstructions at the Penn State NCT system were successful. The object used in the initial tests was a 7.8 cm diameter by 12.7 cm high aluminum tube and with 0.5 cm thick walls with copper tubing wrapped around it. The aluminum tube had holes drilled into it at various heights and angles to demonstrate the ability for the system to reconstruct such details. A total of 361 projections were taken every 0.5 (2 projections/degree) throughout 180 with an image integration time of 0.2s/projection. Figure 3.18 and 3.19 show a photograph of the object with the corresponding radioscopic image and detailed cross-sectional slice using Octopus, respectively. After successfully calculating the detailed cross section information using Octopus, the cross-sectional slice information was loaded into VG-Studio software to visualize the object internal information. Figure 3.20 shows the volume reconstruction results using VG-Studio. 50
Figure 3.18: A photograph of the test object (left) and the corresponding 2-D radioscopic image (right) from the initial NCT system. Figure 3.19: A photograph of the test object (left) and the corresponding crosssectional slice reconstruction using 361 projections in 180 at 0.2s/frame from the initial NCT system. The red line indicates the location of the reconstructed slice. 51
Figure 3.20: The Volume reconstruction results (left) and isolated copper tubing (right) from the initial NCT system. Note that the reconstruction of the holes in the aluminum cylinder. As can be seen, the volume reconstruction result of the aluminum cylinder with copper tubing was successful, but there are still some artifacts in the projection slices that appear in the 3-D reconstructions. This is likely due to insufficient integration times per projection (noisy images) and possible misalignment of the rotary table. 52
3.2.4: Summary and New Data from Upgraded Neutron Computed Tomography System Major enhancements were made to the initial computed tomography system to improve the spatial resolution, to improve the imaging system detector efficiency for low neutron flux and radiation effects reduction in the projected images, including: 1) better radiation shielding, 2) better object alignment, 3) a larger neutron converter, and 4) a more advanced cooled CCD camera. The Cohu 7700 CCD camera which processing a 1004 1004 pixel array with 10- bit gray scale depth in the initial tomography system was replaced with a QImaging Retiga 4000RV. The Retiga 4000 RV has a sensor array format of 2048 2048 pixels, a 12 bit gray scale depth, sensitivity to low-light levels and allows integration times of 10 µ s to 18 min. Onboard three stages Peltier cooling provides cooling to -30 o C, reducing the dark current and random noise, which can occur during the long integration times needed in low neutron flux environments. A lens well suited to imaging on such large CCD sensors was fitted to the camera and allows zooming through adjustment of focal length from 70mm through 200mm. The 23-cm diameter Thomson tube with its GdO 2 S scintillation screen was replaced with a 25.4cm 25.4cm square 6 Li based scintillation screen, which provides a 57% larger viewing area compared to the original Thomson tube. In order to reduce the radiation effects on the CCD chips, a special two mirrors system has been devised and is lay out as the following Figure 3.21. The two front-surface mirrors are 94% reflective and the front mirror allows the positioning CCD camera to receive direct neutron beam light from outside. A turntable was included in the 53
CCD camera mount, allowing a remote alignment of the camera with the object in the collected images. Thermal neutrons Scintillation material Light photons D 2 O Tank Front Surface Mirrors Reactor Examined object and object turntable Turntable to adjust CCD camera angle Beam-Port Wall CCD Camera Figure 3.21: Sketch of the upgraded neutron computed tomography system [29]. In addition to this, the energy spectrum of radiation was measured at the location of the CCD and used to measure the effective radiation shielding. Gamma photons were the main cause that induced the white spots in the collected radioscopic images by interacting with the CCD camera sensor array. Lead bricks were put around the camera system to effectively shield the gamma photons that come from ( n,γ ) interaction with hydrogen in the reactor pool. The other concern was the radiative capture reaction of thermal neutrons. Most neutrons entering a shield system would be absorbed by atoms of the shield material, any kinetic energy of the neutron plus its binding energy in the resulting compound nucleus (usually 7 to 9 MeV) leaves the compound nucleus in a highly excited state. The excited nucleus usually decays within 1ps of the capture, often 54
through several intermediated states and thereby emitting one or more energetic gamma photons that is called capture gamma photons from ( n,γ ) reactions [32]. While the cross section for radiative capture is very small for high-energy neutrons (typically less than a few hundred millibarns for neutrons with energy between 20 KeV and 10 MeV), ( n,γ ) reactions are not of a concern in fast-neutron shielding applications, but greater concern are caused by thermal neutrons which have been slowed by scattering and come into equilibrium with the thermal motion of atoms in the shield. The ( n,γ ) cross section for thermal neutrons can be very large, may reach to thousands of barns for certain nuclides [32]. The neutrons used for neutron imaging work at Penn State are very well thermalized, the thermal neutrons interacted with the lead atoms and produced additional gamma photons, which was not desired for this system. Different layers of radiation shielding materials were used for shielding purpose. Boraflex and borated aluminum plates, two neutron absorbers, were wrapped around the lead bricks, which prevented thermal neutrons interacting with lead bricks. The radiation for the CCD camera chips was greatly reduced with this shielding design, subsequently the white noises in the collected images were substantially reduced. A total of 361 projections were taken every 0.5 (2 projections/degree) throughout 180 with an image integration time of 54 s/projection. Figure 3.22 to 3.26 below show the results. The NCT experiment procedure described in Section 3.2.3 was repeated with this upgraded system. One can see that the resolution of the image was much improved with longer integration time and white noise from the radiation to the CCD camera was greatly reduced. 55
Figure 3.22: A photograph of the test object (left) and the corresponding 2-D radioscopic image (right) from the upgraded NCT system. Figure 3.23: A photograph of the test object (left) and the corresponding 2-D radioscopic image (right) after beam shape and white noise correction from the upgraded NCT system. 56
Figure 3.24: A photograph of the test object (left) and the corresponding crosssectional slice reconstruction using 361 projections in 180 at 54 s/frame from the upgraded NCT system. The red line indicates the location of the reconstructed slice. Figure 3.25: A photograph of the test object (left) and the corresponding 3-D volume reconstruction result using 361 projections in 180 at 54 s/frame from the upgraded NCT system. 57
Figure 3.26: Isolated copper tubing (right) and aluminum cylinder (left) from the upgraded NCT system. Note that the reconstruction of the holes in the aluminum cylinder. From the above new NCT data, it can be seen that the white noises and artifacts in the projection data and the volume reconstruction results have been greatly reduced and the quality of the 3-D reconstructed images has been greatly improved. 58
CHAPTER 4 MODULATION TRANSFER FUNCTION ANALYSIS FOR THE NEUTRON IMAGING SYSTEM 4.1: Modulation Transfer Function Technique for Radioscopic System Characterization The modulation transfer function (MTF) has been recognized as an important tool for the measurement and predication of an imaging system performance. The MTF of the system is determined by the ratio of the magnitude of the system output to the magnitude of the system input [22]. Because of the band-limited nature of MTF curve, MTF can be used to analyze the true resolution parameter of the imaging system. A modulation transfer function technique has been developed to predict the resolving characteristics of the real time system. Two radioscopic systems have been used. One was a Thomson neutron intensifier with an active diameter of 23cm (9inch) used to collect the radioscopic images. A Cohu 7700 CCD camera that had a sensory array of 1004 1004 pixels (picture element) 10bit deep image and was updated at a rate of 30 frames per second was used to capture the intensified images. The upgraded system utilized a 25.4cm 25. 4 cm square scintillation screen that provides 57% larger viewing area compared to the original Thomson tube. A Retiga 4000 RV cooled CCD camera with a sensor array format of 2048 2048 12bit gray scale depth was used for the upgraded system and allows integration time at a rate of 10µ s to 18 min. For a real time system, it is desired that a direct method can be developed to measure the overall system MTF. The result can be used to make a comparative judgment about the overall imaging plane MTF values. 59
Unfortunately, the MTF method is not directly applicable in many contemporary sampled imaging systems (e.g., electrooptical line-scan and sensor-array devices) because sampling causes these systems to have a particular kind of local shift variance. That is, whenever these systems form the (sampled) image of a point source, the appearance of the (reconstructed) image will depend on the location of the point source relative to the sampling (i.e., pixel)grid [23]. Over a span of many pixels in a scene this effect is not noticed. However, over small distances, locally about a small image feature, the geometry involved may be sampled in a number of shifted ways. A small impulse input signal to the imaging system may be partially sampled by several (e.g. 4 pixels) while slightly shifting the input feature may result in a sampling of only one pixel [24]. Figure 4.1 below shows this effect of impulse variance. 60
Figure 4.1: Sampling shift for an impulse input to a real time video system (a): input to the system, (b): possible output to the system, (c): other possible output to the system [24]. 4.1.1: Modulation Transfer Function Theory Normally, the MTF analysis is performed in one dimension. 1-D analysis is sufficient for this problem since the results are applied in a comparative manner. This allows the MTF measurement to be done through scanning a single line of data across an elongate slit or a sharp edge. The 1-D MTF function is normally defined through the following functions: f (x) = system input function, 61
g(x) = system response to the input f (x), (x) h B h I = system impulse response to the imaging beam, (x) = system impulse response to the intensifier/ video hardware, and (x) h i = system impulse response to the component i where i is some arbitrary component of interest [24]. From the above definition, the MTF(u) is defined as G( u) MTF(u)= = H B ( u) H I ( u) H i ( u), (4-1) F( u) where in the above function, G(u) is the Fourier transformation of the function g (x). Equation (4-1) indicates that MTF is calculated as the ratio of known input to the system F (u), and the corresponding output to the system, G (u). If choosing an impulse function as the input function for the imaging system, then the MTF(u) can be determined through the following MTF(u)= G( u) = H ( u) H ( u) H ( u). (4-2) B Technically, it is difficult to make a slit aperture that will provide approximately an impulse function in one dimension for the real time neutron radiography/tomography system that can transmit a sufficient intensity. An alternative way of achieving the same goal is to use a knife-edge aperture [25]. In order to quantify the true resolution characteristics for the neutron imaging system, this aperture can be created with a sharp piece of gadolinium foil or cadmium or similar thin high-contrast material. I The thin piece of knife edge input function for the system can be simplified by a basic step function which is represented as i 62
0, x < 0, f ( x) = (4-3) 1, x 0. Equation (4-2) can also be written as a function of a differentiated system response function g(x) respect to the step function of f (x) written as, and then the system MTF can be d MTF(u)= F[ g( x)] = H B ( u) H I ( u) H i ( u), (4-4) dx from this equation, it is not difficult to see that the MTF(u) is a function of different components of the whole imaging system(i.e. beam, intensifier and other components which are connected to the system). Therefore the MTF(u) can be written as MTF( u) = MTF ( u) MTF ( u) MTF ( u). (4-5) B I The intensity distribution at the intensifier surface due to the 1-D impulse sharp edge effect is a rectangular function with a width of 2 l tan( θ ), where θ is the effective divergence of the neutron beam, l is the distance between the aperture (or in some cases sharp edge) and the intensifier surface [26]. The impulse response function for the neutron beam is then modeled as 1 x x0 h B ( x) = rect, 2 tan( θ ) 2 tan( ) (4-6) l l θ where in the above equation, x0 is the center of the rectangle function rect [ ], and the Fourier transform of h B (x) results in the beam transfer function (u) as H B ( u, l, θ ) = sinc[2 l tan( θ ) u] exp( j 2 π x0 u). (4-7) i H B 63
In the above equation, the parameters ( l, θ ) are included in the function, also it is oblivious that if the knife edge is put directly against the intensifier surface ( l = 0), which is the most sharp case. The MTF can be determined independently of the beam effect d MTF( u,0, θ ) = F[ g( x)] = H I ( u) H i ( u). (4-8) dx The value of H I (u) can be directly measured if there is no additional electronic components connect to the system. The function g(x) corresponds to the measured knife edge data which contains noise which associated with the imaging system. The noise is generated as thermal electronic noise and from the statistical variation in neutrons. The noise effect can create spurious information to the signal that affects the MTF measurement result [24]. The MTF of the neutron radioscopic system can be written as where in the equation, h(x) H ( u) = h( x)exp( iux) dx, (4-9) = line spread function, x = spatial position, and u = spatial frequency. The function h(x) can be obtained by using the system output g(x) of a stepwise input function in the form of a sharp-out gadolinium or cadmium sheet. The line spread function is given by the following equation h ( x) = dg( x). (4-10) dx 64
Harms and Wynam indicated a function for the normalized edge data can be best presented as the following [6] 1 1 1 g( x) = + tan [ β ( x x0 )], 2 π (4-11) where in the equation, β = System resolution parameter, and x0 = x coordinate of knife edge center. Based on Harms and Wyman s theoretical calculations and Wrobel and Greim s experimental investigations [31], Equation (4-10) correctly describe the density variation across the knife edge when the inherent unsharpness is dominant. When geometrical unsharpness dominates, the Gaussian equation 1 1 g( x) = + erf { α ( x x0 )}, (4-12) 2 2 will best fit the density variation across the knife edge, the relationship between β in Equation (4-10) and α in Equation (4-11) is as β = 2.5863 α. (4-13) The first derivative of Equation (4-11) yields the following equation dg( x) dx β / π = 2 1+ β ( x x 0 ) 2. (4-14) A sharp-cut gadolinium edge was placed directly on the intensifier/imaging converter surface and imaged in the real time neutron radioscopic system. The location of the edge was estimated to be at location x 0 by assuming x x f / f, (4-15) = i 0 i i 65
where in the equation, x i f i ' = i th spatial coordinate, and = luminance value corresponding to the i ' th location. The parameters x0 and β are calculated using the method of nonlinear least squares fit [33]. The method optimizes a goodness fit of parameters by interacting between β and x 0. If the resulted curve g (x) fits the experiment data excellently, then the value β and x0 will be assumed to be correct for that particular edge position. After the edge parameter of β and x0 is determined, the impulse response function of Equation (4-10) is determined by taking the first derivative of g (x). Taking the Fourier transformation of Equation (4-14) gives the following transfer function for the intensifier/video system H I β ( u, β ) = exp( j2πx 0u) exp( 2π u / β ). (4-16) π Then the normalized MTF for the neutron imaging system, which combines the beam effects and the intensifier/video components, is written as MTF ( u, l, θ, β ) = sinc[2 l tan( θ ) u] exp( 2π u / β ). (4-17) The first zero value associated with the sinc function in Equation (4-17) is related to the effective beam divergence as compared to the parallel beamθ. According to the sinc function, the function MTF ( u, l, θ, β ) will be equal to zero when 2 l u tan( θ ) = 1, from this analysis point, the beam divergence angle is calculated by setting 1 θ = tan 1 ( ). (4-18) 2 l u 66
The parameter β I for the intensifier/video system varies as a function of spatial positioning of the edge within the field of view. A periodic shifting pattern of the edge gray level will be observed. This phenomenon occurs due to the fact that the edge position is shifting relative to the pixel sampling locations. The parameter β can be determined for component i which is connected to the system. In order to avoid the discrepancies from the calculation, performing analysis at constant location is necessary. The value of β i can be determined mathematically by removing measurement location, where β I, i component I and i are serially connected I i β from β in the same denotes the system resolution parameter when 1 1 1 + =. (4-19) β β β i I I,i The parameter β I is determined by imaging a sharp edge material that is placed directly against the intensifier surface, thus making MTF( u, l, θ, β ) =1. The edge data is recorded and processed, and the value of MTF I ( u, β I ) is determined. The component to be measured is placed in the system and the edge data is recorded using the same spatial location resulting value of MTFI, i ( u, β I, i). The value of MTF i (u) is calculated as the following MTFI, i ( u, β I, i ) MTFi ( u, β i ) = = exp( 2π u / β i ), for u, β i > 0. (4-20) MTF ( u, β ) I I In order to define the physical meaning of β i, a parameter of f N has been introduced in order to quantify the amount of image degradation that can be attributed by component i. For a real time neutron radiography/tomography system that is based on a discrete sampling procedure, the system s resolution is primarily dependent on the sampling I, i 67
distance or the Nyquist sampling frequency. The quantity of f N is defined as the fraction of information passed through the system between zero frequency and the Nyquist frequency is u N = 1/ 2 x, (4-21) where in the above equation x is the sampling space or the pixel width. The value is proportional to the integral of the MTF over the frequency interval of (0, u N ) and is given by f N = un 0 MTF ( u, β ) du i un 0 du i = xβ i [1 exp( π / xβ i )], π (4-22) once the individual component parameters are known, the fundamental component value must be determined. The local constant β, was measured as mentioned above so the component parameter I β i can be calculated using Equation (4-19). The value of β I is varied spatially within the field of view due to the local shift-variant nature. Sampling the edge data to estimate the 1-D impulse response, it is easily to see that the edge within the field view of the CCD camera has an important effect on the MTF calculation. Also, the variability in the recorded data is related to the shift variant impulse response of the digitizer. The current imaging system uses a deep-cooled, high-dynamic range digital CCD camera, which samples the data along the edge in between the center sampling and the off center sampling. The MTF results that uses the discrete Fourier transform are periodic with one-half period occurring at the Nyquist sampling frequency, u N = 1/ 2 x. 68
Let s(x) be defined as the noise-free impulse data and then the discrete center sampled data s c (x) is given by s c ( x) = s( x) comb( x / x), (4-23) and the Fourier transformation of the above equation is as S ( u) = F[ s( x)]* F[ comb( x / x)] = S( u 2 ), (4-24) c nu N n the off-center discrete data are given by s o The Fourier transformation of the above equation is x + x / 2 ( x) = s( x) comb( ). (4-25) x n s ( u) = ( 1) S( u 2nu ). (4-26) o n Averaging Equation (4-24) and Equation (4-26), gives the following Expansion of the above equation gives N 1 S m ( u) = [ Sc ( u) + So ( u)] = S( u 4nu N ). (4-27) 2 u S ( u) S( u) + S( u 4u ) + S( u + 4u ) +... (4-28) m = N N This implies that the data over the frequency interval of 2u N,2u ) will be valid in representing the system MTF, i.e., by the discrete method, ( N 1 MTF( u) = Sc ( u) + So ( u), 2u N < u < 2u N. (4-29) 2 An equivalent technique is developed and averaged to give the spatially invariant result based on two constants, β c (on-center data) and β o (off-center data) [27]. The normalized (to one) analog center sampled relation s cβ (x) corresponding to Equation (4-23) is 69
1 sc β ( x) =. (4-30) 2 2 1+ β ( x x ) The Fourier transformation of the above equation is given by S c cβ ( o c o u) = exp( j2π ux ) exp( 2π u / β ). (4-31) By analogy, the corresponding off-center relation and its Fourier transformation are given by the following 1 soβ ( x) =, (4-32) 2 2 1+ β ( x x ) S o oβ ( o o o u) = exp( j2π ux ) exp( 2π u / β ). (4-33) By averaging Equation (4-31) and Equation (4-33), the resulted expression is analogy to Equation (4-27) S mβ 1 = exp( j2π ux0 )[exp( 2π u / β c ) + exp( 2π u / β 0 )]. (4-34) 2 Using the above equation, the fundamental component of MTF, as a function of β c and β 0, is given by 1 MTF( u, β c, β 0 ) = [exp( 2π u / β c ) + exp( 2π u / β 0 )]. (4-35) 2 Equations (4-34) and (4-35) are valid for over all frequencies due to the fact that no periodic results are associated with the analogy Fourier transformation. And also note that this technique is only valid on the condition that the edge data approximation of Equation (4-11) faithfully represents the actual experimental data. Since the MTF ( β, β, u) is known for the basic system, the combined effect of I c o having other imaging components can be calculated using the values of β given by i 70
Equation (4-19). Then the total system transfer function MTF( l, θ, β, β, β, u), including the beam effects, is calculated by the following equation c o i MTF ( l, θ, β c, βo, βi, u) = MTFB MTFI MTFi = 1 sinc(2 2 l tan( θ ) u) exp( 2π u i i 1 u u ) [exp( 2π ) + exp( 2π )], u 0, β c, β o, β i 0 β i β c β o (4-36) The complete characteristics of radiography/tomography system will be estimated if the values of θ, β c, β o, and β i are known. Equation (4-36) can be also used to estimate the resolution limit of the system x min. This is done by determining a cutoff limit between 2% to 5% for MTF( l, θ, β, β, β, u). This value is determined somewhat subjectively c o i for the system which is under investigation but only allows for a rigid, reproducible technique for comparing components. A distinguishable lp/mm corresponds roughly to spatial frequencies where MTF is between 2% to 5% (0.02 to 0.05). This number varies with the observer, most of whom stretch it as far as they can [30]. Equation (4-36) can be solved transcendentally for the cutoff value for MTF l, θ, β, β, β, u ). Then the resolution parameter xmin is given by ( c o i cut 1 x =. (4-37) min u cut The corresponding value of the image information degradation factor for each component f N can be determined for the intensifier/video or converter/camera plus other components (assumes no beam effect, l = 0cm) 71
72 ]. ) 1 1 ( )]} 1 1 ( exp[ {1 ) 1 1 ( )]} 1 1 ( exp[ {1 [ 2 + + + + + = i i o i i o i i c i i c N x x x f β β β β π β β β β π π (4-38)
4.2: MTF Experimental Measurement Procedure 4.2.1: MTF Measurement for the Original Imaging System The neutron imaging system at Pennsylvania State University was analyzed using a sharp-edge gadolinium foil. A fellow graduate student, Cory Luke Trivelpiece, characterized the sharpness scale of the edge by using white light optical microscopy in the Material Characterization Laboratory at Penn State. Figure 4.2 to Figure 4.4 show the results of the characterization of the gadolinium foil edge. From the images, one can see that the average roughness along the gadolinium foil edge is approximately 10% of 50 µ m (5 to10 µ m ). Based on the current resolution of the imaging system (approximately 101.0 µ m to 157.0 µ m ), the edge of the gadolinium foil can be seen as a perfect sharp edge for the MTF analysis purpose. Figure 4.2: Left side of gadolinium foil edge image using white light optical microscopic characterization. 73
Figure 4.3: Middle of gadolinium foil edge image using white light optical microscopic characterization. Figure 4.4: Right side of gadolinium foil edge image using white light optical microscopic characterization. 74
The edge was placed at a slight angle to the line which perpendicular to the horizontal pixels of the CCD imaging system suggested by Fujita et al [28]. The gray level density values across the edge were measured by discrete intervals and a non-linear least square curve fit technique was used to fit the parameters mentioned above. Figures 4.5 through 4.8 below show the knife-edge gadolinium foil placed directly against intensifier and corresponding blank beam image at magnification setting Mag0 and Mag1, respectively when the reactor power is 1MW. Figure 4.5: Blank beam image at Mag0 condition when the reactor power is 1 MW. 75
Figure 4.6: Knife-edge gadolinium foil at Mag0 condition when the reactor power is 1 MW. Figure 4.7: Blank beam image at Mag1 condition when the reactor power is 1MW. 76
Figure 4.8: Knife-edge gadolinium foil image at Mag1 condition when the reactor power is 1MW. Figure 4.9 below shows the relative density across a sharp gadolinium foil edge as a function of spatial position and using Gaussian fit function at Mag0 condition. 77
1.2 1 Normalized intensity 0.8 0.6 0.4 0.2 0-0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Distance(cm) experiment data curve fit data Figure 4.9: A sample gadolinium foil knife edge data and the curve fit for that data for the old neutron imaging system at Mag0 condition. After a non-linear least square curve fit is used to find β and x 0, the edge parameter β is determined, then the impulse response is calculated by taking the first order derivative of g (x). For the edge shown above, the corresponding impulse response is depicted in Figure 4.10. 78
1.2 Normalized intensity 1 0.8 0.6 0.4 0.2 LSF 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Distance(cm) Figure 4.10: A sample response function calculated from gadolinium foil edge data at Mag0 condition for the old neutron imaging system. The MTF(u) was calculated using the system impulse response function, Figure 4.11 below shows the MTF calculation result. MTF 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 30 35 40 45 50 Sampling frequency(cycles/cm) MTF result Figure 4.11: A sample MTF calculation result of the old neutron imaging system at Mag0 condition. 79
Similarity, Figures 4.12, 4.13, and 4.14 below show the gadolinium foil edge data and system impulse response as well as the corresponding MTF simulation result at Mag1 condition. 1.2 1 Normalized intensity 0.8 0.6 0.4 0.2 Experiment data Curve fit data 0 0 0.1 0.2 0.3 0.4 0.5-0.2 Distance(cm) Figure 4.12: A sample gadolinium foil knife edge data and curve fit for that data for the old neutron imaging system at Mag1 condition. 1.2 1 Normalized intensity 0.8 0.6 0.4 LSF 0.2 0 0 0.1 0.2 0.3 0.4 0.5 Distance (cm) Figure 4.13: A sample system response function calculated from gadolinium foil edge data at Mag1 condition for the old neutron imaging system. 80
MTF 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 MTF result 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 Sampling frequency (cycles/cm) Figure 4.14: A sample MTF calculation result of the old neutron imaging system at Mag1 condition. From above MTF calculation analysis, when the intensifier is at Mag0 and Mag1 conditions, the cutoff limit 5% transmission of information corresponds to approximately 46 cycles/cm and 69 cycles/cm. It is important to estimate the total system output. The constants β and measured for the system by using a knife-cut edge aperture as previously described. c β o were Once the knife edge was positioned, data were taken across the edge at one location and moved along the angled edge. This process was repeated until cyclic criterion was met, i.e. the values of β i along the edge appear cyclically repeated sampling results. These maximum and minimum values for β correspond to β and β. Figure 4.15 shows sample cyclical variation of the resolution parameter for the original 1004 1004 pixel real time neutron radioscopic system at Mag1 condition. c o 81
140 Resolution parameter (1/cm) 120 100 80 60 40 20 Resolution parameter 0 410 415 420 425 430 435 440 445 Edge location (pixels) Figure 4.15: Calculated values of β along the edge as a function of edge location at Mag1 condition for the old neutron imaging system. For this case, the measured values for the maximum and minimum curve parameters were found to be: β c = 130.6292 / cm, β o = 61.5919 / cm And the corresponding family of the MTF curves for the above edge is plotted in Figure 4.16. 82
MTF 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 Frequency(cycles/cm) MTFfamily1 MTFfamily2 MTFfamily3 MTFfamily4 MTFfamily5 MTFfamily6 MTFfamily7 Figure 4.16: MTF family calculation results using the edge resolution parameters at Mag1 condition for the original neutron imaging system. 4.2.2: MTF Measurement for the Upgraded Imaging System Using the same experimental procedure discussed above, a sharp knife-edge gadolinium foil was placed directly against the neutron converter and imaged when the reactor power was 1 MW. Figures 4.17 and Figure 4.18 show the blank beam and the corresponding gadolinium knife-edge foil raw images. 83
Figure 4.17: Blank beam image for the upgraded neutron imaging system when the reactor power is 1MW. Figure 4.18: Knife-edge gadolinium foil image for the upgraded neutron imaging system when the reactor power is 1MW. 84