NEUTRON imaging is a technique for the non-destructive

IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 3, JUNE 2009 1629 Investigating Phase Contrast Neutron Imaging for Mixed Phase-Amplitude Objects Kaushal K. Mishra and Ayman I. Hawari, Member, IEEE Abstract Phase contrast imaging is an imaging modality that has been extensively applied in X-ray imaging and was demonstrated using neutrons over the past few years. In this case, contrast in the image, especially at edges, is enhanced due to phase shifts that take place as the neutron wave passes through regions in the sample that differ in the coherent scattering length density. Usually, a pure phase object approximation is used to formulate the problem, whereas realistic samples represent mixed phase-amplitude objects. In this work, a formulation for mixed phase-amplitude objects with moderate neutron attenuation coefficients and its effect on the neutron image is presented. A computational simulation technique has been devised to study this effect on different types of samples. Using simulations, it is observed that the pure phase object approximation results in over enhancement of edges for a phase-amplitude object, with significant variation (in the case of neutron imaging) depending upon the edge forming material characteristics. The total contrast for the mixed phase-amplitude object is less than the sum of the individual attenuation and phase contrast components. The difference depends on the scalar product of the gradient of the coherent scattering length density and the attenuation coefficient. The presented formulation can aid in predicting and optimizing the performance characteristics of neutron phase contrast imaging experiments. Index Terms Neutron imaging, phase-amplitude objects, phase contrast. I. INTRODUCTION NEUTRON imaging is a technique for the non-destructive testing (NDT) of materials. It is characterized by its sensitivity to low atomic number (Z) elements such as hydrogenous materials. Therefore, it represents a modality that is complementary to more traditional techniques such as X-ray imaging. In recent years, to enhance contrast in the images taken with neutrons, the use of the wave-particle dual nature of neutrons in imaging was demonstrated in what is known as Phase Contrast Neutron Imaging [1] [3]. This is an imaging modality that has been extensively applied in X-ray imaging and electron microscopy [4] [6]. In this case, contrast formation in the image, especially at edges, is enhanced due to phase shifts that take place as the neutron wave passes through the regions in the sample that differ in the coherent scattering length density. The advantage which this technique offers over the usual diffraction imaging techniques is that there is no requirement for the beam to be monochromatic and only spatial coherence is required. Obtaining a monochromatic neutron beam usually requires use of Manuscript received September 11, 2008; revised December 05, 2008. Current version published June 17, 2009. This work was supported by the U.S. Department of Energy under Grant DE-FG07-03ID14532. The authors are with the Department of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695 USA (e-mail: kkmishra@ncsu.edu; ayman. hawari@ncsu.edu). Digital Object Identifier 10.1109/TNS.2009.2016962 monochromators, like bent sapphire or silicon crystals, or beam choppers and leads to significant reduction in neutron beam intensity. Thus, phase contrast neutron imaging offers a simple and elegant alternative for obtaining the higher contrast needed in the neutron images of samples. Typically, this technique is presented using formulations that imply its suitability for achieving contrast in images of almost pure phase objects (i.e., objects with negligible neutron attenuation). The technique is formulated using the transport of intensity equation and the diffraction propagation function [7] [9]. However, edge enhancement among moderately attenuating objects with similar attenuation coefficients is also desirable and possible. In this work, the pure phase contrast formulation is extended to describe moderately neutron attenuating objects. These objects are referred to as mixed phase-amplitude objects. The extension of the formulation to these objects requires consideration of a phase-amplitude interaction term which mathematically is of the same order of magnitude as the pure phase contrast contributing term. This extension provides a more accurate description of the image contrast formation process as well as extends its applicability to various different objects which are not pure phase. A similar relationship in the context of X-ray phase contrast imaging has been obtained, but the phase-amplitude interaction term was neglected assuming that the variation of the attenuation coefficient is insignificant in most of the X-ray samples [9]. A technique for computationally simulating the images using the obtained formulation has also been developed. Simulations of phase contrast neutron images of various relevant materials forming different edges in the artificially designed phantoms was performed using this technique in order to study their numerical significance in characterizing the edge contrast in neutron images. This investigation can lead to more accurate understanding of edge effects in a phase contrast neutron image. II. PHASE IMAGE FORMULATION A. Mathematical Formulation For a plane neutron wave incident on a thin pure phase object the intensity distribution at a distance from the object plane is given by where is the intensity at the contact plane before the object, is the wavelength of the neutron beam and is the real part of the refractive index that depends on the coherent scattering length densities of the materials present in the object and the wavelength of the beam [10]. A schematic of the setup is shown (1) 0018-9499/$25.00 2009 IEEE

1630 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 3, JUNE 2009 B. Simulation Technique For computational simulations, (5) needs to be discritized over a grid and all the parameters involved in it need to be determined. The discretized Laplacian can be written as (6) Fig. 1. A schematic of phase contrast neutron imaging. in Fig. 1. The above equation is well known in the literature of phase contrast imaging. For a mixed phase-amplitude object the transmission function can be written as, where. Here, is the imaginary part of the refractive index that accounts for the attenuation of the neutron beam as it passes through the object. The intensity distribution at a distance for this object can be obtained by convolving the transmission function with the Fresnel propagator given by where. The first order approximation of the obtained intensity distribution can be written as Comparing (3) with (1), the appearance of an additional term, which depends on the scalar product of the gradient of and, can be observed for mixed phase-amplitude objects. This additional term can be referred to as the phase-amplitude interaction effect. It vanishes as becomes zero for a pure phase object reducing (3) to (1). Furthermore, it can also be observed from (3) that the phase-amplitude interaction effect counters the pure phase induced intensity variation. In obtaining (3) it has been assumed that the functions and are at least twice differentiable with respect to and. Although in most of the practical cases we have an edge function which is not differentiable, the above formulation is applicable because the edge function is smoothed by the convolution with the point spread function thus making it differentiable. The final neutron intensity image obtained on the screen is given by For a polychromatic neutron beam and a material with varied composition an approximate approach to obtain the intensity image would be to use the expectation value taken over the neutron energy spectrum as (2) (3) (4) where and the gradient can be descritized using a central differencing scheme. The descritized Laplacian with becomes the regular second order central differencing scheme which is isotropic. Other values of make anisotropic. is a reasonable choice to make in the absence of any information on the anisotropy of in the sample. For a polychromatic neutron beam, calculation of the expected value requires information about the neutron energy spectrum at the image plane. The energy spectrum can be obtained, e.g., through radiation transport calculations. Details on performing such calculations have been already published [11]. These calculations must be performed with the pinhole geometry required for phase contrast imaging. The PSF can be measured either using a gadolinium edge or it can be obtained through a radiation transport calculation using a computer code such as MCNP [12]. The calculation should be performed at the same object to image-plane distance at which the images need to be simulated. C. Beam Requirements The requirement imposed on the neutron beam to obtain the phase induced intensity variation in the image as given by (3) is a high degree of transverse spatial coherence. On the other hand, high degrees of temporal and chromatic coherence are not required. The degree of spatial coherence is a measure of the correlation which exists between the phases of the radiation at two points and is measured by the visibility of fringe patterns in the image. is defined as where, and are the maximum and minimum intensities in the obtained fringe patterns. In the case of perfect coherence it is unity, and for complete incoherence it is zero. For a circular source of uniform intensity and diameter the degree of coherence is given by, where, is the wavelength of the neutron beam, is the first order Bessel function of the first kind, is the distance between the two points whose spatial coherence is of interest (located on the object), is the effective diameter of the source and is the object to image-plane distance. This defines the coherence patch. The radius of the first zero of the above Bessel function is taken as a measure of the width of this patch. This radius is given by (7) where is the expected value of [.] over all wavelengths. (5) (8) which is also referred to as the transverse coherence length (i.e. ). The extent of the observable phase contrast effect is given by the width of this coherence patch which is dependent on beam energy. From (8) it can be observed that

MISHRA AND HAWARI: INVESTIGATING PHASE CONTRAST NEUTRON IMAGING FOR MIXED PHASE-AMPLITUDE OBJECTS 1631 the spatial coherence will increase if the expected wavelength of the neutron beam spectrum is increased. Also, the transverse coherence length is directly proportional to the well-known nondimensional quantity of the beam. Upon the substitution of (from (8)) for, the expression is obtained. The degree of coherence for a circular aperture defined by the visibility thus becomes (9) From (9) it can be observed that the degree of spatial coherence does not depend on the wavelength of the beam or the source diameter. Further, it can also be observed from (9) that is dependent only on the ratio and thus for a given, (obtained using (8) or limited by the beam line length) to observe the coherence effect the object-to-image-plane distance must be large enough (within the Fresnel limit). For a thermal neutron beam the wavelength of the neutrons is of the order of. Thus, to achieve a of the order of micrometers (i.e. the order of detection resolution for neutron radiographs) is required which is much higher than the for conventional neutron imaging. To achieve such a high, the neutron source effective diameter d must be reduced to a pinhole. But, the available neutron flux at the image plane which determines the exposure time is inversely proportional to the square of the as given by (10) Thus, a delicate balance between these two counteracting requirements needs to be achieved. To achieve the optimum between maximizing the coherence length and the neutron flux radiation transport simulations can be used as a design tool. The formulation and the simulation technique presented in Sections II-A and II-B respectively will aid in the design by predicting the observable phase contrast effect for any particular object exposed for a fixed exposure time defined by the neutron flux resulting from a given design. Conversely, if the exposure time and hence the flux requirement is dictated by some other factors e.g. frame speed necessary to observe some particular dynamical effect, then the formulation along with the simulation technique can be used to study and anticipate the expected contrast enhancement. This can be used as a decision making tool for choosing this technique in any particular setting and for some specific application. III. PHASE IMAGE SIMULATION Using designed phantoms, simulations of neutron radiographs were performed in order to observe the contribution of each term in (3) separately in the combined neutron image. The characteristics of the neutron imaging beamline located at the North Carolina State University (NCSU) PULSTAR reactor were used for this purpose [11]. Monte Carlo radiation transport calculations were performed using MCNP5 [12] to obtain a of 1.47 (at 6 meter aperture-to-detector distance) and a reasonable neutron flux using a 0.5 mm diameter gadolinium aperture and a 3 inch long sapphire filter. Sapphire acts as a fast neutron filter [13] and thus, helps in increasing the average wavelength of the beam, thereby increasing the transverse Fig. 2. The neutron energy spectrum (d=d(ln(e)) at the image plane with a 3 inch single crystal sapphire filter. coherence length. The beam divergence was reduced from 2 degrees to 0.5 degrees. This provides a beam size close to 10 cm at the image plane. The wavelength used in the simulations is the average wavelength of the PULSTAR imaging beam and is 1.3. The associated energy spectrum is shown in Fig. 2. For such conditions, the exposure time at the PULSTAR reactor is expected to be between three to four hours depending upon the type of sample. The phantoms were designed to depict various edges that are possible in objects characterized by the difference in the transmission functions of the edge forming materials. Phantom 1 was selected to include various engineering, structural, ornamental, nuclear and aerospace materials with moderate but almost similar attenuation contrast. For example, steel is a common structural material, titanium is an aerospace material, nickel-silver composites are used for electrical contacts and electrodes, zirconium and zircaloy is used in nuclear fuel cladding. It is these types of materials on which non destructive testing and non-invasive measurements need to be performed through imaging to locate possible cracks, voids, delaminations in composites, corrosion sites, misalignment etc. Precisely locating the edges constitutes an important step in performing any of the above tasks using the obtained images. Phantom 2 was selected to include organic and biological materials to investigate phase contrast enhancement in such samples. These materials are either present in plants or animal bodies or are very similar to them. For example, ICRP soft tissue and ICRP bone are equivalent to biological tissue and bone, sapphire specks are used in phantoms to simulate micro-calcifications, nylon fibers are used to simulate fibrous structures, acrylic is used to simulate breast tissues [14], urea is used as fertilizer for plants etc. The geometry of the phantoms and the material composition are shown in Fig. 3 and Table I, respectively. To simulate the neutron images of the phantoms, the values were calculated at different object to image plane distances assuming a 0.5 mm diameter point source and a beam divergence of 0.5. The attenuation radiograph simulation of the selected phantoms were performed using

1632 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 3, JUNE 2009 For each image simulation particles were run in four cycles with each cycle run on 24 processors for two hours. To further improve the counting statistics pinhole detectors were used to bring the relative error in every pixel inside the field of view of the pinhole down to 0.6%. The magnification of the attenuation images were obtained accurately using the Sobel edge detection algorithm [15]. The calculated magnification and the obtained PSFs were used to obtain the pure phase and the phase-amplitude interaction images of the corresponding phantom at the same object to image plane distance. From (3) it is clear that the final image is the algebraic sum of these three images. The coherent scattering length densities for different materials required for simulating phase images were calculated using the data obtained from the National Institute of Standards and Technology (NIST) database [16]. For the purpose of simulation was used in (6) to approximate the Laplacian performing isotropic filtering. A pixel size of 50 was selected in the image simulation. A. Simulation Results Fig. 3. The geometry of (a) phantom 1 (b) phantom 2. Table I gives the material composition of the phantoms. TABLE I THE MATERIAL COMPOSITION OF THE SIMULATED NEUTRON RADIOGRAPHY PHANTOMS. THE GEOMETRY OF THE PHANTOMS IS SHOWN IN Fig. 3 MCNP5 at the same object to image plane distances. The simulations were executed on a Linux cluster available in the High Performance Computing (HPC) Center at NCSU. The cluster has 612 dual Xeon nodes. Each node has two Xeon processors (mix of single-, dual- and quad-core) that have 2 Gigabytes (GBs) of memory per core and 36 73 GBs of disk-space. The simulated radiographs of Phantom 1 are shown in Fig. 4. All the radiographs are on the same gray scale. Figs. 4(a), 4(b) and 4(c) depict the attenuation only, pure phase effect and phase-amplitude interaction effect radiographs respectively corresponding to the first, second and the third terms in (3). Fig. 4(d) depicts the total phase effect which is the algebraic sum of the pure phase and the phase-amplitude interaction effects. Figs. 4(e) and 4(f) depict the radiographs of the object considering it as a pure phase object and as a mixed phase-amplitude object respectively. Clearly, the attenuation only radiograph does not provide enough contrast for all the edges of the object to be distinguishable in the image. As is clear from the phase images, the phase effect does not provide different intensities in different materials which may be important for obtaining material makeup information of the object but, it enhances the edges between the materials providing the geometry information of the object. Comparing Fig. 4(b) and Fig. 4(c), it can be observed that the phase-amplitude interaction term provides less contrast than the pure phase term but still is not negligible. This is because the pure phase term involves a higher order derivative which increases the frequency and amplitude of intensity oscillation and hence higher contrast on the gray scale is visible. Also, the magnitude of the phase-amplitude interaction effect depends on the difference in the coherent scattering length densities of the edge forming materials. For example, the nickel-silver edge has more interaction effect than other edges even with a little difference in their attenuation coefficients. The difference in the contrast due to these interaction effects can also be observed when Fig. 4(e) and Fig. 4(f) are compared. This indicates that for some material edges the interaction effect (i.e., third term in (3)) can become really important. Comparing Fig. 4(a) and Fig. 4(f), it can be observed that features like the crack and material intrusions, which were not visible in the attenuation image due to similar attenuation coefficients, become visible in the phase enhanced image because of the difference in their scattering length densities.

MISHRA AND HAWARI: INVESTIGATING PHASE CONTRAST NEUTRON IMAGING FOR MIXED PHASE-AMPLITUDE OBJECTS 1633 Fig. 4. Phantom 1 simulation results: (a) normal attenuation radiograph (1st term in Eq. (3)) (b) the pure phase image (2nd term in Eq. (3)) (c) the phase amplitude interaction image (3rd term in Eq. (3)) (d) the phase image with phase-amplitude interaction which is the difference of (b) and (c) as given by Eq. (3) (e) phase contrast radiograph with pure phase approximation (1st and 2nd term in Eq. (3)) (f) Phase contrast radiograph with phase-amplitude interaction (all terms in Eq. (3)). The object to detector distance R=80cm. The grayscale for all six images is the same. Figs. 5(a) and 5(b) depict the normalized intensity profiles taken across the horizontal slice a-a and the vertical slice b-b that are shown in Figs. 4(e) and 4(f). These figures illustrate the sharp intensity changes at the edges because of the phase effect. The quantitative effect of the phase-amplitude interaction term can also be observed in these figures. A significant interaction effect appears at the nickel-silver edge in the profiles. In addition, an edge overlap effect arises in the profiles due to the spread

1634 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 3, JUNE 2009 Fig. 5. An intensity profile taken across (a) slice a-a of Fig. 4(e) and Fig. 4(f) and (b) slice b-b of Fig. 4(e) and Fig. 4(f). The profiles compare the pure phase and the mixed phase-amplitude approximations. The circled regions show the edge overlap effect. (c) The shape of the pure phase term with odd symmetry about a material edge in a sample, the phase-amplitude interaction term with even symmetry about the edge in the sample and their difference which is asymmetric about the edge. of the contrast from adjacent silver edges as silver intrusion in the nickel is considerably thin. Moreover, it can be observed that at the edges the mixed phase-amplitude interaction image profile has a higher maximum and a lower minimum, which is due to the asymmetry introduced by the interaction term in (3). This observation can be interpreted with the aid of Fig. 5(c). This figure shows the shape of a generic intensity variation introduced by the pure phase and the phase-amplitude interaction terms of (3) for a material edge in a given sample. As it can be seen, the pure phase consideration leads to an intensity redistribution that has odd symmetry about the mean intensity at the edge. The minimum appears in the material with higher neutron Fig. 6. Phantom 2 simulation results: (a) normal attenuation radiograph (b) phase contrast radiograph with pure phase approximation (c) phase contrast radiograph with phase-amplitude interaction. The object to detector distance R=50cm. scattering length density. The phase-amplitude interaction term leads to an intensity profile with even symmetry about the edge. Thus, the difference of these two terms (as given in (3)) offsets the intensity symmetry by adding intensity on one side and subtracting on the other side of the edge, which results in the profile shown in Fig. 5(c). The simulated radiographs of Phantom 2 are shown in Fig. 6 and Fig. 7. Figs. 6(a), 6(b) and 6(c) depict the attenuation only radiograph, pure phase approximation radiograph and mixed phase-amplitude approximation radiograph respectively. The gray scale of the radiographs is also shown along with the figures. Fig. 6(a) clearly indicates that there is small attenuation contrast present in the sample except at the sapphire specks. On the other hand, Figs. 6(b) and 6(c) indicate the presence of significant phase contrast in these organic and biological materials

MISHRA AND HAWARI: INVESTIGATING PHASE CONTRAST NEUTRON IMAGING FOR MIXED PHASE-AMPLITUDE OBJECTS 1635 nevertheless the combined phase-amplitude analysis will always perform better. Phantom 2 constitutes an example where there are some material edges present which have significant phase-amplitude interaction and other edges where it is not significant. In such samples the interaction term will decrease the contrast at those edges that already have high attenuation contrast and thus virtually expand the gray scale for the lesser contrast edges to become more prominent in the digital image. This effect can be observed in Fig. 7(c) where the sapphire edges are suppressed by the interaction term to make other edges more prominent. Thus, the phase-amplitude interaction term can increase the contrast at those edges where it is needed by providing rescaling of the gray scale in the digital images. IV. CONCLUSION Fig. 7. Phantom 2 simulation results: (a) phase image using pure phase approximation (b) phase-amplitude interaction image (c) the phase image with phase-amplitude interaction which is the difference of (a) and (b) as given by Eq. (3). The object to detector distance R=50cm. The images were normalized between zero and one. especially between ICRP soft tissue and urea, which was invisible in attenuation contrast. The presence of such a high phase contrast among these materials makes neutron phase contrast imaging suited for imaging biological and organic samples. Observing the phase images obtained for Phantom 2 in Fig. 7 it is clear that little phase-amplitude interaction is present in these biological materials except at the sapphire-tissue edge. At this edge the interaction is large because of the comparatively large difference in the attenuation coefficient and not so much due to the large difference in the neutron scattering length densities as was the case for the nickel-silver edge in Phantom 1. The effect of this interaction is also clearly visible in the final phase image in Fig. 7(c). Therefore, for these biological materials it can be concluded that for thin samples the pure phase approximation works fine, In the present work a phase contrast formulation for mixed phase-amplitude objects was presented. A phase-amplitude interaction term, which depends on the scalar product of the gradients of the real and imaginary part of the transmission functions of the edge forming materials, needs to be included to extend the pure phase object approximation to mixed phase-amplitude objects. Also, it was observed that the phase-amplitude interaction effect counters the edge contrast effect provided by the pure phase contribution and leads to an asymmetry in the intensity distribution about the edge. A technique for computational simulation of the phase contrast images has also been developed through which contrast mechanisms in various samples could be understood and expected phase contrast could be predicted, which can help in examining this technique for a particular application. Parameters were identified based on which the feasibility and performance of phase contrast imaging experiments can be studied using neutron transport calculations. The transverse coherence length along with the neutron flux at the image plane, given the source characteristics and other constraints, can be optimized through computational simulations. Through computational simulations, effects of the pure phase as well as phase-amplitude interaction terms on contrast enhancement in the images (for different mixed phase-amplitude materials) were investigated. It was observed that the interaction effect is less than the pure phase effect in terms of contrast enhancement; nevertheless it can be significant depending on the type of material edges present in the sample. The interaction effect can also lead to a higher overall contrast in samples having some material edges with significant phase-amplitude interaction and others where it is not significant. This is achieved by decreasing the contrast at those edges that already have high attenuation contrast thereby virtually expanding the gray scale in the digital image making other edges more prominent. ACKNOWLEDGMENT The authors would like to thank Mr. Andrew Cook for providing information on the NCSU PULSTAR reactor. The authors would also like to thank Ms. Emily Miller for her contribution to the design of phantoms.

1636 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 56, NO. 3, JUNE 2009 REFERENCES [1] B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, Phase radiography with neutrons, Nature (Brief Communications), vol. 408, pp. 158 159, Nov. 2000. [2] P. J. McMahon, B. E. Allman, K. A. Nugent, D. L. Jacobson, M. Arif, and S. A. Werner, Contrast mechanisms for neutron radiography, Appl. Phys. Lett., vol. 78, no. 7, pp. 1012 1013, Feb. 2001. [3] N. Kardjilov, E. Lehmann, E. Steichele, and P. Vontobel, Phase-contrast radiography with a polychromatic neutron beam, Nucl. Instrum. Methods Phys. Res. A, vol. A527, pp. 519 530, 2004. [4] F. Arfelli et al., Low-dose phase contrast X-ray medical imaging, Phys. Med. Biol., vol. 43, pp. 2845 2852, June 1998. [5] K. Kanaya and H. Kawakatsu, Experiments on the electron phase microscope, J. Appl. Phys., vol. 29, pp. 1046 1049, 1958. [6] C. J. Kotre and I. P. Birch, Phase contrast enhancement of X-ray mammography: A design study, Phys. Med. Biol., vol. 44, pp. 2853 2866, 1999. [7] T. E. Gureyev, A. Roberts, and K. A. Nugent, Partially coherent fields, the transport of intensity equation, and phase uniqueness, J. Opt. Soc. Amer. A, vol. 12, no. 9, pp. 1942 1946, Sept. 1995. [8] D. Paganin and K. A. Nugent, Noninterferometric phase imaging with partially coherent light, Phys. Rev. Lett., vol. 80, pp. 2586 2589, Mar. 1998. [9] A. V. Bronnikov, Theory of quantitative phase-contrast computed tomography, J. Opt. Soc. Amer. A, vol. 19, no. 3, Mar. 2002. [10] J. M. Cowley, Diffraction Physics. New York: North-Holland, 1981. [11] K. K. Mishra, A. I. Hawari, and V. H. Gillette, Design and perfromance of a thermal neutron imaging facility at the North Carolina State University PULSTAR reactor, IEEE Trans. Nucl. Sci., vol. 53, no. 6, pp. 3904 3911, Dec. 2006. [12] X-5 Monte Carlo Team, MCNP A General Monte Carlo N-Particle Transport Code, Version 5, Los Alamos National Laboratory, Los Alamos, NM, Tech. Rep. LA-UR-03-1987, 2003. [13] A. I. Hawari, I. I. Al-Qasir, and K. K. Mishra, Accurate simulation of thermal neutron filter effects in the design of research reactor beam applications, in PHYSOR-2006: Advances in Nuclear Analysis and Simulation, Vancouver, BC, Canada, 2006. [14] Mammography Accreditation Phantom Gammex 156 GAMMEX RMI, Middleton, WI, 2002. [15] S. J. Lim, Two-Dimensional Signal and Image Processing. Englewood Cliffs, NJ: Prentice-Hall, 1990, pp. 478 488. [16] V. F. Sears, Neutron scattering lengths and cross sections, Neutron News, vol. 3, no. 3, pp. 29 37, 1992 [Online]. Available: http://www. ncnr.nist.gov/resources/n-lengths