IOP PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
doi:10.1088/0957-0233/18/11/019
Meas. Sci. Technol. 18 (2007) 3391–3398
Feasibility of single-view coded source neutron transmission tomography K J Coakley1 and D S Hussey2 1 2
National Institute of Standards and Technology, Boulder, CO, USA National Institute of Standards and Technology, Gaithersburg, MD, USA
E-mail:
[email protected] and
[email protected]
Received 11 May 2007, in final form 8 July 2007 Published 27 September 2007 Online at stacks.iop.org/MST/18/3391 Abstract In a simulation experiment, we study the feasibility of single-view coded source neutron transmission tomography for imaging water density in fuel cells at the NIST neutron imaging facility. In standard two-dimensional transmission tomography, one reconstructs a spatially varying attenuation image based on many projections or views of an object. Here, we consider the limiting case where only one view is available. Rather than parallel beam sources, the projection data are produced by multiple pinhole sources. For a high-count case where the object is near the sources and the object magnification is approximately 200, and attenuation varies very smoothly in the object, we demonstrate that a penalized maximum likelihood method yields a reconstruction of attenuation that has a fractional root-mean-square prediction error of 5.8%. We determine the regularization parameter in the penalized likelihood method using a statistical learning method called two-fold cross-validation. As the object-to-source distance increases and object magnification in the detector plane decreases, the quality of the reconstruction deteriorates. At the NIST neutron imaging facility, the object magnification in a single-view coded source neutron imaging experiment would be only about 4. Due to this low magnification, even for the favorable case considered where attenuation varies very smoothly and we observe high-count projection data, we conclude that single-view coded source neutron transmission tomography is not a promising method for quantifying the spatial distribution of water in a fuel cell. (Contributions by staff of NIST, an agency of the US Government, are not subject to copyright.) Keywords: computed tomography, computer modeling and simulation, inverse problems, neutron scattering, probability theory, stochastic processes and statistics, proton exchange membrane fuel cells
1. Introduction In a proton exchange membrane fuel cell, effective water management is a key to improved fuel cell performance and durability. Water enters the fuel cell in the form of humidified gas, and is a by-product of the oxidation reaction in the cathode. Proper membrane hydration is required for proton conduction, and as the cathode gas diffusion layer fills with water, the fuel cell can become starved of oxygen, causing the oxygen stoichiometry ratio to fall below 1. Because thermal neutrons have a small scattering cross-section for 0957-0233/07/113391+08$30.00
© 2007 IOP Publishing Ltd
many materials common to fuel cell fabrication, but have a large scattering cross-section for water, neutron imaging has proved to be an invaluable tool for measuring the water content of a fuel cell. In standard two-dimensional transmission tomography, one reconstructs a spatially varying attenuation image based on many projection data sets of the object of interest. Typically, projection data sets are collected at many equally spaced rotation angles between 0◦ and 180◦ . In the standard implementation, for each angle, one measures the ratio of transmitted to incident radiation for many
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nonoverlapping parallel beams. Based on the central slice theorem, one can derive the filtered backprojection (FBP) algorithm for reconstructing an attenuation image [1–3]. In statistical approaches, one typically estimates unknown attenuation parameters based on the principle of maximum likelihood. Unlike FPB, statistical methods account for counting statistics variation and yield nonnegative attenuation parameter estimates. Based on the iterative expectation maximization (EM) algorithm [4], one can estimate the attenuation parameters [5–7]. Due to the large number of parameters to be estimated, the variability of the maximum likelihood estimate is high. To get a more regular estimate of the attenuation parameters, one can maximize a penalized loglikelihood function (or posterior probability density function in the Bayesian approach) [8–16]. Another way to impose regularity is to represent the image with a sparse set of basis functions such as wavelets [17–23]. In the general signal processing community, reconstruction methods that exploit sparse representations of data are of great interest [24, 25]. The standard FBP method is designed for cases where the projection data are sampled over the full 180◦ range. When a relatively small number of projections are sampled over the full 180◦ range, one might interpolate the projection data at the unobserved angles before implementation of FBP, as discussed in [26]. Since smoothing surely introduces systematic error, this method should be used with caution. A more serious problem occurs when projection data are observed over a very limited angular range [27, 28]. For cases where a very small number of projection data sets are available, low resolution reconstructions have been obtained by Bayesian methods [13, 15, 16], maximum entropy regularization methods [29] and wavelet methods [22]. Here, we focus on the limiting case of just one projection. The idea of reconstructing an object from just one projection is not new. In early single-view coded source tomography [1, 2, 30], objects were reconstructed from a single projection data set produced by multiple spherical sources. Assuming that the attenuating object was weak, and that the sources had a delta function autocorrelation, approximate reconstructions were derived using decoding methods involving computationally fast convolutions. In general, these early methods yielded complex out-of-focus reconstructions where reconstructed attenuation coefficients could be negative. In this work, we do not estimate attenuation using decoding methods such as those in single-view coded source imaging or coded aperture imaging [31–34]. Instead, we reconstruct attenuation images with a penalized log-likelihood method designed for overlapping transmission beams [35]. In [34], single-view coded aperture imaging of emitting sources was described as coded aperture laminography. In the language of [34], our single-view coded source transmission tomographic method could be described as a coded source laminography method. For a case where attenuation varies very smoothly and a high number of counts are collected in each of two projection data sets, we demonstrate that a penalized maximum likelihood method can yield a reconstruction with a fractional root-meansquare prediction error (FRMSE) of 5.8%, provided that the object is very close to the sources (about 1 cm) and the object magnification in the detector plane is about 200. For this 3392
smooth phantom, reconstruction accuracy quickly deteriorates as object magnification is reduced. Since the distance between the neutron sources and the object at the NIST neutron imaging facility yields object magnifications that are much less than 200, we conclude that single-view coded source tomography is not a promising method for the simple case where water density varies very smoothly within a fuel cell.
2. Statistical model The measured intensity of a neutron beam passing through a fuel cell is reduced by s-wave scattering [36]. According to Beer’s law, the probability that a neutron with a given energy will pass through an attenuating object without scattering is (1) p = exp − µ dl , l
where µ is the attenuation coefficient. Neutron radiography is an excellent technique to quantify the integrated water density in a fuel cell, because neutrons are readily scattered by water [37]. For the same reason, given that one collects projection data at many rotation angles that cover the full 180◦ range, neutron transmission tomographic imaging based on standard methods such as filtered back projection or likelihood based methods have the potential to quantify the spatial distribution of water density in a fuel cell [38, 39]. In this work, we focus on the feasibility of single-view coded source neutron transmission tomography. Because other materials scatter neutrons (though less strongly than water), one typically estimates a residual attenuation image µ = µw − µd , where µw and µd are the wet and dry attenuation images. Ideally, the water density in a fuel cell is proportional to this residual attenuation. In principle, scattered neutrons can be detected. However, when the detector and fuel cell are separated by large distances, scattering effects should not be significant because of solid angle effects. In practice, neutron beams have a spread of energies and equation (1) is not exact [40]. However, in this work we neglect beam hardening effects associated with energy dependent attenuation. In our simulation experiment, we assume that the dry state does not attenuate neutrons and assume perfect knowledge of the expected numbers of counts that would have been observed at each detector for a nonattenuating object. In our simulation experiments, we reconstruct an attenuation image corresponding to a two-dimensional crosssection of a three-dimensional object (figure 1). In the simulation, the trajectories (rays) of detected neutrons are constrained to lie in the plane containing the sources, the attenuating object and the detectors. This is an approximate model for an idealized experiment where neutrons are collimated by a mask with a very narrow transmitting slit. We assume that any neutron that undergoes s-wave scattering is scattered out of the plane of interest and not detected. We denote the locations of the mth pinhole source and ith detector as sm and di . For the case of no attenuation, we model the expected number of detected neutrons contributed by the mth source to the ith detector during an experiment of duration T as bim , where nˆ · (di − sm ) bim = λs T , (2) |di − sm |3
Feasibility of single-view coded source neutron transmission tomography
The log-likelihood of the observed data is M Nd log L(µ, Y ) = hi uim , Yi − log Yi !,
sources mask
i=1
(8)
m=1
where
object
hi (t, Yi ) = Yi log(t) − t.
(9)
We maximize the penalized log-likelihood , where = log L(µ, Y ) − βR(µ),
as a function of µ, where the adjustable regularization parameter β is positive and R(µ) is a roughness penalty function. Here, we consider a quadratic roughness penalty function (11) R(µ) = ckl (µk − µl )2 ,
detectors z x
Figure 1. Simulation experiment setup: transmitted neutrons through a rectangular attenuating object are detected. An absorbing mask is placed between the pinhole sources and the object. The area of the attenuation object is Lx × Lz where Lx = Lz = 2 mm. The distance between the line containing the sources and the line containing the detectors is Ld = 200 cm. The distance between the midpoint of the object and the lines containing the sources is o .
where nˆ is the detector normal and λs is the emission rate of neutrons by the source. Following [35], we denote the attenuation coefficient parameter for the ith pixel as µi and the attenuation coefficient parameters for all pixels in the region of interest as a matrix µ. Similarly, the observed projection data at the j th detector are Yj and the vector of all observed values is Y. We assume that the attenuation coefficient is constant within any of Np rectangular pixels that partition the attenuating object. Following [35], we define the line integral between the mth source and the ith detector through the attenuating object as lim =
Np
aijm µj
(3)
j =1
where aijm is the line-length of the path through the j th pixel and µj is the attenuation in the j th pixel. We model the observed data at the ith detector as a realization of a Poisson process with expected value λd,i : Yi ∼ P oi(λd,i ),
(4)
where λd,i =
M
(5)
uim ,
m=1
uim = bim exp(−lim ) + ri /M,
(6)
where M is the number of sources and ri is the expected background at the ith detector. In this work, we assume that ri and bim are known. The unknown parameters that we estimate are the Np attenuation parameters. The likelihood of the observed data is L(µ, Y ) =
Nd exp(−λd,i )λYd,ii i=1
Yi !
.
(10)
(7)
where ckl is 1 when the kth and lth pixels share a side boundary, √1 when the kth and lth pixels are diagonal neighbors, and 0 2 for all other cases. Since it is difficult to directly maximize the objective function , we maximize a parabolic surrogate for the objective function, as discussed in [35]. The surrogate is a parabolic function of the current and next parameter estimates. Each parameter is updated individually in an iterative fashion with a Newton method. If the update yields a negative value, the attenuation parameter is set to zero. We determine β by cross-validation [41]. Cross-validation is a procedure to determine adjustable parameters in estimation problems including image estimation problems [42–48]. We select β by a two-fold cross-validation method similar to methods described in [42–44]. For each source-to-object distance of interest, we simulate two statistically similar projection data sets. From one of these two projection data sets, we estimate attenuation images for each of many values of β on a grid. We select β by finding the approximate minimum of the crossvalidation function CV , where CV =
Nd 1 ( Yi − Yv i )2 /Yv i , Nd i=1
(12)
Yv i is the observed number of counts in the ith pixel of the ‘validation’ data, Yˆ i is the predicted number of counts computed by projecting the reconstruction obtained from the ‘estimation’ projection data and Nd = 1000 is the number of pixels in the detector. In other words, we select β so that the predicted projection data based on the forward projected reconstruction of the ‘estimation’ projection is most consistent with an independent ‘validation’ projection data set. Since we model the projection data as a multivariate Poisson random process, the CV function in equation (12) has the form of a chisquared goodness-of-fit statistic. In general, as β increases, the reconstruction becomes smoother. We switch the roles of the two projection data sets, repeat this procedure and average the two reconstructions. For each projection data set, we set β = 0 and perform a fixed number of iterations of the parabolic surrogate optimization code to get a β = 0 reconstruction. This reconstruction serves as the initial guess for reconstructions corresponding to nonzero values of β. Although there is a proof that the objective function (equation (10)) should monotonically increase as a function of the number of 3393
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14 sources
x (cm)
Figure 2. Pinhole sources are nonuniformly spaced on a line segment of length 2 cm. Locations of the sources are shown above.
iterations [35], after a large number of iterations, the objective function decreases. We halt the iterative algorithm when this occurs. We attribute this ‘nonmonotonicity’ to numerical precision problems [49].
3. Example To study the feasibility of single-view coded source neutron transmission tomography, we consider an idealized twodimensional reconstruction experiment (figure 1). We simulate high-count projection data for various source-to-object distances by sampling from a multivariate Poisson process where the expected number of neutron counts for each detector is given by equation (5). Based on these simulated projection data, we reconstruct a two-dimensional 32 by 32 pixel attenuation image within an object with physical dimensions being Lx × Lz , where Lx = Lz = 0.2 cm. Given that the center of the object is (x0 , z0 ), we estimate attenuation in the region defined by −0.5Lx x 0.5Lx and zo − 0.5Lz z zo + 0.5Lz . A mask is placed on the upper surface of the object. At this surface, the mask transmits neutrons when |x − x0 | 0.5Lx . Otherwise, the mask absorbs neutrons. The purpose of the mask is to limit the region that the coded source is encoding, in a manner analogous to coded aperture imaging with modified uniformly redundant arrays. On the line at z = zs , the sources lie between −δs x δs , where δs = 1 cm. This value of δs = 1 cm is based on an estimate of the closest approach to a divergent neutron field produced on a neutron pinhole mask currently in use at NIST Center for Neutron Research. The nonuniformly spaced pinhole locations (figure 2) are derived from a onedimensional uniformly redundant array (URA) generated by a quadratic residue scheme [34, 50]. The sequence is mapped into an interval −δs to δs by an affine transformation. The distance between the line containing the pinhole sources and the detectors is Ld = 200 cm. On the detector line, 1000 detectors are uniformly spaced. The detector spacing is chosen so that the 1000 detectors nearly cover the range over which neutrons can be detected. Thus, the detector spacing is d = Nd2−1 (M(δs + 0.5Lx ) − δs ), where Nd = 1000 and the object magnification in the line containing the detectors is 3394
M = Ld /o − 1, where o is the distance from the midpoint of the object to the source and Ld = 200 cm is the distance between the lines containing the sources and detectors. In all our simulation experiments, the true attenuation image has the same two-dimensional Gaussian shape. For all cases, we scale the attenuation image so that the mean transmission of a parallel beam is exp(−Lz µ) ¯ = 0.67, where µ ¯ is the mean attenuation in the image. In other words, we require that µ ¯ = 2 cm−1. In fuel cells, a 67% transmission factor is a typical value. The minimum and maximum value of the true attenuation parameters across all pixels are 0.696 cm−1 and 2.97 cm−1, respectively. In operating fuel cells, attenuation parameters will surely have a less smooth spatial variation and a broader range. In general, smoother images are easier to reconstruct than images that have more complex variation. Hence, if single-view coded source transmission tomography fails to provide an accurate reconstruction for this simple case, it is unlikely to provide accurate reconstructions for realistic cases of interest. In a real experiment for a fixed observation time, the expected value of the total number of events due to neutrons would rapidly decrease as the source-to-object distance increases. In our simulation experiments, the expected total number of neutron events contributed to projection data is 7.2 × 108 , 9.7 × 108 , 1.10 × 109 , 1.16 × 109 and 1.22 × 109 for source-to-object distances of 1 cm, 2 cm, 10 cm, 20 cm, and 30 cm, respectively. We allow the expected number of neutron events to increase as a function of source-to-object distance because we are interested primarily in the feasibility of single-view coded source tomography from a theoretical perspective. For the above cases, we show that the accuracy of the reconstruction deteriorates rapidly as source-to-object distance increases. In our estimation algorithm, we assume perfect knowledge of the expected number of counts contributed by background (ri in equation (6)) at each detector. The expected number of background counts per detector is 5 × 105 for all cases. The background parameter ri is treated as a known constant in our penalized likelihood method. As a caveat, in an actual experiment, there surely will be systematic uncertainties that are not accounted for in our simulation study, due to a variety of effects including beam hardening unknown backgrounds, uncertainties in the neutron intensity parameters bim (equation (2)) and possible detection of scattered neutrons. In our first simulation experiment the distance between the midpoint of the object and line containing the sources is o = 1 cm. In figure 3, we display the simulated projection data and CV as a function of β for one of the two permutations of the data for this case. For both permutations of the projection data for this case, the cross-validation statistic takes its approximate minimum value at β = 15. In general, for any pair of ‘estimation’ and ‘validation’ projection data sets, the optimal value β would depend on the true attenuation image, the geometry of the sources and detectors, and random and systematic errors in the observed projection data. i At the ith pixel, the fractional prediction error is µˆ iµ−µ , i where the estimated and true attenuation for the ith pixel are µ ˆ i and µi . The fractional root-mean-square prediction error (FRMSE) and root-mean-square prediction error (RMSE)
Feasibility of single-view coded source neutron transmission tomography
2.5
x 10
6
counts
2 1.5 1 0.5 0 0
200
400 600 detector index
800
1000
CV
1.14 1.13 1.12 1.11 1.1
5
10
15
20
β
25
30
35
40
Figure 3. Top: a simulated high-count projection data set for the case where the source-to-object distance is 1 cm. Bottom: CV as a function of β. Table 1. Root-mean-square prediction error (RMSE) and fractional-root-mean-square prediction error (FRMSE). The object magnification is M = Ld /o − 1, where o is the distance from the midpoint of the object to the source and Ld = 200 cm is the distance between the lines containing the sources and detectors. o (cm)
M
RMSE (cm−1)
FRMSE
1 2 10 20 30
199 99 19 9 5.67
0.101 0.099 0.171 0.289 0.407
0.058 0.060 0.103 0.196 0.299
computed from all Np = 322 pixels are
Np
1 µ ˆ i − µi 2 , FRMSE = Np i=1 µi and
Np
1 RMSE = (µ ˆ i − µi )2 . Np i=1
As the distance between the coded sources and the object increases, magnification decreases and reconstruction accuracy deteriorates (table 1, figures 4, 5). This makes sense because the rays that pass through the object from different pinholes have a tighter angular distribution and are closer to being parallel as magnification decreases. At the NIST neutron imaging facility, the distance between the plane containing a coded neutron source and the object would be about 40 cm, and the distance from the sources to the detector plane would be about 200 cm. Hence, the magnification would be approximately 4. Since this is a low magnification, we do not expect single-view coded source tomography to yield useful results for imaging water in fuel cells because high uncertainties in the reconstruction
are expected (based on the results of our simulation study). Moreover, in our simulation experiment, we assumed that a relatively large number of neutron events would be detected at low magnifications. In realistic experiments, the actual number of detected neutrons would rapidly decrease as the source-to-object distance increases. In our simulation, we allow the expected number of events due to transmitted neutrons to slightly increase as the source-to-object distance increases. Hence, in real experiments of fixed duration, we expect the relative accuracy of reconstructions obtained from low magnification data (compared to reconstructions from high magnification data) to be worse than what we observe in our simulation experiments. In this work, we consider a particular mask geometry. In other studies with different phantoms where we did not use a mask, reconstruction accuracy also deteriorated rapidly as object magnification decreased. For the specific case where the object magnification was M = 199, we simulated projection data for the case of no mask for the same phantom shown in figure 4. The associated reconstruction of this mask-free projection data was similar to the corresponding reconstruction shown in figure 4. How the particular choice of mask geometry affects results, and whether one needs a mask to obtain optimal reconstructions, are topics beyond the scope of this study.
4. Summary We studied the feasibility of single-view coded source neutron transmission tomography for imaging water in fuel cells in a simulation experiment. We restricted attenuation to a simplified two-dimensional case where attenuation varied very smoothly according to a Gaussian function of position. Based on a Poisson likelihood model for projection data, we simulated single-view projection data sets corresponding to a variety of source-to-object distances that yielded object magnifications in the detector plane ranging from 5.7 to 199. For a high-count case where object magnification was 199, a penalized maximum likelihood method yielded a reconstruction that had a fractional root-mean-square prediction error of 5.8%. We selected the value of the regularization parameter in the penalized maximum likelihood method by two-fold cross-validation. However, as the object-to-source distance increased and object magnification decreased, reconstruction accuracy deteriorated (figures 4, 5, table 1) for high-count projection data sets. For object magnifications in the neighborhood of those achievable at the NIST neutron imaging facility (about 4), the reconstruction quality was very low and the fractional rootmean-square prediction error was about 30%. In general, one expects reconstruction quality to deteriorate as magnification decreases because neutron rays from the sources become more parallel to one another as object magnification decreases. In our simulation, we allowed the expected number of events due to transmitted neutrons to slightly increase as the sourceto-object distance increased. For real experiments of fixed duration, we expect the relative accuracy of reconstructions obtained from low magnification (compared to reconstructions from high magnification data) to be worse than what we observed in our simulation experiments because the expected number of neutron events contributed to the projection 3395
K J Coakley and D S Hussey true
z (cm)
z (cm)
magnification = 199
magnification = 19
magnification = 9
z (cm)
x (cm)
z (cm)
x (cm)
x (cm)
x (cm)
0.20 0.10 0.00
z (cm)
FRMSE
0.30
magnification = 5.7
5 x (cm)
10
20
50
100
200
magnification
Figure 4. Contour plots of true and estimated attenuation parameters (in units of cm−1) for different source-to-object distances, and FRMS as a function of object magnification.
data would rapidly decrease as the source-to-object distance increased. We conclude that single-view coded source neutron transmission tomography is not promising for imaging water density in fuel cells even for the simple case where the water density variation in the fuel cell is very smooth, such as that considered in this study. In general, for cases where attenuation varies smoothly, such as the case studied here, two-dimensional single-view coded source neutron transmission tomography may yield useful reconstructions provided that the sources and object are very close and object magnification is approximately 200 and sufficiently high signal-to-noise projection data are collected. As a caveat, for cases where the variation of attenuation is more complex than the phantom studied here, but object magnification is around 200 and high-count projection data are collected, we do not expect reconstruction quality to be as high as that for the simple phantom studied here. In other words, single-view coded source transmission tomography is likely to yield low resolution reconstruction when true attenuation is more complex than the simple case studied here. For tomographic imaging of water in fuel cells, rather than single-view coded source transmission tomography, we 3396
recommend more traditional approaches such as filtered back projection or penalized likelihood where parallel beam projection data are collected at many rotation angles that cover the full 180◦ range. The methods presented in this work may be relevant to applications where only a few views are collected such as plasma diagnostics [29] and where (smooth) reconstructions are typically obtained. In such cases, one might use multiple spherical sources rather than parallel beams and reconstruct images using an extended version of the single-view coded source tomographic method developed here. For the fuel cell problem, such a fewview coded source reconstruction method would probably yield better reconstructions than the single-view coded source method studied here. Finally, if projection data were generated by parallel beams in a two-view experiment, one could reconstruct images using a sparse angle implementation of filtered back projection or a penalized likelihood method such as the one developed here. We speculate that a penalized likelihood method would yield superior results since filtered back projection can yield negative estimates of the attenuation coefficient parameters and filtered back projection does not account for Poisson counting statistics variation in the data. How well a two-view filtered back
Feasibility of single-view coded source neutron transmission tomography
magnification = 19
0
50
100
0 20 40 60 80 100
150
magnification = 199
magnification = 9
magnification = 5.7
60 40 0
0
20
50 100 150 200
fractional prediction error
80
fractional prediction error
fractional prediction error
0.0 0.5 1.0 fractional prediction error
Figure 5. Histograms of fractional prediction error.
projection reconstruction of parallel beam projection data would compare to a penalized likelihood reconstruction of single-view projection data produced by coded sources is an open question.
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