·Table of Contents
·Conservation and Restoration in Art and Architecture
Compton Scattering Elemental Imaging of a Deep Layer Performed with the Principal Component AnalysisA.Tartari
Department of Physics, University of Ferrara, I44100 Ferrara, Italy
ENEA, Applied Physics Division, Via Don Fiammelli,2, I-40129 Bologna, Italy
E. Lodi, C. Bonifazzi
Department of Biomedical Science, University of Ferrara, I44100 Ferrara, Italy
|Fig 1: Layout of the axial symmetric configuration. D: detector; S: shielded source; L: tested layer; Dq: annular collimator.||Fig 2: Test on sample of increasing thickness.|
The axial symmetric assembly consists of a large-area NaI(Tl) scintillation detector (40 mm diameter).A point source (2 mm diameter) of 241Am emitting 59.54 keV photons is posed in a cylindrical shielding head at the center of the detector. In this configuration, the sensitive volume is approximated by a cylinder - 3 mm diameter - entering the sample for all its depth. The annular shape of the detector collimation, is a consequence of the source head position and a further circular shield. A thick layer of Plexiglas (6x6x1cm3), with a circular central hole (6 mm diameter), was used to simulate a sample with homogeneous composition and a density variation in a limited volume. By this phantom the ECOSP efficiency in finding the presence of a void at different depth was tested. In order to evaluate the sensitivity of the experiment to density variation, some experimental situations have been considered.
The background radiation coming from 25 voxel sited linearly 1 mm one from another has been measured from layer 1 in Fig.2. Then, two further layers of Plexiglas (each of them 1 cm thick) were added, and the scanning repeated independently for each of the new sample configuration. An analogous experiment was then repeated placing the layers 2 and 3 before the layer 1, and, finally, with the layer 1 in central position. Another series of experiments was performed on a phantom simulating inhomogeneous materials, where the presence of elements of different atomic number was provided by introducing in the central hole a small cylinder of Aluminum or Brass. In the following discussion, the scattered photons coming from scanning done on the Plexiglas, eventually in the presence of the central hole, will be referred as the background, while the scanning results in the presence of Aluminum, and Brass, as signal.
|Y = UX,||(1)|
where X is the matrix of backscattered photons associated to each tested position; Y is the matrix of principal components, and U is the N x N unitary matrix deduced from the variance-covariance matrix of X, CX = XTX [ The variables , xj, j = 1, K, N are mean centered] , the rows of matrix U being the N normalized eigenvectors of Cx. The covariance matrix of the principal components is then Cy = UCxUT, and the variances of the PCs are the eigenvalues of Cx ordered such that l1 > l2 > ... > lN. Finally, since U is a unitary transformation, the total data variance is preserved, i.e.,
This redistribution of variance is important in information recovery; indeed, since the PCs are uncorrelated, and each yj has variance less than the previous components, few PCs would contain a larger percentage of total variance. In other words, it is expected that, for a layer with homogeneous composition, with o without density variation, a very large fraction of the total data variance be described by a single PCs. On the contrary, the presence of insertions with different atomic numbers would induce two kinds of data variation: i) the backscattering photons coming from region with uniform composition, and ii) the phtons coming from the reduced number of SVs containing materials of different elemental composition. This complex structure of the matrix, Cx, would results (Eqn.2) in the presence of two relevants terms, two PCs, discribing the larger fraction of total variance.
|Fig 3: Density profile measurements.|
A. Estimate of Thickness effects.
As previously outlined, the backscattering configuration gives raise to cylindrical SVs entering the sample, whose dimensions along the central axis are limited by the depth of the sample itself or, more properly, by the mean-free-path of the source-emitted photons. Therefore, in a sample of high thickness, the increased number of backscattered photons from deep layers may, in principle, attenuate the difference in the total backscattering arising from regions belonging to layers near the surface. At the same time, the presence of flaws and insertions in a deep layer could be masked by photons scattered in superficial layers.
The experimental configuration adopted for the first series of trials is shown in Fig.2. The total number of backscattered photons detected, in order of increasing thickness, are shown in Fig.3A to 3C. The continuous line is the background density profile calculated from Eqn.1 where only the first PC has been taken into account (corresponding to about the 75 percent of the total variance). The PCA was carried out on all the seven scanning. At increasing depth, the total number of detected photons increases in a not-linear way, and - as clearly shown by the measured data - at 3 cm thickness the presence of the central hole is masked by the statistical fluctuations (see Fig.3C). In Tab.1 a comparison between the seven experiments is shown, where the first column indicates the position of layers 2 and 3 with respect to the inspected layer. All the data comes from the density profile reconstructed by the first PC.
In all the experiment, the baseline values show that the number of backscattered photons increase when the sample thick augments; moreover in the experiment identified as Before, appears the greater backscattering. The former result is a consequence of the increased SV dimension; the later may be explained, at all the sample thickness, by the relative position of the inspected layer with respect to the profile of the detector solid angle. The difference, D , between the value at the baseline and at the peak was used to test the efficiency of the ECOSP to estimate the density variation at increasing thickness. It appears evident that the value of D decreases at increasing thickness, but for a radiation path-length up to the radiation mean-free-path, a correct evaluation of the density profile is still possible. Finally, as shown in Fig.3, the dimension of the central hole, evaluated in terms of width at half maximum, is correctly evaluated at each thickness an for all the experimental configuration.
|Table 1: Comparison between background estimate in sample of different thickness.|
Further insight on the sensitivity of the ECOSP may be obtained by inspecting the data in Fig.3. It is worth noticing that, at all of sample thickness, the background profile exhibits the same behaviour in every measurements all the seven experiment. This finding is noticeable considering that the shape delineated by the first PC has the higher SNR for the data under testing ; i.e. the common fluctuation pattern in all the profiles may really represent local density variations, made evident by the PCA.
B. Elemental Composition Evaluation.
|Fig 4: Elemental composition evaluation|
Fig 5: PCA description elemental composition of the layer (see text).|
In order to better understand the results, some detail on the physical involved processes are to be introduced. Since a thick sample is concerned, in correspondence of each voxel the backscattered photons are produced by twofold action: that due to the electronic density, re, and that relevant to the attenuation factor, m . Each of them acts, respectively, increasing and decreasing the total backscattering contribution. The backscattering photons detected during the 'Plexiglas' scanning show a constant value, within the experimental errors, since both re and m , remain unchanged. Correspondingly, the drop of re, makes clear the reduction in total backscattering observed in the 'Void' scanning. Finally, the scanning profile obtained with the Aluminium and Brass insertion may be immediately understood if we consider that, for the former the increase of re is slightly prevalent with respect to the increase of the attenuation, and viceversa for the later measurement where the increase of the m coefficient strongly predominates onto the increase of re. As for the possible correlation, Cx , between data coming from adjacent voxel, the action of the PCA in discriminating regions with different elemental composition is evident. In the scanning performed on regions with homogeneous composition (Plexiglas or Void) the occasional drop of electron density, re, does not modify the correlation between the total backscattering coming from the SVs, even if density variations are present. This means that the background scanning may be described by means of a single PC, while the drop in total backscattering is taken in account (Eqn.1) by the changes in the elements of the 'loading' vector, u1 . Otherwise, when the variation in the number of backscattered photons is due to the presence of heterogeneous elements, the concurrent variation of both re and m produces correlations on some entries of Cx and, as consequence, the variation of the data set must be described in the right-hand-side of Eqn.2 by means at least two relevant eigenvalues. The result of the PCA applied to the four scanning are shown in Fig.5. In panel A the pixel density image obtained by the raw data is shown, where in the scanning 2 and 4 the backscattered photons obtained in the presence of the Aluminium and Brass insertion, respectively, are show. In panel B, the density image obtained by the reconstruction with the first PC alone is shown and, as evident, the only evident variations belong to the 'Aluminium' and 'Brass' scanning. Finally, the backscattering photons corresponding to the 'Plexiglas' and 'Void' scanning, respectively, are shown in panel C, as obtained by reversing the Eqn.1 with the second PC alone. In all the density image the variations with respect the background level are shown. It is important to highlight the ability of the PCA, in describing the correlation between adjacent SVs. Indeed, while in the Brass scanning the high value of correlation, cxij , between the voxel 3 to 7 might be deduced directly from Fig.4, the same is not true for the 'Aluminium' scanning where the small variation in total backscattering and the presence of noise make difficult any inference about the dimension of the insert. Conversely, from panel B and C the similarity between the dimension of the flaw in 'Void', and the dimension of the inserts of Brass and Aluminium may be easily deduce.
In the linear model describing the heterogeneous composition of the tested sample, Eqn.1, the first PC ( the l 1 value) gives reason of about the 80% of the total variation, and the second PC, the l 2 value, of about the 17%, thus providing an explanation of the strong correlation between the neighbouring voxels, induced by the presence of Brass and Aluminium.
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