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Abstract
In spectroscopy, the compositional analysis of the spectrum is important, such as extracting information about the species of spectral objects contributing to spectral data from an emission spectrum of photon energy. A quantitative spectral component analysis method based on Maximum Likelihood Estimation using Expectation Maximization (MLEM) is developed, which could quantitatively decompose out the components of the measured spectrum of low counts and surpass conventional techniques which belong to classification or regression. Abundant experimental and simulated spectra data on gamma-ray spectrum of radionuclides are presented to demonstrate and evaluate this method, while the ingredient radionuclides in the mixed spectrum are identified accurately with high precision. It will be a powerful and alternative method recommended for the circumstances needing fast and quantitative spectral analysis, including radionuclide identification (gamma-ray spectra), biomass or mineral composition (near-infrared spectra), laser-induced breakdown spectra and other spectroscopy scenarios.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spectroscopy is an indispensable observing tool to reveal further information of illuminating objects involving multiple disciplines with different photon energy bands or wavelength ranges. The visible and near-infrared reflectance spectroscopy (NIRS) of asteroid and meteorite are used to determine the mineralogy and petrology of the object in astrophysics [1], and NIRS has also been used extensively in lignocellulose analysis of forages and predicting the chemical composition of biomass feedstocks [2], etc [3]. Laser-induced breakdown spectroscopy (LIBS) has been successfully applied to gas, liquid, and solid samples for performing multielement real-time analysis [4]. Gamma spectroscopic analysis is a typical representative in spectroscopy, and a set of radionuclide spectra are employed in this current work as a demonstration of the proposed method. The gamma-ray spectrum is generated by radioactive sources or materials which emit gamma-rays of various energies and intensities ranging from tens of keV to MeV [5]. A nuclide’s characteristic gamma-ray emission spectrum denotes the isotope’s “fingerprint” in the realm of gamma radiations and can be detected by spectroscopic portal monitors mainly based on scintillator or semiconductor detectors. In the field of nuclear spectroscopy, the relevant study of gamma-ray spectrum component analysis is “Radionuclide identification” and it is of great interest in the following vital scenarios: Scanning of cargo for radiological and nuclear material in border control [6], Compton imaging for gamma camera [7], trace analysis of radionuclides in environmental science [8,9], food radiation security in customs [10], etc.
Strategies of Gamma spectroscopic identification of radionuclides rely upon methodologies that recognize signature patterns in a measured spectrum, and the main inference goal is to identify all isotopes present in the source, and also quantify the isotopes. Challenges in spectroscopic analysis have led to development of various signature extracting techniques for the aim of radionuclide identification [11]. Common radionuclide detection and identification algorithms were based primarily on peak search algorithms in the past but mostly missed the information of other parts of spectrum [12,13]. More recently, full spectrum analysis techniques are more accurate in spectroscopic analysis, such as template matching including multiple linear regression [14], nonlinear least-square fitting [15], multiobjective optimization [16] and Artificial Neural Networks [17], and multivariate analysis such as Principal Component Analysis [18]. Previous work using MLEM on gamma-ray spectroscopy is to achieve the deconvolution of gamma-ray spectra in order to filter the detector response of gamma-ray energy to obtain an original energy-loss spectrum of specific gamma-ray sources, and MLEM method was shown to be superior [19].
However, a practical measured spectrum often exhibits a high degree of complexity because it consists of a combination of counts from more than one radionuclide. Previous practical applications of radionuclide identification used to do a qualitative work since that the focus on whether the target nuclides are detected or not [20], and the quantification of components is ignored. Due to the Poisson statistics of the spectrum [11], it is a loose way to affirm the existence of certain nuclides and more rigorous to estimate the contributions of all potential nuclides in probability. In order to achieve quantitative analysis conventional approaches used to identify the radionuclide existed in the sample spectrum and then determine the amount of each component by extracting the peak counting which requires a large number of spectrum count. Recent quantitative attempts to analyze the spectrum is seen in [21] by adopting a variety of algorithms and computational tools derived from different scientific and engineering fields, and the contribution of detected components of relevant isotopes is determined through constituent gamma-ray peaks in the measured spectrum with large counts up to 18571 in the demonstrating case.
In order to achieve the goal of quantitative analysis of the spectrum, a novel quantitative spectral component analysis method inspired from MLEM is developed here. This paper details the development of a quantitative spectral component analysis method based on Maximum Likelihood Estimation using Expectation Maximization (MLEM) for extracting anomalous gamma-ray signature components from a measured or test emission spectrum of radionuclides. Abundant experimental and simulated results imply that the proposed method quantitatively identifies the components of a testing spectrum with high precision even it is count-starved, and the work could be rated on technical merit, originality and innovation benefitting from its quantitative analysis ability.
2. Theory
2.1 Inspiration from emission tomography (ET)
MLEM is a successful methodology applied to the emission tomography such as SPECT or PET [22], and its strict mathematical principle and logical structure makes itself accurate, qualitative and credible in complex image reconstruction [23]. The inspiration of the present methodology roots from the physical abstract and procedure of the MLEM’s application in nuclear medicine image reconstruction. Since the spectrum is a unique characteristic of each radionuclide, the specific problem is that how to identify the components of measured low-count spectrum composed of one or more radionuclides which is included in the normalized spectra of 25 different signatures. In ET, the reconstruction target is the position of the gamma-emitting source and the basic premise is that each position has its own detector response, in other words, detector matrix has a detecting probability density on each pixel. In fact, each position has a special projection and it is unique. In gamma-ray spectral analysis, the signature of each type of radionuclide is its spectrum detected by one detector, and the spectrum also can be seen as a distinct projection of a specific radionuclide. In spectral analysis, the reconstruction target is the species of radionuclides relative to the position in ET image reconstruction. And it is a consistent one-to-one match between each nuclide and each distinct response so that it is possible to infer a maximum likelihood estimator of existing nuclides according to the back-projection calculation of the measured detector response, i.e. spectrum. Intuitive comparison of detector response between ET and spectral component analysis is presented in Fig. 1.
Fig. 1 The origin of idea of the proposed method. The sun-like spot is the gamma-ray emitting source, and in ET imaging the position of source is of interest while the species of source is the target we are searching in spectral analysis.
2.2 Algorithm process
The algorithm depicted in Fig. 2 assumes that all relevant isotopes of interest, including the signature of background are in the spectral library so-called “template library” presented here. The preselected template spectra compose of background (bg), 133Ba, 241Am, 57Co, 22Na, 137Cs, 60Co, 67Ga, 192Ir, 152Eu, 131I, 40K, 75Se, 18F, 111In, 138La, 51Cr, 103Pd, 133Xe, 226Ra, 237Np, 237Pu, 232Th, 233U and 235U. These radioisotopes are typical standard gamma-ray sources widely applied in nuclear medicine, industry and environmental science. The present method utilizes the template library, but it is not the conventional method that uses template matching based on the linear comparison between the sample spectrum and a set of combinations of templates to find out whether one template or a composition of few template spectra fit the sample data significantly well. The present methodology focuses on inferring the constituents of a measured spectrum from a set of predefined signature spectra declared as a template library and there is no procedure for a linear combination of templates. The method is proceeded by running the following steps as a demonstration of Fig. 2:
- 1) Prepare a set of preselected template spectra. Specific process of generating template library is depicted in Fig. 3.
- 2) Normalize the template spectra to the relative probability density spectra as “template library” and won’t be changed in future for radionuclide identification.
- 3) Normalize the measured spectrum to a relative probability density spectrum and the “measured spectrum” in the block diagram is the probability spectrum actually.
- 4) Define an “initial list” contains 25 elements denoting the total species of preselected template spectra and initial value of each element is 1/25.
- 5) Execute the iteration loop in the form of formula as defined in Eq. (4), and P is the list, Y represents the measured spectrum, T is the template library.
- 6) The iteration loop is stopped within finite number of iterations depending on the convergence of the outcome of P.
Fig. 3 Block diagram of generation process of template library of high count and test spectra of low count
As a special note here, the normalization of the specific spectrum is described in Eq. (12). All the variables needed in the analysis calculation process is given as
where S = 25 denoting the total species of the template library is 25, and C = 2048, denoting the total number of channels of a spectrum is 2048. The specific statement is that P is the list of 25 species of spectra with pi meaning the proportion of the i-th radionuclide of total counts of the measured spectrum, and Y is the measured spectrum with yi meaning the normalized probability density of the i-th channel of spectrum, and T is the template library matrix with Tij meaning the normalized probability density of the j-th channel of the i-th radionuclide. The formula of the iterative scheme based on the EM algorithm for renewing the result of each loop step iswhere the algorithm is started with an initial list P satisfying pi = 1/S, i = 1,2,3···S, and then in each iteration, if Pold denotes the current estimate of P, Pnew is the new estimate. The specific calculation process is as followswhere the calculation of Y’ is the convolution procedure so-called forward projection and Y’ is the computed spectrum statistically. And the comparison procedure of Y and Y’ is given byand the back projection procedure isFinally, the old estimate of P is modified with A and the result of each iteration isNo extra parameters but only a normalized measured spectrum, an initial list and a preset template library are needed to proceed the spectral component analysis.2.3 Radionuclide Quantification Performance
To be specific, the output outcome (P) of the spectral analysis procedure is a final list containing 25 elements with each value of element quantifying the radionuclide’s contribution to the measured spectrum. The total sum of 25 elements is 1 meaning that the absolute contribution of 25 signature spectra is 1, and the actual meaning of the value of each element is the proportion of the corresponding radionuclide in the total count of the measured spectrum. Abundant demonstrating cases are shown in Figs. 3-12. In order to quantitatively assess the ability of the proposed method to extract the information of spectral components, the quantification precision (Fi) of the test specific radionuclide is shown in Eq. (9) as
where pi is the proportion value of the i-th relative radionuclide in the final list, and pi’ is the “ground truth” proportion value respectively. Higher quantification precision value means that the results of identification match better to the true spectral composition. High quantification precision implies that recognizing what radioisotopes are existed in a detected source or radioactive material appears to be a quantification process in component analysis of spectrum.Fig. 5 Template library of 25 characteristic spectra are in a thumbnail and the spectrum of 60Co is indicated while the ordinates of them are the same.
Fig. 6 Set 1: An original measured 60Co spectrum with 100 counts (up); The quantitative analysis result of 100 MLEM iterative times (down).
Fig. 7 Quantification precision results for single radionuclide spectrum of 60Co with 100 counts or 1000 counts under various iterative times. Quantification precision is 86.55% for 60Co-100 and 93.70% for 60Co-1000 under 100 MLEM iterative times.
Fig. 8 Set 2: (a) An original mixed spectrum of 57Co with 1000 counts and 137Cs with 1000 counts, and the identification results is shown in upper right corner under 100 MLEM iterative times. Quantification precision is 96.50% for 57Co and 96.18% for 137Cs correspondingly. (b) Quantification precision results of Set 2 under various iterative times.
Fig. 9 Set 3: (a) An original mixed spectrum of 241Am with 600 counts, 22Na with 400 counts and 60Co with 200 counts, and the identification results is shown in upper right corner under 600 MLEM iterative times. Quantification precision is 96.88% for 241Am, 96.70% for 22Na and 72.24% for 60Co correspondingly. (b) Quantification precision results of Set 3 under various iterative times.
Fig. 10 Set 4: (a) An original mixed spectrum of 57Co with 1000 counts, 133Ba with 1000 counts,22Na with 1000 counts, 137Cs with 1000 counts and 60Co with 1000 counts, and the identification results is shown in upper right corner under 1000 MLEM iterative times. Quantification precision is 98.21% for 137Cs, 98.20% for 241Am, 95.97% for 22Na, 92.56% for 57Co, 91.33% for 60Co and 84.56% for 133Ba correspondingly. (b) Quantification precision results of Set 4 under various iterative times.
Fig. 11 Quantification precision results of 25 single spectrum of 25 characteristic signatures with counts from 50 to 5000 under 100 iterations.
3. Experimental setup, simulation configuration and data preparation
Each spectrum applied in the present work has 2048 channels on X-axis with each channel linearly corresponding to the deposited energy value of gamma-ray and the value of Y-axis is the counts received by the detector at a certain channel. After the normalization of the spectrum, the value of a given channel implies the probability that the measured radionuclide emits γ-ray detected with a certain energy. One practical challenge in constructing such a comprehensive spectral library is that many nuclides of interest are hard to access and corresponding measurements could only be performed in special facilities or labs which would cost a lot. And so the reference spectra (template library) is generated by experimental measurements and simulating with computer tools. The MonteCarlo (MC) based simulation toolkit GEANT4 [24] was chosen as the simulation tool according to the recent investigations. Data in the entire analysis were produced by experiments and GEANT4 simulation including template library and test spectra and was detailed in schematic subdivision of Fig. 3 [25]. The first step is to measure a set of spectra of the existing radioactive source (133Ba, 241Am, 57Co, 22Na, 137Cs and 60Co) in the laboratory and their measurements in-field are influenced by a set of scene conditions as the scenario and operational performance of the device. The spectra are measured by a scintillator detector and the crystal used is a 6 millimeter cube of cerium-doped Gd3Al2Ga3O12 (Ce:GAGG) which is a relatively new single crystal scintillator with several properties that make it interesting for applications such as gamma spectroscopy [26]. The GAGG scintillator is coupled to one Hamamatsu H1949-50 photomultiplier (PMT) shown in Fig. 4 (up) and PMT is supplied with a voltage of 1000V. The measurement results of the relative available radionuclide source are summarized in Table 1. In order to obtain a stable spectrum the large counts are needed to reduce Poisson statistical error, and the background of each radionuclide’s spectrum was subtracted according to the specific measuring time shown in Table 1. The spectra of these 6 radionuclides and spectrum of the background are put in the “Reference Spectra” shown in Fig. 3. After collecting these experimental spectra including the signature of background, relative detector response of the current scenario and device will be quantified and calculated depending on these spectral characteristic such as energy and energy resolution calibration in Eqs. (10) and (11) as
Table 1. Species of Radioactive Sources in the Laboratory and Measurements for Standard Spectraa
Since the output from the GEANT4 simulation does not reflect the Gaussian statistics of the deposition of the photo’s energy response that is caused by the statistical fluctuation of the photo-electrons in the PMT and electronic noise in the measuring equipment, the purpose of energy and energy resolution calibration for the current detector response is to simulate the energy spectrum in GEANT4 instead of the actual experimental situation. Figure 4 (down) shows that the geometry model of GEANT4 is constructed with reference to Fig. 4 (up) and the physical process occurred in the simulation are achieved by MC. When some energy E is deposited into the scintillator detector, a corresponding channel of the spectrum is recorded while the relationship between energy and channel is Eq. (10). And the resolution calibration in Eq. (11) is to broaden the MC calculated energy deposition spectrum to the realistic spectrum by executing a convolution procedure which uses the finite resolution of the current detector [27]. After the other spectra of radionuclides are generated with no background by GEANT4, the reference spectra is completed and will be normalized to a template library as shown in Fig. 5 including 25 different spectra descripted in the method demonstration in detail. The background-stripped spectrum of 60Co is depicted in an entire frame. To be specified, the channel number of X-axis is 2048 and the energy calibration makes the detected energy and the channel site one-to-one. The label of Y-axis comes to probability density after the normalization of the spectrum and the probability density value of each channel is calculated by Eq. (12)
where pd(i) is the probability density value of the i-th channel, C(i) is the count value of the original spectrum before normalization. The test spectra consists of experimental low-count spectra of various measuring scenarios and Poisson statistics extracted from different radionuclides obeying respective probability density distribution. The prerequisites for the test spectra is that all components in the spectrum belong to the radionuclide or background consisted in the preselected template library. A detailed description of the test spectra and their identification results will be discussed in the section 4.4. Radionuclide identification results and methodological evaluation
The spectral component analysis based on MLEM is applied to the decomposition of a variety of representative test spectra generated through experimental measurements and GEANT4 simulation and individual evaluations are given by analyzing the results. The actual experimental spectra to be tested are obtained within measuring time as depicted in Table 2. There are 4 specified scenarios as typical cases to be analyzed: Set 1: An original measured 60Co spectrum with only 100 counts; Set 2: A spectrum combines 1000 counts from 60Co and 1000 counts from 137Cs; Set 3: A spectrum combines 600 counts from 241Am, 400 counts from 22Na and 200 counts from 60Co; Set 4: A spectrum combines 1000 counts from 241Am, 1000 counts from 57Co, 1000 counts from 133Ba, 1000 counts from 22Na, 1000 counts from 137Cs and 1000 counts from 60Co. These 4 sets of data are all produced by experimental spectra.
Table 2. Measuring Time (s) of the Experimental Scenariosa
In scenario 1, an original measured 60Co spectrum with only 100 counts in 0.2s obtained by multichannel spectrometer and there is no extra subsequent processing, i.e. it is an initial spectrum with no background subtraction. The result of spectral component analysis of Fig. 6 (down) is displayed to illuminate the final list of 100 MLEM iterative times in the form of a histogram, and it reveals that 60Co accounts for 86.55% of the test spectrum and corresponding radionuclide identification (RI) quantification precision is also 86.55%. In expectation, the result is not as same as the “ground truth” constituents of the test spectrum perfectly because of the Poisson statistics error at low counts. Analytically, the specified spectrum likes Fig. 6 (down) is almost impossible to be identified reaching a same level in other radionuclide identification works. The present method not only correctly identifies the 60Co signature but also achieves a high quantification precision. In scenarios from 2 to 4, the test spectra are multiple combinations of different radionuclides with specific proportions and corresponding analytical results are demonstrated intuitively. In these scenarios, all actual components in the test spectra are accurately identified and it quantifies the proportion of spectral components with high precision results shown in corresponding Figs. 8, 9 and 10.
Because the present spectral component analysis strategy stems from an innovative methodology inspired by ET and there is no relevant inheritance from previous work. According to the characteristics of this method, what needs to be specially concerned is the iterative scheme based on the Expectation Maximization (EM) algorithm [28] adopted in the methodological iterative realization process and relevant evaluations are very concerned with the iterative properties shown in Fig. 7. Figure 7 illustrates that for a spectrum containing only one radionuclide signature, as the number of iterations increases, the results of the identification tend to be stable and convergent in both scenarios of 100 counts and 1000 counts, and the 1000-counts scenario reaches a higher quantification precision actually. All the analysis charts show that the identification results will reach stability and convergence under a limited number of iterations and a definite number of iterations is chosen when the resulting performance increases very slowly, i.e., the relevant number of iterations in above scenarios is large enough to make the result infinitely close to the final convergence. For the test spectrum containing multiple nuclides with different proportions, the results identify their true proportion without distortion which confirms the reliability and robustness of the proposed methodology. The powerful performance of the present method should benefit from lots of merits belonging to the Maximum Likelihood estimator such as accuracy, stability, convergence and consistency [28,29,30].
Rich experimental data of each single radionuclide is shown in Table 2 with the corresponding quantification precision results shown in Table 3, and Fig. 11 shows the results of all single radionuclide Poisson statistics with counts from 50 to 5000. All these results demonstrate that the behavior of the proposed spectral component analysis method is excellent especially in the case where the spectrum sampling count is very low. As a more detailed discussion, the worst performer of all 25 radionuclides including background is 22Na, the specific reason is that the shape of spectrum of 22Na is similar to which of 18F in a wide range and the characteristic peak of 511 keV generates the main part of their spectra. The more similar two signatures are, the harder it is tantamount to distinguish in mathematical probability. The only process that requires manual intervention during algorithm execution is the choice of the number of iterations depending on the convergence of output result, so that alternate methods to determine the appropriate iterations is worth studying based on the gradient since it is available from the quantification precision curve. In Fig. 9(b), the trend of 60Co is different from others. However, the curve of precision goes to flat and stable when the iterative times increases. Actually, final identification result is determined when the precision (estimated pi) is not fluctuating, i.e. a large number of iterative times is more reliable and the computing time under 1000 times could still be neglected. Furthermore, the identification results of each spectrum is not exactly correct since that there is false detection of other nonexistent radionuclides shown in Figs. 8-10, and theory of limits of detection in spectroscopy may be considered to eliminate the incorrect detected ones of low proportion in future [31].
Table 3. Single Radionuclide quantification precision (%) of the Experimental Scenariosa
5. Conclusion
A robust approach is developed for spectral analysis based on MLEM, and it is verified for quantitative spectral component analysis and reaches high precision in inferring the constituents of the test spectrum. In the study set forth, exact radionuclides are detected correctly with the corresponding proportion inferred with high precision in a test spectrum. The analysis procedure is concise and efficient due to the fact that the test spectrum is forwarded to the iteration process directly after normalization with no background subtraction or feature extraction process. It is incredible that there are no extra parameters or indirect judgment process while achieving a high sensitivity of spectral component analysis of low-count scenarios.
Funding
The Key Research Program of Chinese Academy of Sciences (ZDRW-CN-2018-101) and the Instrument Developing Project of the Chinese Academy of Sciences (29201707).
Acknowledgments
We acknowledge Li Li, Yan-Tao Liu, Hao-Hui Tang, Ying-Jie Wang, Xiao-Rou Han and Guo-Fu Cao for helpful discussion.
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