An algorithm for ²⁵²Cf-Source-Driven neutron signal denoising based on Compressive Sensing

An algorithm for 252 Cf-Source-Driven neutron signal denoising based on Compressive Sensing 1 corresp first-author Corresponding author, cqujwclpc@163.com Corresponding author, cqujwclpc@163.com cqujwclpc@163.com 1 1 1 1 Key Laboratory of Opto-electronics Technology and System, Ministry of Education, Chongqing University, Chongqing 400044, China Key Laboratory of Opto-electronics Technology and System, Ministry of Education, Chongqing University, Chongqing 400044, China As photoelectrically detected 252 Cf-source-driven neutron signals always contain noise, a denoising algorithm is proposed based on compressive sensing for the noised neutron signal. In the algorithm, Empirical Mode Decomposition (EMD) is applied to decompose the noised neutron signal and then find out the noised Intrinsic Mode Function ( IMF ) automatically. Thus, we only need to use the basis pursuit denoising (BPDN) algorithm to denoise these IMF s. For this reason, the proposed algorithm can be called EMDCSDN (Empirical Mode Decomposition Compressive Sensing Denoising). In addition, five indicators are employed to evaluate the denoising effect. The results show that the EMDCSDN algorithm is more effective than the other denoising algorithms including BPDN. This study provides a new approach for signal denoising at the front-end. 252 Cf-source-driven neutron signal Empirical mode decomposition Compressive sensing Denoising I. INTRODUCTION 1 The photoelectric detection and imaging technology have been widely used in Nuclear Material Identification System (NMIS) to detect neutron signal and nuclear components. Time-frequency method has been used to analyze neutron signal and identify properties of nuclear material [ 1 1 , 2 2 ], while tomographic imaging has been used to provide more information about geometry of nuclear components, which leads to high accuracy of NMIS [ 3 3 - 5 5 3-5 ]. However, it is inevitable that there are noises in the measurement process due to external environment, detector and electronic device, and the noises may confuse some useful but weak information. Thus, we should denoise the neutron signals first to improve accuracy of NMIS. Except the circuit and technic neutron signal denoising methods, traditional denoising algorithms mostly represent the noised signal in transform domain and then threshold for the purpose of denoising. As a result, FFT filter and wavelet shrinkage has been applied in neutron signal denoising [ 6 6 ]. Besides, a new denoising algorithm named EMDSD (Empirical Mode Decomposition Double Smoothness Detecting) has been proposed for neutron signal [ 7 7 ]. However, these denoising algorithms cannot adjust decomposition base automatically. In recent years, the paradigm of sparse sampling and reconstruction, called compressive sensing (CS), is a state-of-the-art research [ 8 8 , 9 9 ]. CS approaches have opened up many new research avenues in the field of under-determined systems, and found many practical applications in image processing, wireless communication, data-streaming, and medical resonance. Also, CS has been applied in image denoising [ 10 10 - 14 14 10-14 ], and some new denoising algorithms [ 15 15 , 16 16 ] are derived from basis pursuit denoising (BPDN) algorithm [ 17 17 ]. Although these algorithms are proposed for all signal frequency, the compressive reconstruction error will still increase and signal may be noised again. Fortunately, the Empirical Mode Decomposition (EMD) can decompose the signal into Intrinsic Mode Function ( IMF ), from finer temporal scales (high frequency IMF s) to coarser ones (low frequency IMF s). As the noise is always in high frequency IMF s, BPDN algorithm can be adopted to denoise them. In this paper, we propose a new denoising algorithm named EMDCSDN (Empirical Mode Decomposition Compressive Sensing Denoising). Several denoising algorithms are compared with EMDCSDN algorithm to verify its robustness. II. METHODOLOGY 1 A. Materials 2 Neutron signals from 252 Cf-source-driven nuclear material fission are detected by photoelectric detectors. We designed a 252 Cf-source-driven verification system to identify properties of 235 U ( Fig. 1(a) Fig. 1(a) ). It consists of a 252 Cf neutron source, an ionization chamber, scintillation neutron detectors, workstation with high-speed (1 GHz) data acquisition card with a large-capacity disk array, verification software and user interface (UI) system. Three detectors are placed around the fissile material. The target-detector distance d and the angle α between detectors can be adjusted according to the measurement requirements ( Fig. 1(b) Fig. 1(b) ). Figure 2 Figure 2 shows the noised neutron pulse signal acquired by the 252 Cf-source-driven verification system. For experiment purpose, the EMDCSDN algorithm is adopted to denoise this signal. As the pure neutron signal is unknown, the curve fitting of noised neutron signal is regarded as the pure signal, so as to evaluate the denoising effect. All the materials are used without further purification. B. Basis pursuit denoising 2 Compressive sensing, the paradigm of sparse sampling and reconstruction, is an advanced theory. Specifically, let x = x 1 , x 2 ,, x N , x R N with x = α , where α has only K non-zero elements and K N . The x is called as K -sparse with respect to the transform . So the random measurements y = { y 1 , y 2 ,, y N }, y R M are generated by <math display="block" id="M1"><mrow><mi>y</mi><mo>=</mo><mi>Φ</mi><mi>x</mi><mn>,</mn><mtext> </mtext><mtext>where</mtext><mi>Φ</mi><mo>∈</mo><msup><mi>R</mi><mrow><mi>M</mi><mo>×</mo><mi>N</mi></mrow></msup><mn>,</mn></mrow></math> where, Φ is a randomly sampling matrix, and the number of measurement M N . It is an ill-posed problem recovering x from y . In the compressive sensing theory, it testifies that K -sparse signal x can be recovered by M = O [ K log( N/K )] measurements as long as Φ satisfies the restricted isometry property (RIP), and the reconstruction can be achieved with probability close to one by solving the following convex optimization. <math display="block" id="M2"><mrow><mi>α</mi><mo>=</mo><mi>arg</mi><mi>min</mi><mo>∥</mo><mi>α</mi><msub><mo>∥</mo><mn>1</mn></msub><mn>,</mn><mtext> </mtext><mtext>subject to </mtext><mi>y</mi><mo>=</mo><mi>Φ</mi><mi>ψ</mi><mi>α</mi><mn>,</mn></mrow></math> where | α | 1 denotes the l 1 -norm of the vector α . Thus, the compressive sensing process of neutron signal can be described as follows. It assumes that the noised neutron signal x = s + n = ψα + n , where s is the original signal, n denotes additive Gaussian white noise and ψ is the sparse basis. Consequently, the compressive sampling can be defined as, <math display="block" id="M3"><mrow><mi>y</mi><mo>=</mo><mi>Φ</mi><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Φ</mi><mi>s</mi><mo>+</mo><mi>Φ</mi><mi>n</mi><mo>=</mo><mi>Φ</mi><mi>s</mi><mo>+</mo><mi>z</mi><mo>=</mo><mi>Φ</mi><mi>ψ</mi><mi>α</mi><mo>+</mo><mi>z</mi><mn>,</mn></mrow></math> where z is the noise in sampling and | z | 2 ε . The reconstruction can be achieved by Eq. (2). With noisy or imperfect data, it is impossible to fit the linear system exactly. Instead, the constraint in BP is relaxed to obtain the basis pursuit denoising problem. <math display="block" id="M4"><mrow><munder><mstyle mathsize="140%" displaystyle="true"><mrow><mi>min</mi></mrow></mstyle><mi>α</mi></munder><mo>∥</mo><mi>α</mi><msub><mo>∥</mo><mn>1</mn></msub><mtext> </mtext><mtext>subject to</mtext><mo>∥</mo><mi>y</mi><mo>−</mo><mi>Φ</mi><mi>ψ</mi><mi>α</mi><msub><mo>∥</mo><mn>2</mn></msub><mo>≤</mo><mi>ε</mi><mn>.</mn></mrow></math> Therefore, an effective algorithm using Spectral Projected Gradient (SPG) is used to solve BPDN [ 17 17 ]. It can solve the BPDN problem much more quickly, and the memory requirement s are constant throughout all the iterations. However, most of the noised signal frequencies are pure and there is no need to denoise them. So, if we use the BPDN algorithm for all frequencies, the signal will be noised again and some useful information may be filtered. Therefore, based on compressive sensing, a new denoising algorithm for the noised frequencies only is proposed. C. EMDCSDN algorithm 2 Hilbert-Huang Transform (HHT) is a new method of time-frequency analysis technique, which was proposed by Huang [ 18 18 ], and the essence of this method is EMD. The EMD is advantageous in that the base functions are derived from the signal itself. Hence, the analysis is more adaptive, in contrast to the wavelet method where the base functions are fixed. Any signal can be decomposed into a finite and a small number of IMF s to give meaningful instantaneous frequency, as described by Eq. (5). <math display="block" id="M5"><mrow><mi>x</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><mtext> </mtext><mi>I</mi><mi>M</mi><msub><mi>F</mi><mi>j</mi></msub><mo>+</mo><mi>r</mi><mi>e</mi><mi>s</mi><mn>,</mn></mrow></math> where res means a residual of EMD. Figure 3 Figure 3 shows the IMF s of neutron signal decomposed by EMD. It seems that only the first few high frequencies of IMF s are noised, and rest of the IMF s are pure. Although the EMD itself has characteristics of filter, the noised IMF s cannot be filtered directly. Therefore, we need to find the noised IMFs, to use BPDN algorithm for denoising. Thus, the EMDCSDN algorithm is proposed as shown in Fig. 4 Fig. 4 . In this EMDCSDN algorithm, the IMF selector is designed according to the characteristics of white noise [ 19 19 ]. One of the characteristics is that i = E i P i is a constant, where E i is the energy density and P i is period of the IMF s. <math display="block" id="M6"><mrow><msub><mi>E</mi><mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><msup><mrow><mo stretchy="false">[</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>2</mn></msup><mn>,</mn><mtext> </mtext><msub><mi>P</mi><mi>i</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mi>N</mi></mrow><mrow><msub><mi>n</mi><mrow><mi>max</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>n</mi><mrow><mi>min</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where, N , C , n min and n max are the length, amplitude, the number of maxima and the number of minima of the IMF s, respectively. If <math id="M7"><mrow><msub><mi>R</mi><mi>i</mi></msub><mo>=</mo><mo>|</mo><mo stretchy="false">(</mo><msub><mi>Γ</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>Γ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>/</mo><mover accent="true"><mi>Γ</mi><mo>¯</mo></mover><mo>|</mo><mo>≥</mo><mn>1</mn></mrow></math> , then Γ j ( j = 1,2,, i -1) is a constant and the first i -1 IMF s are noised, where <math id="M8"><mover accent="true"><mi>Γ</mi><mo>¯</mo></mover></math> is the mean of j . Thus, we can use this IMF selector to find out which IMF is noised. Figure 5 Figure 5 shows that R 3 > 1, so for the noised neutron signal, IMF 1 and IMF 2 need to be denoised. Thereby, the EMDCSDN algorithm can narrow the denoising scope and reduce the compressive sensing reconstruction error, and the denoising effect will be improved. In the next section, how to evaluate the denoising effect will be discussed. D. Denoising evaluation 2 In general, the signal-to-noise ratio (SNR), peak signal-to-noise ratio (PSNR) and mean squared error (MSE) are adopted to evaluate the denoising effect. However, only the three indicators are not enough because high SNR means that some useful information has been filtered. So, another two indicators, smoothness of curve ( SOC ) and correlation coefficient ( CC ) should also be involved. The SOC and CC can be defined as: <math display="block" id="M9"><mrow><mi>S</mi><mi>O</mi><mi>C</mi><mo>=</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msup><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>2</mn></msup></mrow><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msup><mrow><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>x</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M10"><mrow><mi>C</mi><mi>C</mi><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mover accent="true"><mi>x</mi><mo>^</mo></mover><mi>i</mi></msub><mo>−</mo><mover accent="true"><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mrow><msqrt><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><msup><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>⋅</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><msup><mrow><mo stretchy="false">(</mo><msub><mover accent="true"><mi>x</mi><mo>^</mo></mover><mi>i</mi></msub><mo>−</mo><mover accent="true"><mover accent="true"><mi>x</mi><mo>^</mo></mover><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mn>,</mn></mrow></math> where x is the original noised signal and <math id="M11"><mover accent="true"><mi>x</mi><mo>^</mo></mover></math> is the denoising signal. For CC indicator, the bigger the better. However, for SOC indicator, smaller SOC means better denoising effect. Hence, in order to evaluate the denoising effect, we need to consider all the indicators above. III. RESULTS AND DISCUSSION 1 For the noised neutron signal x , x { R } N with elements x [ n ], n = 1,2,, N and N = 1024. The number of compressive sampling is regarded as M , then the corresponding compressive sampling rate is M/N . As mentioned before, the first two IMF s ( IMF 1 , IMF 2 ) are noised. Therefore, BPDN algorithm is used to denoise the two IMFs. The results are shown in Fig. 6 Fig. 6 , where DN is the abbreviation of denoising and M/N = 0.65. Figure 7 Figure 7 shows the EMDCSDN algorithm denoising result of neutron signal. However, this cannot fully reflect the denoising effect, it should be evaluated by the five indicators of SNR , PSNR , MSE , SOC and CC . Different compressive sampling rates M/N shall lead to different denoising effects. Figure 8 Figure 8 shows the denoising effects of EMDCSDN algorithm for different M/N . Figure 8 Figure 8 has double Y axis. The left represents the MSE , SOC and CC , and the right one is SNR . SNR and CC increase with M/N , while MSE and SOC decrease with increasing M/N . The best denoising effect can be seen at M/N =0.75, where SNR and CC are the biggest, and MSE and SOC are the smallest, and the denoising effect is the best. However, if we want a lower compressive sampling rate, M/N =0.65 may be a better choice because its denoising effect is very close to M/N =0.75. Here, we choose M/N =0.75. The EMDCSDN algorithm is compared with four denoising algorithms: (1) EMD Filter (EMDF) that filters the noised IMF s directly; (2) Basis Pursuit Denoising (BPDN), compressive sensing denoising problem solved by SPGL1 algorithm; (3) Wavelet Threshold Denoising (WTDN) that denoise signal in wavelet domain with threshold; (4) EMD Wavelet Denoising (EMDWTDN), which uses WTDN to denoise the noised IMF s and the reconstruct signal with all IMF s. The denoising effects are given in Table 1 Table 1 . 1 1 Algorithm 1 1 SNR 1 1 PSNR 1 1 MSE 1 1 SOC 1 1 CC 1 1 Processing time (s) 1 1 noised neutron signal 1 1 24.24559 1 1 39.93027 1 1 5.67538 1 1 5.28919 1 1 0.99813 1 1 1 1 EMDF 1 1 18.16586 1 1 33.57995 1 1 11.42829 1 1 1.12880 1 1 0.99249 1 1 2.377 1 1 BPDN ( M/N = 0.75) 1 1 27.59472 1 1 42.93629 1 1 3.85956 1 1 2.46778 1 1 0.99917 1 1 1.307 1 1 WTDN 1 1 28.74912 1 1 43.98175 1 1 3.37923 1 1 0.96627 1 1 0.99938 1 1 1.588 1 1 EMDWTDN 1 1 29.48255 1 1 44.98517 1 1 3.10561 1 1 1.99482 1 1 0.99944 1 1 2.726 1 1 EMDCSDN ( M/N = 0.75) 1 1 30.62113 1 1 45.89118 1 1 2.72407 1 1 1.57471 1 1 0.99957 1 1 3.019 From Table 1 Table 1 , the EMDF algorithm is a simple but less effective algorithm because it also filters out some useful information. As the EMDWTDN algorithm is better than the WTDN algorithm due to the EMD, we think that EMD can improve the denoising effect and the effect of EMDCSDN algorithm confirms this view. Considering all the five indicators, the EMDCSDN algorithm is more effective than the other algorithms due to the EMD and the noised IMF s automatic selector, although its processing time is a little longer. IV. CONCLUSION 1 A new denoising algorithm, EMDCSDN, has been proposed for noised neutron signal. In fact, it is a modified BPDN algorithm because we only need to denoise the noised IMF s that are selected automatically. Thus, it makes our denoising algorithm more specific. It also reduces the compressive sensing reconstruction error. Using five indicators, the EMDCSDN algorithm is more effective than the other denoising algorithms. Moreover, it can reduce the sampling costs. The EMDCSDN algorithm can be used in other denoising applications. References S Avdic , P Marinkovic , S A Pozzi , et al . Identification of neutron sources by spectral analysis of pulse height distributions . IEEE T Nucl Sci , 2009 , 56 : 2796 - 2801 . 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DOI: 10.1098/rspa.2003.1221 http://doi.org/10.1098/rspa.2003.1221 A study of the characteristics of white noise using the empirical mode decomposition method 10.13538/j.1001-8042/nst.26.060404 Supported by the National Natural Science Foundation of China (Nos. 61175005 and 61401049) 2014-12-24 2015-03-06 2015-12-20 2015-12-20 2021-04-11 2015 Nuclear Science and Techniques 6 26