A. Materials

Neutron signals from ²⁵²Cf-source-driven nuclear material fission are detected by photoelectric detectors. We designed a ²⁵²Cf-source-driven verification system to identify properties of ²³⁵U (Fig. 1(a)). It consists of a ²⁵²Cf 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.Full-screen View

(Color online) Prototype of ²⁵²Cf-source-driven verification system (a) and position of detectors (b).

Figure 2 shows the noised neutron pulse signal acquired by the ²⁵²Cf-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.

Fig. 2.Full-screen View

(Color online) The noised neutron signal.

B. Basis pursuit denoising

Compressive sensing, the paradigm of sparse sampling and reconstruction, is an advanced theory. Specifically, let x=x₁, x₂,, xN, x RN 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 = {y₁, y₂,, yN }, y RM are generated by

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.

where | α |₁ denotes the l₁-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,

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.

Therefore, an effective algorithm using Spectral Projected Gradient (SPG) is used to solve BPDN [17]. It can solve the BPDN problem much more quickly, and the memory requirements 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

Hilbert-Huang Transform (HHT) is a new method of time-frequency analysis technique, which was proposed by Huang [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 IMFs to give meaningful instantaneous frequency, as described by Eq. (5).

where res means a residual of EMD.

Figure 3 shows the IMFs of neutron signal decomposed by EMD. It seems that only the first few high frequencies of IMFs are noised, and rest of the IMFs are pure. Although the EMD itself has characteristics of filter, the noised IMFs cannot be filtered directly. Therefore, we need to find the noised IMFs, to use BPDN algorithm for denoising.

Fig. 3.Full-screen View

(Color online) IMFs of neutron signal decomposed by EMD.

Thus, the EMDCSDN algorithm is proposed as shown in Fig. 4.

Fig. 4.Full-screen View

The EMDCSDN algorithm.

Full-screen View

Algorithm 1: EMDCSDN algorithm for neutron signal

In this EMDCSDN algorithm, the IMF selector is designed according to the characteristics of white noise [19]. One of the characteristics is that i = EiPi is a constant, where Ei is the energy density and Pi is period of the IMFs.

where, N, C, n_min and n_max are the length, amplitude, the number of maxima and the number of minima of the IMFs, respectively.

If $R_i=|({\Gamma }_i-\overline{\Gamma })/\overline{\Gamma }|\ge 1$, then Γj (j = 1,2,,i-1) is a constant and the first i-1 IMFs are noised, where $\overline{\Gamma }$ is the mean of j. Thus, we can use this IMF selector to find out which IMF is noised. Figure 5 shows that R₃>1, so for the noised neutron signal, IMF₁ and IMF₂ 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.

Fig. 5.Full-screen View

(Color online) The IMF selector results of neutron signal.

D. Denoising evaluation

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:

where x is the original noised signal and $\widehat{x}$ 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.