Jiangmei Zhang, Haibo Ji, Qingping Zhu, Hongsen He and Kunpeng Wang(.Department of Automation, University of Science and Technology of China, Hefei 3007, China; .School of Information Engineering, Southwest University of Science and Technology, Mianyang 600, Sichuan, China)

In recent years, radioactive materials are widely used in a number of application fields, such as medical detection and industrial manufacture. A typical characteristic of this class of materials is radioactivity, which can easily bring about potential radioactive leakage in production and transportation[1]. Thus, once the radiation leakage happens[2], radioactivity is an important issue for fast detecting and localizing the radiation sources to reduce the adverse effects. Due to the presence of environmental noise and the performance limit of the detector, it is difficult to effectively detect and identify the radiation sources.

In radioactive environment, signals captured by corresponding sensors are a series of pulses. Usually, these radioactive pulse signals are heavily contaminated by noise. To determine the existence of the radiation signals in the sensor signals, there are two tasks to be done. One is the radioactive pulse signals enhancement that can decrease the negative effect of noise. The other is the pulse signal shaping that is employed to identify the nuclides. In this paper, we focus on the recovery of weak radioactive signal from noisy environments. For the pulse signal detection, the traditional approaches first determine the energy spectra of detector signals, which are then preprocessed by the popular methods such as Kalman filter[3], smoothing filter[4]. Then, some feature peaks are sought from the energy spectra for matching nuclides identification. These methods obtain good performances in dealing with the dead time and overlapping pulse peaks. These strategies, however, are not robust to restore the radioactive pulse amplitude and its location.

Sparseness is often considered as priori information to improve the robustness in signal processing fields. Sparse representation, which is closely related to compressed sensing[5], is extensively applied to signal processing and image processing. Using such technology, the original signal can be reconstructed by exploiting a few measured signals. Since an obvious feature of the pulse signals is that they are sparse in the time domain, we propose a sparse representation model to recover the radioactive pulse signals contaminated by noise in this paper. A Gabor dictionary is built, which uses only a few of atoms to reconstruct the original signal. This sparse optimization model is solved by a matching pursuit algorithm. Simulations and experiments demonstrate the effectiveness of the proposed method.

1 Pulse Signal Recovery via Sparse Representation

1.1 Signal model

It is known that a radioactive material radiates energy in the form of pulses. The radioactive decay is a random process. The intervaltbetween the radiation particles is often small, which is also a random variable. The statistical property of the amplitude of the radioactive pulses can be modeled by a Gaussian distribution. For a single radioactive pulse, an exponential decay function can be involved to define it as

(1)

whereEis the amplitude of the pulse,τ,aandbare constants, respectively.

In a real environment, the radioactive pulse signal is affected by noise in most cases. We assume that the noise obeys a Gaussian distribution. Then, the noisy pulse signal can be modeled as

h(t)=f(t)+N(t)

(2)

whereN(t) is the Gaussian noise. We assume thatf(t) andN(t) are independently distributed variables.

After checkin long time statistics of the radioactivity, it is found that the radioactive signal is of an evident sparse feature. That is, the most energy of the radioactive signal focuses on the dominant pulse peaks. If the amplitudes and locations of these peaks can be effectively estimated in the noisy environments, the radioactive pulse signal is easily recovered. To this end, the signals captured by the detectors are formulated as a vector form of sizenas follows

y=x+b

(3)

where b is a noise vector of sizen, x is ann-dimensional vector with

x≈Dα

(4)

αis a coefficient vector of sizem, and

D=[d1d2… dn]

(5)

whereαis a dictionary matrix of sizen×m. The basic element of a complete dictionary, which is called an atom, can be used to represent the useful signal. In this work, this dictionary is produced by the Gabor function on the basis of the double exponential decay feature of the radioactive pulse signal. It can be seen from Eq.(4) that the ideal radioactive signal is a linear map of the vectorα, which indicates that if the dictionary matrix is designed well and noise is free, the radioactive pulse signal can be recovered by solving the following equation[6]

y≈Dα

(6)

1.2 Optimization criterion

As mentioned previously, radioactive pulse signals are sparse. The purpose of the sparse representation is to select a small amount of atoms from the complete dictionary to represent the original signal. In this section, this property is exploited to construct the optimization criterion. The accurate metric of sparseness is counting the nonzero numbers of a vector. To describe the sparseness,l0-norm is most commonly used in the literature since it is perfectly in accord with the definition of sparseness. Therefore, when noise does not exist there, Eqs. (3) (4) can be used to form a sparse optimization criterion as follows:

(7)

where ‖‖0denotes thel0-norm of a vector. However, thel0-norm is not convex and its minimization is non-deterministic polynomial-time (NP)-hard. To relax the NP-hard problem, the basis pursuit (BP) is proposed and it adoptsl1-norm instead ofl0-norm as follows[7]

min ‖α‖1s.t. y=Dα

(8)

where ‖‖1denotes thel1-norm of a vector. Thel1-norm is convex, and is a good approximation of thel0-norm, which has been testified by a large number of applications.

When noise is considered, the equation constraintsin Eqs.(7) (8) do not hold any longer. To achieve the robustness to noise, an inequality constraint is introduced into this optimization problem as follows

(9)

(10)

whereλis a Lagrange multiplier.

1.3 Matching pursuit algorithm

The sparse optimization problem can be solved by many existing approaches, such as the linear programming[9], the interior point method[10], the primal-dual interior point method[11], augmented Lagrangian alternating direction method[12]. Unlike the previously mentioned techniques, we use the matching pursuit method to solve this problem due to its simplicity[13]. The matching pursuit method is one of the most commonly used greedy pursuit strategies for the practical applications[14]. A large number of variations of this algorithm are developed according to different requirements, such as the orthogonal matching pursuit[15], the high resolution greedy algorithm[16]. As for the matching pursuit algorithm, the basic idea is to iteratively choose the best atom group from the dictionary according to a certain similarity measurement to approximately obtain the sparse solution[13]. The proof on the convergence of the matching pursuit algorithm is shown in Ref.[13]. The main steps of the matching pursuit algorithm for the sparse signal recovery are listed in the following algorithm 1.

Algorithm1. Steps of pulse signal recovery via the matching pursuit algorithm.

Task: To solve the constraint problem:

Input: The measurement signal vector y and dictionary matrix D=[d1d2… dn]

Initialization:

r0=y,α=0,D0=?, index setΛ0=? where ? denotes an empty set.

Fort=1, 2, …

Step 1: Find the index of the best matching vector by calculating the inner product between rt-1and dj(j?Λt-1)

Step 2: Update the index setΛt=Λt-1∪λt, and reconstruct the matrix

Dt=[Dt-1,dλt]

Step 3: Compute the sparse coefficient via

=argmin ‖y-Dt‖1

Step 4: Update the representation residual via

rt=y-Dt

Step 5: If ‖rt‖2<ε, go to Step 1; otherwise, stop iteration.

End

2 Simulations

In this section, the effectiveness of the proposed method on the pulse signal recovery is investigated by simulations. For comparison, the finite impulse response (FIR)[3]filtering and smoothing filtering[4]methods are also employed to recover the pulse signals.

We use five random pulses to simulate the short-time radioactive pulse signal that is random in time and amplitude. In the following simulation, the results are obtained by averaging over 250 runs. The iteration number of the proposed algorithm is 30.

Fig.1 illustrates the result of the pulse signal recovery via the proposed method in the absence of noise. In Fig.1,Xrepresents the location of the pulse peaks andYdenotes the amplitude of the pulse peaks. As can be seen, the proposed method can effectively recover the pulse signal in the corresponding location and comparable amplitude, which is almost the same as the original signal. To validate the performance of the new method in noisy environments, a similar simulation is conducted at the signal-to-noise ratio (SNR) of 0 dB. The result is shown in Fig.2. As we can see from Fig.2, the FIR filtering method effectively recovers the amplitude of the pulses, however, there exists an excursion for the location of the pulses, and the level of noise reduction is finite. The ability of the smoothing method for noise reduction is better, but the peaks of the pulses basically lose the sharpness and the exponential feature. As compared to the first two methods, the proposed approach obtains the best performance for the recovery of the pulse signals. The shape and location of the pulse peaks are well reconstructed, which demonstrates the effectiveness of the new strategy.

Fig.1 Pulse signal recovery when noise is absent

Fig.2 Comparison between three different reconstruction methods for the pulse signal recovery

Fig.3 shows the mean square error (MSE) of the three methods for reconstruction of the pulse signals at different SNRs. It can be seen from Fig.3 that the smoothing method does not gain the robustness to noise for the pulse signal recovery since some information about the pulse peaks is lost by this approach. The FIR filtering technique improves the reconstruction performance at different SNRs due to its recovery ability to the pulse peaks. In comparison, the proposed approach outperforms the other two methods at different SNRs since its robustness is embodied at reliable restoration to the amplitude and location of the pulse peaks.

Fig.3 MSE of three methods for reconstruction of the pulse signals at different SNRs

3 Experiments

3.1 Experimental environments

To verify the effectiveness of the proposed method in a practical environment, experiments on the real radioactive pulse signal are conducted in this section. The measurement scene of the radioactive pulse signals is shown in Fig.4. The CsI detector produced by Japanese ALOKA company is implemented for Eu155, Am241, and Pu239 detection. The temperature and humidity of the laboratory are 25 degrees and 50%, respectively. The iteration number of the proposed algorithm is 100.

Fig.4 Measurement scene of the radioactive pulse signals

3.2 Experimental results

The first experiment is to investigate the effectiveness of the proposed method in recovery of the pulse signals of a single muclide.

Fig.5 Recovery of the pulse signal of Eu155 in practical environments

Fig.5 illustrates the recovery of the pulse signal of Eu155 in practical environments by the three different approaches. As shown in Fig.5, the measured radioactive pulse signal is contaminated by noise. This noise, however, is not easily estimated or measured due to its uncertainty. So, the corresponding SNR is not given. Note that the actual radioactive pulse signal is indeed sparse. Similar to the case of simulations, the FIR filtering method does not produce a good reconstruction result since it is sensitive to noise. The smoothing filtering technique is robust to noise, but it cannot recover ideal radioactive pulse peaks. Among the three reconstruction methods, the proposed approach obtains the best performance by the use of 20 atoms, especially for the amplitude and location of the radioactive pulses. In addition, the height of the pulse recovered by the proposed method is comparable to that of the measured signals. The superiority comes from the fact that the proposed strategy effectively exploits the sparseness of the radioactive pulse signal.

Fig.6 Recovery of the pulse signal of Eu155, Am241 and Pu239 in practical environments

The second experimcent is carried out to check the superiority of the presented approach for recovering the pulse signals of multiple nuclides. Fig.6 describes the recovery of the pulse signal of Eu155, Am241, and Pu239 in practical environments by the three different approaches. It can be found from Fig.6 that the FIR filtering method cannot accurately estimate the location of the radioactive pulses, which yield a grave estimate error. The smoothing filtering approach, however, cannot effectively reconstruct the sharp shape of the radioactive pulses. In comparison with the previous methods, the proposed scheme achieves the best estimate result, either for the location of the radioactive pulses or its sharp shape. Similarly, the height of the pulse recovered by the proposed approach is almost equal to that of the measured signals. These results demonstrate that the proposed approach is effective on the recovery of the pulse signals produced by mixing multiple nuclides.

4 Conclusion

In this paper, a pulse signal recovery approach is proposed. From the perspective of the sparseness of the pulse signals, we construct the problem base on a sparse optimization model, which is then solved via the matching pursuit algorithm. Numerical simulations show that the proposed approach can effectively reconstruct the pulse signals when noise is present or absent. Furthermore, the experiments in practical radioactive environments are conducted to investigate the performance of the proposed method. The results demonstrate that in noisy environments, the proposed strategy can effectively recover the radioactive pulse signals for either a single nuclide or multiple nuclides. In particular, the sharp shape and location of the radioactive pulses can be accurately reconstructed using the proposed algorithm even in practical noisy environments.

[1] Chan K S, Li J, Eichinger W, et al. A new physics-based method for detecting weak nuclear signals via spectral decomposition[J]. Nuclear Instruments & Methods in Physics Research, 2012, 667(5): 16-25.

[2] Albert F, Anderson S G, Anderson G A, et al. Isotope-specific detection of low-density materials with laser-based monoenergetic gamma-rays[J]. Optics Letters, 2010, 35(3): 354.

[3] Haykin S. Kalman filtering and neural networks[M].New York: John Wiley & Sons, Inc, 2001.

[4] Savitzky A, Golay M J E. Smoothing and differentiation of data by simplified least squares procedures[J]. Analytical Chemistry, 1964, 36(8): 1627-1639.

[5] Donoho D L. Compressed sensing[J]. IEEE Transactions on Information Theory, 2006, 52(4): 1289-1306.

[6] Murray J F, Kreutz-Delgado K. An improved FOCUSS-based learning algorithm for solving sparse linear inverse problems[C]∥Asilomar Conference on IEEE, 2001:347-351.

[7] Chen S, Saunders M A, Donoho D L. Atomic decomposition by basis pursuit[J]. Siam Review, 2006, 43(1): 129-159.

[8] Kostic Z, Sezan M I, Titlebaum E L. Estimation of the parameters of a multipath channel using set-theoretic deconvolution [J]. IEEE Trans Commun, 1992, 40(6): 1006-1011.

[9] Jiang X, Kirubarajan T, Zeng W J. Robust sparse channel estimation and equalization in impulsive noise using linear programming[J]. Signal Processing, 2013, 93(5): 1095-1105.

[10] Kim S J, Koh K, Lustig M, et al. An interior-point method for large-scale l1-regularized least squares[J]. Journal of Machine Learning Research, 2007, 1(4): 606-617.

[11] Wright S J. Primal-dual interior-point methods[M]. [S.l.]:Society for Industrial and Applied Mathematics, 1997: 995-996.

[12] He H, Yang T, Chen J. On time delay estimation from a sparse linear prediction perspective[J]. Journal of the Acoustical Society of America, 2015, 137(2): 1044-1047.

[13] Mallat S G, Zhang Z. Matching pursuits with time-frequency dictionaries[J]. IEEE Transactions on Signal Processing, 1993, 41(12): 3397-3415.

[14] Chen F Y, Shang Y S, Yang C C. A general description of matching pursuits decomposition method[J]. Progress in Geophysics, 2007, 22(5): 1466-1473.

[15] Needell D, Vershynin R. Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit[J]. IEEE Journal of Selected Topics in Signal Processing, 2007, 4(2): 310-316.

[16] Akhtar N, Shafait F, Mian A. SUnGP: a greedy sparse approximation algorithm for hyperspectral unmixing[C]∥International Conference on Pattern Recognition, IEEE Computer Society, 2014:3726-3731.