Locating microseismic sources in time is a challenging problem in microseismic monitoring. In order to improve the accuracy and efficiency of locating sources, this paper presents a method for locating microseismic sources using deep reinforcement learning. We first construct and train a convolutional autoencoder to preprocess the seismic records in the microseismic waveform database. Then, the problem of locating the source is described as a Markov decision process for the application of deep reinforcement learning. We decompose the task of locating the source into three subtasks and design the critical elements of deep reinforcement learning. Three agents independently learn optimal policies for their respective subtasks in the framework of a deep Q-network (DQN) and jointly determine the precise location of the microseismic source. Finally, we evaluate the proposed method using synthetic data generated from the Marmousi model and the 3D velocity model. The experiment results indicate that the proposed method can locate microseismic sources efficiently and accurately.

Abstract Image denoising is a very important problem in image processing field. In order to improve denoising effects and meanwhile keep image structures, a novel weighted total variation (WTV) model is proposed in this paper. The WTV model consists of data fidelity and norm based regularisation terms. In the WTV model, a weight function in exponential form is incorporated into the regularisation term, which only depends on the given image itself without extra parameters. The nonlinearly monotone formulation of helps to increase gaps between lower and higher frequencies of images, which is effective to highlight edges and keep textures. For solving the proposed model, the alternating direction method of multipliers is explored and the according convergence is analysed. Compared experiments of TV, HOTV, ATV and models are conducted and the results show the effectiveness and efficiency of the proposed model.

The linear canonical transform (LCT) is a kind of integral transforms with wide applications in signal analysis. There have been numerous studies in the literature to generalize the LCT by making use of the quaternion algebra. In this paper, we first define the octonion linear canonical transform (OCLCT). Based on the definition of OCLCT, we extend the relationship between the LCT and the Fourier transform (FT) to the OCLCT and the octonion Fourier transform (OFT). Then explore related properties for the OCLCT such as shift property, inversion formula, isometry and Riemann-Lebesgue lemma. The relation between OCLCT and 3-D LCT is also builded. Moreover, based on these properties, we obtain Heisenberg’s uncertainty principle and Donoho-Stark’s uncertainty principle associated with the OCLCT. Finally, some potential applications are presented to show the effectiveness of the OCLCT.

The images used in various practices are often disturbed by noise, such as Gaussian noise, speckled noise, and salt and pepper noise. Images with noise are one of the challenges for segmentation, since the noise may cause inaccurate segmented results. To cope with the effect of noise on images during segmentation, a novel active contour model is proposed in this paper. The newly proposed model consists of fitting term, regularization term and penalty term. The fitting term is designed using a Gaussian kernel function and fractional order differentiation with an adaptively defined fractional order, which applies different orders to different pixels. The regularization term is applied to maintain the smoothness of curves. In order to ensure stable evolution of curves, a penalty term is added into the proposed model. Comparison experiments are conducted to show the effectiveness and efficiency of the proposed model.

The classical uncertainty principle plays an important role in quantum mechanics , signal processing and applied mathematics. With the development of novel signal processing methods, the research of the related uncertainty principles has gradually been one of the most hottest research topics in modern signal processing community. In this paper, the weighted Heisenberg–Pauli–Weyl uncertainty principles for the linear canonical transform (LCT) have been investigated in detail. Firstly, the Plancherel–Parseval–Rayleigh identities associated with the LCT are derived. Secondly, the weighted Heisenberg–Pauli–Weyl uncertainty principles in the LCT domain are investigated based on the derived identities. The signals that can achieve the lower bound of the uncertainty principle are also obtained. The classical Heisenberg uncertainty principles in the Fourier transform (FT) domain are shown to be special cases of our achieved results. Thirdly, examples are provided to show that our weighted Heisenberg–Pauli–Weyl uncertainty principles are sharper than those in the existing literature. Finally, applications of the derived results in time frequency resolution analysis and signal energy concentrations are also analyzed and discussed in detail.

An uncertainty principle (UP), which offers information about a function and its Fourier transform (FT) in the time-frequency plane, is particularly powerful in the field of signal processing. In this paper, based on the fundamental relationship between the quaternion linear canonical transform (QLCT) and quaternion Fourier transform (QFT), we propose two different UPs related to the two-sided QLCT. Different from existing results in the spatial and frequency domains, new derived consequences can be regarded as a general form of the UP of the QLCT, which present lower bounds for the product of spreads of a quaternion-valued function in two different QLCT domains.

In this study, the properties of noise after a discrete linear canonical transform (DLCT) are analysed. First, the authors prove that the DLCT of noise can be modelled by a Gaussian distribution with very weak assumptions on the noise in the time domain. Then, the mean and covariance matrix of this Gaussian distribution is derived and the general trend of the noise after DLCT is described. In addition, they find that the properties of noise in the LCT domain are the generalisation of the properties of noise in the Fourier transform domain. What is more, the authors prove that the additive white Gaussian noise (AWGN) is still an AWGN after performing the DLCT. Finally, the simulations are performed to verify the effectiveness of the obtained results.

Firefly algorithm (FA) is one of the newly developed nature inspired optimisation algorithm, inspired by the flashing behaviour of fireflies that a firefly tends to be attracted towards other fireflies with higher brightness. Thus FA has two advantages: local attractions and automatic regrouping. Based on these good properties, a novel image watermarking method based on FA in discrete wavelet transform (DWT)‐QR transform domain is proposed in this study. Structural similarity index measure and bit error rate are used in the objective function to trade‐off invisibility and robustness. The experiment results show that the proposed image watermarking method not only meet the need of invisibility, but also has better or comparable robustness as compared with some related methods.

Inspired by the fact that wavelet transform can be written as a classical convolution form, a new linear canonical wavelet transform (LCWT) based on generalised convolution theorem associated with linear canonical transform (LCT) is proposed recently. The LCWT not only inherits the advantages of multi‐resolution analysis of wavelet transform (WT), but also has the capability of image representations in the LCT domain. Based on these good properties, the authors propose a novel image watermarking method using LCWT and QR decomposition. Compared with the existing image watermarking methods based on discrete WT or QR, this novel image watermarking method provides more flexibility in the image watermarking. Peak signal‐to‐noise ratio, normalised cross and structural similarity index measure are used to verify the advantages of the proposed method in simulation experiments. The experiment results show that the proposed method is not only feasible, but also robust to some geometry attacks and image processing attacks.

Convolution and correlation operations are very important in signal processing community, as well as in sampling, filter design and applications. In this study, the authors derive the convolution and correlation theorems for the two‐dimensional linear canonical transform (2D LCT). Moreover, they utilise the convolution theorem to investigate the sampling theorem for the band limited signal in the 2D LCT domain. They also discuss multiplicative filter for the band limited signal in the 2D LCT domain which has much lower computational load than the method in the 2D LCT domain.

The aliased polyphase sampling in the linear canonical transform (LCT) domain has been proposed in this study. The result shows that the sampling theorem can be achieved by using parallel samplers to obtain a higher sampling rate in the LCT domain. Further analysis shows that the LCT spectrum of x(nM+m) replicates |X(a,b,c,d)(u)| with a period bMFs along with linear phase modulation in the LCT domain, where b is one parameter of the LCT, M is the number of samplers and Fs is the sampling frequency at which these samplers are operated in the Fourier transform domain. Finally, the simulations are also proposed to verify the correctness of the derived results.