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Biomedical signals are usually contaminated with interfering noise, which may result in misdiagnosis of diseases. Additive white Gaussian noise (AWGN) is a common interfering noise, and much work has been proposed to suppress AWGN. Recently, hierarchical multiresolution analysis-based empirical mode decomposition (EMD) denoising method is proposed and shows potential performance. In order to further improve performance of hierarchical multiresolution analysis-based EMD denoising, this paper combines hierarchical multiresolution analysis-based EMD, thresholding operation and averaging operation together. In particular, EMD is applied to the first intrinsic mode function (IMF) in the first level of decomposition to obtain IMFs in the second level of decomposition. The first IMF in the second level of decomposition is chosen as the noise component. For each realization, this noise component is segmented into various pieces, and these segments are permutated. By summing up this permutated IMF to the rest of IMFs in both the first level of decomposition and the second level of decomposition, new realization of the noisy signal is obtained. Next, for original signal and each realization of newly generated noisy signal, EMD is performed again. IMFs in the first level of decomposition are obtained. Then, consecutive mean squared errors-based criterion is used to classify IMFs in the first level of decomposition into the information group of IMFs or the noise group of IMFs. Next, EMD is applied to IMFs in the noise group in the first level of decomposition and IMFs in the second level of decomposition are obtained. Detrended fluctuation analysis is used to classify IMFs in the second level of decomposition into the information group of IMFs or the noise group of IMFs. After that, thresholding is applied to IMFs in the noise group in the second level of the decomposition to obtain denoised signal. Finally, the above procedures are repeated, and several realizations of denoised signals are obtained. Then, denoised signal obtained by applying thresholding to each realization is averaged together to obtain final denoised signal. The extensive numerical simulations are conducted and the results show that our proposed method outperforms existing EMD-based denoising methods.

The existing methods for performing the super-resolution of the three-dimensional images are mainly based on the simple learning algorithms with the low computational powers and the complex deep learning neural network-based learning algorithms with the high computational powers. However, these methods are based on the prior knowledge of the images and require a large database of the pairs of the low-resolution images and the corresponding high-resolution images. To address this difficulty, this paper proposes a method based on the joint generalized singular value decomposition and tensor decomposition for performing the super-resolution. Here, it is not required to know the prior knowledge of the pairs of the low-resolution images and the corresponding high-resolution images. First, an image is represented as a tensor. Compared to the three-dimensional singular spectrum analysis, the spatial structure of the local adjacent pixels of the image is retained. Second, both the generalized singular value decomposition and the Tucker decomposition are applied to the tensor to obtain two low-resolution tensors. It is worth noting that the correlation between these two low-resolution tensors is preserved. Also, these two decompositions achieve the exact perfect reconstruction. Finally, the high-resolution image is reconstructed. Compared to the de-Hankelization of the three-dimensional singular spectrum analysis, the required computational complexity of the reconstruction of our proposed method is much lower. The computer numerical simulation results show that our proposed method achieves a higher peak signal-to-noise ratio than the existing methods.

This paper generalizes the discrete Fourier transform matrix to an energy-preserved transform matrix as well as introduces the idea of the generalized mean frequency and the idea of the generalized frequency variance to perform the tachycardias classification. In particular, these two physical quantities are employed as the features and the random forest is employed as a classifier for performing the tachycardias classification. The computer numerical simulation results show that the performance based on both the generalized mean frequency and the generalized frequency variance is better than that based on both the conventional mean frequency and the conventional frequency variance.

This paper proposes a fractional singular spectrum analysis (SSA)-based method for performing the fractional delay. First, the input sequence is divided into two overlapping sequences with the first sequence being the input sequence without its last point and the second sequence being the input sequence without its first point. Then, the singular value decompositions (SVD) are performed on the trajectory matrices constructed based on these two sequences. Next, the designs of both the right unitary matrix and the left unitary matrix for generating the new trajectory matrix are formulated as the quadratically constrained quadratic programing problems. The analytical solutions of these quadratically constrained quadratic programing problems are derived via the SVD approach. Finally, the fractional SSA components are obtained by performing the diagonal averaging operation, and the fractional delay sequence is obtained by summing up all the fractional SSA components together. Since the fractional SSA operations are nonlinear and adaptive, our proposed method is a kind of nonlinear and adaptive approach for performing the fractional delay. Besides, by discarding some fractional SSA components, the joint fractional delay operation and the denoising operation can be performed simultaneously.

This paper proposes a singular spectrum analysis (SSA)-based hierarchical multiresolution analysis (HMA) with the exploitation of the frequency selectivities of the desirable grouped functions. To perform the HMA, the SSA components are grouped based on the desirable grouped functions. Similar procedures are applied to the sum of the SSA components in a group in the previous level of decomposition. Computer numerical simulation results show that the SSA components in the next level of decomposition are localized within the passband of the sum of the SSA components in the corresponding group in the previous level of decomposition if its intrinsic mode functions (IMFs) or the ideal filters are employed as the desirable grouped functions. Moreover, unlike the empirical mode decomposition (EMD)-based HMA, the total number of the SSA components in each level of decomposition can be chosen.

Continuous monitoring of the blood glucose levels is essential and critical for controlling diabetes and its complications. With the improvement of the measurement accuracy of the acquisition devices developed in recent decades, developing the optical-based methods for performing the non-invasive blood glucose estimation for the consumer applications becomes very important. The authors’ previous work is based on the heart rate variability of the electrocardiogram and the existing method is based on applying the random forest to the features extracted from the photoplethysmogram. However, the accuracies of these two methods are not very high. In this study, a joint empirical mode decomposition and exponential function estimation approach is proposed for estimating the mean value of a photoplethysmogram acquired from a wearable non-invasive blood glucose device. Also, the exponential function fitting approach is employed for estimating the blood glucose levels via an L 1 norm formulation. The computer numerical simulation results show that the estimation accuracy based on their proposed method is higher than that based on the state-of-the-art methods. Therefore, their proposed method can be employed for performing blood glucose estimation effectively.

The quaternion-valued signals consist of four signal components. The discrete quaternion Fourier transform is to map these four signal components in the time domain to that in the frequency domain. These four signal components in the frequency domain are called the discrete quaternion Fourier transform components. There are a total of 16 inner products among any two discrete quaternion Fourier transform components. The total orthogonal error among the discrete quaternion Fourier transform components is defined based on these 16 inner products. This study aims to find the optimal quaternion number in the discrete quaternion Fourier transforms so that the total orthogonal errors among the discrete quaternion Fourier transform components are minimised. It is worth noting that finding the optimal quaternion number in the discrete quaternion Fourier transform is equivalent to finding the optimal rescaling factors. Since the discrete quaternion Fourier transform components are expressed in terms of the high-order polynomials of the trigonometric functions of the rescaling factors, this optimisation problem is non-convex. To address this problem, a two-stage approach is employed for finding the solution to the optimisation problem. The comparison results show that the authors proposed method outperforms the existing methods in terms of achieving the low total orthogonal error among the discrete quaternion Fourier transform components.

This paper proposes a spherical coordinate-based kernel principal component analysis (PCA). Here, the kernel function is the nonlinear transform from the Cartesian coordinate system to the spherical coordinate system. In particular, first, the vectors represented in the Cartesian coordinate system are expressed as those represented in the spherical coordinate system. Then, certain rotational angles or the radii of the vector are set to their corresponding mean values. Finally, the processed vectors represented in the spherical coordinate system are expressed back in the Cartesian coordinate system. As the degrees of the freedoms of the processed vectors represented in the spherical coordinate system are reduced, the dimension of the manifold of the processed vectors represented in the Cartesian coordinate system is also reduced. Moreover, since the conversion between the vectors represented in the Cartesian coordinate system and those represented in the spherical coordinate system only involves some elements in the vectors, the required computational power for the conversion is low. Computer numerical simulation results show that the mean squares reconstruction error via the spherical coordinate-based kernel PCA is lower than that via the conventional PCA. Also, the required computational power is significantly reduced.

The conventional perceptron with the sign type activation function can be used for performing the linearly separable pattern recognition with its weight vector being found by the conventional perceptron training algorithm. On the other hand, the perceptron with the multi-pulse type activation function can be used for performing the piecewise linearly separable pattern recognition. This paper proposes a training algorithm for finding its weight vector. Moreover, some application examples of this perceptron are given for the demonstration purpose.

The conventional singular spectrum analysis is to divide a signal into segments where there is only one non-overlapping point between two consecutive segments. By putting these segments into the columns of a matrix and performing the singular value decomposition on the matrix, various one dimensional singular spectrum analysis vectors are obtained. Since different one dimensional singular spectrum analysis vectors represent different parts of the signal such as the trend part, the oscillation part and the noise part of the signal, the singular spectrum analysis plays a very important role in the nonlinear and adaptive signal analysis. However, as the length of each one dimensional singular spectrum analysis vector is the same as that of the original signal, there is a redundancy among these one dimensional singular spectrum analysis vectors. In order to reduce the required computational power and the required units for the memory storage for performing the singular spectrum analysis, this article proposes a method to reduce the total number of the elements of all the one dimensional singular spectrum analysis vectors. In particular, the length of the shift block for generating the trajectory matrix is increased from one to a positive integer greater than one under a certain criterion. In this case, the total number of the columns of the trajectory matrix is reduced. As a result, the total number of the off-diagonals of all the two dimensional singular spectrum analysis matrices is reduced. Hence, the total number of the elements of all the one dimensional singular spectrum analysis vectors is reduced. In order to guarantee exact perfect reconstruction, this article reformulates the de-Hankelization process. In particular, the first element of the off-diagonals of all the two dimensional singular spectrum analysis matrices is taken as the elements of the one dimensional singular spectrum analysis vectors. Exact perfect reconstruction condition is derived. Simulations show that exact perfect reconstruction can be achieved while the total number of the elements of all the one dimensional singular spectrum analysis vectors is significantly reduced.

Denoising is the first step in both the QRS complex detection and the beat classification. However, infinite impulse response filters usually exhibit nonlinear phase responses. As a result, the group delays of the output signals based on the infinite impulse response filtering are different at different time instants. This causes the inaccuracies of the estimations of the RR intervals of the electrocardiograms. In this paper, the denoising is performed based on the singular spectrum analysis approach. The linear phase properties of the singular spectrum analysis components are investigated. Finally, the RR intervals of the denoised electrocardiograms are estimated and those obtained based on the infinite impulse response filtering approach are compared. The computer numerical simulation results show that the estimation performance based on the singular spectrum analysis approach outperforms that based on the infinite impulse response filtering approach.

For a sparse optimization problem with an L 1 norm objective function subject to an L ∞ norm inequality constraint, this paper finds that there is a piecewise linear relationship between the L 1 norm objective functional values and the L ∞ norm constraint specifications. This piecewise linear relationship is proved mathematically. Also, computer numerical simulations on a set of signal vectors verify the validity of this result. This result can provide a guidance for system analysts to define the specification on the L ∞ norm specification.

Nearly bandlimited signals play an important role in the biomedical signal processing community. The common method to analyze these signals is via the empirical mode decomposition approach which decomposes the non-stationary signals into the sums of the intrinsic mode functions. However, this method is computational demanding. A natural idea to reduce the computational cost is via the block processing. However, the severe boundary effect would happen due to the discontinuities between two consecutive blocks. In order to solve this problem, this paper proposes to realize the parallel implementation via polyphase representation. That is, the empirical mode decomposition is implemented on each polyphase component of the original signal. Then each sub-signals are combined after upsampling. The simulation results show that our proposed method achieves the approximate intrinsic mode functions both qualitatively and quantitatively very close to the true intrinsic mode functions. Besides, compared with the conventional block processing method which significantly suffered from the boundary effect problem, our proposed method does not have this issue.

In recent years, the signal processing opportunities with the multi-channel recording and the high precision detection provided by the development of new extracellular multi-electrodes are increasing. Hence, designing new spike sorting algorithms are both attractive and challenging. These algorithms are used to distinguish the individual neurons’ activity from the dense and simultaneously recorded neural action potentials with high accuracy. However, since the overlapping phenomenon often inevitably arises in the recorded data, they are not accurate enough in practical situations, especially when the noise level is high. In this paper, a spike feature extraction method based on the wavelet packets’ decomposition and the mutual information is proposed. This is a highly accurate semi-supervised solution with a short training phase for performing the automation of the spike sorting framework. Furthermore, the evaluations are performed on different public datasets. The raw data are not only suffered from multiple noises (from 5% level to 20% level) but also includes various degrees of overlapping spikes at different times. The clustering results demonstrate the effectiveness of our proposed algorithm. In addition, it achieves a good anti-noise performance with ensuring a high clustering accuracy (up to 99.76%) compared with the state-of-the-art methods.

Moving object detection is a fundamental and necessary step in many computer vision algorithms. These algorithms are built in many intelligent devices such as in the smartphones, the tachographs and the personal video recorders. Recently, the methods for performing the moving object detection based on the low-rank representation have been proposed. For these methods, it is assumed that the background is represented by a low-rank matrix. On the other hand, the foreground objects cannot be represented by low-rank matrices. They are seen as the outliers. Hence, detecting the contiguous outliers in the low-rank representation (DECELOR) can be formulated as an extension of the robust principal component analysis problem. This method fully utilizes the spatial continuity of the foreground regions. To achieve a more accurate detection, this paper integrates both the concave penalty function and the priori target rank information into a single optimization problem based on the DECELOR formulation. The optimization problem is efficiently solved by an alternating direction scheme. The computer numerical simulation results on the real-world scenes demonstrate the superiority of our method in terms of the effective handling of a wide range of complex scenarios.

A signal can be represented as the sum of the intrinsic mode functions via performing the empirical mode decomposition. For the discrete time signals, the lengths of the intrinsic mode functions are equal to the lengths of the input signals. As there is usually more than one intrinsic mode function, the total numbers of discrete points of all the intrinsic mode functions are usually more than the lengths of the input signals. In other words, the empirical mode decomposition is an oversampled representation. For some applications such as the compression application, the oversampled representation is not preferred. Therefore, this paper proposes an optimal adaptive nonuniform filter bank design approach for performing the decimations on the intrinsic mode functions. In particular, the passbands of the intrinsic mode functions are estimated based on an adaptive gradient algorithm. Then, the intrinsic mode functions are nonuniformly downsampled and upsampled with the sampling integers derived based on their estimated passbands. Next, a bank of filters is employed to reconstruct the original signal. Here, the passbands of the filters are also derived based on those of the intrinsic mode functions. After that, the nonuniform filter bank design problem is formulated as a semi-infinite programming problem such that the total ripple energies of all the filters are minimized subject to the specifications on the absolute maximum values of both the real part and the imaginary part of the reconstruction error. The semi-infinite programming problem is approximated by a semi-definite programming problem. Computer numerical simulation results show that our proposed system could achieve a very small reconstruction error at a very small oversampling ratio.

This study proposes an augmented Lagrange multiplier-based method to perform the singular spectral analysis-based denoising without computing the singular values. In particular, the one-dimensional (1D) signal is first mapped to a trajectory matrix using the window length L . Second, the trajectory matrix is represented as the sum of the signal dominant matrix and the noise-dominant matrix. The determination of these two matrices is formulated as an optimisation problem with the objective function being the sum of the rank of the signal dominant matrix and the norm of the noise-dominant matrix. This study employs the Schatten q-norm operator with and the double nuclear-norm penalty for approximating the rank operator as well as the minimum concave penalty (MCP)-norm operator for approximating the -norm operator. Third, some auxiliary variables are introduced and the augmented Lagrange multiplier algorithm is applied to find the optimal solution. Finally, the 1D denoised signal is obtained by applying the diagonal averaging method to the obtained signal dominant matrix. Computer numerical simulation results show that the authors' proposed method outperforms the existing methods.

An approximated empirical mode decomposition generates a set of approximated intrinsic mode functions via a linear, nonadaptive but iterative approach. The decomposition was found to be very useful for a content-independent pattern recognition application. As the process is characterized by a system kernel matrix and performed iteratively, the approximated intrinsic mode functions can be understood as the original signals processed by a set of mask operations. Here, some properties of the decomposition are studied and an optimal design of the system kernel matrix is proposed. It is found that there is only one approximated intrinsic mode function if the exact perfect reconstruction condition is satisfied. Obviously, the approximated intrinsic mode function is the original signal. Therefore, the decomposition is practically not meaningful. To address this issue, the infinite number of iterations in the algorithm is truncated to a finite number of iterations. Also, the design of the system kernel matrix is formulated as an optimization problem. In particular, the exact perfect reconstruction error between the sum of the approximated intrinsic mode functions and the original signal is minimized and the total absolute sum of the difference between any two different eigenvalues of the iterative matrix is maximized subject to a stability condition. Here, the stability condition refers to the eigenvalues of the iterative matrix being between zero and one. Since the optimization problem is nonsmooth and nonconvex, a genetic algorithm is employed for finding its near global optimal solution. Compared to the conventional approximated empirical mode decomposition, computer numerical simulations show that our proposed approach can achieve more than one approximated intrinsic mode function with each approximated intrinsic mode function corresponding to an output of a more meaningful mask operation.

Time series usually consist of an underlying trend and the irregularities. Therefore, the underlying trend extraction plays an important role in the analysis of the time series. This paper proposes a method which combines the ensemble intrinsic timescale decomposition (EITD) algorithm and the matching pursuit (MP) approach for performing the underlying trend extraction. In order to extract the underlying trend, the EITD algorithm is applied to obtain a set of components. Then, the first component which contains most of the noise is removed. Next, some appropriate components are selected by the MP approach. In particular, the total number of components that composites of the underlying trend is minimized. To guarantee that the underlying trend tracks the signal, it is required to impose a constraint on the maximum absolute error between the underlying trend and the denoised signal. As the selection of the EITD components is binary, this component selection problem is formulated as an $$ L_{0} $$ -norm binary programming problem subject to the specification on the maximum absolute error between the denoised signal and the underlying trend. This optimization problem is further solved by the MP approach. Finally, the underlying trend is constructed. Compared with the conventional empirical mode decomposition algorithm and the ensemble empirical mode decomposition algorithm, our proposed approach could extract a better underlying trend for some random time series.

This paper proposes to employ the L1 norm criterion to perform the de-Hankelization in the singular spectrum analysis (SSA). In particular, the represented values of the off-diagonals in the two-dimensional SSA matrices are found via minimizing the L1 norm errors of the vectors defining as the absolute differences between the off-diagonal vectors and the vectors with all their elements being the represented values. This results to reduce the total number of the large-valued elements in the error vectors. Also, this paper guarantees to achieve the exact perfect reconstruction of the original signal. As the formulated problem is a standard linear programming problem, the solution could be efficiently found via the simplex method. The computer numerical simulations verify the results.

This study proposes a computer cryptographic system through performing the chaotic modulation on the intrinsic mode functions with a non-dyadic number of the encrypted signals. First, the empirical mode decomposition is applied to an input signal to generate a set of intrinsic mode functions. Then, these intrinsic mode functions are categorised into two groups of signals. Next, a type 1 polyphase is employed to represent each group of signals. These polyphase components are combined to generate a non-dyadic number of polyphase components. Second, the chaotic modulation is applied to these combined polyphase components for performing the encryption in the time frequency domain. To reconstruct the original signal, first, the chaotic demodulation is applied to the encrypt components to reconstruct the combined polyphase components. Then, the original groups of intrinsic mode functions are reconstructed through the type 2 polyphase representation and the original signal is reconstructed. Compared with the chaotic filter bank system, the proposed approach enjoys the nonlinear and adaptive property of the empirical mode decomposition. Therefore, a better security performance can be achieved particularly for the non-stationary signals. Compared with the conventional chaotic modulation approach, the proposed system allows performing the cryptography in the time frequency domain.

The concept of mask operation in fractional Fourier domains is a generalization of the conventional Fourier-based filtering in the frequency domain. It is known that simultaneously employing multiple mask operations in multiple different fractional Fourier domains can lead to significant performance advantages when compared with just employing a single mask operation in a single fractional Fourier domain. However, there is no systematic scheme for optimal joint design of the discrete fractional Fourier transform (DFrFT) matrices and the corresponding sets of mask coefficients. In this paper, we consider this design problem and construct a formulation that does not depend on the knowledge of noise statistics. We then develop an iterative algorithm, which is a hybrid descent (HD) approach, to solve the formulated optimization problem. For this HD approach, a gradient descent method is supplemented by a modified simulated annealing algorithm. It is employed to find the global optimal rotation angles of the DFrFT matrices. During the iterative process, the corresponding sets of mask coefficients can be constructed analytically. Simulation results demonstrate that the proposed scheme is highly effective.

This paper proposes to use a set of discrete fractional Fourier transform (DFrFT) matrices with different rotational angles to construct an overcomplete kernel for sparse representations of signals. The design of the rotational angles is formulated as an optimization problem as follows. The sum of the L1 norms of both the real part and the imaginary part of transformed vectors is minimized subject to different values of the optimal rotational angles. In order to avoid all the optimal rotational angles within a small neighbourhood, constraints on the sum of the L1 norms of both the real part and the imaginary part of the product of the individual optimal DFrFT matrices and training vectors being either stationary or nondifferentiable are imposed. Solving this optimization problem is very challenging not only because of the nonsmooth and the nonconvex nature of the problem, but also due to expressing the optimization problem in a nonstandard form. To solve the problem, first it is shown in this paper that this design problem is equivalent to an optimal sampling problem as follows. The absolute sum of the L1 norms of both the real part and the imaginary part of the frequency responses of a set of filters at the optimal sampling frequencies is minimized subject to similar constraints. Second, it is further shown that the optimal sampling frequencies are the roots of a set of harmonic functions. As the frequency responses of the filters are required to be computed only at frequencies in a discrete set, the globally optimal rotational angles can be found very efficiently and effectively.

This study proposes an integer quadratic programming method for grouping and selecting the singular spectral analysis components based on the empirical mode decomposition for performing the denoising. Here, the total number of the grouped singular spectral analysis components is equal to the total number of the intrinsic mode functions. The singular spectral analysis components are assigned to the group indexed by the corresponding intrinsic mode function where the two norm error between the corresponding intrinsic mode function and the sum of the grouped singular spectral analysis components is minimum. Actually, this assignment of the singular spectral analysis components to a particular group is an integer quadratic programming problem. However, the required computational power for finding the solution of the integer quadratic programming problem is high. On the other hand, by representing the integer quadratic programming problem as an integer linear programming problem and employing an existing numerical optimisation computer aided design tool for finding the solution of the integer linear programming problem, the solution can be found efficiently. Computer numerical simulation results are presented.

This paper proposes a threshold-free method for grouping and selecting the singular spectrum analysis (SSA) components for performing the signal denoising via the empirical mode decomposition (EMD) approach. First, the total number of the groups of the SSA components is selected to be the same as the total number of the intrinsic mode functions (IMFs) of the signal. The SSA components are assigned to the group where the absolute correlation coefficient between the IMF and the SSA component is the highest. This grouping method is implemented using the matching pursuit algorithm. Then, the groups of the SSA components are selected based on the selection criterion used in an existing EMD denoising method. As the EMD denoising approach is a time-domain approach and the SSA components are represented in the transformed domain, our proposed method exploits both the time-domain and the transformed-domain information for performing the denoising. Computer numerical simulation results show that the signal-to-noise ratios of common practical signals denoised by our proposed method are higher than those denoised by the existing methods.

Brain computer interface (BCI) is a medium that converts the brain activity signals such as the electroencephalograms (EEGs) to the motion control signals such as the wrist movement signals. Recently, the magnetoencephalograms (MEGs) are used to record the brain activities representing the wrist movements of the healthy right handed subjects. Here, there are four types of the wrist movements. They are the right movements, the forward movements, the left movements and the back movements of the wrist. Since the MEGs are not the monotonic frequency signals, one of the major challenges is the difficulty to extract the features from the pieces of a finite duration of the MEGs for classifying the wrist movements. In order to overcome this challenge, the noise assisted multivariate empirical mode decomposition (NA-MEMD) is proposed. There are three major steps for our proposed NA-MEMD based algorithm. The first step is to employ the NA-MEMD for performing the multi-channel and the multi-scale signal denoising. The second step is to extract the statistical features such as the mean and the variance as well as the time frequency features such as the marginal spectrum entropy, the multi-scale permutation entropy and the multi-scale fuzzy entropy. Finally, some typical classifiers such as the random forest (RF), the back propagation neural network (BPNN) and the support vector machine (SVM) as well as some advanced classifiers such as the extreme learning machine (ELM), the random vector functional link (RVFL) and the long short term memory network (LSTM) are employed for performing the classification. The computer numerical simulation results show that our proposed NA-MEMD based algorithm achieves a higher classification accuracy compared to the state of the art methods without the multi-channel and the multi-scale analysis and the semi-improved methods with either the multi-channel or the multi-scale analysis.

Sleeping stage classification plays an important role in the diagnosis and the treatment of the sleeping related diseases. This paper proposes a joint quaternion valued singular spectrum analysis (QSSA) and ensemble empirical mode decomposition (EEMD) based method for performing the sleeping stage classification. First, the fast Fourier transform (FFT) is employed for decomposing the electroencephalograms (EEGs) into various waves. Then, both the QSSA and the EEMD are applied for denoising these waves. Finally, the features are extracted from the selected components and the bootstrap aggregating (bagging) classifier is employed for performing the sleeping stage classification. Sixteen sleeping records obtained from both the Sleep-EDF database and the Sleep-EDF expanded database in the Physio bank are employed for performing the evaluation. A statistical analysis is conducted. The computer numerical simulation results show that the accuracies achieved by our proposed algorithm for the 6 state stage classification to the 2 state stage classification based on the EEGs in the Sleep-EDF database are 89.39%, 90.66%, 94.09%, 94.17%, and 97.50%, respectively. Moreover, the classification accuracies in the Sleep-EDF expanded database are also very high. Also, the classification accuracies achieved by our proposed method are higher than those achieved by the existing methods. This demonstrates the effectiveness of our proposed method.

The Fisher linear discriminant analysis (FLDA)-based method is a common method for jointly optimizing the intraclass separation and the interclass separation of the projected feature vectors by defining the objective function as the ratio of the intraclass separation over the interclass separation. To address the eigenproblem of the FLDA, a quadratic equality constraint is imposed on the square of the $$l_{2}$$ norm of the decision vector. However, the constrained optimization problem is highly nonconvex. This paper proposes to reformulate the objective function as a weighted sum of the intraclass separation and the interclass separation subject to the same quadratic equality constraint on the square of the $$l_{2}$$ norm of the decision vector. Although both the objective function and the feasible set of the optimization problem are still nonconvex, this paper shows that the global minimum of the objective functional value is equal to the minimum singular value of the Hessian matrix of the objective function. Also, the globally optimal solution of the optimization problem is in the null space of the Hessian matrix minus this singular value multiplied by the identity matrix. As it is only required to find the singular value of the Hessian matrix, no numerical optimization-based computer aided design tool is required to find the globally optimal solution. Therefore, the globally optimal solution can be found in real time. Experimental results demonstrate the effectiveness of our proposed method.

This study presents a new property of the filter bank-based multicarrier (FBMC) system. Also, an efficient iterative algorithm for designing the system with a large number of subcarriers and a prototype filter with a very long length are proposed. For the system, the compact from conditions are derived for both the intersymbol interference free and the interchannel interference (ICI) free. Based on these new conditions, the design of the prototype filter is formulated as an unconstrained optimisation problem where the objective function is the weighted sum of the total distortion of the system and the stopband energy. By deriving the gradient vector of the objective function, an efficient iterative algorithm is proposed for finding the solution of the optimisation problem. In addition, an efficient matrix inversion approach is presented to greatly reduce the computational complexity of the iterative algorithm. As a result, it is feasible to design the FBMC system with thousands of subcarriers. The convergence of the iterative algorithm is proved. Computer numerical simulation results with the comparisons to the existing methods are presented. It is shown that the proposed design algorithm is more effective and efficient than the existing methods.

This paper proposes a continuous time irrational filter structure via a set of the fractional order Gammatone components instead of via a set of integer order Gammatone components. The filter design problem is formulated as a nonsmooth and nonconvex infinite constrained optimization problem. The nonsmooth function is approximated by a smooth operator. The domain of the constraint functions is sampled into a set of finite discrete points so the infinite constrained optimization problem is approximated by a finite constrained optimization problem. To find a near globally optimal solution, the norm relaxed sequential quadratic programming approach is applied to find the locally optimal solutions of this nonconvex optimization problem. The current or the previous locally optimal solutions are kicked out by adding the random vectors to them. The locally optimal solutions with the lower objective functional values are retained and the locally optimal solutions with the higher objective functional values are discarded. By iterating the above procedures, a near globally optimal solution is found. The designed filter is applied to perform the denoising. It is found that the signal to noise ratio of the designed filter is higher than those of the filters designed by the conventional gradient descent approach and the genetic algorithm method, while the required computational power of our proposed method is lower than those of the conventional gradient descent approach and the genetic algorithm method. Also, the signal to noise ratio of the filter with the fractional order Gammatone components is higher than those of the filter with the integer order Gammatone components and the conventional rational infinite impulse response filters.

Most of the existing works on the studies of the virus are focused on finding its genomic sequences and its clinical presentations as well as its effects on the patients via monitoring the computer tomography images. However, the molecular properties of the drugs for suppressing the virus have rarely been studied. This paper addresses this issue. In particular, the total number of the hydrogen bonds in a drug in which these hydrogen bonds have their dominant intermolecular forces bonded to some strongly electronegative atoms is classified. Here, 10 molecular properties are extracted from each atom in the drug and they are used as the features to generate a feature matrix for performing the classification. However, as different drugs have different total numbers of the atoms in the molecules, the dimensions of the feature matrices are not unified. This paper proposes to employ the discrete cosine transform based zero padding method to tackle this issue. Moreover, as the dimensions of the feature matrices are very large, the traversal based feature selection method is employed to reduce the dimensions of the feature matrices. Finally, the random forest based classifier is employed for performing the classification. It is found that our proposed method can achieve the accuracy and the macro F1 score up to 81.69 % and 78.93 %, respectively. The obtained results are useful for the drug discovery and the development of the vaccine for suppressing the virus.

Thinning algorithms often cause stroke distortions at the crosses or intersections of strokes, which lead to bad results in pattern recognition tasks. In order to overcome these drawbacks, this paper proposes a parallel thinning algorithm based on stroke continuity detection. In the algorithm, before it uses the conditions of parallel algorithms to delete a boundary point, it first detects whether the boundary point is a reserved point to keep stroke’s continuity or not. Consequently, it can produce a skeleton with good symmetry, control the large deformation at the cross or intersection of strokes, and make a better skeleton more quickly. Moreover, it is practically immune to noise.

Nowadays, signals are transmitted using different digital modulation systems according to different channel conditions. Different digital modulation systems transmit signals using different frequency bands. To demodulate the signals and decode the corresponding digital symbols, different filters with different frequency bands are employed. This paper explores the possibility of performing the digital demodulation in a general energy preserved transform domain instead of in the conventional frequency domain. In particular, instead of performing the multiplication in the frequency domain for performing the conventional filtering, this paper proposes to perform the multiplication in an optimal Hermitian transform domain for performing the optimal mask operation. The joint design of the Hermitian transform matrix and the corresponding mask coefficients is formulated as a least squares optimization problem subject to the Hermitian condition. This is actually a complex-valued quadratic matrix equality constrained least squares optimization problem. A condition on an optimal solution is derived via a singular value decomposition approach. Based on the derived condition, an iterative approach is proposed for finding the solution of the optimization problem. Computer numerical simulation results show that our proposed approach outperforms the conventional filtering approach.

Discrete fractional Fourier transform (DFrFT) is a powerful signal processing tool. This paper proposes a method for DFrFT-based image retrieval via Fisher discriminant and 1-NN classification rule. First, this paper proposes to extend the conventional discrete Fourier transform (DFT) descriptors to the DFrFT descriptors to be used for representing the edges of images. The DFrFT descriptors extracted from the training images are employed to construct a dictionary, for which the corresponding optimal rotational angles of the DFrFTs are required to be determined. This dictionary design problem is formulated as an optimization problem, where the Fisher discriminant is the objective function to be minimized. This optimization problem is nonconvex (Guan et al. in IEEE Trans Image Process 20(7):2030---2048, 2011; Ho et al. in IEEE Trans Signal Process 58(8):4436---4441, 2010). Furthermore, both the intraclass separation and interclass separation of the DFrFT descriptors are independent of the rotational angles if these separations are defined in terms of the 2-norm operator. To tackle these difficulties, the 1-norm operator is employed. However, this reformulated optimization problem is nonsmooth. To solve this problem, the nondifferentiable points of the objective function are found. Then, the stationary points between any two consecutive nondifferentiable points are identified. The objective function values are evaluated at these nondifferentiable points and these stationary points. The smallest L objective function values are picked up and the corresponding rotational angles are determined, which are then used to construct the dictionary. Here, L is the total number of the rotational angles of the DFrFTs used to construct the dictionary. Finally, an 1-NN classification rule is applied to perform the image retrieval. Application examples and experimental results show that our proposed method outperforms the conventional DFT approach.

This paper proposes a novel content removal technique for enhancing the camera identification performance. Here, very low bit rate videos with the overall noise patterns having time-varying statistics are considered. First, different two-dimensional discrete fractional Fourier transforms with different rotational angles are applied to the overall noise pattern of each frame of each video. Second, the modulus of each element of each transformed matrix is normalized to one if the rotational angles of the transforms are not equal to the integer multiples of $$\pi $$ . Third, the corresponding two-dimensional inverse discrete fractional Fourier transform is applied to each normalized matrix, and the corresponding real part is taken out for the further processing. Fourth, the absolute values of the elements in each normalized real-valued matrix are bounded by certain threshold values. Here, different threshold values are employed for different rotational angles. Finally, the processed matrices are averaged over all the rotational angles and all the frames of the videos of the same camera. To evaluate the performance, the correlation function is employed. Extensive computer numerical simulations are preformed. The obtained results show that the proposed method outperforms existing methods (Kang et al. in IEEE Trans Inf Forensics Secur 7(2):393–402, 2012; Li in IEEE Trans Inf Forensics Secur 5(2):280–287, 2010).

The paper proposes an efficient method for solving a one- norm equality constrained optimization problem. In fact, this kind of optimization problems is nonconvex. First, the problem is formulated as the least absolute shrinkage and selection operator (LASSO) optimization problem. Then, it is solved by iterative shrinkage algorithms such as the fast iterative shrinkage thresholding algorithm. Next, the solution of the LASSO optimization problem is employed for formulating the constraint of the corresponding least-squares constrained optimization problem. The solution of the least-squares constrained optimization problem is taken as a near globally optimal solution of the one-norm equality constrained optimization problem. The main advantage of this proposed method is that a solution with both lower one-norm constraint error and two-norm reconstruction error can be obtained compared to those of the LASSO problem, while the required computational power is significantly reduced compared to the full search approach. Computer numerical simulation results are illustrated.

This study proposes an iterative method to approximate an N-dimensional optimisation problem with a weighted L p and L 2 norm objective function by a sequence of N independent one-dimensional optimisation problems. Inspired by the existing weighted L 1 and L 2 norm separable surrogate functional (SSF) iterative shrinkage algorithm, there are N independent one-dimensional optimisation problems with weighted L p and L 2 norm objective functions. However, these optimisation problems are non-convex. Hence, they may have more than one locally optimal solutions and it is very difficult to find their globally optimal solutions. This paper proposes to partition the feasible set of each approximated problem into various regions such that the sign of the convexity of the objective function in each region remains unchanged. Here, there is no more than one stationary point in each region. By finding the stationary point in each region, the globally optimal solution of each approximated optimisation problem can be found. Besides, this study also shows that the sequence of the globally optimal solutions of the approximated problems converge to the globally optimal solution of the original optimisation problem. Computer numerical simulation results show that the proposed method outperforms the existing weighted L 1 and L 2 norm SSF iterative shrinkage algorithm.

This study proposes an iterative method to approximate an N-dimensional optimisation problem with a weighted Lp norm and L2 norm objective function by a sequence of N independent one-dimensional optimisation problems. This iterative method is inspired by the existing weighted L1 norm and L2 norm separable surrogate functional (SSF) iterative shrinkage algorithm. However, as these independent one-dimensional optimisation problems consist of weighted Lp norm and L2 norm objective functions, these optimisation problems are non-convex and they may have more than one locally optimal solutions. In general, it is very difficult to find their globally optimal solutions. To address this difficulty, this study proposes to partition the feasible set of each approximated problem into various regions such that the sign of the convexity of the objective function in each region remains unchanged. In this case, there is no more than one stationary point in each region. By finding the stationary point in each region, the globally optimal solution of each approximated optimisation problem can be found. Besides, this study also shows that the sequence of the globally optimal solutions of the approximated problems converge to the globally optimal solution of the original optimisation problem.

This paper studies the effectiveness of applying the bio-electronic stimulation therapy to suppress the syndromes of the Parkinson’s diseases. Here, eight patients suffering from the Parkinson’s diseases are invited for performing the investigation. The patients wear the wearable devices before and after applying the bio-electronic stimulation therapy for acquiring the corresponding resting tremor signals. First, the DC means of these signals are removed. Then, these zero mean signals are normalized to the unit energy signals. Second, the singular spectrum analysis is performed and the sets of singular spectrum analysis components are obtained. Next, the singular spectrum analysis components corresponding to the voluntary movements and the noises from various sources are removed. Third, the fast Fourier transforms are computed on the sums of the rest of singular spectrum analysis components. Then, the signal components between 3 Hz and 8 Hz are extracted out. Next, the energies of the signal components are computed for performing the analysis. The computer numerical simulation results show that the energies of the tremors after applying the bio-electronic stimulation therapy are lower than those before applying the bio-electronic stimulation therapy for all these eight patients. Compare to the processing without performing the singular spectrum analysis, the energies of the tremors after applying the bio-electronic stimulation therapy are lower than those before applying the bio-electronic stimulation therapy only for seven patients. This verifies that the singular spectrum analysis approach is effective for performing the analysis and the bio-electronic stimulation therapy is effective for suppressing the syndromes of the Parkinson’s diseases.

Electroencephalograms (EEGs) play a significant role in both the detection and the prediction of the epileptic seizures. This paper proposes to employ the logarithmic normalized functional singular values as the features for performing the classification of both the two class problem (normal and seizure set) and the three class problem (normal, seizure free and seizure set). Here, the EEGs are taken from two well known datasets. First, each EEG is decomposed via the singular spectrum analysis (SSA) to obtain the singular values. Then, the logarithmic normalized functional values of these singular values are calculated to form the feature vectors for performing the classification of the epileptic seizure. Next, different classifiers including the support vector machine (SVM), the k nearest neighbor classifier, the extreme learning machine (ELM) and the artificial neural network (ANN) are employed for performing the classification. Finally, a ten fold cross validation procedure is employed to ensure the reliability and the stability of the classifiers. The computer numerical simulation results show that the k nearest neighbor classifier achieves the best performance compared to other classifiers for performing both the two class epileptic classification and the three class epileptic classification. Our proposed method also achieves the higher classification accuracies compared to the case without performing the logarithmic normalized operation. Moreover, the performance of our proposed method is evaluated on the original EEGs under the white noise environment at different signal to noise ratios. Similar results are obtained.

This paper proposes a design of a nonuniform transmultiplexer with block samplers and single-input single-output linear time-invariant filters. First, the perfect reconstruction condition of the nonuniform transmultiplexer is derived. Then, the design problem is formulated as an optimization problem. In particular, the perfect reconstruction error is minimized in the $$L_{1}$$L1 norm sense subject to the frequency selectivities of the filters. Computer numerical simulation results show that the designed nonuniform transmultiplexer with block samplers is robust to the channel noise and achieves a small reconstruction error.

Although the expected patch log likelihood (EPLL) achieves good performance for denoising, an inherent nonadaptive problem exists. To solve this problem, an adaptive learning is introduced into the EPLL in this paper. Inspired from the structured sparse dictionary, an adaptive Gaussian mixture model (GMM) is proposed based on patch priors. The maximum a posteriori estimation is employed to cluster and update the image patches. Also, the new image patches are used to update the GMM. We perform these two steps alternately until the desired denoised results are achieved. Experimental results show that the proposed denoising method outperforms the existing image denoising algorithms.