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Fourier methods used in two‐ and three‐dimensional image reconstruction can be used also in reconstructability analysis (RA). These methods maximize a variance‐type measure instead of information‐theoretic uncertainty, but the two measures are roughly collinear and the Fourier approach yields results close to that of standard RA. The Fourier method, however, does not require iterative calculations for models with loops. Moreover, the error in Fourier RA models can be assessed without actually generating the full probability distributions of the models; calculations scale with the size of the data rather than the state space. State‐based modeling using the Fourier approach is also readily implemented. Fourier methods may thus enhance the power of RA for data analysis and data mining.

In this paper, two transform methods, the Fourier transform (FT) and the wavelet transform (WT) methods, are utilized to process moiré fringes for the strain analysis of electronic packaging. With the introduction of fringe carriers, those transform techniques need only one fringe pattern for each deformation state. The strain modulation to the carrier frequency can be subtracted by filtering as the pattern is transformed into spectrum domain by the fast‐FT processing, and the deformation field can thus be restored by the inverse FT transform after spectral shifting. The WT method expands the pattern information involved in the fringe carrier in both spatial domain and spectral domain to analyze the deformation distribution in this combined space. By changing the transform scales in the processing, the wavelet transform offers multi‐resolution analysis for the deformation field with high gradients.

An introduction is given to the generation and use of new transform techniques which have important applications in binary control and processing methods. A comparison is made between the fast Fourier transform and the equivalent fast Walsh transform together with the steps required to produce a transform algorithm and computer program. Some applications of the transform are then discussed and which include spectral analysis, filtering, non‐linear control and communications uses. 18 references to current work in these applications areas are included.

This paper aims to explore a new way to extract the fault feature of a rolling bearing signal on the basis of a combinatorial method.

By combining local mean decomposition (LMD) with Teager energy operator, a new feature-extraction method of a rolling bearing fault signal was proposed, called the LMD–Teager transform method. The principles and steps of method are presented, and the physical meaning of the time–frequency power spectrum and marginal spectrum is discussed. On the basis of comparison with the fast Fourier transform method, a simulated non-stationary signal was processed to verify the effect of the new method. Meanwhile, an analysis was conducted by using the recorded vibration signals which include inner race, out race and bearing ball fault signal.

The results show that the proposed method is more suitable for the non-stationary fault signal because the LMD–Teager transform method breaks through the difficulty of the Fourier transform method that can process only the stationary signal. The new method can extract more useful information and can provide better analysis accuracy and resolution compared with the traditional Fourier method.

Combining the advantage of the local mean decomposition and the Teager energy operator, the LMD–Teager method suits the nature of the fault signal. A marginal spectrum obtained from the LMD–Teager method minimizes the estimation bias brought about by the non-stationarity of the fault signal. So, the LMD–Teager transform has better analysis accuracy and resolution than the traditional Fourier method, which provides a good alternative for fault diagnosis of the rolling bearing.

The paper aims to build a new objective evaluation method of fabric pilling by combining an integrated image analysis technology with a deep learning algorithm.

Series of image analysis techniques were adopted. First, a Fourier transform transformed images into the frequency domain. The optimal resolution matrix of an exponential high-pass filter was determined by combining the energy algorithm. Second, the multidimensional discrete wavelet transform determined the optimal division level. Third, the iterative threshold method was used to enhance images to obtain a complete and clear pilling ball images. Finally, the deep learning algorithm was adopted to train data from pilling ball images, and the pilling levels were classified according to the learning features.

The paper provides a new insight about how to objectively evaluate fabric pilling grades. Results of the experiment indicate that the proposed objective evaluation method can obtain clear and complete pilling information and the classification accuracy rate of the deep learning algorithm is 94.2%, whose structures are rectified linear unit (ReLU) activation function, four hidden layers, cross-entropy learning rules and the regularization method.

Because the methodology of the paper is based on woven fabric, the research study’s results may lack generalizability. Therefore, researchers are encouraged to test other kinds of fabric further, such as knitted and unwoven fabrics.

Combined with a series of image analysis technology, the integrated method can effectively extract clear and complete pilling information from pilled fabrics. Pilling grades can be classified by the deep learning algorithm with learning pilling information.

The paper deals with the development of an improved fast Fourier transform (FFT)-based numerical method for computing the effective properties of composite conductors. The convergence of the basic FFT-based methods is recognized to depend drastically on the contrast between the phases. For instance, the primal formulation is not suited for solving the problems with high conductivity whereas the dual formulation is computationally costly for problems with high resistivity. Consequently, it raises the problem of computing the properties of composites containing both highly conductive and resistive inclusions.

In the present work, the authors' propose a new iterative scheme for solving that kind of problems which is formulated in term of the polarization.

The capability and relevance of this iterative scheme is illustrated through numerical implementation in the case of composites containing squared inclusions. It is shown that the rate of convergence is increased and thus, particularly when the case of high contrasts is considered. The predominance of the polarization based iterative scheme (PBIS) over existing ones is also illustrated in the case of a composite containing both highly conductive and highly resistive inclusions.

The method is easy to implement and uses the same ingredients as the basic schemes: the FFT and the exact expression of the Green tensor in the Fourier space. Moreover, its convergence conditions do not depend on the conductivity properties of the constituents, which then constitutes the main difference with other existing iterative schemes. The method can then be applied for computing the effective properties of composites conductors with arbitrary contrasts.

The purpose of this paper is to introduce the double-periodic lattice, composed of bending-resistant fibers. The essence of the model is that the filaments are of infinite length and withstand tension and bending. The constitutive equations of the lattice in discrete and differential formulations are derived. Two complementary systems of loads, which cause different deformation two orthogonal families of fibers, occur in the lattice. The fracture behavior of the material containing a semi-infinite crack is investigated. The crack problem reduces to the exactly solvable Riemann-Hilbert problem. The solution demonstrates that the behavior of material cardinally depends upon the tension in the orthogonal family of fibers. If tension in fibers exists, opening of the crack under action of loads in two-dimensional lattice is similar to those in elastic solid. In the absence of tension, contrarily, there is a finite angle between edges at the crack tip.

The description of stress state in the crack vicinity is reduced to the solution of mixed boundary value problem for simultaneous difference equations. In terms of Fourier images for unknown functions the problem is equivalent to a certain Riemann-Hilbert problem.

The analytical solution of the problem shows that fracture behavior of the material depends upon the presence of stabilizing tension in fibers, parallel to crack direction. In the presence of tension in parallel fibers fracture character of two-dimensional lattice is similar to behavior of elastic solid. In this case the condition of crack grows can be formulated in terms of critical stress intensity factor. Otherwise, in the absence of stabilizing tension, the crack surfaces form a finite angle at the tip.

Linear behavior of fibers until rupture. Small deflections. Perfect two-dimensional lattice.

The model provides exact analytical estimation of stresses on the crack tip as the function of fibers’ stiffness.

The model is the extension of known lattice models, taking into account the semi-infinite crack in the lattice. This is the first known closed form solution for an infinite lattice model with the crack.

This paper aims to present the spherical entropy image (SEI), a novel global descriptor for the scan registration of three-dimensional (3D) point clouds. This paper also introduces a global feature-less scan registration strategy based on SEI. It is advantageous for 3D data processing in the scenarios such as mobile robotics and reverse engineering.

The descriptor works through representing the scan by a spherical function named SEI, whose properties allow to decompose the six-dimensional transformation into 3D rotation and 3D translation. The 3D rotation is estimated by the generalized convolution theorem based on the spherical Fourier transform of SEI. Then, the translation recovery is determined by phase only matched filtering.

No explicit features and planar segments should be contained in the input data of the method. The experimental results illustrate the parameter independence, high reliability and efficiency of the novel algorithm in registration of feature-less scans.

A novel global descriptor (SEI) for the scan registration of 3D point clouds is presented. It inherits both descriptive power of signature-based methods and robustness of histogram-based methods. A high reliability and efficiency registration method of scans based on SEI is also demonstrated.

This study examines the tracking error (TE) of a sample of sector exchange traded funds (ETFs) using spectral techniques.

TE is examined by computing its power spectrum using the wavelet transform. The wavelet transform maps the TE time series from the time domain to the time–frequency domain. Albeit the wavelet transform is a more complicated mathematical tool compared with the Fourier transform, it also has important advantages such as that it allows to analyze non-stationary data and to detect transient behavior.

Results show that changes in the TE of a sample of sector ETFs are captured by the wavelet transform. Moreover, the authors also find that the wavelet coherence function can be used as a measure of TE in the time–frequency domain.

The study shows that the wavelet coherence function can be used as a reliable measure of TE.

The purpose of this paper is to consider the time-splitting Fourier spectral (TSFS) method to solve the fractional coupled Klein–Gordon–Schrödinger (K-G-S) equations. A time-splitting spectral approach is applied for discretizing the Schrödinger-like equation and along with that, a pseudospectral discretization has been accurately utilized for the temporal derivatives in the Klein–Gordon-like equation. Furthermore, the time-splitting scheme is proved to be unconditionally stable. Numerical experiments guarantee high accuracy of the TSFS scheme for the K-G-S equations. Here, the derivative of fractional order is taken in the Riesz sense.

The focus of this paper is to study the Riesz fractional coupled K-G-S equations using the TSFS method. This method is dependent on evaluating the solution to the given problem in small steps, and treating the nonlinear and linear steps separately. The nonlinear step is made in the time domain, while the linear step is made in the frequency domain, which necessitates the use of Fourier transform back and forth. It is a very effective, powerful and efficient method to solve the nonlinear differential equations, as in previous works (Bao et al., 2002; Bao and Yang, 2007; Muslu and Erbay, 2003; Borluk et al., 2007), the initial and boundary-value problem is decomposed into linear and nonlinear subproblems. Summarizing the technique of the TSFS method, it can be stated that first the Schrödinger-like equation is solved in two splitting steps. Then, the Klein–Gordon-like equation is solved by discretizing the spatial derivatives by means of the pseudospectral method.

The utilized method is found to be very efficient and accurate. Moreover, the time-splitting spectral scheme is found to be unconditionally stable. By means of thorough study, it is found that the spectral method is time-reversible, is gauge-invariant and also conserves the total charge. Moreover, the results have been graphically presented to exhibit the accuracy of the proposed methods. Apart from that, the numerical solutions have been also compared with the exact solutions. Numerical experiments establish that the proposed technique manifests high accuracy and efficiency.

To the authors’ best knowledge, the Riesz fractional coupled K-G-S equations have been for the first time solved by using the TSFS method.