International Journal of Wavelets, Multiresolution and Information Processing

In this paper, we study the two-parameter spectral element method based on weighted shifted orthogonal polynomials for solving singularly perturbed diffusion equation on an interval [0, 1] which are modeled with singular parameters. We continue our study to estimate the lower bound of the weighted orthogonal polynomial coefficient and the upper bound of a posteriori error estimates of the method through different weighted norms to minimize the computational cost. Numerical examples are implemented to study the applicability and efficiency of the technique. The obtained error bounds for the coefficient of orthogonal polynomials and the posteriori estimates fall within the bounds derived in the theoretical section. It is also observed that the two weighted norms decreases when the values of $N_1$ and $N_2$ increases for the three choices of $𝜖$ and for different values of x and y. The quality and accuracy of the solution can be realized through figures and tables.

Structured meshes are becoming increasingly popular as a discretization technique in industries where solution accuracy is of paramount importance. However, generating structured meshes directly on the model surface presents a challenge. Consequently, many researchers are now exploring mapping methods for structured mesh generation. In this context, parameterization algorithms are vital as they map the model surface from 3D space to a 2D parametric domain. Existing parameterization algorithms, despite their advancements, still face limitations in preserving symmetry information from the input model. To address this issue, we propose a symmetry preserving parameterization algorithm based on geometric constraint. This algorithm utilizes plane reflection symmetry transformation to identify the main symmetry plane of the model and calculate the symmetry factor. Based on the fixed points and the symmetry factor, a conformal parametric symmetry plane is obtained. Furthermore, we develop a surface structured mesh generation system to provide engineers with a tool that can rapidly generate meshes. Experimental results demonstrate that our proposed algorithm achieves higher quality results compared to other parameterization algorithms.

Spatio-temporal action detection (STAD) aims to classify the actions present in a video and localize them in space and time. It has become a particularly active area of research in computer vision because of its explosively emerging real-world applications, such as autonomous driving, visual surveillance and entertainment. Many efforts have been devoted in recent years to build a robust and effective framework for STAD. This paper provides a comprehensive review of the state-of-the-art deep learning-based methods for STAD. First, a taxonomy is developed to organize these methods. Next, the linking algorithms, which aim to associate the frame- or clip-level detection results together to form action tubes, are reviewed. Then, the commonly used benchmark datasets and evaluation metrics are introduced, and the performance of state-of-the-art models is compared. At last, this paper is concluded, and a set of potential research directions of STAD are discussed.

Numerous studies have demonstrated the potential of robust principal component analysis (RPCA) in image recovery. However, conventional RPCA methods for color image recovery apply RPCA independently to each color channel, which ignores the correlation information of the red, green and blue channels. To improve the performance of RPCA-based methods and draw inspiration from the success of quaternion representation in color image processing, we propose an improved quaternion RPCA (IQRPCA) method for color image recovery. The IQRPCA method treats all color channels holistically and considers the correlation information of different color channels naturally. In addition, we have developed a quaternion nuclear norm known as improved quaternion Cauchy nuclear norm, which produces an even more effective and robust approach to color image recovery task. Compared to the RPCA method in the quaternion setting, the IQRPCA method treats the singular values differently, which shows better recovery performance than competing methods. Furthermore, we provide a convergence analysis of the proposed method. Our experiments confirm the effectiveness of IQRPCA in the application of color image recovery.

The goal of the paper is to establish quadrature formulas on combinatorial graphs. Three types of quadrature formulas are developed. Quadrature formulas of the first type are obtained through interpolation by variational splines. This set of formulas is exact on spaces of variational splines on graphs. Since bandlimited functions can be obtained as limits of variational splines we obtain quadrature formulas which are approximately exact on spaces of bandlimited functions. Accuracy of this type of quadrature formulas is given in terms of geometry of the set of nodes of splines and in terms of smoothness of functions which is measured by means of the combinatorial Laplace operator. Quadrature formulas of the second type are obtained through point-wise sampling for bandlimited functions and based on existence of certain frames in appropriate subspaces of bandlimited functions. The third type quadrature formulas are based on the average sampling over subgraphs. Our quadrature formulas which are based on sampling are exact on a relevant subspaces of bandlimited functions. The results of the paper have potential applications to problems that arise in data mining.

In this paper, we propose a novel integral transform coined as quaternion quadratic-phase wavelet transform (QQPWLT) by invoking the elegant convolution structure associated with the quaternion quadratic-phase Fourier transform. First, we explore some mathematical properties of the QQPWLT, including the orthogonality relation, inversion formula, reproducing kernel and some notable inequalities. Second, we study Heisenberg’s uncertainty principles and the logarithmic uncertainty principle associated with the quadratic-phase wavelet transform in quaternion domain. We culminate our investigation by presenting some illustrative examples.

Recently, Gabor analysis on locally compact abelian (LCA) groups has become the focus of an active research. In practice, the time variable cannot be negative. The half real line ${ℝ}_{+}=(0,\infty )$ is an LCA group under multiplication and the usual topology, with the Haar measure $d\mu =\frac{dx}{x}$. This paper addresses Gabor frame multipliers and Parseval duals for $L^2({ℝ}_{+},d\mu )$. We introduce and characterize Gabor frame multipliers and Parseval Gabor frame multipliers based on Zak transform matrices. Our Zak transform matrix is essentially different from the conventional Zibulski–Zeevi matrix. It allows us to define Gabor frame generators by designing suitable matrix-valued functions of finite size. We also prove that an arbitrary Gabor frame ${𝒢}(g,a,b)$ admits a Parseval dual frame/tight dual frame whenever $lna\cdot lnb$ are rational numbers not greater than $\frac{1}{2}$.

In this paper, we introduce the continuous and discrete version of the linear canonical shearlet transform (LCST). The authors begin with the definition of the LCST and then establish a relationship between the linear canonical transform (LCT) and the LCST. Next, the paper derives several basic properties like Parseval’s Formula, inversion formula, and the characterization of the transform’s range. In addition to the continuous version of the transform, the authors also present a discrete version of the LCST. This discrete version allows for practical implementation and efficient computation of the transform in digital signal processing systems. Lastly, the paper establishes a frame condition for the discrete LCST, which thereby helps in establishing the reconstruction formula for the discrete LCST. The paper ends with a conclusion.

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield $𝔽$ of $ℂ$ that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters.

Image denoising is a significant procedure in image processing, and a good denoising method can improve the image quality. This study focuses on enhancing the estimation model of wavelet coefficients by improving the threshold function. Thus, a general construction method for a class of threshold functions is introduced, and sufficient conditions are given to ensure that the threshold functions are continuous, first-order differentiable and higher-order differentiable. It makes an existing class of improved threshold functions a particular case and enables the construction of other needed threshold functions based on this general approach. Experiments have shown that using the new threshold functions yields better denoising results for images containing salt and pepper noise, Gaussian noise and speckle noise. Moreover, as the smoothness of the threshold function is enhanced, the denoising effect is also improved. The improved wavelet coefficients estimation model can effectively boost the quality of image denoising, further enhancing the accuracy and effectiveness of computer vision algorithms.

For any parameter vector $(r,t,\omega )$ in $ℍ:=ℝ\times {ℝ}^n\times {ℝ}^n$, the representation matrix $S(r,t,\omega )$ of the Heisenberg group $(ℍ,\ast )$ is extended to a new form $S_{B,W,A}(r,t,\omega )$ with nonsingular matrices $B,W,A$ which are related to the directional translation ${{𝒯}}_{Bt}:f\to f(\cdot -Bt)$, directional modulation ${{ℳ}}_{W\omega }:f(x)\to {\mathrm{\text{e}}}^{i{x}^{\prime }W\omega }$ and dilation ${{𝒟}}_A:f\to |\mathrm{\text{det}}(A){|}^{-\frac{1}{2}}f(A^{-1}\cdot )$ if $B^{\prime }W=I_n$, the identity matrix of order $n$. In the case $B^{\prime }W\ne I_n$, an intrinsic group structure under the matrix $S_{B,W,A}(r,t,\omega )$ is investigated and a unitary operator representation is offered.

This note designs a statistic model involving TM system with Gaussian noise. Based on adaptive approximation of the parameters in TM system and greedy choices for the generalized Fourier coefficients, a regression function is determined for this model from the finite observed data. Finally its statistical properties are proved.

In this paper, we propose a new image denoising technique which consists in applying a stationary wavelet transform (SWT)-based image denoising technique, in the domain of 2D dual-tree discrete wavelet transform (DWT). In fact, this proposed technique consists first of applying the 2D dual-tree DWT to the noisy image. Then, the noisy wavelet coefficients obtained from this application are denoised by applying to each of them, a SWT-based image denoising technique. Finally, the denoised image is reconstructed by applying the inverse of the 2D dual-tree DWT to the obtained denoised wavelet coefficients. For applying this SWT-based image denoising technique, we use the soft thresholding, the Daubechies 4 as the mother wavelet, and the decomposition level is equal to 5. The performance of this image denoising technique proposed in this work is proven by its comparison to three other denoising techniques existing in the literature. These three techniques are the denoising technique based on the soft thresholding in the SWT domain, the image denoising technique based on soft thresholding in the domain of 2D dual-tree DWT and the image denoising approach using deep neural network. All the previously mentioned techniques, including our proposed denoising approach, are applied to a number of noisy images, and the obtained results are in terms of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM). Those results show that this proposed denoising technique outperforms the other three denoising techniques used in this evaluation.

This study considers the issue for recognizing unknown source within a time-space fractional parabolic equation. This particular issue is characterized by severe ill-posedness, where the solution does not depend continuously on the data. To tackle this issue, the wavelet dual least squares method is extended to handle ill-posedness of the issue under priori rule. Additionally, a posterior wavelet regularization parameter selection rule is introduced to address the Cauchy problem. Furthermore, error estimates between the exact solution and its approximation are established using both a priori and a posteriori methodologies. Numerical examples are given to validate the effectiveness and stability of the proposed method.