International Journal of Wavelets, Multiresolution and Information Processing

A multiwavelet is typically constructed starting from a vector-valued function satisfying a matrix refinement equation. The approximation order of such a refinable function vector is related to the sum rules of order $p$ satisfied by the corresponding refinement mask. A refinable function vector can be obtained using cascade algorithm by constructing a refinement mask which satisfies the sum rules of order $1.$ A standard pair associated with a refinement mask gives information about its spectral properties. In this paper, we present a procedure for constructing refinement masks satisfying the sum rules of order $1$, starting from standard pairs. How this helps in the construction of asymmetric multiwavelets using standard pairs is illustrated through examples. A sufficient condition on a standard pair and a necessary and sufficient condition on a left standard pair are established so that the corresponding refinement mask satisfies the sum rules of order $1$.

Removing the ambient noise and increasing the signal-to-noise ratio are very important for detecting defects and corrosions of conductive material by using the electromagnetic acoustic transducer. It is still an issue to remove the ambient noise without losing the original signal information. The aim of this paper is to solve the issue by using a new closed-form shrinkage function based on Gauss–Laplace mixture distribution in wavelet domain. First, we prove that Gauss–Laplace mixture distribution is well fitted to the statistical model for wavelet coefficients of noise-free signal of electromagnetic acoustic transducer. As well, we use Gauss–Laplace mixture distribution and Gauss distribution for statistical modeling on the wavelet coefficients of noise-free signal and ambient noise, respectively. Using these distributions, we derive a new closed-form shrinkage function that is an analytical solution of a Bayesian maximum a posteriori estimator. Next, we evaluate the denoising performance of new shrinkage function compared with various shrinkage functions in terms of the improved signal-to-noise ratio, root mean squared error and entropy. The experiment results show that the wavelet denoising method using the proposed shrinkage function effectively removes the ambient noise than the other existing denoising methods for noisy signal of electromagnetic acoustic transducer.

In this paper, the inversion formula and the continuity of the continuous Bessel wavelet transform of distributions in $D_{L^2}^{\prime }$- type space are investigated. For the uniqueness of the inversion formula, the Bessel wavelet kernel of nonvanishing moments is considered.

Motivated by Jin–Ke–Wang’s work (J. Jin, Z. T. Ke and W. Wang, Ann. Statist. 45(5) (2017) 2151–2189), this paper studies estimation of misclassification rate in the Asymptotic Rare and Weak (ARW) model. In contrast to Jin–Ke–Wang’s theorem, we measure the performance of the estimator by the misclassification rate instead of Hamming distance, and extend the Gaussian noise to sub-Gaussian’s. The probability estimation with convergence rate is first given under some conditions. Then we prove that condition necessary as well. A direct corollary of our estimation can be compared with Jin–Ke–Wang’s theorem. It turns out that our statistical limit coincides with theirs.

A robust algorithm for source number estimation based on the formation of the Hankel covariance matrix is presented. First, multiple data snapshots are taken successively from overlapped subarrays in a way similar to the forward spatial smoothing method to construct the special Hankel covariance matrix and for the total number of subarrays, these special covariance matrices are generated. Then, the average of these matrices is employed in singular value decomposition to generate the corresponding eigenvalues. Finally, the resulting eigenvalues are evaluated via the rule presented in this paper as the Moving Gradient Criterion (MGC) to estimate the number of sources by detection of the largest singular values. The greatest difference between the proposed algorithm and the other conventional methods is the form of the covariance matrix with the observed signal that can handle both non-coherent as well as fully coherent sources. Also, the proposed MGC rule adopted with this form of the covariance matrix is the strength of this work. Numerical simulations demonstrate the high superiority of the proposed approach over the competing methods such as MDL, AIC, SORTE, RAE and MSEE methods, especially in the cases of very closely spaced sources, low SNR values, low sensors number and low snapshots number.

When we wanted to further investigate the results of Bourgain–Pavlović, Koch–Tataru and Li–Xiao–Yang, we encountered the problem of stability of the parameter flow. For initial data in some space $X$, norm inflation implies that the solution does not belong to $C(X)$. For ${\mathrm{\text{BMO}}}^{-1}$, Auscher–Dubois–Tchamitchian proved that Koch–Tataru’s solution is stable in their defined space $C_0$. In this paper, we consider Koch–Tataru’s iteration space and construct a non-Gauss flow function to show that $C_0$-stability may have meaning different to $L^{\infty }({({\mathrm{\text{BMO}}}^{-1})}^n)$, which supports Chemin and Gallagher’s point: well-posedness and norm inflation may have no conflict.

Convolutional neural network (CNN) has been widely applied to hyperspectral image (HSI) classification and shows good performance. In this paper, a novel 3D residual attention network (3D-RAN) based on CNN is proposed for HSI classification. The main contributions and novelties of the method are as follows: (1) the 3D-RAN can directly process the 3D HSI data and effectively preserve the data structure of HSI; (2) the constructed residual module can supplement the loss of some important information during the process of information transmission, so that it makes the extracted features contain richer information; (3) the embedded attention module can strengthen the important features and suppress the unimportant features by adjusting weights, which effectively improves the classification performance. Moreover, weight sharing makes the attention module require fewer parameters. The experiments on three public HSI data sets demonstrate that the proposed 3D-RAN outperforms several state-of-the-art methods.

Siamese trackers, coupled with an efficient cross-correlation layer and benefiting from potent convolutional network technology, draw continuous interest in the field of visual object tracking. However, previous Siamese trackers may suffer from inconsistency of predictions that results in degraded performance during tracking: most of the prevalent Siamese networks employ two parallel branches for different subtasks whereas the corresponding outputs may mismatch with each other to some extent. To attack this issue, we advance a two-stage Siamese tracker named SiamPA for accurate object tracking. It employs center-based anchor-free heads in the first stage for preliminary predictions, meanwhile taking the carefully designed Prediction Alignment and Refinement module (PAR) as the second stage to refine the first-stage output. The PAR module is designed for Alignment and Refinement of multi-branch prediction, which works subtly in a mini-Siam manner. It is equipped with two different prediction branches: one used to align the multiple predictions induced in the first stage and the other to adjust coordinates of proposals. Extensive experiments are conducted to demonstrate the effectiveness of our SiamPA, showing that it achieves favorable performance on several prevalent benchmark datasets. Particularly, SiamPA achieves desirable performance while running at 67 FPS, which is far beyond real-time speed.

In signal processing, rational $\frac{p}{q}$-wavelets are preferable than the wavelets corresponding to dyadic MRA because it allows more variations in scale factors of signal components. In this paper, for a rational number $\frac{p}{q},p,q\in \mathbb{N},p>q\ge 1,(p,q)=1$ and $\alpha \in (0,\frac{q}{p-1})$, we consider a collection ${{𝒜}}_{\alpha }^{p/q}$, the space of all continuous functions in $L^2(ℝ)$ that are linear on $[pqk+pl\alpha ,pqk+p(l+1)\alpha ]$ and $[pqk+(p-1)p\alpha ,pqk+pq]$ for all $l=0,1,\dots ,p-2,k\in \mathbb{Z}$. For $\frac{p}{q}=\frac{3}{2},\frac{4}{3}$, under certain conditions, we prove that, if $\frac{1}{\sqrt{q}}\varphi (\frac{\cdot }{q})\in {{𝒜}}_{\alpha }^{p/q}$ generates a $\frac{p}{q}$-MRA, then $\alpha =\frac{q}{p}$. Also, we show that if $\alpha =\frac{q}{p}$, there exists a function $\frac{1}{\sqrt{q}}\varphi (\frac{\cdot }{q})\in L^2(ℝ)$, satisfying the above conditions, that generates $\frac{p}{q}$-MRA. In addition, we construct orthonormal $\frac{p}{q}$-wavelets corresponding to $\frac{p}{q}$-MRA.

Object detection from underwater sea images based on deep learning techniques provides preferable results in a controlled environment. Yet, these techniques experience some challenges in detecting underwater objects due to color distortion, noise, and scattering. Hence, enhancing the underwater imaginary is important for accurately determining the objects under water. This research presents a deep learning approach for perceiving underwater objects from enhanced underwater images. Very Deep Super-Resolution Network (VDSR), which exhibits a higher visual quality, is utilized for improving the underwater image quality, thereby it is sufficient for object detection. Then, the object is detected from the enhanced underwater image through the proposed Border Collie Flamingo optimization-based deep CNN classifier (BCFO-based deep CNN). The developed BCFO-based algorithm is the main highlight of the research, which effectively tunes the classifier’s hyperparameter. The evaluation is established using the UIEB and DUO datasets on the basis of performance standards, such as specificity, accuracy, and sensitivity. When the training percentage is 80 and the $K$-fold is 10, the suggested model achieved accuracy results of 93.89% and 95.24%, sensitivity results of 95.93 and 97.29%, and specificity results of 98.64% and 99%, which is very efficient compared to some existing approaches.