International Journal of Wavelets, Multiresolution and Information Processing

Ensemble mean forecast errors during a tropical cyclone event are probed with a spherical wavelet transform constructed by the lifting scheme. Coefficient spectra and associated filtered error components are examined during the forecast, with an emphasis on feature detection, for mean sea level pressure and wind components. Leading wavelet coefficients within a reference circle centered on the estimated cyclone track demonstrate a clear affinity for local error extrema, reflecting the transform’s feature detection capacity. Compression performance of the transform is also demonstrated by truncated wavelet expansions, which exhibit contrasting behavior reflecting fundamental structural differences between the wind and pressure error fields.

For the non-stationary signal denoising, an effective method for dropping ambient noise is based on discrete wavelet transform. Also, in order to minimize the loss of useful signal and get high SNR in the wavelet denoising, it is very important that the thresholding is suitable for the characteristics of signal. In this paper, we propose new thresholding method to reduce an ambient noise and to detect effectively the useful signal. First, we analyze four kinds of previous wavelet threshold functions (Hard, Soft, Garrote and Hyperbola) and propose new wavelet threshold function compromised between Garrote and Hyperbola threshold functions. Next, a threshold value is selected by value to reduce exponentially according to the wavelet decomposition level. We also analyze a continuity and monotonicity, and prove the logic of new threshold function. The results of theoretical analysis show that new threshold function solves the problems of constant error and discontinuity of previous threshold functions, and minimizes the information loss of useful signal. The results of experiment show that SNR of new thresholding method is highest and RMSE and Entropy are smallest. The results of theoretical analysis and experiment show that new thresholding method is more appropriate to wavelet denoising for dropping ambient noise than previous methods.

The natural resources require continuous monitoring of pipelines to avoid algae blooms, corrosion, leakages, damages at joints for better management. Monitoring is performed through Wireless Sensors Network in collaboration with various communication methods using image sensors, but they face limitations like time consumption, color variations, blurred data, bit or data losses. To overcome the limitations, the paper proposes a Panoramic Image Transmission and Refinement Technique that uses acoustical sensors, Hybrid-Orthogonal Frequency Division Multiplexing (Hybrid-OFDM) and Mosaicing technique, targets to sense, capture and forward data to the surface station. It smooths the data using Random Sample Consensus (RANSAC) algorithm. The proposed method is tested and validated through MATLAB simulations. The obtained results support technique’s efficiency with respect to less energy consumption, communication, maximum delivery of data at low cost.

The relationship between frames and Parseval frames is an important topic in frame theory. In this paper, we investigate Parseval transforms, which are linear transforms turning general finite frames into Parseval frames. We introduce two classes of transforms in terms of the right regular and left Parseval transform matrices (RRPTMs and LPTMs). We give representations of all the RRPTMs and LPTMs of any finite frame. Two important LPTMs are discussed in this paper, the canonical LPTM (square root of the inverse frame operator) and the RGS matrix, which are obtained by using row’s Gram–Schmidt orthogonalization. We also investigate the relationship between the Parseval frames generated by these two LPTMs. Meanwhile, for RRPTMs, we verify the existence of invertible RRPTMs for any given finite frame. Finally, we discuss the existence of block diagonal RRPTMs by taking the graph structure of the frame elements into consideration.

Fusion frames or frames of subspaces are an extension of frames that have many applications in science. The paper presents non-orthogonal fusion frames with real valued bounds and $C^{\ast }$-valued bounds in Hilbert $C^{\ast }$-modules which they are called NOFF and NO${}^{\ast }$FF, respectively. Their elementary properties are studied, for example, the relations between NOFF and NO${}^{\ast }$FF in Hilbert $C^{\ast }$-modules are established. More precisely, we investigate these fusion frames in Hilbert $C^{\ast }$-modules over different $C^{\ast }$-algebras.

In this paper, we propose a novel scheme to learn high-level representative features and conduct classification for hyperspectral image (HSI) data in an automatic fashion. The proposed method is a collaboration of a wavelet-based extended morphological profile (WTEMP) and a deep autoencoder (DAE) (“WTEMP-DAE”), with the aim of exploiting the discriminative capability of DAE when using WTEMP features as the input. Each part of WTEMP-DAE is ingenious and contributes to the final classification performance. Specifically, in WTEMP-DAE, the spatial information is extracted from the WTEMP, which is then joined with the wavelet denoised spectral information to form the spectral-spatial description of HSI data. The obtained features are fed into DAE as the original input, where the good weights and bias of the network are initialized through unsupervised pre-training. Once the pre-training is completed, the reconstruction layers are discarded and a logistic regression (LR) layer is added to the top of the network to perform supervised fine-tuning and classification. Experimental results on two real HSI data sets demonstrate that the proposed strategy improves classification performance in comparison with other state-of-the-art hand-crafted feature extractors and their combinations.

The results in this paper can be divided into three parts. First, we generalize the recent results of Benavente, Christensen and Zakowicz on approximately dual generalized shift-invariant frames on the real line to generalized translation-invariant (GTI) systems on locally compact abelian (LCA) groups. Second, we explain in detail how GTI frames can be realized as $g$-frames. Finally, the known results on perturbation of $g$-frames and results on perturbation of generalized shift-invariant systems are applied and extended to GTI systems on LCA groups.

The fusion of infrared and visible images of the same scene aims to generate a composite image which can provide a more comprehensive description of the scene. In this paper, we propose an infrared and visible image fusion method based on convolutional neural networks (CNNs). In particular, a siamese convolutional network is applied to obtain a weight map which integrates the pixel activity information from two source images. This CNN-based approach can deal with two vital issues in image fusion as a whole, namely, activity level measurement and weight assignment. Considering the different imaging modalities of infrared and visible images, the merging procedure is conducted in a multi-scale manner via image pyramids and a local similarity-based strategy is adopted to adaptively adjust the fusion mode for the decomposed coefficients. Experimental results demonstrate that the proposed method can achieve state-of-the-art results in terms of both visual quality and objective assessment.

In the last few years, many known works in learning theory stepped over the classical assumption that samples are independent and identical distribution and investigated learning performance based on non-independent samples, as mixing sequences (e.g., $\alpha$-mixing, $\beta$-mixing, $\varphi$-mixing etc.), they derived similar results with the investigation based on classical sample assumption. Negative association (NA) sequence is a kind of significant dependent random variables and plays an important role in non-independent sequences. It is widely applied to various subjects such as probability theory, statistics and stochastic processes. Therefore, it is essential to study the learning performance of learning process for dependent samples drawn from NA process. Obviously, samples in this learning process are not independent and identical distribution. The results in classical learning theory are not applied directly. In this paper, we study the consistency of least-square regularized regression with NA samples. We establish the error bound of least-square regularized regression for NA samples, and prove that the learning rate of least-square regularized regression for NA samples is $m^{𝜀-1}$, which is tend to $m^{-1}$ when $𝜀$ arbitrarily close to 0, where $m$ denote the number of the samples. The simulation experiment of convergence rate on NA samples reveals that the least-square regularized regression algorithm for NA samples is consistent. This result generalizes the classical result of independent and identical distribution.

Composite dilation wavelets generate bases or frames using two groups of dilations. One group is associated with a set $\{a^i,a\in G{L}_n(ℝ),i\in \mathbb{Z}\},$ where $a$ is an expanding matrix with $\mathrm{\text{det}}(a)$ an integer. The second group is associated with the subgroup $B\subseteq S{L}_n(ℝ)$ of matrices $b\in B$ with $|\mathrm{\text{det}}(b)|=1.$ Here, we present crystallographic Haar type composite dilation wavelets generated by such two dilation groups associated with $P_4=C_4⋉{\mathbb{Z}}^2.$ Further, we provide matricial filters and reconstruction formulae at various scales.

The windowed-Shannon wavelet is not recommended generally as the window function will destroy the partition of unity of Shannon mother wavelet. A novel windowing scheme is proposed to overcome the shortcoming of the general windowed-Shannon function, and then, a novel and efficient Shannon–Cosine wavelet spectral method is provided for solving the fractional PDEs. Taking full advantage of the waveform of sinc function to hold the partition of unity, Shannon–Cosine wavelet is constructed, which is composed of Shannon wavelet and the trigonometric polynomials. It was proved that the proposed wavelet function meets the requirements of being a trial function and possesses many other excellent properties such as normalization, interpolation, two-scale relations, compact support domain, and so on. Therefore, it is a real wavelet function instead of a general Shannon–Gabor wavelet which is a kind of quasi-wavelet. Next, by means of the Shannon–Cosine wavelet collocation method, the corresponding algebraic equation system of the fractional Fokker–Planck equation can be obtained. Approximate solutions of the fractional Fokker–Plank equations are compared with the exact solutions. These calculations illustrate that the accuracy of the Shannon–Cosine wavelet collocation solutions is quite high even using a small number of grid points.

A WH-packet is a system of vectors which is analogous to Aldroubi’s model for explicit expression of vectors (including frame vectors) in terms of a series associated with a given frame. In this paper, we study frame properties of WH-packet type system for matrix-valued wave packet frames in the function space $L^2({ℝ}^d,{ℂ}^{s\times r})$. A necessary and sufficient condition for WH-packets of matrix-valued wave packet frames in terms of a bounded below operator is given. We present sufficient conditions for both lower and upper frame conditions on scalars associated with WH-packet of matrix-valued wave packet frames. Finally, a Paley–Wiener type perturbation theorem for WH-packet of matrix-valued wave packet frames is given. Several examples are given to illustrate the results.