Single Wavelet Research Articles

Stochastic wavelets from minimizers of an uncertainty principle: An example

This paper proposes a method through which a family of wavelets can be obtained. This is done by choosing each member based on a random variable. The method is preferred in situations where a single mother wavelet proves inadequate and an evolving sequence of mother wavelets is needed but a priori the next member in the sequence is uncertain. The adopted approach is distinct from the way spatiotemporal wavelets are used or even the way stochastic processes have been studied using spatiotemporal wavelets.

Deep ResNet Based Remote Sensing Image Super-Resolution Reconstruction in Discrete Wavelet Domain

We present a single-image super-resolution (SR) method for Remote Sensing Image based on deep learning within Discrete Wavelet Domain in this paper. Our method is inspired Residual Learning. Firstly, an input image is decomposed by single level 2D Discrete wavelet transform to get four sub-bands. The four sub-bands coefficients are feeding into the Deep Learning Residual Network to predict correspondingly residual images; Adding four sub-band images and residual images as the new sub-bands of 2D wavelet transform; Finally, uses the inverse 2D Discrete wavelet transform to get the final output Super Resolution HR image. Our proposed method performs better than existing methods in accuracy and visual improvements in our results are easily noticeable.

Time-frequency Connectedness between Coal Market Prices, New Energy Stock Prices and CO2 Emissions Trading Prices in China

This paper aims to examine whether there is inherent dynamic connectedness among coal market prices, new energy stock prices and carbon emission trading (CET) prices in China under time- and frequency-varying perspectives. For this purpose, we apply a novel wavelet method proposed by Aguiar-Conraria et al. (2018). Specifically, utilizing the single wavelet power spectrum, the multiple wavelet coherency, the partial wavelet coherency, also combined with the partial phase difference and the partial wavelet gains, this paper discovers the time-frequency interaction between three markets. The empirical results show that the connectedness between the CET market price and the coal price is frequency-varying and mainly occur in the lower and higher frequency bands, while the connectedness between the CET market price and the new energy stock price mainly happen in the middle and lower frequency bands. In the high-frequency domain, the CET market price is mainly affected by the coal price, while the CET market price is dominated by the new energy stock price in the middle frequency. These uncovered frequency-varying characteristics among these markets in this study could provide several implications. Main participants in these markets, such as polluting industries, governments and financial actors, should pay close attention to the connectedness under different frequencies, in order to realize their goal of the production, the policymaking, and the investment.

Partial Discharge Signal Denoising Based on Wavelet Pair and Block Thresholding

In electrical engineering, it is of great importance to monitor partial discharge (PD) of high voltage apparatus. However, on-site PD signal easily gets corrupted by white noise, and hence denoising the measured PD signal is necessary to acquire pure PD. In traditional method, single wavelet and regular thresholding are utilized. By combining wavelets of the same family, dual-tree complex wavelet pairs can be constructed. Specific combinations are approximately translation invariant, and they are also able to achieve better denoising effect without sophisticated computation. Meanwhile, considering the correlation between adjacent wavelet coefficients, the block thresholding is employed. In this paper, it is proposed to combine wavelet pair and block thresholding for PD denoising. There are four combinations based on single wavelet/wavelet pair and regular thresholding/block thresholding, and they are compared with each other. Based on a numerical study, the results demonstrate that the proposed algorithm outperforms the traditional method.

A New Boubaker Wavelets Operational Matrix of Integration

Many fields of science and engineering have used wavelet functions. They are established from expansion of a single mother wavelet function. Boubaker wavelet functions are presented in this paper based on the important properties of Boubaker polynomials. The research goal of this article is to drive a Boubaker wavelets operation matrix of integration in general formulas. Then an approximate solution method for solving a singular initial value problem is presented using Boubaker wavelets along the obtained operational matrix of integration. The importance of this method is that it converts a singular initial value problem in order to solve algebraic examples as a system. The process is based on reducing by means of integration the original problem into integral equations using a Boubaker wavelets operation matrix of integration to predict the integral equation. Illustrative experiments are included. In addition, computational results obtained by a Boubaker wavelets operation matrix of integration are compared with the exact solutions and other existing methods.

Recognition of Bangla Handwritten Number Using Combination of PCA and FIS with the Aid of DWT

The structure of any Bangla numerical character is more complex compared to English numerical character. Two pairs of numerical character in Bangla resembles to be closed and they are: “one and nine” and “five and six”. We found that, handwritten Bangla numerical character cannot be recognized using single machine learning algorithm or discrete wavelet transform (DWT). Above phenomenon motivated us to use combination of DWT, Fuzzy Inference System (FIS) and Principal Component Analysis (PCA) to recognize numerical characters of Bangla in handwritten format. The four lowest spectral components of a preprocessed image are taken using DWT, which is considered as the feature vector to recognize the digits in first phase. The feature vector is then applied to FIS and PCA separately. The combined method provides recognition accuracy of 95.8% whereas application of individual method gives less rate of accuracy. Instead of storing the images itself in a folder, if we can store the feature vector of images achieved from DWT in tabular form. The records of table can be applied in FIS, PCA or other object detection algorithm. Although the technique used in the paper can detect objects with moderate rate of accuracy but can save huge storage against a benchmark database of images. If a tradeoff is made between storage requirements and accuracy of recognition, the model of the paper is preferable compared to other present state-of-art. Another finding of the paper is that, the spectral components of images acquired by DWT only matched with FIS and PCA for classification but do not match properly with unsupervised (K-mean clustering) and supervised (support vector machine) learning.

MRI image compression using multiple wavelets at different levels of discrete wavelets transform

Medicine is benefited in large amount with the utilization of imaging. Magnetic Resonance Imaging (MRI) is an important imaging modality that is used for better diagnosis. Since the number of images generated by MRI is in large number per patient, the requirement for storage space and transmission bandwidth will be larger. In the area of image processing, image compression finds an important application that reduces storage space and transmission bandwidth. One way to compress the image is by using transform based techniques. In this work, we have used DWT (Discrete Wavelet Transform) with multiple wavelets at different levels to transform the MRI images. In the first few levels CDF (Cohen–Daubechies–Feauveau) 9/7 wavelet is used for transformation and Haar wavelet is used for remaining levels. After transformation SPIHT (Set Partitioning In Hierarchical Trees) algorithm is used for compression of the images. We have observed better compression in terms of MSE (Mean squared Error) and PSNR (Peak Signal to Noise Ratio) with the proposed method in comparison with the single wavelet transformation.

Image compression using fractal multiwavelet transform

Multiwavelets occur in wavelet theory for more general multiresolution analysis (MRA) using several scaling functions. The muliwavelets feature orthogonality, short support, symmetry, approximation order and regularity at a time which is not possible by using only a single wavelet system. In the present work, theory of MRA and wavelet using several scaling functions is reviewed to define Discrete Multiwavelet Transform and Fractal Multiwavelet Transform in matrix form, using a multifilter. As an application the defined transforms are applied to an image for its compression and the resulting entropy and PSNR values are compared with those occurring with earlier used transforms like Haar and D4 transforms.

Time-varying wavelet estimation and its applications in deconvolution and seismic inversion

Wavelet holds an essential role in seismic data processing and characterization, for examples deconvolution and seismic inversion. Unfortunately, wavelet is an unknown data. Several existing methods attempt to estimate and extract the wavelet from seismic data. However, the methods give only a single wavelet from one seismic trace. When seismic data are non-stationer, single wavelet usage will cause a problem, that is raising the error. This paper proposes a time-varying wavelet estimation method to accommodate this problem. It uses matrix diagonalization to estimate a set of wavelets. Next, the time-varying wavelet is applied to deconvolution and seismic inversion. The experiment shows that time-varying wavelet improves the results in both deconvolution and seismic inversion. The errors decreased and spectrum bandwidth broadened.

Reduced-reference image quality assessment method based on wavelet feature extraction and fusion

Virtual reality has many characteristics such as multi-sensation, interactivity, and presence. Virtual reality images, also known as 360-degree panoramic images, provide users with an immersive experience and have been widely used in education, medical, military and many other fields in recent years. 360-degree panoramic images have high resolution and are subject to varying degrees of distortion during transmission and storage. Therefore, the evaluation of virtual reality images has practical significance. To this end, we propose a method for evaluating the quality of images in the new CVIQD database. The proposed image quality assessment model is developed based on Haar wavelet and Db4 wavelet feature extraction and fusion. The method is compared with a single wavelet feature extraction method. Experiments show that the method has better performance.

Content based medical image retrieval based on new efficient local neighborhood wavelet feature descriptor.

This paper presents a new class of local neighborhood based wavelet feature descriptor (LNWFD) for content based medical image retrieval (CBMIR). To retrieve images effectively from large medical databases is backbone of diagnosis. Existing wavelet transform based medical image retrieval methods suffer from high length feature vector with confined retrieval performance. Triplet half-band filter bank (THFB) enhanced the properties of wavelet filters using three kernels. The influence of THFB has employed in the proposed method. First, triplet half-band filter bank (THFB) is used for single level wavelet decomposition to obtain four sub-bands. Next, the relationship among wavelet coefficients is exploited at each sub-band using 3 × 3 neighborhood window to form LNWFD pattern. The novelty of the proposed descriptor lies in exploring relation between wavelet transform values of pixels rather than intensity values which gives more detail local information in wavelet sub-bands. Thus, proposed feature descriptor is robust against illumination. Manhattan distance is used to compute similarity between query feature vector and feature vector of database. The proposed method is tested for medical image retrieval using OASIS-MRI, NEMA-CT, and Emphysema-CT databases. The average retrieval precisions achieved are 71.45%, 99.51% of OASIS-MRI and NEMA-CT databases for top ten matches considered respectively and 55.51% of Emphysema-CT database for top 50 matches. The superiority in terms of performance of the proposed method is confirmed by the experimental results over the well-known existing descriptors.

Effects of different wavelet filters on correlation and diagnostic performance of radiomics features

To investigate the effects of different wavelet filters on correlation and diagnostic performance of radiomics features. Methods: A total of 143 colorectal cancer (CRC) patients (64 positive in lymph node metastasis and 79 negative) with contrast-enhanced CT examination were recruited. After labeling the tumor area by experienced radiologists, radiomics wavelets features based on 48 different wavelets were extracted using in-house software coded by Matlab. The correlation coefficients of the features with same names between different wavelets were calculated and got the distribution of high-correlation features between each wavelet. The least absolute shrinkage and selection operator (LASSO) was used to build signatures between lymph node metastasis and wavelet features data set based on different wavelets. The numbers of features in signatures and diagnostic performance were compared using Delong's test. Results: With the difference of wavelet order increased, the number of high-correlation features between two wavelets decreased. Some features were prone to high correlation between different wavelets. When building radiomics signature based on single wavelet, signatures built from 'rbio2.2', 'sym7' and 'db7' did well in predicting lymph node metastasis. The signature based on Daubechies wavelet feature set had the highest performance in predicting lymph node metastasis, while the signature from Biorthogonal wavelet features was worst. Improvement was significant in diagnostic performance after excluding the high-correlation features in the whole features set (P=0.004). Conclusion: In order to reduce the data redundancy of features, it is recommended to select wavelets with large differences in wavelet orders when calculating radiomics wavelet features. It is necessary to remove high correlation features for improving the diagnostic performance of radiomics signature.

Identification of power quality events: selection of optimum base wavelet and machine learning algorithm

This study comprehensively investigates power quality (PQ) identification problem and proposes the optimum combination of base wavelet and machine learning algorithm (MLA) which would yield the highest classification accuracy. Although this problem has been studied by various researchers in the recent past, the selection of appropriate base wavelet and MLA, which would give better classification accuracy, have received comparatively less attention. This study bridges this gap by investigating the classification performance of 110 wavelets and 7 well-known MLAs across various noise levels using over 3500 PQ events generated as per IEEE Standard 1159. The results of this investigation demonstrate that the choice of base wavelet does significantly affect the classification performance. Further, it was observed that a single base wavelet does not provide optimum performance across all MLAs at various noise levels. In contrast, each MLA gives the maximum accuracy with a distinct base wavelet. The robustness of MLA against noise is studied which establishes that the simple MLAs, such as decision tree and Naive-Bayes, are more robust against noise compared to the other intricate MLAs. Finally, several recommendations are drawn for the selection of base wavelet and MLA which yields the best possible accuracy.

Detecting gravitational wave bursts with LISA in the presence of instrumental glitches

The Laser Interferometer Space Antenna (LISA) will open a rich discovery space in the milli-Hertz gravitational wave band. In addition to the anticipated signals from many millions of binary systems, this band may contain new and previously un-imagined sources for which we currently have no models. To detect unmodeled and unexpected signals we need to be able to separate them from instrumental noise artifacts, or glitches. Glitches are a regular feature in the data from ground based laser interferometers, and they were also seen in data from the LISA Pathfinder mission. In contrast to the situation on ground, we will not have the luxury of having multiple independent detectors to help separate unmodeled signals from glitches, and new techniques have to be developed. Here we show that unmodeled gravitational wave bursts can be detected with LISA by leveraging the different way in which instrument glitches and gravitational wave bursts imprint themselves in the time-delay interferometery data channels. We show that for signals with periods longer than the light travel time between the spacecraft, the "breathing mode" or Sagnac data combination is key to detection. Conversely, for short period signals it is the time of arrival at each spacecraft that aids separation. We investigate the conditions under which we can distinguish the origin of signals and glitches consisting of a single sine-Gaussian wavelet and determine how well we can characterize the signal. We find that gravitational waves bursts can be unambiguously detected and characterized with just a single data channel (four functioning laser links), though the signal separation and parameter estimation improve significantly when all six laser links are operational.

Wavelet Collocation Method for Solving Elliptic Singularly Perturbed Problem

Wavelet collocation method is used to solve an elliptic singularly perturbed problem with two parameters. The B-spline function is used as a single mother wavelet, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problem and the result shows the reliability and efficiency of the method.

Denoising and Baseline Drift Removal Method of MEMS Hydrophone Signal Based on VMD and Wavelet Threshold Processing

Aiming at the problem that the signals received by MEMS vector hydrophones are mixed with a large amount of external environmental noise, and inevitably produce baseline drift and other distortion phenomenons which made it difficult for the further signal detection and recognition, a joint denoising method (VMD-NWT) based on variational mode decomposition (VMD) and nonlinear wavelet threshold (NWT) processing is proposed. The main frequency of the noisy signal is first obtained by Fourier transform. Then the noisy signal is decomposed by VMD to obtain the IMF components. The center frequency and correlation coefficient of each IMF component further determine that the IMF components belong to noise IMF components, noisy IMF components or pure IMF components. Then the pure IMF components are reserved, the noise IMF components are removed, and the noisy IMF components are denoised by NWT processing method with new threshold function as a whole. Finally, the denoised IMF components and the pure IMF components are reconstructed to obtain the denoised signal to realize the extraction of useful signals and baseline drift removal. Compared with complete ensemble empirical mode decomposition with adaptive noise combined with wavelet threshold processing method (CEEMDAN-WT), ensemble empirical mode decomposition combined with wavelet threshold processing method (EEMD-WT), and single wavelet threshold processing method with compromised function between hard and soft threshold (ZWT), the VMD-NWT in this paper has the advantages of less calculation and simple implementation. The simulation comparison experiments verify that the VMD-NWT is superior to CEEMDAN-WT, EEMD-WT, and ZWT. Then VMD-NWT is applied to process the measured data obtained from the Fenji experiment conducted by the North University of China. Simulation and lake trial results show that VMD-NWT has better denoising effect and can realize baseline drift removal. So the proposed VMD-NWT has certain practical research value.

Adaptive tuning piecewise cubic Hermite interpolation with Wiener filter in wavelet domain for scanning electron microscope images.

Image processing is introduced to remove or reduce the noise and unwanted signal that deteriorate the quality of an image. Here, a single level two-dimensional wavelet transform is applied to the image in order to obtain the wavelet transform sub-band signal of an image. An estimation technique to predict the noise variance in an image is proposed, which is then fed into a Wiener filter to filter away the noise from the sub-band of the image. The proposed filter is called adaptive tuning piecewise cubic Hermite interpolation with Wiener filter in the wavelet domain. The performance of this filter is compared with four existing filters: median filter, Gaussian smoothing filter, two level wavelet transform with Wiener filter and adaptive noise Wiener filter. Based on the results, the adaptive tuning piecewise cubic Hermite interpolation with Wiener filter in wavelet domain has better performance than the other four methods.

Constructing Symmetric-Antisymmetric Multi-wavelets and Transforming Digital Image by Multi-wavelets

Multi-wavelet is an effective tool for image processing. In order to provide more wavelet choices for image denoising, coding and enhancement, a symmetric-antisymmetric multi-wavelet algorithm based on symmetric or nearly symmetric single wavelet is proposed, and the two-fold symmetric-antisymmetric multi-wavelet constructed by this algorithm is applied to digital image processing. Theoretically, this algorithm can construct arbitrary multiple symmetric-antisymmetric multi-wavelets and obtain quantitative coefficients of multi-wavelet filters. Multi-wavelets constructed by this algorithm are used to decompose and reconstruct digital images. The results show that the low-frequency components of the decomposed images account for 99.52% of the total, and the MSNR of the restored images can reach 88.4096 dB, 18.2928 dB higher than the classical multi-wavelet CL 2.

Fusion of noisy images based on joint distribution model in dual‐tree complex wavelet domain

AbstractSource images are frequently corrupted by noise before fusion, which will lead to the quality decline of fused image and the inconvenience for subsequent observation. However, at present, most of the traditional medical image fusion scheme cannot be implemented in noisy environment. Besides, the existing fusion methods scarcely make full use of the dependencies between source images. In this research, a novel fusion algorithm based on the statistical properties of wavelet coefficients is proposed, which incorporates fusion and denoising simultaneously. In the proposed algorithm, the new saliency and matching measures are defined by two distributions: the marginal statistical distribution of single wavelet coefficient fit by the generalized Gaussian Distribution and joint distribution of dual source wavelet coefficients modeled by the anisotropic bivariate Laplacian model. Additionally, the bivariate shrinkage is introduced to develop a noise robust fusion method, and a moment‐based parameter estimation applied in the fusion scheme is also exploited in denoising method, which allows to achieve the consistency of fusion and denoising. The experiments demonstrate that the proposed algorithm performs very well on both noisy and noise‐free images from multimodal medical datasets (computerized tomography, magnetic resonance imaging, magnetic resonance angiography, etc.), outperforming the conventional methods in terms of both fusion quality and noise reduction.

Numerical solution of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions via Chebyshev wavelet method

ABSTRACTIn this paper, the fourth kind Chebyshev wavelets collocation method (FCWM) is applied for solving a class of fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the fourth kind. Moreover, upper bound of error of the fourth kind Chebyshev wavelets expansion is given. Based on the collocation technique, the fourth kind Chebyshev wavelets together with Gaussian integration are used to reduce the problem to the solution of a system of algebraic equations. During the process of establishing the expression of the solution, the boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Some examples are provided to confirm the reliability and effectiveness of the proposed method.