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Consider the nonparametric curve estimation problem. Wavelet shrinkage methods are applied to the data in the wavelet domain for noise reduction. After denoising, the function can be estimated by w...

Wavelet thresholding (or shrinkage) attempts to remove the noises existing in the signals while preserving inherent pattern characteristics in the reconstruction of true signals. For data-denoising purpose, we present a new wavelet thresholding procedure which employs the step-down testing idea of identifying active contrasts in unreplicated fractional factorial experiments. The proposed method employs bootstrapping methods to a step-down test for thresholding wavelet coefficients. By introducing the concept of a false discovery error rate in testing wavelet coefficients, we shrink the wavelet coefficients with $p$ -values higher than the error rate. The error rate controls the expected proportion of wrongly accepted coefficients among chosen wavelet coefficients. Bootstrap samples are used to approximate the $p$ -value for computational efficiency. We also present some guidelines for selecting the values of hyper-parameters which affect the performance in the step-down thresholding procedure. Based on some common testing signals and an air-conditioner sounds example, the comparison of our proposed procedure with other thresholding methods in the literature is performed. The analytical results show that the proposed procedure has a potential in data-denoising and data-reduction in a variety of signal reconstruction applications.

We describe the lifting dual tree complex wavelet transform (LDTCWT), a type of lifting wavelets remodeling that produce complex coefficients by employing a dual tree of lifting wavelets filters to get its real part and imaginary part. Permits the remodel to produce approximate shift invariance, directionally selective filters and reduces the computation time (properties lacking within the classical wavelets transform). We describe a way to estimate the accuracy of this approximation and style appropriate filters to attain this. These benefits are often exploited among applications like denoising, segmentation, image fusion and compression. The results of applications shrinkage denoising demonstrate objective and subjective enhancements over the dual tree complex wavelet transform (DTCWT). The results of the shrinkage denoising example application indicate empirical and subjective enhancements over the DTCWT. The new transform with the DTCWT provide a trade-off between denoising computational competence of performance, and memory necessities. We tend to use the PSNR (peak signal to noise ratio) alongside the structural similarity index measure (SSIM) and the SSIM map to estimate denoised image quality.

To remove the speckle noise of synthetic aperture radar (SAR) images, a novel denoising algorithm based on Bayes wavelet shrinkage and a fast guided filter is proposed. According to the statistical properties of SAR images, the noise-free signal and speckle noise in the wavelet domain are modeled as Laplace and Fisher-Tippett distributions respectively. Then a new wavelet shrinkage algorithm is obtained by adopting the Bayes maximum a posteriori estimation. Speckle noise in the high-frequency domain of SAR images is shrunk by this new wavelet shrinkage algorithm. As the wavelet coefficients of the low-frequency domain also contain some speckle noise, speckle noise in the low-frequency domain can be further filtered by the fast guided filter. The result of the denoising experiments of simulated SAR images and real SAR images demonstrate that the proposed algorithm has the ability to better denoise and preserve edge information.

In this paper, we revisit the effects of principal component analysis (PCA) on hyperspectral imagery denoising. Our previous work combined PCA with wavelet shrinkage and particularly good denoising results has been achieved. We debate that any denoising methods can be used to replace wavelet shrinkage in our PCA+wavelet shrinkage algorithm. The major difference between this work and our previous PCA-based denoising method is that we consider a mixture of Gaussian and shot noise in this work whereas our previous methods studied Gaussian white noise alone. In addition, we retain [Formula: see text] [Formula: see text] PCA output components in our forward PCA transform in this paper whereas we keep all PCA output components [Formula: see text] in our previous works. The [Formula: see text] above is the number of spectral bands in the original hyperspectral imagery data cube. In addition, PCA is much better than nonlinear PCA for hyperspectral imagery denoising when Gaussian white noise and shot noise are introduced as demonstrated in this paper. Extensive experiments demonstrate that the method proposed in this paper outperforms the existing methods significantly in terms of signal-to-noise ratio for two testing hyperspectral imagery data cubes.

Rapid switching of the power switches inside a grid-connected power converter causes a huge number of high-frequency spikes, which could degrade the power converters' functioning. The power spike smoothing algorithm is a digital solution to removing these switching spikes from the sensor measurement. It uses the wavelet decomposition to reveal the high-frequency components so that specific suppression policy could apply. Reconstruction from the original low-frequency components and reduced high-frequency components result in the smoothing effect in the output waveform, and the degree of the smoothing effect is adjustable depending on the variance of each power converter through the parameter shrinkage degree.

We develop a data-driven approach for signal denoising that utilizes variational mode decomposition (VMD) algorithm and Cramer Von Misses (CVM) statistic. In comparison with the classical empirical mode decomposition (EMD), VMD enjoys superior mathematical and theoretical framework that makes it robust to noise and mode mixing. These desirable properties of VMD materialize in segregation of a major part of noise into a few final modes while majority of the signal content is distributed among the earlier ones. To exploit this representation for denoising purpose, we propose to estimate the distribution of noise from the predominantly noisy modes and then use it to detect and reject noise from the remaining modes. The proposed approach first selects the predominantly noisy modes using the CVM measure of statistical distance. Next, CVM statistic is used locally on the remaining modes to test how closely the modes fit the estimated noise distribution; the modes that yield closer fit to the noise distribution are rejected (set to zero). Extensive experiments demonstrate the superiority of the proposed method as compared to the state of the art in signal denoising and underscore its utility in practical applications where noise distribution is not known a priori. (C) 2020 Elsevier Inc. All rights reserved.

The method of detrended fluctuation analysis (DFA) is useful in revealing the extent of long-range dependence, it has successfully been applied to different fields of interest. In this paper we proposed a smoothed detrended fluctuation analysis method based on the principle of wavelet shrinkage. The procedure is illustrated and compared with the DFA method by Monte Carlo simulations on fractional Gaussian noise models.

Partial Discharge (PD) measurements of large generators are extremely affected and hampered by noise, making the denoising of PD signal an inevitable issue. Wavelet shrinkage is the most commonly employed method for PD signal denoising. The appropriate mother wavelet and decomposition level are critically important for the denoising performance. In consideration of the PD signal characteristics of large generators, a novel wavelet shrinkage scheme for PD signal denoising is presented. In the scheme, a scale dependent wavelet selection method is proposed, the core idea is that the optimum wavelet at each scale is selected as the one maximizing the energy ratio of coefficients beside and inside the range formed by the threshold, which correspond to the signal to be reserved and noise to be removed, respectively. In addition, taking into account the influence of mother wavelet at each scale on the decomposition level, an approach for decomposition level determination is put forward based on the energy composition after decomposition at each scale. The application results on the simulated signals with different SNR obtained by combining the various pulses and measured signal on-site show the effectiveness of the proposed scheme. Besides, the denoising results are compared with that of the existing wavelet selection methods and the proposed wavelet selection method shows an obvious advantage.

Magnetocardiograms recorded in unshielded environments avoid the necessity of especially designed chambers, but are more sensitive to external interferences. Reliable denoising techniques have to be developed, which should not distort the clinical measurements. In the present paper, two filtering algorithms based on wavelet shrinkage, using intrinsic properties of the signal, have been developed and compared. The algorithm is fast computing and displays good results, enabling further improvements of the magnetocardiogram tehnique.

The extraction of Fetal ECG (FECG) is still a difficult task in non-invasive approach since the frequency components of dominant maternal ECG and fetal ECG signals are generally overlapped. Besides, the baseline wander and high-frequency noise make the clear FECG difficult to be extracted. In this paper, we proposed a new combination of independent component analysis (ICA), improved complete ensemble empirical mode decomposition (ICEEMDAN), and wavelet shrinkage (WS) de-noising (ICA-ICEEMDAN-WS) to extract FECG while reducing the noise. ICA algorithm as the first step of our proposed method separated the mixed abdominal ECG signal to obtain the noisy FECG. Then for further denoisng and obtain higher SNR, noisy FECG was decomposed by the newly proposed ICEEMDAN that is more accurate for non-linear and non-stationary bio-signal processing. The informative extracted components were then determined based on the statistical significance test. WS de-noising removed the remained noisy subcomponents and finally the baseline wander was reduced by partial reconstruction. The performance of ICA-ICEEMDAN-WS method was evaluated using simulated and real data sets. Our results showed that our proposed method outperformed ICA-EEMD-WS, the recently proposed algorithm based on ensemble empirical mode decomposition

A previsao de series temporais e largamente utilizada nas diversas areas do conhecimento humano, principalmente no planejamento e direcionamento estrategico das empresas. O sucesso desta tarefa depende das tecnicas de previsoes aplicadas. Neste artigo, e proposta uma metodologia hibrida para se projetar series temporais. Para a validacao da metodologia foi escolhida uma serie de tempo ja modelada por outros autores, possibilitando a comparacao dos resultados. A metodologia proposta integra as seguintes tecnicas: encolhimento wavelet, decomposicao wavelet de nivel r e redes neurais artificiais (RNAs). Primeiramente, uma serie temporal a ser prevista e submetida ao metodo de filtragem wavelet proposto, o qual a decompoe em componentes de tendencia e de residuo linear. Em seguida, ambas sao decompostas via decomposicao de nivel r, gerando, para cada uma, r+1 componentes wavelet (CWs); e, em seguida, cada CW e individualmente modelada por uma RNA. Finalmente, as previsoes para todas as CWs sao linearmente combinadas, produzindo as previsoes para a serie temporal supracitada. Para avalia-lo, a serie temporal de Canadian Lynx foi usada e todos os resultados alcancados pelo metodo proposto foram melhores do que outros existentes na literatura.

Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given function f and is widely used in data compression, signal processing and statistics, etc. For Calderon-Zygmund operators T, it is interesting to construct estimator of Tf, based on wavelet shrinkage estimator of f. With the help of a representation of operators on wavelets, due to Beylkin et al., an estimator of Tf is presented in this paper. The almost everywhere convergence and norm convergence of the proposed estimators are established.

In this paper, the generalized Bayes estimator of elliptical distribution parameter’s under asymmetric Linex error loss function is considered. The new shrinkage generalized Bayes estimator by applying wavelet transformation is investigated. We develop admissibility and minimaxity of shrinkage estimator on multivariate normal distribution.We present the simulation in order to test validity of purpose estimator.

У роботі запропоновано ефективний метод відновлення сигналів SIMS від сильно розмитих дискретних піків. Ця методика грунтується на регуляризації Тихонова-Міллера, де включена апріорна модель розв’язку. Останній – це шумопригнічуючий сигнал, отриманий при використанні фільтра Калмана. Це цікавий методом оцінки, але він може бути використаний тільки тоді, коли ми можемо точно описати наш зразок. Порівнюючи результати запропонованої методики з результатами літератури, наш алгоритм дає найкращі результати без артефактів і коливань, пов'язаних з шумом, і значного поліпшення глибинного аналізу, у той час як коефіцієнт підсилення менш поліпшений, ніж коефіцієнт, отриманий методом вейвлетів. Таким чином, цей новий алгоритм може розширити межі вимірювань SIMS до граничної роздільної здатності. This paper presents an efficient method for recovery of SIMS signals from strongly noised blurred discrete data. This technique is based on Tikhonov-Miller regularization where a priori model of solution is included. The latter is a denoisy signal obtained using the Kalman filter. This is an interesting estimation method, but it can only be used when the system is described precisely. By comparing the results of the proposed technique with those of the literature, our algorithm gives the best results without artifacts and oscillations related to noise and significant improvement of the depth resolution. While, the gain in FWHM is less improved than those obtained by the wavelet technique. Therefore, this new algorithm can push the limits of SIMS measurements towards its ultimate resolution.

In view of the low signal to noise ration and the difficulty on de-noising for ECG signal collected under unconstrained environment, a GA(Generation Algorithm) based wavelet method of finger ECG de-noising is proposed after a systematic study of wavelet threshold de-noising and generation algorithm. This new algorithm can dynamically get the optimal solution to fit the threshold in accordance with the characteristic of finger ECG. Then the algorithm is verified by fitting ECG and finger ECG. Compared with other algorithm, results show that the proposed method get the better performance according to the performance index including the output SNR, correlation coefficient and root mean square. Keywords-finger ECG; generation algorithm; wavelet de-noising

In our previous work, we proposed wavelet shrinkage estimation (WSE) for nonhomogeneous Poisson process (NHPP)-based software reliability models (SRMs), where WSE is a data-transform-based nonparametric estimation method. Among many variance-stabilizing data transformations, the Anscombe transform and the Fisz transform were employed. We have shown that it could provide higher goodness-of-fit performance than the conventional maximum likelihood estimation (MLE) and the least squares estimation (LSE) in many cases, in spite of its non-parametric nature, through numerical experiments with real software-fault count data. With the aim of improving the estimation accuracy of WSE, in this paper we introduce other three data transformations to preprocess the software-fault count data and investigate the influence of different data transformations to the estimation accuracy of WSE through goodness-of-fit test.

On the basis of wavelet theory, we propose an outlier-detection algorithm for satellite gravity gradiometry by applying a wavelet-shrinkage-de-noising method to some simulation data with white noise and outliers. The result shows that this novel algorithm has a 97% success rate in outlier identification and that it can be efficiently used for pre-processing real satellite gravity gradiometry data.

In Medical diagnosis operations such as feature extraction and object recognition will play the key role. These tasks will become difficult if the images are corrupted with noise. So the development of effective algorithms for noise removal became an important research area in present days. Developing Image denoising algorithms is a difficult task since fine details in a medical image embedding diagnostic information should not be destroyed during noise removal. Many of the wavelet based denoising algorithms use DWT (Discrete Wavelet Transform) in the decomposition stage are suffering from shift variance and lack of directionality. To overcome this in this paper we are proposing the denoising method which uses dual tree complex wavelet transform to decompose the image and shrinkage operation to eliminate the noise from the noisy image. In the shrinkage step we used semi-soft and stein thresholding operators along with traditional hard and soft thresholding operators and verified the suitability of dual tree complex wavelet transform for the denoising of medical images. The results proved that the denoised image using DTCWT (Dual Tree Complex Wavelet Transform) have a better balance between smoothness and accuracy than the DWT and less redundant than UDWT (Undecimated Wavelet Transform). We used the SSIM (Structural similarity index measure) along with PSNR (Peak signal to noise ratio) to assess the quality of denoised images.

By substituting an anisotropic diffusion operator for the isotropic Laplace operator in the directed diffusion equation, adding two different coefficients in the two diffusion terms, and choosing the image denoised by soft wavelet shrinkage as the initial approximate image, an improved directed diffusion equation model is proposed. Experiments show that the new model can solve the problem that edges and details will be rapidly blurred during the diffusion process in the directed diffusion equation.

The field of terahertz (THz) science and technology has achieved significant progress over the last decades. Research interest focuses on THz spectroscopy and imaging. The signal-to-noise ratio (SNR) of the THz system is of great importance in the application of THz spectroscopy and imaging. In this letter, the wavelet de-noising technology is used to improve SNR and increase the speed of the THz time-domain spectroscopy system by reducing the repeating times.

In this paper, we introduce a novel hybrid variational model which gen- eralizes the classical total variation method and the wavelet shrinkage method. An alternating minimization direction algorithm is then employed. We also prove that it converges strongly to the minimizer of the proposed hybrid model. Finally, some nu- merical examples illustrate clearly that the new model outperforms the standard total variation method and wavelet shrinkage method as it recovers better image details and avoids the Gibbs oscillations. AMS subject classifications: 90C25, 52A41, 28E99, 62P35

While wireless sensors are increasingly adopted in various applications, the need of developing data reduction methods to alleviate data transmission rate issue between the sensors and the data interpretation unit becomes more urgent. This article presents a new data reduction method for sensors used in structural health monitoring application. Our goal is to achieve an effective data reduction capability while maintaining adequate power for damage detection. We propose to establish an explicit measure of damage detection capability for the features in the response signals and use this measure to select the subset of the features that balance between the degree of data reduction and the damage detection capability. We also explore a computationally efficient procedure searching for the best subset of the features. This new method is tested on experimentally obtained Lamb wave signals for beam damage detection. Performance comparisons with respect to the existing methods demonstrate the strength of the proposed method.

Cross-validation has been successfully used in various areas of statistics. However, it has not been used much in wavelet shrinkage estimation because fast wavelet methods cannot be applied to deleted data. In this paper, we show this problem can be avoided by using a fast imputation of data. This allows level-dependent cross- validation which is attractive to data with different sparseness. The proposed methods can be easily extended to higher dimensional problem such as image. Results from simulation and examples demonstrate the promising empirical properties of the procedure. In particular, the methods proposed in this work provide outstanding results for non-Gaussian noises because cross-validation is not based on normality assumptions.

This paper considers the problem of selecting a robust threshold of wavelet shrinkage. Previous approaches reported in literature to handle the presence of outliers mainly focus on developing a robust procedure for a given threshold; this is related to solving a nontrivial optimization problem. The drawback of this approach is that the selection of a robust threshold, which is crucial for the resulting fit is ignored. This paper points out that the best fit can be achieved by a robust wavelet shrinkage with a robust threshold. We propose data-driven selection methods for a robust threshold. These approaches are based on a coupling of classical wavelet thresholding rules with pseudo data. The concept of pseudo data has influenced the implementation of the proposed methods, and provides a fast and efficient algorithm. Results from a simulation study and a real example demonstrate the promising empirical properties of the proposed approaches.

In this paper, we present a new class of iterative regularization methods in the setting of Besov spaces, which can be seen as generalizations of J. Xu's method. By incorporating translation invariant wavelet transform, minimizers of the new methods can be understood as the alternative to translation invariant wavelet shrinkage with weight that is dependent on the wavelet decomposition scale and the Besov smooth order. And we generalize the iterative regularization methods to a new class of nonlinear inverse scale spaces with scale and Besov smooth order dependent weight. The numerical results show an excellent denoising effect and improvement over J. Xu's method.

We consider the denoising of a function (an image or a signal) containing smooth regions and edges. Classical ways to solve this problem are variational methods and shrinkage of a representation of the data in a basis or a frame. We propose a method which combines the advantages of both approaches. Following the wavelet shrinkage method of Donoho and Johnstone, we set to zero all frame coefficients with respect to a reasonable threshold. The shrunk frame representation involves both large coefficients corresponding to noise (outliers) and some coefficients, erroneously set to zero, leading to Gibbs-like oscillations in the estimate. We design a specialized (nonsmooth) objective function allowing all these coefficients to be selectively restored, without modifying the other coefficients which are nearly faithful, using regularization in the domain of the restored function. We analyze the well-posedness and the main properties of this objective function. We also propose an approximation of this method which...

Consider an asymmetric Linex loss. We provide the soft wavelet shrinkage estimation of a Bayesian interpretation under such a loss.

Computer signal processing is used for quantitative data analysis (QDA) in magnetic resonance spectroscopy (MRS). The main difficulty in QDA is that MRS signals appear to be contaminated with random noise. Noise reduction can be achieved by coherent averaging, but it is not always possible to average many MRS waveforms. Wavelet shrinkage de-noising (WSD) is a technique that can be employed in this case. The potentialities of WSD in MRS, alone and combined with the Cadzow algorithm, are analyzed through computer simulations. The results can facilitate an appropriate application of WSD, as well as a deeper understanding of this technique. © 2002 Wiley Periodicals, Inc. Concepts Magn Reson 14: 388–401, 2002

The current work addresses topics related to signal filtering via wavelet shrinkage. Wavelets techniques applied to signal filtering go back to the 80’s and an overview about the main results in the area are presented, as well as the most recent contributions, concerning applications and methodological aspects involved. A short discussion about wavelets and a detailed presentation about the threshold operation are given. The main methods to define the threshold values are also described and their performances are tested through numerical simulations for 1D and 2D data. The presented results highlight the potential of associating wavelet methods to signal denoising techniques.

In this paper, we propose a new patch-based image denoising algorithm using graph signal processing. The concept of this algorithm is to take advantage of the redundancy of the BM3D transform and the edge preservation property of graph-based image processing. More specifically, we collect similar patches in the image, and construct a graph by connecting obtained patches. Then we apply a graph wavelet filter bank on graph signals to attenuate additive white gaussian noise by shrinking derived coefficients. We apply our proposed algorithm to depth map denoising. The experimental results demonstrate significant performance gains for the edge preservation and the noise reduction.

Filtering time series is an important task in many domains as biomedical signal processing. The Heart Rate Variability (HRV) is the analysis of the RR time series obtained as intervals between heart beats. Usually the RR time series obtained are non-stationary and are present ectopic beats, which prevents extraction of useful information, therefore must often be filtered in order to emphasize different characteristics or properties. This paper presents a method to filter R-R time series, using wavelet shrinkage techniquein order to have a new and more adapted visualization of the HRV’s graphical interpretation as Lorenz plot.

The Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.

We propose methods of both slow and rapid post-processing of signals for erasure of artifacts that arise in the process of thresholding and quantization. We use wavelets as tools to define constraints and variational functionals as measures of complexity of signals. The methods come from analyses of different possibilities of blending variational calculus and wavelet multiresolution in ways that appear to be natural.

Wavelet shrinkage estimators are obtained by applying a shrinkage rule on the empirical wavelet coefficients. Such simple estimators are now well explored and widely used in wavelet-based nonparametrics. Results of Tao (1996, Appl. Comput. Harmon. Anal.3, 384–387) demonstrated that hard and soft thresholding shrinkage estimators absolutely converge almost everywhere to the original function when the threshold value goes to zero. Such natural and intuitive behavior of threshold estimators is expected, yet this result does not translate to the Fourier expansions. In this paper we show that almost everywhere convergence of shrinkage estimators holds for a range of shrinkage rules, not necessarily thresholding, subject to some mild technical conditions. Comments about the norm convergence are provided as well.