Wavelet Shrinkage Method Research Articles

Simulation studies to compare bayesian wavelet shrinkage methods in aggregated functional data

Simulation studies to compare bayesian wavelet shrinkage methods in aggregated functional data

Performance Enhancement of INS and UWB Fusion Positioning Method Based on Two-Level Error Model

In GNSS-denied environments, especially when losing measurement sensor data, inertial navigation system (INS) accuracy is critical to the precise positioning of vehicles, and an accurate INS error compensation model is the most effective way to improve INS accuracy. To this end, a two-level error model is proposed, which comprehensively utilizes the mechanism error model and propagation error model. Based on this model, the INS and ultra-wideband (UWB) fusion positioning method is derived relying on the extended Kalman filter (EKF) method. To further improve accuracy, the data prefiltering algorithm of the wavelet shrinkage method based on Stein's unbiased risk estimate-Shrink (SURE-Shrink) threshold is summarized for raw inertial measurement unit (IMU) data. The experimental results show that by employing the SURE-Shrink wavelet denoising method, positioning accuracy is improved by 76.6%; by applying the two-level error model, the accuracy is further improved by 84.3%. More importantly, at the point when the vehicle motion state changes, adopting the two-level error model can provide higher computational stability and less fluctuation in trajectory curves.

Gamma-Minimax Wavelet Shrinkage for Signals with Low SNR

In this paper, we propose a method for wavelet denoising of signals contaminated with Gaussian noise when prior information about the ${L^{2}}$-energy of the signal is available. Assuming the independence model, according to which the wavelet coefficients are treated individually, we propose simple, level-dependent shrinkage rules that turn out to be Γ-minimax for a suitable class of priors. The proposed methodology is particularly well suited in denoising tasks when the signal-to-noise ratio is low, which is illustrated by simulations on a battery of some standard test functions. Comparison to some commonly used wavelet shrinkage methods is provided.

Jump detection in high-frequency financial data using wavelets

The presence of spikes or cusps in high-frequency return series might generate problems in terms of inference and estimation of the parameters in volatility models. For example, the presence of jumps in a time series can influence sample autocorrelations, which can cause misidentification or generate spurious ARCH effects. On the other hand, these jumps might also hide relevant heteroskedastic behavior of the dependence structure of a series, leading to identification issues and a poorer fit to a model. This paper proposes a wavelet-shrinkage method to separate out jumps in high-frequency financial series, fitting a suitable model that accounts for its stylized facts. We also perform simulation studies to assess the effectiveness of the proposed method, in addition to illustrating the effect of the jumps in time series. Lastly, we use the methodology to model real high-frequency time series of stocks traded on the Brazilian Stock Exchange and OTC and a series of cryptocurrencies trades.

An Efficient Noise Elimination Method for Non-stationary and Non-linear Signals by Averaging Decomposed Components

In this paper, a moving-average method of smoothing noise based on complex exponential decomposition is applied to eliminate noise of a non-stationary signal and a non-linear signal produced by Bouc–Wen model, which are added to white Gaussian noise to simulate the noise in measured signal. The method uses a sliding window cutting the entire non-stationary and/or non-linear signal into small segments and considers that the small segments are stable and linear. The segments are decomposed into a series of components via complex exponential decomposition, and the high-energy components are reserved to reconstruct de-noised signal. Then, due to the overlap of the reconstructed segments, the average value at the same time point of reconstruction signal is regarded as the de-noised data. A non-stationary signal and a non-linear signal are selected to investigate the performance of the proposed method, the results show that the proposed method has better de-noising efficiency compared with the wavelet shrinkage method and the Savitzky–Golay filter method based on EMD (EMD-SG) for dealing with the signals with SNR of 10 dB, 15 dB, and 20 dB, and de-noised signal using the proposed method has the highest signal-to-noise ratio (SNR) and the least root mean square error (RMSE).

Comparison of Time Series Models before and after Using Wavelet Shrinkage Filtering to Forecast the Amount of Natural Gas in Iraq

The procedure of reducing noise or reduction before analyzing the time series is important to get accurate and reliable outcomes when building models. Wavelet Shrinkage consisting of wavelets with thresholding is a powerful mathematical method used to reduce noise that can be exposed. It has time-series observations and selects the cut-off threshold level to be suitable for removing most noise. In this paper, natural gas production data in Iraq were used during (1981-2019), which included non-stationary data or outliers, so the researcher must treat this problem before starting the analysis. Thus, the researcher selected the way of time series analysis using Box- Jenkins (ARIMA (p,d,q) models and the method of wavelet shrinkage in the two methods of cutting the threshold before and after filtering wavelet shrinkage to choose the most suitable mathematical model that defines the study data. The paper was concerned with finding an efficient model by comparing the (Box-Jenkins) linear ARIMA (p, d, q) models estimated from the time series data before and after filtering the wavelet shrinkage, and then decreasing the rank of the estimated model from the candidate observations (While maintaining the accuracy and suitability of the estimated models) and re-comparing it with the estimated linear models of the original observations, and then measuring the most efficient model based on some statistical criteria, including the AIC (Akaike information criterion), SBIC (Schwarz information criterion, Bayesian information criterion BIC)and HQIC (Hannan-Quinn information criterion). Statistical programs such as Statgraphics XVII - X64 and MATLAB were used to analyze the data. The paper reached the efficiency of wavelet shrinkage filters in treating the noise problem and getting efficient estimated models, specifically the wavelet shrinkage filter (Daubechies(db8)) with a soft-threshold cut-off estimated by the fixed-form method, and the possibility of obtaining linear models with lower orders and higher efficiency for the filtered observations compared to the corresponding models estimated from the original observations, that is, the suitable model for the data is a model ARIMA(0,1,1) for the data paved in a Daubechies method, i.e., it is preferable to do the waveform analysis of the data before analyzing the time series concerning the observations of natural gas production in Iraq The preference was given to the model estimated by Box and Jenkins ARIMA (0,1,1) model using wavelet shrinkage. Daubechies (db8) wavelet reduction

Image Noise Reduction Based on a Fixed Wavelet Frame and CNNs Applied to CT.

Radiation exposure in CT imaging leads to increased patient risk. This motivates the pursuit of reduced-dose scanning protocols, in which noise reduction processing is indispensable to warrant clinically acceptable image quality. Convolutional Neural Networks (CNNs) have received significant attention as an alternative for conventional noise reduction and are able to achieve state-of-the art results. However, the internal signal processing in such networks is often unknown, leading to sub-optimal network architectures. The need for better signal preservation and more transparency motivates the use of Wavelet Shrinkage Networks (WSNs), in which the Encoding-Decoding (ED) path is the fixed wavelet frame known as Overcomplete Haar Wavelet Transform (OHWT) and the noise reduction stage is data-driven. In this work, we considerably extend the WSN framework by focusing on three main improvements. First, we simplify the computation of the OHWT that can be easily reproduced. Second, we update the architecture of the shrinkage stage by further incorporating knowledge of conventional wavelet shrinkage methods. Finally, we extensively test its performance and generalization, by comparing it with the RED and FBPConvNet CNNs. Our results show that the proposed architecture achieves similar performance to the reference in terms of MSSIM (0.667, 0.662 and 0.657 for DHSN2, FBPConvNet and RED, respectively) and achieves excellent quality when visualizing patches of clinically important structures. Furthermore, we demonstrate the enhanced generalization and further advantages of the signal flow, by showing two additional potential applications, in which the new DHSN2 is used as regularizer: (1) iterative reconstruction and (2) ground-truth free training of the proposed noise reduction architecture. The presented results prove that the tight integration of signal processing and deep learning leads to simpler models with improved generalization.

Bayesian wavelet shrinkage with logistic prior

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 wavelet basis expansion. The present paper proposes a Bayesian approach for wavelet shrinkage with the use of a mixture of a point mass function at zero and the logistic distribution symmetric around zero as the prior distribution to the wavelet coefficients of an unknown function in models with additive Gaussian errors. Statistical properties such as bias, classical and Bayesian risks of the rules are analyzed and performances of the proposed rules are obtained in simulations studies involving the Donoho-Johnstone test functions. Application in a mass spectrometry real data set is done.

K - Shrinkage Function for ECG Signal Denoising.

ECG signals is a graphical way of recording the electrical actions of the heart for the various diagnostic purposes. ECG signals are affected by various noises such as Electrode Contact, Baseline Wandering, Motion artifact, Power-line interference, Muscle Contractions, and Electrosurgical noise during data acquisition. Denoising is a technique which is used for removing the noise in ECG signals which keeps the useful information. In this paper, a new category of Wavelet shrinkage methods is proposed. The white Gaussian noise is mixed with the ECGs for simulation and tested with the new class of shrinkage function and is compared with the other wavelet shrinkage functions such as hard and soft shrinkage. The performance measures such as Signal to Noise Ratio (SNR) and Percent Root mean - square Difference (PRD) etc. are used to examine the performance of various shrinkage functions. The experimental result shows that it gives better MSE over conventional shrinkage functions.

Two-stage wavelet shrinkage and EEG-EOG signal contamination model to realize quantitative validations for the artifact removal from multiresource biosignals

Abstract Electroencephalogram (EEG) data inevitably contain large amounts of noise, particularly from ocular potentials in tasks with eye movements, known as electro-oculography (EOG) artifact, which has been a crucial issue in brain–computer-interface studies. This time-frequency characteristic has been substantially dealt with in previous proposed denoising algorithms that relied on a consistent assumption based on the single-noise component model. However, the traditional model is not applicable for biomedical signals that consist of multiple signal components, such as weak EEG signals that are easily recognized as noise because of the signal amplitude with respect to the EOG signal. In consideration of the realistic signal contamination, we designed a novel EEG-EOG signal contamination model to quantitatively validate artifact removal from EEGs. We then proposed the two-stage wavelet shrinkage method with the decomposition of the undecimated wavelet transform (UDWT), which is suitable for signal structure. Open-source clinical intracranial EEGs with the hundred dataset in each behavioral condition were introduced to the validation as “true EEG” before the contamination of artificial EOGs. The quality of the reconstructed EEG signal has evaluated in the frequency spectrum, which represents how much the original specific brain-state characteristic can be reconstructed. Numerical analyses demonstrated that the first stage pursued abrupt changes with high amplitudes provided by assumed EOGs, and the second stage provided the EEG frequency spectrum as observed in the original signal. What the performance exceeded the conventional shrinkage suggests that threshold values is required to be set properly depending on individual amplitudes of contaminated bio-signal sources as our proposed method demonstrated. This result focused on the actual amplitude-frequency structure in the polygenetic signal. It not only provided the decomposition performance, but also revealed how bio-signals are mixed together by using a new standard model for robust validation in the EEG-EOG signal contamination.

An improved time–frequency noise reduction method using a psycho-acoustic Mel model

This paper addresses the problem of noise reduction in non-stationary signals. The paper first describes a human physiology based time–frequency (TF) representation (HPTF) using Mel filterbanks. It is then used to improve a noise reduction algorithm that does not require any a priori information about the signal of interest and the noise. This algorithm is efficiently implemented using an original wavelet shrinkage method. The overall method results in an original TF denoising procedure that yields a denoised HPTF (DHPTF). From this representation, one can reconstruct a denoised time-domain signal and therefore define a new improved noise reduction algorithm, whose performance is evaluated and compared with other state-of-the-art methods. The performance assessment uses several criteria: (1) signal-to-noise-ratio (SNR), (2) segmental SNR (SSNR) and (3) mean square error (MSE). The results indicate an improvement of up to 4.72 dB with respect to (w.r.t.) SNR, 2.79 dB w.r.t. SSNR and 4.72 dB w.r.t. MSE for a speech database signals corrupted with four different noises. In addition, other applications such as EEG signal enhancement show promising results.

Spatially adaptive denoising for X-ray cardiovascular angiogram images

Abstract The X-ray angiogram image denoising is always one of the most popular research in the field of computer vision. While the methods removed the noise, the useful structure (such as peripheral vascular) had also been smoothed, the fundamental reason is that the denoising methods cannot efficiently distinguish structural areas from flat areas. In this paper, we have proposed a spatially adaptive image denoising (SAID) method which contains two steps: spatially adaptive gradient descent (SAGD) image denoising and dual-domain filter (DDF). The SAGD denoising method contains the following parts: first of all, the wavelet shrinkage method is used to estimate redundant information which is composed of the noise and useful structures; secondly, according to the characteristic of second order matrix, a spatially adaptive gradient factor (SAGF) has been constructed to distinguish the structure from flat areas; finally, the SAGF replaces the original gradient factor and then the SAGD image denoising method is formed. To further improve the quality of the SAGD image, the SAGD image is re-denoised by a modified DDF which is guided with a rotationally invariant non-local filter (RINLF) in spatial domain and gets structural details by wavelet shrinkage in frequency domain. The results of simulation experiments verify that the proposed SAID method can get well quantitative and qualitative results which are even superior to those using the state-of-the-art denoising methods. Even more, the fluctuation of peak signal-to-noise ratio (PSNR) value is very small with a small disturbance of SAGF, which illustrates that our algorithm is more robust than the prior progressive image denoising (PID) method. Moreover, the comparison results of the extensive experiments on clinical X-ray cardiovascular angiogram images further illustrate that our method can yield clearer cardiovascular images which can provide more useful vascular information for clinicians to analyze and diagnose the cardiovascular diseases.

Spectral Subtraction Denoising Preprocessing Block to Improve Slow Cortical Potential Based Brain–Computer Interface

The measured electroencephalographic (EEG) signals used in brain–computer interface (BCI) are usually contaminated with several additive random and physiological noise components. Even though several preprocessing strategies were proposed in the literature to extract useful EEG signal components for further processing and analysis, their performance is yet to meet practical needs to boost BCI applications performance. Of particular interest among those methods are the ones addressing random noise suppression that could help enhance the performance of low-cost BCI systems/headsets. A preprocessing block for signal denoising of slow cortical potential (SCP) that provides efficient noise removal and better classification accuracy is presented. This method is a variant of the spectral subtraction signal denoising whereby the noise power spectrum is estimated and removed adaptively. This method is applied to each EEG channel separately thus providing a preprocessing block that could be integrated with the existing spatial/temporal preprocessing methods. The impact of this method on classification accuracy is studied and compared to the conventional wavelet shrinkage method. The proposed method is verified using experimental data from the BCI competition II data set whereby a performance boost was obtained with the new preprocessing block. The improvement was quantitatively assessed using mean square error and mean absolute error measures and was also shown to be statistically significant. Moreover, spectral subtraction denoising is shown to have less computational complexity than wavelet shrinkage based methods. The new preprocessing block for SCP-based BCI signals based on spectral subtraction provides significant improvement in performance when added and offers an adaptive yet less computationally expensive alternative to existing methods such as wavelet shrinkage based denoising method. Also, given its independent channel processing, it shows potential for seamless integration into conventional processing chain for different BCI applications.

Mixed Hölder matrix discovery via wavelet shrinkage and Calderón–Zygmund decompositions

This paper concerns two related problems in the analysis of data matrices whose rows and columns are equipped with tree metrics. First is the problem of recovering a matrix that has been corrupted by additive noise. Under the assumption that the clean matrix exhibits a specific regularity condition, known as the mixed Hölder condition, we adapt the well-known Donoho–Johnstone wavelet shrinkage methods from classical nonparametric statistics to obtain estimators that are within a logarithmic factor of the minimax error rate with respect to mean squared error loss. The second part of this paper develops a theory of Besov spaces on products of tree geometries. We show that matrices with small Besov norm can be written as a sum of a mixed Hölder matrix and a matrix with small support. Such decompositions are known as Calderón–Zygmund decompositions and are of general interest in harmonic analysis. The decompositions we establish impose fewer conditions on the function with small support than previous decompositions of this type while maintaining the same guarantees on the mixed Hölder matrix. As such, they are applicable to a greater variety of matrices and should find use in many data organization problems. As part of our analysis, we provide characterizations of the underlying Besov spaces using wavelets and other multiscale difference operators that are analogous to those from the classical Euclidean theory.

Fast generation of computer-generated holograms using wavelet shrinkage

Computer-generated holograms (CGHs) are generated by superimposing complex amplitudes emitted from a number of object points. However, this superposition process remains very time-consuming even when using the latest computers. We propose a fast calculation algorithm for CGHs that uses a wavelet shrinkage method, eliminating small wavelet coefficient values to express approximated complex amplitudes using only a few representative wavelet coefficients.

Generalized Wavelet Shrinkage of Inline Raman Spectroscopy for Quality Monitoring of Continuous Manufacturing of Carbon Nanotube Buckypaper

Process monitoring and quality control is essential for continuous manufacturing processes of carbon nano- tube (CNT) thin sheets or buckypaper. Raman spectroscopy is an attractive inline quality characterization and quantification tool for nanomanufacturing because of its nondestructive nature, fast data acquisition speed, and ability to provide detailed material information. However, there is signal-dependent noise buried in the Raman spectra, which reduces the signal-to-noise (S/N) ratio and affects the accuracy, efficiency, and sensitivity for Raman spectrum-based quality control approaches. In this paper, a signal analysis model with signal-dependent noise for Raman spectroscopy is developed and validated based on experimental data. The wavelet shrinkage method is used for denoising and improving the S/N ratio of raw Raman spectra. Based on the validated signal-noise relationship, a novel generalized wavelet shrinkage approach is introduced to remove noise in all wavelet coefficients by applying individual adaptive wavelet thresholds. The effectiveness of this method is demonstrated using both simulation and experimental case studies of inline Raman monitoring of continuous buckypaper manufacturing. The proposed method allows for a significant reduction of Raman data acquisition time without much loss of S/N ratio, which inherently enables Raman spectroscopy for inline monitoring and control for continuous nanomanufacturing processes.

Optimized Wavelet Denoising for Self-Similar ffα-Stable Processes

We investigate the performance of wavelet shrinkage methods for the denoising of symmetric- ${\alpha }$ -stable ( $\text{S} {\alpha }\text{S}$ ) self-similar stochastic processes corrupted by additive white Gaussian noise (AWGN), where $ {\alpha }$ is tied to the sparsity of the process. The wavelet transform is assumed to be orthonormal and the shrinkage function minimizes the mean-square approximation error (MMSE estimator). We derive the corresponding formula for the expected value of the averaged estimation error. We show that the predicted MMSE is a monotone function of a simple criterion that depends on the wavelet and the statistical parameters of the process. Using the calculus of variations, we then optimize this criterion to find the best performing wavelet within the extended family of Meyer wavelets, which are bandlimited. These are compared with the Daubechies wavelets, which are compactly supported in time. We find that the wavelets that are shorter in time (in particular, the Haar basis) are better suited to denoise the sparser processes (say, $ {\alpha ), while the bandlimited ones (including the Held and Shannon wavelets) offer the best performance for $ {\alpha >1.6}$ , the limit corresponding to the Gaussian case (fBm) with $ {\alpha =2}$ .

Patch-based anisotropic diffusion scheme for fluorescence diffuse optical tomography—part 1: technical principles

Fluorescence diffuse optical tomography (fDOT) provides 3D images of fluorescence distributions in biological tissue, which represent molecular and cellular processes. The image reconstruction problem is highly ill-posed and requires regularisation techniques to stabilise and find meaningful solutions. Quadratic regularisation tends to either oversmooth or generate very noisy reconstructions, depending on the regularisation strength. Edge preserving methods, such as anisotropic diffusion regularisation (AD), can preserve important features in the fluorescence image and smooth out noise. However, AD has limited ability to distinguish an edge from noise. In this two-part paper, we propose a patch-based anisotropic diffusion regularisation (PAD), where regularisation strength is determined by a weighted average according to the similarity between patches around voxels within a search window, instead of a simple local neighbourhood strategy. However, this method has higher computational complexity and, hence, we wavelet compress the patches (PAD-WT) to speed it up, while simultaneously taking advantage of the denoising properties of wavelet thresholding. The proposed method combines the nonlocal means (NLM), AD and wavelet shrinkage methods, which are image processing methods. Therefore, in this first paper, we used a denoising test problem to analyse the performance of the new method. Our results show that the proposed PAD-WT method provides better results than the AD or NLM methods alone. The efficacy of the method for fDOT image reconstruction problem is evaluated in part 2.

Bayesian Wavelet Shrinkage of the Haar-Fisz Transformed Wavelet Periodogram.

It is increasingly being realised that many real world time series are not stationary and exhibit evolving second-order autocovariance or spectral structure. This article introduces a Bayesian approach for modelling the evolving wavelet spectrum of a locally stationary wavelet time series. Our new method works by combining the advantages of a Haar-Fisz transformed spectrum with a simple, but powerful, Bayesian wavelet shrinkage method. Our new method produces excellent and stable spectral estimates and this is demonstrated via simulated data and on differenced infant electrocardiogram data. A major additional benefit of the Bayesian paradigm is that we obtain rigorous and useful credible intervals of the evolving spectral structure. We show how the Bayesian credible intervals provide extra insight into the infant electrocardiogram data.

Wavelet base selection for de‐noising and extraction of partial discharge pulses in noisy environment

Wavelet-based de-noising is used to separate partial discharge (PD) signals from the noises resulting from measurement circuits or the surrounding environment. PD de-noising by using the wavelet shrinkage method is capable of separating the noise component to some extent, but the selection of the wavelet base has a remarkable effect on the de-noising results. The wavelet base is directly related to the distortion of the PD waveform and quality of the de-noising process. Although there are applications on PD noise separation in the literature, the selection of the wavelet base, which affects the evaluation of the PD characteristics, is still challenging. Instead of using correlation-based wavelet base selection for de-noising PD data, in this study a novel wavelet base selection method based on the most informative sub-band energy and entropy for separating noise from PD pulses is introduced and successfully applied to raw data obtained from the PD measurement set-up. The advantage of the proposed method is that the wavelet base selection solution is automatic and independent of the original noise-free pulse waveform. This study shows that the proposed method is useful for the extraction of noisy PD pulses by describing the basic discharge parameters such as discharge amplitude and the duration and time of occurrence more clearly.