Soft-thresholding Scheme Research Articles

High dynamic range land wavefield reconstruction from randomized acquisition

Compressive sensing (CS) is an alternative to regular Shannon sampling that captures similar information from reduced measurements. It relies on randomized sampling patterns and a sparse data representation to reconstruct the regularly sampled object. CS is an important ingredient in affordable seismic acquisition, which can lead to improvements in the near-surface mapping and noise suppression for land data. However, the near surface traps most of the source-generated energy, resulting in data that are rich in high-wavenumber content and have amplitudes spanning several orders of magnitude. When dealing with such high dynamic range nonstationary data, the Fourier domain is not optimal for providing a sparse representation — a necessary condition for successful application of CS. In contrast, a discrete complex wavelet transform can localize high-energy features, has good directional selectivity, and is near-shift invariant. Combined, these properties allow complex wavelets to represent detail-rich wavefields in a compact form. To leverage these features and achieve good CS reconstructions, we develop a scale- and orientation-dependent iterative soft thresholding (IST) scheme for reconstructing high dynamic range wavefields. Our approach requires little parameterization, is easy to implement, and is robust to reconstructed wavefield sampling grid and dynamic range. We test IST on different wavefields with randomly missing traces and compare the data reconstructions with the spectral projected gradient solver and projection onto convex sets. We quantify the reconstructions by a direct comparison of Fourier coefficients between fully sampled and reconstructed wavefields. Taking log10 of Fourier coefficients prior to computing the quality metric deemphasizes the importance of magnitude match while highlighting Fourier coefficient support accuracy, which usually translates into good structural fidelity of reconstructed data. We find that IST performs consistently among all examples, yielding good phase match while performing gentle denoising.

Hybrid Thresholding for Image Deconvolution in Expectation Maximization framework

ABSTRACT In this study, we proposed an image deconvolution method in the expectation maximization (EM) framework. This method involves two steps: (i) E-step: utilizing the fast Fourier transform (FFT) for computationally efficient inversion of the convolution operator and (ii) M-step: employing the discrete wavelet transform (DWT) for estimating the original image from the image obtained in the E-step. In M-step, we proposed a modified L1-clipped penalty resulting in a hybrid thresholding scheme that integrates conventional hard and soft thresholds. This hybrid threshold ameliorates the inherent bias-variance trade-offs associated with traditional hard and soft thresholding schemes. The mathematical expressions for the risk, bias, and variance of the proposed hybrid threshold are derived and the performance is evaluated through simulation. Experimental results show that the proposed method achieves optimal values for variance and bias, thereby minimizing the risk. Moreover, the proposed method outperforms state-of-the-art methods in terms of performance metrics: PSNR, ISNR, and SSIM.

Soft thresholding schemes for multiple signal classification algorithm.

Multiple signal classification algorithm (MUSICAL) exploits temporal fluctuations in fluorescence intensity to perform super-resolution microscopy by computing the value of a super-resolving indicator function across a fine sample grid. A key step in the algorithm is the separation of the measurements into signal and noise subspaces, based on a single user-specified parameter called the threshold. The resulting image is strongly sensitive to this parameter and the subjectivity arising from multiple practical factors makes it difficult to determine the right rule of selection. We address this issue by proposing soft thresholding schemes derived from a new generalized framework for indicator function design. We show that the new schemes significantly alleviate the subjectivity and sensitivity of hard thresholding while retaining the super-resolution ability. We also evaluate the trade-off between resolution and contrast and the out-of-focus light rejection using the various indicator functions. Through this, we create significant new insights into the use and further optimization of MUSICAL for a wide range of practical scenarios.

TFisher: A powerful truncation and weighting procedure for combining $p$-values

The $p$-value combination approach is an important statistical strategy for testing global hypotheses with broad applications in signal detection, meta-analysis, data integration, etc. In this paper we extend the classic Fisher’s combination method to a unified family of statistics, called TFisher, which allows a general truncation-and-weighting scheme of input $p$-values. TFisher can significantly improve statistical power over the Fisher and related truncation-only methods for detecting both rare and dense “signals.” To address wide applications, analytical calculations for TFisher’s size and power are deduced under any two continuous distributions in the null and the alternative hypotheses. The corresponding omnibus test (oTFisher) and its size calculation are also provided for data-adaptive analysis. We study the asymptotic optimal parameters of truncation and weighting based on Bahadur efficiency (BE). A new asymptotic measure, called the asymptotic power efficiency (APE), is also proposed for better reflecting the statistics’ performance in real data analysis. Interestingly, under the Gaussian mixture model in the signal detection problem, both BE and APE indicate that the soft-thresholding scheme is the best, the truncation and weighting parameters should be equal. By simulations of various signal patterns, we systematically compare the power of statistics within TFisher family as well as some rare-signal-optimal tests. We illustrate the use of TFisher in an exome-sequencing analysis for detecting novel genes of amyotrophic lateral sclerosis. Relevant computation has been implemented into an R package TFisher published on the Comprehensive R Archive Network to cater for applications.

An efficient ECG denoising methodology using empirical mode decomposition and adaptive switching mean filter

Abstract Electrocardiogram (ECG) is a widely employed tool for the analysis of cardiac disorders. A clean ECG is often desired for proper treatment of cardiac ailments. However, in the real scenario, ECG signals are corrupted with various noises during acquisition and transmission. In this article, an efficient ECG denoising methodology using combined empirical mode decomposition (EMD) and adaptive switching mean filter (ASMF) is proposed. The advantages of both EMD and ASMF techniques are exploited to reduce the noises in the ECG signals with minimum distortion. Unlike conventional EMD based techniques, which reject the initial intrinsic mode functions (IMFs) or utilize a window based approach for reducing high-frequency noises, here, a wavelet based soft thresholding scheme is adopted for reduction of high-frequency noises and preservation of QRS complexes. Subsequently, an ASMF operation is performed to enhance the signal quality further. The ECG signals of standard MIT-BIH database are used for the simulation study. Three types of noises in particular white Gaussian noise, Electromyogram (EMG) and power line interference contaminate the test ECG signals. Three standard performance metrics namely output SNR improvement, mean square error, and percentage root mean square difference measure the efficacy of the proposed technique at various signal to noise ratio (SNR). The proposed denoising methodology is compared with other existing ECG denoising approaches. A detail qualitative and quantitative study and analysis indicate that the proposed technique can be used as an effective tool for denoising of ECG signals and hence can serve for better diagnostic in computer-based automated medical system.

Seismic random noise attenuation using modified wavelet thresholding

<p>In seismic exploration, random noise deteriorates the quality of acquired data. This study analyzed existing denoising methods used in seismic exploration from the perspective of random noise. Wavelet thresholding offers a new approach to reducing random noise in simulation results, synthetic data, and real data. A modified wavelet threshold function was developed by considering the merits and demerits of conventional soft and hard thresholding schemes. A MATLAB (matrix laboratory) simulation model was used to compare the signal-to-noise ratios (SNRs) and mean square errors (MSEs) of the soft, hard, and modified threshold functions. The results demonstrated that the modified threshold function can avoid the pseudo-Gibbs phenomenon and produce a higher SNR than the soft and hard threshold functions. A seismic convolution model was built using seismic wavelets to verify the effectiveness of different denoising methods. The model was used to demonstrate that the modified thresholding scheme can effectively reduce random noise in seismic data and retain the desired signal. The application of the proposed tool to a real raw seismogram recorded during a land seismic exploration experiment located in north China clearly demonstrated its efficiency for random noise attenuation.</p>

Seismic data reconstruction via weighted nuclear-norm minimization

The low-rank matrix completion theory based on nuclear-norm minimization is becoming increasingly popular in practice, and has been applied to seismic data reconstruction of missing traces. In this paper, we investigate the weighted nuclear-norm minimization model on low-rank seismic texture matrix which is obtained with a designed texture-patch pre-transformation. Unlike the previous nuclear-norm model that treats all singular values of the seismic data matrix equally, the weighted nuclear-norm model, based on the idea of iterative reweighting, makes a closer approximation to the ‘counting rank function’ and can promote the low-rank matrices. We apply weighted spectral soft-thresholding iterative scheme and weighted iterative supported detection method to solve the weighted model. Experimental study on synthetic and real seismic data shows that high-quality reconstruction can be obtained by the weighted low-rank methods.

Wavelet-based regularization of the extracted topographic index from high-resolution topography for hydro-geomorphic applications

High-resolution topography, e.g. 1-m digital elevation model (DEM) from light detection and ranging (LiDAR), offers opportunity for accurate identification of topographic features of relevance for hydrologic and geomorphologic modelling. Yet, the computation of some derived topographic properties, such as the topographic index (TI), is characterized by daunting challenges that hamper the full exploration of topography-based models. Particular problems, for example, arise when a distributed (or semi-distributed) rainfall–runoff model is applied to high-resolution DEMs. Indeed, the characteristic dependency between landscape resolution and the computed TI distribution results in the formation of un-physical, unconnected saturated zones, which in turn cause unrealistic representations of rainfall–runoff dynamics. In this study, we present a methodology based on a multi-resolution wavelet transformation that, by means of a soft-thresholding scheme on the wavelet coefficients, filters the noise of high-resolution topography to construct regularized sets of locally smoother topography on which the TI is computed. While the methodology needs a somewhat arbitrary definition of the wavelet coefficients threshold, our study shows that when the information content (entropy) of the TI distribution is used as a filtering efficiency metric, a critical threshold automatically emerges in the landscape reconstruction. The methodology is demonstrated using 1-m LiDAR data for the Elder Creek River basin in California. While the proposed case study uses a TOPMODEL approach, the methodology can be extended to different topography-based models and is not limited to hydrological applications. Copyright © 2012 John Wiley & Sons, Ltd.

A majorization–minimization approach to variable selection using spike and slab priors

We develop a method to carry out MAP estimation for a class of Bayesian regression models in which coefficients are assigned with Gaussian-based spike and slab priors. The objective function in the corresponding optimization problem has a Lagrangian form in that regression coefficients are regularized by a mixture of squared l2 and l0 norms. A tight approximation to the l0 norm using majorization–minimization techniques is derived, and a coordinate descent algorithm in conjunction with a soft-thresholding scheme is used in searching for the optimizer of the approximate objective. Simulation studies show that the proposed method can lead to more accurate variable selection than other benchmark methods. Theoretical results show that under regular conditions, sign consistency can be established, even when the Irrepresentable Condition is violated. Results on posterior model consistency and estimation consistency, and an extension to parameter estimation in the generalized linear models are provided.

A wavelet-based spectral procedure for steady-state simulation analysis

We develop WASSP, a wavelet-based spectral method for steady-state simulation analysis. First WASSP determines a batch size and a warm-up period beyond which the computed batch means form an approximately stationary Gaussian process. Next WASSP computes the discrete wavelet transform of the bias-corrected log-smoothed-periodogram of the batch means, using a soft-thresholding scheme to denoise the estimated wavelet coefficients. Then taking the inverse discrete wavelet transform of the thresholded wavelet coefficients, WASSP computes estimators of the batch means log-spectrum and the steady-state variance parameter (i.e., the sum of covariances at all lags) for the original (unbatched) process. Finally by combining the latter estimator with the batch means grand average, WASSP provides a sequential procedure for constructing a confidence interval on the steady-state mean that satisfies user-specified requirements concerning absolute or relative precision as well as coverage probability. An experimental performance evaluation demonstrates WASSP’s effectiveness compared with other simulation analysis methods.

Fault Diagnosis of Single-Variate Systems Using a Wavelet-Based Pattern Recognition Technique

A pattern recognition-based methodology is presented for fault diagnosis of a single-variate and dynamic system. A group of wavelet coordinates discriminating the classes of events most efficiently among other wavelet coordinates are determined according to the linear discriminant basis (LDB) method and a principal component analysis (PCA) technique. The proposed feature extractor couples the LDB method with the double wavelet packet tree in order to determine the best configuration of pattern windows causing the most discrimination among classes. The lifting scheme-based wavelet filters are used so that the required computation time is reduced significantly without degrading the robustness of the method. To reduce the size of the feature space, the wavelet coordinates are projected into a new low-dimensional space, by using a PCA technique, where minimum correlation exists among the new space variables. The tuning of some parameters, which affect the performance of the approach, is also discussed. The feature classifier is a binary decision tree that employs a soft-thresholding scheme for recognition of a noisy input pattern. The performance of the proposed technique is examined by a classification benchmark problem, and the faults classification problems for the Tennessee Eastman process. It is observed that the proposed pattern recognition methodology succeeds satisfactorily to classify the noisy input pattern into the known classes of events.

Asymptotic behavior of the minimum mean squared error threshold for noisy wavelet coefficients of piecewise smooth signals

This paper investigates the asymptotic behavior of the minimum risk threshold for wavelet coefficients with additive, homoscedastic, Gaussian noise and for a soft-thresholding scheme. We start from N samples from a signal on a continuous time axis. For piecewise smooth signals and for N/spl rarr//spl infin/, this threshold behaves as C/spl radic/(2logN)/spl sigma/, where /spl sigma/ is the noise standard-deviation. The paper contains an original proof for this asymptotic behavior as well as an intuitive explanation. The paper also discusses the importance of this asymptotic behavior for practical cases when we estimate the minimum risk threshold.

A rate-control scheme for video transport over wireless channels

We investigate the scenario of using the automatic repeat request (ARQ) retransmission scheme for two-way video communications over wireless Rayleigh fading channels. Video quality is the major concern of these applications. We show that, during the retransmissions of error packets, due to the reduced channel throughput, the video encoder buffer may fill-up quickly and cause the TMN8 rate-control algorithm to significantly reduce the bits allocated to each video frame. This results in PSNR degradation and many skipped frames. To minimize the number of frames skipped, we propose improved rate-control schemes that take into consideration the effects of the video buffer fill-up, an a priori channel model, and the channel feedback information. We also show that a discrete cosine transform (DCT) coefficient soft-thresholding scheme can be applied to further improve video quality. As a result, our proposed rate-control schemes encode the video sequences with less frame skipping and with higher PSNR compared to TMN8.