Thresholding Of Wavelet Coefficients Research Articles

Limit Theorems for Risk Estimate in Models with Non-Gaussian Noise

The problem of constructing an estimate of a signal function from noisy observations, assuming that this function is uniformly Lipschitz regular, is considered. The thresholding of empirical wavelet coefficients is used to reduce the noise. As a rule, it is assumed that the noise distribution is Gaussian and the optimal parameters of thresholding are known for various classes of signal functions. In this paper a model of additive noise whose distribution belongs to a fairly wide class, is considered. The mean-square risk estimate of thresholding is analyzed. It is shown that under certain conditions, this estimate is strongly consistent and asymptotically normal.

PURE-LET Image Deconvolution.

We propose a non-iterative image deconvolution algorithm for data corrupted by Poisson or mixed Poisson-Gaussian noise. Many applications involve such a problem, ranging from astronomical to biological imaging. We parameterize the deconvolution process as a linear combination of elementary functions, termed as linear expansion of thresholds. This parameterization is then optimized by minimizing a robust estimate of the true mean squared error, the Poisson unbiased risk estimate. Each elementary function consists of a Wiener filtering followed by a pointwise thresholding of undecimated Haar wavelet coefficients. In contrast to existing approaches, the proposed algorithm merely amounts to solving a linear system of equations, which has a fast and exact solution. Simulation experiments over different types of convolution kernels and various noise levels indicate that the proposed method outperforms the state-of-the-art techniques, in terms of both restoration quality and computational complexity. Finally, we present some results on real confocal fluorescence microscopy images and demonstrate the potential applicability of the proposed method for improving the quality of these images.We propose a non-iterative image deconvolution algorithm for data corrupted by Poisson or mixed Poisson-Gaussian noise. Many applications involve such a problem, ranging from astronomical to biological imaging. We parameterize the deconvolution process as a linear combination of elementary functions, termed as linear expansion of thresholds. This parameterization is then optimized by minimizing a robust estimate of the true mean squared error, the Poisson unbiased risk estimate. Each elementary function consists of a Wiener filtering followed by a pointwise thresholding of undecimated Haar wavelet coefficients. In contrast to existing approaches, the proposed algorithm merely amounts to solving a linear system of equations, which has a fast and exact solution. Simulation experiments over different types of convolution kernels and various noise levels indicate that the proposed method outperforms the state-of-the-art techniques, in terms of both restoration quality and computational complexity. Finally, we present some results on real confocal fluorescence microscopy images and demonstrate the potential applicability of the proposed method for improving the quality of these images.

Wavelet ANOVA approach to model validation

Model validation is that critical component in the simulation development process that ensures a model is truly representative of the system that it is meant to model. Although there are numerous validation techniques described in the literature, many of these techniques still require some amount of subjective analysis in order to assess validity. This is particularly true with dynamic simulation output. To reduce or eliminate this subjectivity, this paper proposes a validation process that uses wavelet analysis of variance (WANOVA) as an effective method to statistically accept or reject a model as valid. This WANOVA validation approach performs statistical inference in the time-frequency domain to take advantage of wavelet sparsity and decorrelation. This process uses a test statistic based on thresholded wavelet coefficients to test the null hypothesis that the set of system data and model data are statistically equivalent. The validation technique is illustrated using a simulation study and empirical data from an automobile crash study.

Applied multiresolution analysis to infrared images for defects detection in materials

In this paper, an advanced approach to characterize defects in homogeneous materials based on multiresolution analysis of infrared images is presented. This is mainly a non-destructive evaluation technique based on the flash method in transmission mode. An experimental device using infrared thermography was designed and realized. The equipment includes an infrared imager for following the temporal evolution of the temperature in the rear face of the tested sample. The other front face receives a flash from a halogen lamps excitation source. The thermal images generated at the rear face are segmented using a wavelet transform multiresolution analysis in order to extract all defects zones and subsequently processed to estimate their corresponding surface sizes. The experimental results on the test materials with hidden defects show the advantage of the segmentation technique applied to the response images of the system. High precision of the localization and accurate estimation of the surface sizes of the detected defects are achieved with the advanced method, involving a denoising of the images using a thresholding of wavelet coefficients. The outcome is a significantly improved detection quality, due mainly to the advanced processing, which does not depend on the type, shape and size of the defects.

Stabilized hard thresholding for an unknown noise level

The effect wa ys of estimating noise variance on the statistical characteristics of the stabilized hard thresholding of signal wavelet coefficients is studied. An unbiased estimator of the mean-square risk is analyzed. It is shown that under certain conditions, the estimator distribution tends to a normal law with variance that depends on the type of noise variance estimate.

Speech enhancement using sparse dictionary learning in wavelet packet transform domain

Sparse coding, as a successful representation method for many signals, has been recently employed in speech enhancement. This paper presents a new learning-based speech enhancement algorithm via sparse representation in the wavelet packet transform domain. We propose sparse dictionary learning procedures for training data of speech and noise signals based on a coherence criterion, for each subband of decomposition level. Using these learning algorithms, self-coherence between atoms of each dictionary and mutual coherence between speech and noise dictionary atoms are minimized along with the approximation error. The speech enhancement algorithm is introduced in two scenarios, supervised and semi-supervised. In each scenario, a voice activity detector scheme is employed based on the energy of sparse coefficient matrices when the observation data is coded over corresponding dictionaries. In the proposed supervised scenario, we take advantage of domain adaptation techniques to transform a learned noise dictionary to a dictionary adapted to noise conditions captured based on the test environment circumstances. Using this step, observation data is sparsely coded, based on the current situation of the noisy space, with low sparse approximation error. This technique has a prominent role in obtaining better enhancement results particularly when the noise is non-stationary. In the proposed semi-supervised scenario, adaptive thresholding of wavelet coefficients is carried out based on the variance of the estimated noise in each frame of different subbands. The proposed approaches lead to significantly better speech enhancement results in comparison with the earlier methods in this context and the traditional procedures, based on different objective and subjective measures as well as a statistical test.

Coherent structure extraction in turbulent channel flow using boundary adapted wavelets

ABSTRACTWe introduce boundary adapted wavelets, which are orthogonal and have the same scale in the three spatial directions. The construction thus yields a multiresolution analysis. We analyse direct numerical simulation data of turbulent channel flow computed at a friction Reynolds number of 395, and investigate the role of coherent vorticity. Thresholding of the vorticity wavelet coefficients allows us to split the flow into two parts, coherent and incoherent flows. The coherent vorticity is reconstructed from its few intense wavelet coefficients and the coherent velocity is reconstructed using Biot–Savart's law. The statistics of the coherent flow, i.e. energy and enstrophy spectra, are close to the statistics of the total flow, and moreover, the nonlinear energy budgets of the total flow are very well preserved. The remaining incoherent part, represented by the large majority of the weak wavelet coefficients, corresponds to a structureless, i.e. noise-like, background flow whose energy is equidistributed.

A Synthetic Solution for Identification and Extraction of the Effective Microseismic Wave Component Using Decomposition on Time, Frequency, and Wavelet Coefficient Domains

To reduce noise components from original microseismic waves, a comprehensive fine signal processing approach using the integrated decomposition analysis of the wave duration, frequency spectrum, and wavelet coefficient domain was developed and implemented. Distribution regularities of the wave component and redundant noise on the frequency spectrum and the wavelet coefficient domain were first expounded. The frequency threshold and wavelet coefficient threshold were determined for the identification and extraction of the effective wave component. The frequency components between the reconstructed microseismic wave and the original measuring signal were compared. The noise elimination effect via the scale-changed domain decomposition was evaluated. Interaction between the frequency threshold and the wavelet coefficient threshold in the time domain was discussed. The findings reveal that tri-domain decomposition analysis achieves the precise identification and extraction of the effective microseismic wave component and improves the reliability of waves by eliminating the redundant noise. The frequency threshold and the wavelet coefficient threshold on a specific time window are two critical parameters that determine the degree of precision for the identification of the extracted wave component. This research involves development of the proposed integrated domain decomposition method and provides a diverse view on the fine processing of the microseismic signal.

Wavelet Denoising of Vehicle Platform Vibration Signal Based on Threshold Neural Network

Vehicle Platform Vibration Signal (VPVS) denoising is essential to achieve high measurement accuracy of precise optical measuring instrument (POMI). A method to denoise the VPVS is proposed based on the wavelet coefficients thresholding and threshold neural network (TNN). According to the characteristics of VPVS, a novel thresholding function is constructed, and then its optimized threshold is selected through unsupervised learning of TNN. The original VPVS mixed in trend and random noise is constructed as VPVS model. A VPVS denoising flow is proposed based on the power spectral and energy distribution of the VPVS model. The simulation shows that the proposed denoising method achieves better results, compared to the previous denoising methods using the indexes of SNR and RMSE. The experiment demonstrates that it is efficient for denoising VPVS polluted by the trend and random noise.

Novel Wavelet-Based Segmentation of Prostate CBCT Images with Implanted Calypso Transponders.

Segmentation of prostate Cone Beam CT (CBCT) images is an essential step towards real-time adaptive radiotherapy (ART). It is challenging for Calypso patients, as more artifacts generated by the beacon transponders are present on the images. We herein propose a novel wavelet-based segmentation algorithm for rectum, bladder, and prostate of CBCT images with implanted Calypso transponders. For a given CBCT, a Moving Window-Based Double Haar (MWDH) transformation is applied first to obtain the wavelet coefficients. Based on a user defined point in the object of interest, a cluster algorithm based adaptive thresholding is applied to the low frequency components of the wavelet coefficients, and a Lee filter theory based adaptive thresholding is applied on the high frequency components. For the next step, the wavelet reconstruction is applied to the thresholded wavelet coefficients. A binary (segmented) image of the object of interest is therefore obtained. 5 hypofractionated Calypso prostate patients with daily CBCT were studied. DICE, Sensitivity, Inclusiveness and ΔV were used to evaluate the segmentation result.

Cross-validated wavelet block thresholding for non-Gaussian errors

Wavelet thresholding generally assumes independent, identically distributed normal errors when estimating functions in a nonparametric regression setting. VisuShrink and SureShrink are just two of the many common thresholding methods based on this assumption. When the errors are not normally distributed, however, few methods have been proposed. A distribution-free method for thresholding wavelet coefficients in nonparametric regression is described, which unlike some other non-normal error thresholding methods, does not assume the form of the non-normal distribution is known. Improvements are made to an existing even–odd cross-validation method by employing block thresholding and level dependence. The efficiency of the proposed method on a variety of non-normal errors, including comparisons to existing wavelet threshold estimators, is shown on simulated data.

An efficient microcalcifications detection based on dual spatial/spectral processing

Microcalcifications are tiny deposits of calcium located in breast tissue. They appeared as very small highlighted regions in comparison with their surrounding tissue. Spatial non linear enhancement can be applied for microcalcification detection. However, efficiency of a such approach depends on breast density: in case of extreme breast density, the contrast between microcalcification's details and their surrounding tissue is attenuated leading to a limitation of spatially based approaches. In that case, frequency analysis such as wavelet based analysis can be more relevant for dissociating microcalcifications. The main goal of Computer Aided Detection systems (CAD) is to detect breast cancer at an early stage for all breast density classes by using entropies to enhance and then detect microcalcification details. Accordingly, we combine our approach a spatial Automatic Non Linear Stretching (ANLS) and Shannon Entropy based Wavelet Coefficient Thresholding (SE_WCT). Validation of the proposed approach is done on the Mammographic Image Analysis Society (MIAS) database. The evaluation of the contrast is based on the Second-Derivative-Like measure of enhancement(SDME). Accordingly, it yields to a mean SDME of 78.8dB on the whole database. The performance metric for evaluating our proposed CAD is the Receiver Operating Characteristic(ROC) curve and the free-response ROC (FROC). An area under the ROC curve Az = 0.92 is obtained as well as 97.14 % of True Positives (TP) with 0,48 False positives per image (FP).

Automatic microseismic denoising and onset detection using the synchrosqueezed continuous wavelet transform

Typical microseismic data recorded by surface arrays are characterized by low signal-to-noise ratios (S/Ns) and highly nonstationary noise that make it difficult to detect small events. Currently, array or crosscorrelation-based approaches are used to enhance the S/N prior to processing. We have developed an alternative approach for S/N improvement and simultaneous detection of microseismic events. The proposed method is based on the synchrosqueezed continuous wavelet transform (SS-CWT) and custom thresholding of single-channel data. The SS-CWT allows for the adaptive filtering of time- and frequency-varying noise as well as offering an improvement in resolution over the conventional wavelet transform. Simultaneously, the algorithm incorporates a detection procedure that uses the thresholded wavelet coefficients and detects an arrival as a local maxima in a characteristic function. The algorithm was tested using a synthetic signal and field microseismic data, and our results have been compared with conventional denoising and detection methods. This technique can remove a large part of the noise from small-amplitudes signal and detect events as well as estimate onset time.

SU‐F‐J‐27: Segmentation of Prostate CBCT Images with Implanted Calypso Transponders Using Double Haar Wavelet Transform

Purpose:Segmentation of prostate CBCT images is an essential step towards real‐time adaptive radiotherapy. It is challenging For Calypso patients, as more artifacts are generated by the beacon transponders. We herein propose a novel wavelet‐based segmentation algorithm for rectum, bladder, and prostate of CBCT images with implanted Calypso transponders.Methods:Five hypofractionated prostate patients with daily CBCT were studied. Each patient had 3 Calypso transponder beacons implanted, and the patients were setup and treated with Calypso tracking system. Two sets of CBCT images from each patient were studied. The structures (i.e. rectum, bladder, and prostate) were contoured by a trained expert, and these served as ground truth. For a given CBCT, the moving window‐based Double Haar transformation is applied first to obtain the wavelet coefficients. Based on a user defined point in the object of interest, a cluster algorithm based adaptive thresholding is applied to the low frequency components of the wavelet coefficients, and a Lee filter theory based adaptive thresholding is applied to the high frequency components. For the next step, the wavelet reconstruction is applied to the thresholded wavelet coefficients. A binary/segmented image of the object of interest is therefore obtained. DICE, sensitivity, inclusiveness and ΔV were used to evaluate the segmentation result.Results:Considering all patients, the bladder has the DICE, sensitivity, inclusiveness, and ΔV ranges of [0.81–0.95], [0.76–0.99], [0.83–0.94], [0.02–0.21]. For prostate, the ranges are [0.77–0.93], [0.84–0.97], [0.68–0.92], [0.1–0.46]. For rectum, the ranges are [0.72–0.93], [0.57–0.99], [0.73–0.98], [0.03–0.42].Conclusion:The proposed algorithm appeared effective segmenting prostate CBCT images with the present of the Calypso artifacts. However, it is not robust in two scenarios: 1) rectum with significant amount of gas; 2) prostate with very low contrast. Model based algorithm might improve the segmentation in these two scenarios.

Wavelet-based estimation of regression function with strong mixing errors under fixed design

ABSTRACTWe consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes Bsp, q. The theory is illustrated with some numerical examples.A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.

Convergence rate of wavelet density estimator with data missing randomly when covariables are present

ABSTRACTIn this article, we study global L2 error of non linear wavelet estimator of density in the Besov space Bspq for missing data model when covariables are present and prove that the estimator can achieve the optimal rate of convergence, which is similar to the result studied by Donoho et al. (1996) in complete independent data case with term-by-term thresholding of the empirical wavelet coefficients. Finite-sample behavior of the proposed estimator is explored via simulations.

Image Denoising via Residual Kurtosis Minimization

AbstractA new algorithm for the removal of additive uncorrelated Gaussian noise from a digital image is presented. The algorithm is based on a data driven methodology for the adaptive thresholding of wavelet coefficients. This methodology is derived from higher order statistics of the residual image, and requires no a priori estimate of the level of noise contamination of an image.

Jamming mitigation in global positioning system receivers using wavelet packet coefficients thresholding

In this paper, the authors describe a new pre-correlation mitigation technique for global positioning system (GPS) receivers. Proposed mitigation technique is based on wavelet packet coefficients thresholding. This technique employs adaptive thresholding to mitigate GPS jammers. The authors compare proposed technique against discrete wavelet coefficient thresholding method. In this paper, the interferers are generated by an RF signal generator. The received GPS signal and interference are combined in a combiner. Then, the combined signal is processed by a software GPS receiver which includes the proposed mitigation technique. Simulation results show benefits of the proposed method in terms of signal-to-noise power ratio improvement and mean square error (MSE) in contrary to previous methods. Comparison of presented method and the discrete wavelet method illustrates that the average improvement is 50% and 80% in terms of SNR improvement and MSE, respectively.

Nonparametric regression on random fields with random design using wavelet method

We consider non-linear wavelet-based estimators of spatial regression functions with (known) random design on strictly stationary random fields, which are indexed by the integer lattice points in the \(N\)-dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability.

Wavelet methodology to improve single unit isolation in primary motor cortex cells

The proper isolation of action potentials recorded extracellularly from neural tissue is an active area of research in the fields of neuroscience and biomedical signal processing. This paper presents an isolation methodology for neural recordings using the wavelet transform (WT), a statistical thresholding scheme, and the principal component analysis (PCA) algorithm. The effectiveness of five different mother wavelets was investigated: biorthogonal, Daubachies, discrete Meyer, symmetric, and Coifman; along with three different wavelet coefficient thresholding schemes: fixed form threshold, Stein's unbiased estimate of risk, and minimax; and two different thresholding rules: soft and hard thresholding. The signal quality was evaluated using three different statistical measures: mean-squared error, root-mean squared, and signal to noise ratio. The clustering quality was evaluated using two different statistical measures: isolation distance, and L-ratio. This research shows that the selection of the mother wavelet has a strong influence on the clustering and isolation of single unit neural activity, with the Daubachies 4 wavelet and minimax thresholding scheme performing the best.