Sparsity Of Wavelet Coefficients Research Articles

Denoising of Nifti (MRI) Images with a Regularized Neighborhood Pixel Similarity Wavelet Algorithm.

The recovery of semantics from corrupted images is a significant challenge in image processing. Noise can obscure features, interfere with accurate analysis, and bias results. To address this issue, the Regularized Neighborhood Pixel Similarity Wavelet algorithm (PixSimWave) was developed for denoising Nifti (magnetic resonance imaging (MRI)). The PixSimWave algorithm uses regularized pixel similarity detection to improve the accuracy of noise reduction by creating patches to analyze the intensity of pixels and locate matching pixels, as well as adaptive neighborhood filtering to estimate noisy pixel values by allocating each pixel a weight based on its similarity. The wavelet transform breaks down the image into scales and orientations, allowing a sparse image representation to allocate a soft threshold on its similarity to the original pixels. The proposed method was evaluated on simulated and raw T1w MRIs, outperforming other methods in terms of an SSIM value of 0.9908 for a low Rician noise level of 3% and 0.9881 for a high noise level of 17%. The addition of Gaussian noise improved PSNR and SSIM, with the results indicating that the proposed method outperformed other models while preserving edges and textures. In summary, the PixSimWave algorithm is a viable noise-elimination approach that employs both sparse wavelet coefficients and regularized similarity with decreased computation time, improving the accuracy of noise reduction in images.

Single-trial extraction of event-related potentials (ERPs) and classification of visual stimuli by ensemble use of discrete wavelet transform with Huffman coding and machine learning techniques

BackgroundPresentation of visual stimuli can induce changes in EEG signals that are typically detectable by averaging together data from multiple trials for individual participant analysis as well as for groups or conditions analysis of multiple participants. This study proposes a new method based on the discrete wavelet transform with Huffman coding and machine learning for single-trial analysis of evenal (ERPs) and classification of different visual events in the visual object detection task.MethodsEEG single trials are decomposed with discrete wavelet transform (DWT) up to the 4{th} level of decomposition using a biorthogonal B-spline wavelet. The coefficients of DWT in each trial are thresholded to discard sparse wavelet coefficients, while the quality of the signal is well maintained. The remaining optimum coefficients in each trial are encoded into bitstreams using Huffman coding, and the codewords are represented as a feature of the ERP signal. The performance of this method is tested with real visual ERPs of sixty-eight subjects.ResultsThe proposed method significantly discards the spontaneous EEG activity, extracts the single-trial visual ERPs, represents the ERP waveform into a compact bitstream as a feature, and achieves promising results in classifying the visual objects with classification performance metrics: accuracies 93.60pm {6.5}, sensitivities 93.55pm {4.5}, specificities 94.85pm {4.2}, precisions 92.50pm {5.5}, and area under the curve (AUC) 0.93pm {0.3} using SVM and k-NN machine learning classifiers.ConclusionThe proposed method suggests that the joint use of discrete wavelet transform (DWT) with Huffman coding has the potential to efficiently extract ERPs from background EEG for studying evoked responses in single-trial ERPs and classifying visual stimuli. The proposed approach has O(N) time complexity and could be implemented in real-time systems, such as the brain-computer interface (BCI), where fast detection of mental events is desired to smoothly operate a machine with minds.

On the minimax optimality and superiority of deep neural network learning over sparse parameter spaces

Deep learning has been applied to various tasks in the field of machine learning and has shown superiority to other common procedures such as kernel methods. To provide a better theoretical understanding of the reasons for its success, we discuss the performance of deep learning and other methods on a nonparametric regression problem with a Gaussian noise. Whereas existing theoretical studies of deep learning have been based mainly on mathematical theories of well-known function classes such as Hölder and Besov classes, we focus on function classes with discontinuity and sparsity, which are those naturally assumed in practice. To highlight the effectiveness of deep learning, we compare deep learning with a class of linear estimators representative of a class of shallow estimators. It is shown that the minimax risk of a linear estimator on the convex hull of a target function class does not differ from that of the original target function class. This results in the suboptimality of linear methods over a simple but non-convex function class, on which deep learning can attain nearly the minimax-optimal rate. In addition to this extreme case, we consider function classes with sparse wavelet coefficients. On these function classes, deep learning also attains the minimax rate up to log factors of the sample size, and linear methods are still suboptimal if the assumed sparsity is strong. We also point out that the parameter sharing of deep neural networks can remarkably reduce the complexity of the model in our setting.

Phase harmonic correlations and convolutional neural networks

Abstract A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters that are well localized in the frequency domain, with different phases. We show that a rectifier then acts as a filter on the phase of the resulting coefficients. It computes signal descriptors that are local in space, frequency and phase. The nonlinear phase filter becomes a multiplicative operator over phase harmonics computed with a Fourier transform along the phase. We prove that it defines a bi-Lipschitz and invertible representation. The correlations of phase harmonics coefficients characterize coherent structures from their phase dependence across frequencies. For wavelet filters, we show numerically that signals having sparse wavelet coefficients can be recovered from few phase harmonic correlations, which provide a compressive representation.

Improved Wavelet Denoising by Penalty Function and Regularization Parameter

Wavelet-based convex optimization in sparse signal processing has attracted extensive research interest in recent years. Research demonstrates that the penalty function promotes the sparsity and improves the accuracy better than L1 norm method in dealing with convex and sparse signal reconstruction issues. This paper presents an improved denoising model which utilizes a penalty function to induce stronger sparsity in wavelet domain. In this paper, we have redefined the relationship model between signal noise levels and the regularization parameter to enhance sparsity caused by penalty functions in a wavelet domain. And three penalty functions are employed for wavelet coefficients to induce strong sparse wavelet coefficients under the wavelet domain. We then apply the improved wavelet denoising algorithm for image denoising to get the values of the PSNR are analysed under different noise level conditions. Experimental results clearly show that the improved wavelet denoising model is strength in terms of both quantitative measure and structural similarity quality, and outperforms many widely used denoising algorithms.

Improved Wavelet Denoising by Non-Convex Sparse Regularization Under Double Wavelet Domains

This paper presents a double wavelet denoising (DWAD) method, which can preserve more details of an original signal. Although the noise removal method based on wavelet transform has been widely used, it still performs poorly for the signals with a low signal-to-noise ratio (SNR) or frequency overlap. Different from the wavelet denoising methods based on a single basis function, the DWAD considers filtering the wavelet coefficients of the noisy signal by threshold functions under two different wavelet domains, simultaneously. It considers using the difference of wavelet coefficient distribution and forcing the denoised signals under two wavelet domains to be the same to achieve more retention of details. In addition, the arctangent function is employed as a penalty function for wavelet coefficients to induce strong sparse wavelet coefficients. The DWAD is applied to one-dimensional signals and it is found that some wavelet coefficients which are smaller than the threshold could be retained during noise removal. The experiment results show that the average SNR of different noise levels is improved by at least 4.2 and 2.1 dB compared with the classical soft threshold method for the one-dimensional and image signals, respectively. Besides, the DWAD tends to obtain better performance on the details of original signals.

Model-Based Nonuniform Compressive Sampling and Recovery of Natural Images Utilizing a Wavelet-Domain Universal Hidden Markov Model

In this paper, a novel model-based compressive sampling (CS) technique for natural images is proposed. Our algorithm integrates a universal hidden Markov tree (uHMT) model, which captures the relation among the sparse wavelet coefficients of images, into both sampling and recovery steps of CS. At the sampling step, we employ the uHMT model to devise a nonuniformly sparse measurement matrix $\boldsymbol{\Phi }_{\text{uHMT}}$ . In contrast to the conventional CS sampling matrices, such as dense Gaussian, Bernoulli or uniformly sparse matrices that are oblivious to the signal model and the correlation among the signal coefficients, the proposed $\boldsymbol{\Phi }_{\text{uHMT}}$ is designed based on the signal model and samples the coarser wavelet coefficients with higher probabilities and more sparse wavelet coefficients with lower probabilities. At the recovery step, we integrate the uHMT model into two state-of-the-art Bayesian CS recovery schemes. Our simulation results confirm the superiority of our proposed HMT model-based nonuniform compressive sampling and recovery , referred to as uHMT-NCS , over other model-based CS techniques that solely consider the signal model at the recovery step. This paper is distinguished from other model-based CS schemes in that we take a novel approach to simultaneously integrating the signal model into both CS sampling and recovery steps. We show that such integration greatly increases the performance of the CS recovery, which is equivalent to reducing the required number of samples for a given reconstruction quality.

A Note on Compressed Sensing of Structured Sparse Wavelet Coefficients From Subsampled Fourier Measurements

We consider signal recovery from Fourier measurements using compressed sensing (CS) with wavelets. For discrete signals with structured sparse Haar wavelet coefficients, we give the first proof of near-optimal recovery from discrete Fourier samples taken according to an appropriate variable density sampling scheme. Crucially, in taking into account such structured sparsity—known as sparsity in levels—as opposed to just sparsity, this result yields recovery guarantees that agree with the empirically observed recovery properties of CS in this setting. This result complements a recent theorem in Adcock et al. [Breaking the coherence barrier: A new theory for compressed sensing, arXiv preprint arXiv: 1302.0561, 2014.], which addressed the case of continuous time signals. Moreover, we provide a significantly shorter and more expositional argument, which clearly illustrates the key factors governing recovery in this setting: namely the division of frequency space into dyadic bands corresponding to wavelet scales, the near-block diagonality of the Fourier/wavelet cross-Gramian matrix, and the structured sparsity of wavelet coefficients.

Sparse Signal Representations of Bearing Fault Signals for Exhibiting Bearing Fault Features

Sparse signal representations attract much attention in the community of signal processing because only a few coefficients are required to represent a signal and these coefficients make the signal understandable. For bearing faults’ diagnosis, bearing faults signals collected from transducers are often overwhelmed by strong low-frequency periodic signals and heavy noises. In this paper, a joint signal processing method is proposed to extract sparse envelope coefficients, which are the sparse signal representations of bearing fault signals. Firstly, to enhance bearing fault signals, particle swarm optimization is introduced to tune the parameters of wavelet transform and the optimal wavelet transform is used for retaining one of the resonant frequency bands. Thus, sparse wavelet coefficients are obtained. Secondly, to reduce the in-band noises existing in the sparse wavelet coefficients, an adaptive morphological analysis with an iterative local maximum detection method is developed to extract sparse envelope coefficients. Simulated and real bearing fault signals are investigated to illustrate how the sparse envelope coefficients are extracted. The results show that the sparse envelope coefficients can be used to represent bearing fault features and identify different localized bearing faults.

Wavelet Shrinkage with Double Weibull Prior

In this article, we propose a denoising methodology in the wavelet domain based on a Bayesian hierarchical model using Double Weibull prior. We propose two estimators, one based on posterior mean (Double Weibull Wavelet Shrinker, DWWS) and the other based on larger posterior mode (DWWS-LPM), and show how to calculate them efficiently. Traditionally, mixture priors have been used for modeling sparse wavelet coefficients. The interesting feature of this article is the use of non-mixture prior. We show that the methodology provides good denoising performance, comparable even to state-of-the-art methods that use mixture priors and empirical Bayes setting of hyperparameters, which is demonstrated by extensive simulations on standardly used test functions. An application to real-word dataset is also considered.

Biorthogonal Loop-Subdivision Wavelets

We present a biorthogonal wavelet construction for Loop subdivision, based on the lifting scheme. Our wavelet transform uses scaling functions that are recursively defined by Loop subdivision for arbitrary manifold triangle meshes. We orthogonalize our wavelets with respect to local scaling functions. This way, the wavelet analysis computes locally a least squares fit when reducing the resolution and converting geometric detail into sparse wavelet coefficients. The contribution of our approach is local computation of both, wavelet analysis and synthesis in linear time. We provide numerical examples supporting the stability of our scheme.