Wavelet Dictionary Research Articles

An efficient method for robust projection matrix design

Our objective is to efficiently design a robust projection matrix Φ for the Compressive Sensing (CS) systems when applied to the signals that are not exactly sparse. The optimal projection matrix is obtained by mainly minimizing the average coherence of the equivalent dictionary. In order to drop the requirement of the sparse representation error (SRE) for a set of training data as in [15,16], we introduce a novel penalty function independent of a particular SRE matrix. Without requiring of training data, we can efficiently design the robust projection matrix and apply it for most of CS systems, like a CS system for image processing with a conventional wavelet dictionary in which the SRE matrix is generally not available. Simulation results demonstrate the efficiency and effectiveness of the proposed approach compared with the state-of-the-art methods. In addition, we experimentally demonstrate with natural images that under similar compression rate, a CS system with a learned dictionary in high dimensions outperforms the one in low dimensions in terms of reconstruction accuracy. This together with the fact that our proposed method can efficiently work in high dimension suggests that a CS system can be potentially implemented beyond the small patches in sparsity-based image processing.

Incipient Bearing Fault Feature Extraction Based on Minimum Entropy Deconvolution and K-Singular Value Decomposition

Machinery condition monitoring and fault diagnosis are essential for early detection of equipment malfunctions or failures, which insure productivity, quality, and safety in the manufacturing process. This paper aims at extracting fault features of rolling element bearings at the incipient fault stage. K-singular value decomposition (K-SVD), one technique for sparse representation of signals, is used for study. In K-SVD, its dictionary is trained from data by machine learning techniques, which allows more flexibility to adapt to variation of real signals than the predefined dictionaries. Analysis on simulated bearing signals and real signals shows that K-SVD can give better bearing fault features than the predefined dictionaries such as wavelet dictionaries. However, during our simulation study, K-SVD was found to have large representation error under heavy noise. To reduce the noise effect, minimum entropy deconvolution (MED) is used as a prefilter. The combination of MED and K-SVD is proposed for incipient bearing fault detection. The method is verified by simulation and experimental study. It is shown that the proposed method can effectively extract the impulsive fault feature of the tested bearing at its incipient fault stage.

Single image super-resolution using coupled dictionary learning and cross domain mapping

In this paper, a new algorithm for single image super resolution using coupled wavelet and spatial domain dictionary pairs is proposed. The standard deviation parameter, which is approximately scale invariant for low and high resolution patch pairs is employed for clustering. A pair of online coupled dictionaries is learned for each cluster using a low resolution image. The standard deviation measure of a low resolution patch is used to select the appropriate cluster dictionary pair for reconstructing the high resolution counterpart. Experimental results show that the performance of the proposed algorithm is superior to the existing methods in terms of objective and subjective quality measures. The objective image quality is measured in terms of PSNR and SSIM. This paper also proposed an extended algorithm, based on selective sparse representation over a set of coupled dictionary pair. The extended algorithm applies the coupled dictionary based sparse framework for patches having high standard deviation. Whereas, low complexity patch collaging method is used to super resolve low standard deviation valued patches. It is found empirically that a large percentage of patches have low standard deviation values. Moreover, the selective approach significantly reduces the computational complexity without losing the overall reconstruction quality.

Automatic detection of Hyperreflective Foci in optical coherence tomography B-scans using Morphological Component Analysis.

Hyperreflective Foci (HF) is one of the most common complications distributed in cross-sectional images of patients with Diabetic Macular Edema (DME). Scanning Laser Ophthalmoscope (SLO) images usually consists of several B-scans that represent a cross-sectional reconstruction of a plane through the anterior or posterior regions of retina. In each B-scan, HFs are geometrically distinct constituents in different retinal layers. Since the intensity levels of HFs and many other subjects in B-scans are the same, in this paper we try to separate HFs from other objects by detection of the point and curve singularities in each B-scan. The decomposition algorithm presented in this paper is based on sparse image representation of B-scans using Morphological Component Analysis (MCA) technique. By using curvelet transform and Daubechies wavelet basis, two different over-complete dictionaries are constructed which represent two various aspects of B-scans. The HFs are more distinguished in reconstructed image with wavelet dictionary and other objects are mostly detectable by curvelet dictionary. So, HFs can be detected by applying an optimum threshold criterion on reconstructed image by wavelet atoms. Finally, the false positive points are reduced by removing the candidate points in RNFL and RPE layers, which are automatically segmented based on ridgelet transform. Our simulation results on 1924 HFs show that sensitivity and specificity for HF detection is 91.0% and 100%, respectively.

Geophysical inversion by dictionary learning

Dictionary learning, a form of unsupervised machine learning, has recently been applied to ocean sound speed profile (SSP) data to obtain compact dictionaries of shape functions which explain SSPs using as few as one non-zero coefficient. In this presentation, the results of this analysis and potential applications of dictionary learning techniques to the inversion of real acoustic data are discussed. The estimation of true geophysical parameters from acoustic observations often is an ill-conditioned problem that is regularized by enforcing prior constraints such as sparsity or energy penalities, and by reducing the size of the parameter search. Traditionally, empirical orthogonal functions (EOFs) and overcomplete wavelet and curvelet dictionaries have been used to represent complex geophysical structures with few parameters. Using the K-SVD dictionary learning algorithm, the representation of ocean SSP data is significantly compressed relative to EOF analysis. The regularization performance of these learned dictionaries is evaluated against EOFs in the estimation of ocean sound speed structure from ocean acoustic observations and the limitations of such unsupervised methods are considered.

An adaptive SAR image compression method

The compression algorithm is very important for SAR image processing. In this paper, an adaptive SAR image compression method is presented. The algorithm consists of three stages: training stage, coding stage and decoding stage. In the training stage, the wavelet dictionaries are constructed according to the training samples. In the coding stage, SAR image is transformed into different frequency subbands in the wavelet domain, preserves the low frequency coefficients, represents the high frequency subbands with sparse coefficients by the dictionaries, and codes all the different frequency coefficients. In the decoding stage, the high frequency coefficients are reconstructed by the dictionaries, and do the invert wavelet transformation on all frequency coefficients to reconstruct the image. The experimental results and performance analysis showed that the proposed method has good results in the compression ratio and PSNR aspects, meanwhile, it can suppress the speckle noise of SAR image.

Single-channel and multi-channel orthogonal matching pursuit for seismic trace decomposition

The conventional matching pursuit (MP) algorithm can decompose a 1D signal into a set of wavelet atoms adaptively. As to reflection seismic data, some applicable algorithms based on the MP decomposition has been developed, such as single-channel matching pursuit (SCMP) and multi-channel matching pursuit (MCMP). However, these algorithms cannot always select the optimal atoms, which results in less meaningful decompositions. To overcome this limitation, we introduce the idea of orthogonal matching pursuit into a multi-channel decomposition scheme, which we refer to as the multi-channel orthogonal matching pursuit (MCOMP). Each iteration of the proposed MCOMP might extract a more reasonable atom among a redundant Morlet wavelet dictionary, like the MCMP decomposition does, and estimate the corresponding amplitude more accurately by solving a least-squares problem. In order to correspond to SCMP, we also simplified the MCOMP decomposition to single-channel orthogonal matching pursuit (SCOMP) for decompositions of an individual seismic trace. We tested the proposed SCOMP algorithm on a synthetic signal and a field seismic trace. Then a field marine dataset example showed relative high resolution of the proposed MCOMP method with applications to the detection of low-frequency anomalies. These application examples all demonstrate more meaningful decomposition results and relative high convergence speed of the proposed algorithms.

Medical Image Denoising based on Log-Gabor Wavelet Dictionary and K-SVD Algorithm

Medical image denoising is the main step in medical diagnosis, which removes the noise without affecting relevant features of the image. There are many algorithms that can be used to reduce the noise such as: threshold and the sparse representation. The K-SVD is one of the most popular sparse representation algorithms, which is depend on Orthogonal Matching Pursuit (OMP) and Discrete Cosine Transform (DCT) dictionary. In this paper, an algorithm for image denoising was designed to develop K-SVD by using Regularized Orthogonal Matching Pursuit (ROMP) over log Gabor wavelet adaptive dictionary. To evaluate the performance of the proposed techniques, the results were compared with the results of DCT and Gabor wavelet dictionary. The numerical results show that the performance of our algorithm is more efficient in medical image denoising. Keywordsrepresentation (SR), K-SVD, log-Gabor wavelet dictionary, regularized orthogonal matching pursuit and orthogonal matching pursuit.

A high-resolution processing technique for improving the energy of weak signal based on matching pursuit

<p>This paper proposes a new method to improve the resolution of the seismic signal and to compensate the energy of weak seismic signal based on matching pursuit. With a dictionary of Morlet wavelets, matching pursuit algorithm can decompose a seismic trace into a series of wavelets. We abstract complex-trace attributes from analytical expressions to shrink the search range of amplitude, frequency and phase. In addition, considering the level of correlation between constituent wavelets and average wavelet abstracted from well-seismic calibration, we can obtain the search range of scale which is an important adaptive parameter to control the width of wavelet in time and the bandwidth of frequency. Hence, the efficiency of selection of proper wavelets is improved by making first a preliminary estimate and refining a local selecting range. After removal of noise wavelets, we integrate useful wavelets which should be firstly executed by adaptive spectral whitening technique. This approach can improve the resolutions of seismic signal and enhance the energy of weak wavelets simultaneously. The application results of real seismic data show this method has a good perspective of application.</p>

Analysis of fMRI data based on sparsity of source components in signal dictionary

Blind Source Separation (BSS) methods, like Independent Component Analysis (ICA), show good performance in the analysis of fMRI data. However, the independence assumption used in ICA, may be violated in practice. Hence, it is important to develop algorithm which can fully exploit the characteristics of fMRI data and use more reliable assumptions. In this paper, we propose an fMRI data analysis method which exploits the sparsity of source components in a signal dictionary. The proposed method, derived as a two-stage method, is established by reformulating the blind separation problem as a sparse approximation problem. First, a priori selection of a particular dictionary, in which the source components are assumed to be sparsely representable. By choosing a particular dictionary (like wavelet dictionary), the source components, which can be well sparsified in the selected dictionary, are estimated more accurately. Second, the source components are extracted by exploiting their sparse representability. The extracted signal components are applied to find consistent task related (CTR) component, activation voxels of CTR, and performance of neural decoding. Numerical results show that compared to ICA based method, the proposed method can extract more useful information from fMRI data, and higher performance on voxel selection and neural decoding can be achieved by using the separated sources.

Developing a New Computer-Aided Clinical Decision Support System for Prediction of Successful Postcardioversion Patients with Persistent Atrial Fibrillation.

We propose a new algorithm to predict the outcome of direct-current electric (DCE) cardioversion for atrial fibrillation (AF) patients. AF is the most common cardiac arrhythmia and DCE cardioversion is a noninvasive treatment to end AF and return the patient to sinus rhythm (SR). Unfortunately, there is a high risk of AF recurrence in persistent AF patients; hence clinically it is important to predict the DCE outcome in order to avoid the procedure's side effects. This study develops a feature extraction and classification framework to predict AF recurrence patients from the underlying structure of atrial activity (AA). A multiresolution signal decomposition technique, based on matching pursuit (MP), was used to project the AA over a dictionary of wavelets. Seven novel features were derived from the decompositions and were employed in a quadratic discrimination analysis classification to predict the success of post-DCE cardioversion in 40 patients with persistent AF. The proposed algorithm achieved 100% sensitivity and 95% specificity, indicating that the proposed computational approach captures detailed structural information about the underlying AA and could provide reliable information for effective management of AF.

Dictionary Learning for Analysis-Synthesis Thresholding

Thresholding is a classical technique for signal denoising. In this process, a noisy signal is decomposed over an orthogonal or overcomplete dictionary, the smallest coefficients are nullified, and the transform pseudo-inverse is applied to produce an estimate of the noiseless signal. The dictionaries used is this process are typically fixed dictionaries such as the DCT or Wavelet dictionaries. In this work, we propose a method for incorporating adaptive, trained dictionaries in the thresholding process. We present a generalization of the basic process which utilizes a pair of overcomplete dictionaries, and can be applied to a wider range of recovery tasks. The two dictionaries are associated with the analysis and synthesis stages of the algorithm, and we thus name the process analysis-synthesis thresholding. The proposed training method trains both the dictionaries and threshold values simultaneously given examples of original and degraded signals, and does not require an explicit model of the degradation. Experiments with small-kernel image deblurring demonstrate the ability of our method to favorably compete with dedicated deconvolution processes, using a simple, fast, and parameterless recovery process.

PRISM: Recovery of the primordial spectrum fromPlanckdata

The primordial power spectrum describes the initial perturbations that seeded the large-scale structure we observe today. It provides an indirect probe of inflation or other structure-formation mechanisms. In this letter, we recover the primordial power spectrum from the Planck PR1 dataset, using our recently published algorithm PRISM. PRISM is a sparsity-based inversion method, that aims at recovering features in the primordial power spectrum from the empirical power spectrum of the cosmic microwave background (CMB). This ill-posed inverse problem is regularised using a sparsity prior on features in the primordial power spectrum in a wavelet dictionary. Although this non-parametric method does not assume a strong prior on the shape of the primordial power spectrum, it is able to recover both its general shape and localised features. As a results, this approach presents a reliable way of detecting deviations from the currently favoured scale-invariant spectrum. We applied PRISM to 100 simulated Planck data to investigate its performance on Planck-like data. We also tested the algorithm's ability to recover a small localised feature at $k \sim 0.125$ Mpc$^{-1}$, which caused a large dip at $\ell \sim 1800$ in the angular power spectrum. We then applied PRISM to the Planck PR1 power spectrum to recover the primordial power spectrum. We find no significant departures from the fiducial Planck PR1 near scale-invariant primordial power spectrum with $A_s=2.215\times10^{-9}$ and $n_s = 0.9624$.

Automatic fault feature extraction of mechanical anomaly on induction motor bearing using ensemble super-wavelet transform

Mechanical anomaly is a major failure type of induction motor. It is of great value to detect the resulting fault feature automatically. In this paper, an ensemble super-wavelet transform (ESW) is proposed for investigating vibration features of motor bearing faults. The ESW is put forward based on the combination of tunable Q-factor wavelet transform (TQWT) and Hilbert transform such that fault feature adaptability is enabled. Within ESW, a parametric optimization is performed on the measured signal to obtain a quality TQWT basis that best demonstrate the hidden fault feature. TQWT is introduced as it provides a vast wavelet dictionary with time-frequency localization ability. The parametric optimization is guided according to the maximization of fault feature ratio, which is a new quantitative measure of periodic fault signatures. The fault feature ratio is derived from the digital Hilbert demodulation analysis with an insightful quantitative interpretation. The output of ESW on the measured signal is a selected wavelet scale with indicated fault features. It is verified via numerical simulations that ESW can match the oscillatory behavior of signals without artificially specified. The proposed method is applied to two engineering cases, signals of which were collected from wind turbine and steel temper mill, to verify its effectiveness. The processed results demonstrate that the proposed method is more effective in extracting weak fault features of induction motor bearings compared with Fourier transform, direct Hilbert envelope spectrum, different wavelet transforms and spectral kurtosis.

PRISM: Sparse recovery of the primordial spectrum from WMAP9 and Planck datasets

AbstractThe primordial power spectrum is an indirect probe of inflation or other structure-formation mechanisms. We introduce a new method, named PRISM, to estimate this spectrum from the empirical cosmic microwave background (CMB) power spectrum. This is a sparsity-based inversion method, which leverages a sparsity prior on features in the primordial spectrum in a wavelet dictionary to regularise the inverse problem. This non-parametric approach is able to reconstruct the global shape as well as localised features of the primordial spectrum accurately and proves to be robust for detecting deviations from the currently favoured scale-invariant spectrum. We investigate the strength of this method on a set of WMAP nine-year simulated data for three types of primordial spectra and then process the WMAP nine-year data as well as the Planck PR1 data. We find no significant departures from a near scale-invariant spectrum.

Compressed convolution

We introduce the concept of compressed convolution, a technique to convolve a given data set with a large number of non-orthogonal kernels. In typical applications our technique drastically reduces the effective number of computations. The new method is applicable to convolutions with symmetric and asymmetric kernels and can be easily controlled for an optimal trade-off between speed and accuracy. It is based on linear compression of the collection of kernels into a small number of coefficients in an optimal eigenbasis. The final result can then be decompressed in constant time for each desired convolved output. The method is fully general and suitable for a wide variety of problems. We give explicit examples in the context of simulation challenges for upcoming multi-kilo-detector cosmic microwave background (CMB) missions. For a CMB experiment with detectors with similar beam properties, we demonstrate that the algorithm can decrease the costs of beam convolution by two to three orders of magnitude with negligible loss of accuracy. Likewise, it has the potential to allow the reduction of disk space required to store signal simulations by a similar amount. Applications in other areas of astrophysics and beyond are optimal searches for a large number of templates in noisy data, e.g. from a parametrized family of gravitational wave templates; or calculating convolutions with highly overcomplete wavelet dictionaries, e.g. in methods designed to uncover sparse signal representations.

Basis function selection for compressed sensing and sparse representations of pulsed radar echoes

Compressed sensing theory supposes that a sparse signal can be sampled at a rate much lower than the Nyquist–Shannon rate and reconstructed with high probability. Such lower sampling rate commonly requires finding a set of the optimal basis functions to sparsely represent the signal first. This paper provides a simple and effective working process to select the basis functions for a family of pulsed radar echoes. The selection process is performed in two steps. First,the waveform matching based on the the known array excitation is carried out to select a mother function from a wavelet dictionary. Second, the spectrum matching principle is used to produce a small set of basis functions from the selected mother function. The proposed method is numerically validated by a pulsed radar system equipped with two different dipole arrays. The results demonstrate that the new method is quite effective. With the selected basis functions, all echoes can be under-sampled at a rate lower than of the conventional Nyquist–Shannon rate and reconstructed with the root mean-squared error of less than .

X-ray CT Metal Artifact Reduction Using Wavelet Domain <formula formulatype="inline"><tex Notation="TeX">$L_{0}$</tex></formula> Sparse Regularization

X-ray computed tomography (CT) imaging of patients with metallic implants usually suffers from streaking metal artifacts. In this paper, we propose a new projection completion metal artifact reduction (MAR) algorithm by formulating the completion of missing projections as a regularized inverse problem in the wavelet domain. The Douglas-Rachford splitting (DRS) algorithm was used to iteratively solve the problem. Two types of prior information were exploited in the algorithm: 1) the sparsity of the wavelet coefficients of CT sinograms in a dictionary of translation-invariant wavelets and 2) the detail wavelet coefficients of a prior sinogram obtained from the forward projection of a segmented CT image. A pseudo- L0 synthesis prior was utilized to exploit and promote the sparsity of wavelet coefficients. The proposed L0-DRS MAR algorithm was compared with standard linear interpolation and the normalized metal artifact reduction (NMAR) approach proposed by Meyer using both simulated and clinical studies including hip prostheses, dental fillings, spine fixation and electroencephalogram electrodes in brain imaging. The qualitative and quantitative evaluations showed that our algorithm substantially suppresses streaking artifacts and can outperform both linear interpolation and NMAR algorithms.

Astronomical image denoising using dictionary learning

Astronomical images suffer a constant presence of multiple defects that are consequences of the intrinsic properties of the acquisition equipments, and atmospheric conditions. One of the most frequent defects in astronomical imaging is the presence of additive noise which makes a denoising step mandatory before processing data. During the last decade, a particular modeling scheme, based on sparse representations, has drawn the attention of an ever growing community of researchers. Sparse representations offer a promising framework to many image and signal processing tasks, especially denoising and restoration applications. At first, the harmonics, wavelets, and similar bases and overcomplete representations have been considered as candidate domains to seek the sparsest representation. A new generation of algorithms, based on data-driven dictionaries, evolved rapidly and compete now with the off-the-shelf fixed dictionaries. While designing a dictionary beforehand leans on a guess of the most appropriate representative elementary forms and functions, the dictionary learning framework offers to construct the dictionary upon the data themselves, which provides us with a more flexible setup to sparse modeling and allows to build more sophisticated dictionaries. In this paper, we introduce the Centered Dictionary Learning (CDL) method and we study its performances for astronomical image denoising. We show how CDL outperforms wavelet or classic dictionary learning denoising techniques on astronomical images, and we give a comparison of the effect of these different algorithms on the photometry of the denoised images.

Image Denoising using K-SVD Algorithm based on Gabor Wavelet Dictionary

denoising problem can be addressed as an inverse problem. One of the most recent approaches to solve an inverse problem is a sparse decomposition over overcomplete dictionaries. In sparse representation, images are represented as a linear combination of dictionary atoms. In this paper, we propose an algorithm for image denoising based on Orthogonal Matching Pursuit (OMP) for determining sparse representation over Gabor Wavelet adaptive dictionary by K- SVD algorithm. The results of this algorithm have more efficiency of image recovery than using DCT dictionary.