Random seismic noise attenuation by learning-type overcomplete dictionary based on K-singular value decomposition algorithm

The transformation of basic functions is one of the most commonly used techniques for seismic denoising,which employs sparse representation of seismic data in the transform domain. The choice of transform base functions has an influence on denoising results. We propose a learning-type overcomplete dictionary based on the K-singular value decomposition( K-SVD) algorithm. To construct the dictionary and use it for random seismic noise attenuation,we replace fixed transform base functions with an overcomplete redundancy function library. Owing to the adaptability to data characteristics,the learning-type dictionary describes essential data characteristics much better than conventional denoising methods. The sparsest representation of signals is obtained by the learning and training of seismic data. By comparing the same seismic data obtained using the learning-type overcomplete dictionary based on K-SVD and the data obtained using other denoising methods,we find that the learning-type overcomplete dictionary based on the K-SVD algorithm represents the seismic data more sparsely,effectively suppressing the random noise and improving the signal-to-noise ratio.

Robust Damage Detection and Localization Under Complex Environmental Conditions Using Singular Value Decomposition-based Feature Extraction and One-dimensional Convolutional Neural Network

Ultrasonic guided wave is an attractive monitoring technique for large-scale structures but is vulnerable to changes in environmental and operational conditions(EOC),which are inevitable in the normal inspection of civil and mechanical structures.This paper thus presents a robust guided wave-based method for damage detection and localization under complex environmental conditions by singular value decomposition-based feature extraction and one-dimensional convolutional neural network(1D-CNN).After singular value decomposition-based feature extraction processing,a temporal robust damage index(TRDI)is extracted,and the effect of EOCs is well removed.Hence,even for the signals with a very large temperature-varying range and low signal-to-noise ratios(SNRs),the final damage detection and localization accuracy retain perfect 100%.Verifications are conducted on two different experimental datasets.The first dataset consists of guided wave signals collected from a thin aluminum plate with artificial noises,and the second is a publicly available experimental dataset of guided wave signals acquired on a composite plate with a temperature ranging from 20℃to 60℃.It is demonstrated that the proposed method can detect and localize the damage accurately and rapidly,showing great potential for application in complex and unknown EOC.

Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.

IMPROVED SINGULAR VALUE DECOMPOSITION TECHNIQUE FOR DETECTING AND EXTRACTING PERIODIC IMPULSE COMPONENT IN A VIBRATION SIGNAL

Vibration acceleration signals are often measured from case surface of arunning machine to monitor its condition. If the measured vibration signals display to have periodicimpulse components with a certain frequency, there may exist a corresponding local fault in themachine, and if further extracting the periodic impulse components from the vibration signals, theseverity of the local fault can be estimated and tracked. However, the signal-to-noise ratios (SNRs)of the vibration acceleration signals are often so small that the periodic impulse components aresubmersed in much background noises and other components, and it is difficult or inconvenient for usto detect and extract the periodic impulse components with the current common analyzing methods forvibration signals. Therefore, another technique, called singular value decomposition (SVD), istried to be introduced to solve the problem. First, the principle of detecting and extracting thesignal periodic components using singular value decomposition is summarized and discussed. Second,the infeasibility of the direct use of the existing SVD based detecting and extracting approach ispointed out. Third, the approach to construct the matrix for SVD from the signal series is improvedlargely, which is the key program to improve the SVD technique; Other associated improvement is alsoproposed. Finally, a simulating application example and a real-life application example ondetecting and extracting the periodic impulse components are given, which showed that the introducedand improved SVD technique is feasible.

A SINGULAR VALUE DECOMPOSITION BASED TRUNCATION ALGORITHM IN SOLVING THE STRUCTURAL DAMAGE EQUATIONS

The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditioning of the equations. Based on the singular value decomposition (SVD) of the coefficient matrix, an error based truncation algorithm is proposed in this paper. By rejection of selected small singular values, the influence of noise can be reduced. A simply-supported beam is used as a simulation example to compare the results to other methods. Illustrative numerical examples demonstrate the good efficiency and stability of the algorithm in the nondestructive identification of structural damage through modal data.

A NOTE ON SINGULAR VALUE DECOMPOSITION FOR RADON TRANSFORM IN R^n

The singular value decomposition is derived when the Radon transform is restricted to functions which are square integrable on the unit ball in R-n with respect to the weight W-lambda(x). It fulfilles mainly by means of the projection-slice theorem. The range of the Radon transform is spanned by products of Gegenbauer polynomials and spherical harmonics. The inverse transform of the those basis functions are given. This immediately leads to an inversion formula by series expansion and range characterizations.

An Intercomparison of Rules for Testing the Significance of Coupled Modes of Singular Value Decomposition Analysis

This paper clarifies the essence of the significance test of singular value decomposition analysis （SVD）, and investigates four rules for testing the significance of coupled modes of SVD, including parallel analysis, nonparametric bootstrap, random-phase test, and a new rule named modified parallel analysis. A numerical experiment is conducted to quantitatively compare the performance of the four rules in judging whether a coupled mode of SVD is significant as parameters such as the sample size, the number of grid points, and the signal-to-noise ratio vary. The results show that the four rules perform better with lower ratio of the number of grid points to sample size. Modified parallel analysis and nonparametric bootstrap perform best to abandon the spurious coupled modes, but the latter is better than the former to retain the significant coupled modes when the sample size is not much larger than the number of grid points. Parallel analysis and random-phase test are robust to abandon the spurious coupled modes only when either （1） the observations at the grid points are spatially uncorrelated, or （2） the coupled signal is very strong for parallel analysis and is not weak for random-phase test. The reasons affecting the accuracy of the test rules are discussed.

Estimation of fracture density and orientation from azimuthal elastic impedance difference through singular value decomposition

Accurate estimation of fracture density and orientation is of great significance for seismic characterization of fractured reservoirs.Here,we propose a novel methodology to estimate fracture density and orientation from azimuthal elastic impedance(AEI)difference using singular value decomposition(SVD).Based on Hudson's model,we first derive the AEI equation containing fracture density in HTI media,and then obtain basis functions and singular values from the normalized AEI difference utilizing SVD.Analysis shows that the basis function changing with azimuth is related to fracture orientation,fracture density is the linearly weighted sum of singular values,and the first singular value contributes the most to fracture density.Thus,we develop an SVD-based fracture density and orientation inversion approach constrained by smooth prior elastic parameters.Synthetic example shows that fracture density and orientation can be stably estimated,and the correlation coefficient between the true value and the estimated fracture density is above 0.85 even when an S/N ratio of 2.Field data example shows that the estimated fracture orientation is consistent with the interpretation of image log data,and the estimated fracture density reliably indicates fractured gas-bearing reservoir,which could help to guide the exploration and development of fractured reservoirs.

Analysis of heart rate variability based on singular value decomposition entropy

Assessing the dynamics of heart rate fluctuations can provide valuable information about heart status. In this study, regularity of heart rate variability （HRV） of heart failure patients and healthy persons using the concept of singular value decomposition entropy （SvdEn） is analyzed. SvdEn is calculated from the time series using normalized singular values. The advantage of this method is its simplicity and fast computation. It enables analysis of very short and non-stationary data sets. The results show that SvdEn of patients with congestive heart failure （CHF） shows a low value （SvdEn： 0.056±0.006, p 〈 0.01） which can be completely separated from healthy subjects. In addition, differences of SvdEn values between day and night are found for the healthy groups. SvdEn decreases with age. The lower the SvdEn values, the higher the risk of heart disease. Moreover, SvdEn is associated with the energy of heart rhythm. The results show that using SvdEn for discriminating HRV in different physiological states for clinical applications is feasible and simple.

Source-Generated Noise Suppression Using the Singular Value Decomposition

Source-generated noise, such as air, refracted, guided waves, near-surface multiples, and radial ground roll, is one of the most challenging problems in the land seismic method. The interference of the noise with reflection events often results in a distorted representation of the subsurface and gives rise to interpretation uncertainties. To suppress the noise, geophysicists have devised various techniques in both acquisition and processing stages. Conventional processing methods, such as high-pass, f - k and hyperbolic velocity filters, however, have certain disadvantages when handling actual seismic data. In this study, we present a new hybrid method combining singular value decomposition (SVD) with a special linear transformation of the common-shot gather. The method is aimed at effectively removing the noise while minimizing harm to the signal. As compared with other methods, the SVD-based one gives a denser approximation to source-generated noise before its subtraction from the seismic data, due to the use of more appropriate basis functions. The special transformation applied in advance to the data is intended to align the source-generated noise events horizontally and thus to benefit the subsequent SVD. The effectiveness of the method in suppressing source-generated noise is demonstrated with a synthetic data set. Emphasis is put on the comparison of the performance of the method with that of conventional f - k filtering.

Solutions to the generalized Sylvester matrixequations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.

A Robust Image Watermarking Scheme Using Z-Transform, Discrete Wavelet Transform and Bidiagonal Singular Value Decomposition

Watermarking is a widely used solution to the problems of authentication and copyright protection of digital media especially for images,videos,and audio data.Chaos is one of the emerging techniques adopted in image watermarking schemes due to its intrinsic cryptographic properties.This paper proposes a new chaotic hybrid watermarking method combining Discrete Wavelet Transform(DWT),Z-transform(ZT)and Bidiagonal Singular Value Decomposition(BSVD).The original image is decomposed into 3-level DWT,and then,ZT is applied on the HH3 and HL3 sub-bands.The watermark image is encrypted using Arnold Cat Map.BSVD for the watermark and transformed original image were computed,and the watermark was embedded by modifying singular values of the host image with the singular values of the watermark image.Robustness of the proposed scheme was examined using standard test images and assessed against common signal processing and geometric attacks.Experiments indicated that the proposed method is transparent and highly robust.

Randomized Generalized Singular Value Decomposition

The generalized singular value decomposition(GSVD)of two matrices with the same number of columns is a very useful tool in many practical applications.However,the GSVD may suffer from heavy computational time and memory requirement when the scale of the matrices is quite large.In this paper,we use random projections to capture the most of the action of the matrices and propose randomized algorithms for computing a low-rank approximation of the GSVD.Serval error bounds of the approximation are also presented for the proposed randomized algorithms.Finally,some experimental results show that the proposed randomized algorithms can achieve a good accuracy with less computational cost and storage requirement.

Audio Zero-Watermark Scheme Based on Discrete Cosine Transform-Discrete Wavelet TransformSingular Value Decomposition

Traditional watermark embedding schemes inevitably modify the data in a host audio signal and lead to the degradation of the host signal.In this paper,a novel audio zero-watermarking algorithm based on discrete wavelet transform(DWT),discrete cosine transform(DCT),and singular value decomposition(SVD) is presented.The watermark is registered by performing SVD on the coefficients generated through DWT and DCT to avoid data modification and host signal degradation.Simulation results show that the proposed zero-watermarking algorithm is strongly robust to common signal processing methods such as requantization,MP3 compression,resampling,addition of white Gaussian noise,and low-pass filtering.

Atomic Decompositions of Weak Hardy Spaces of B-Valued Martingales

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities for B-valued martingales are established in the case 0〈r≤1. Here the results are connected closely with the p-uniform smoothness and q-uniform convexity of Banach spaces which the martingales take values in.

Super-resolution reconstruction of synthetic-aperture radar image using adaptive-threshold singular value decomposition technique

A super-resolution reconstruction approach of radar image using an adaptive-threshold singular value decomposition (SVD) technique was presented,and its performance was analyzed,compared and assessed detailedly.First,radar imaging model and super-resolution reconstruction mechanism were outlined.Then,the adaptive-threshold SVD super-resolution algorithm,and its two key aspects,namely the determination method of point spread function (PSF) matrix T and the selection scheme of singular value threshold,were presented.Finally,the super-resolution algorithm was demonstrated successfully using the measured synthetic-aperture radar (SAR) images,and a Monte Carlo assessment was carried out to evaluate the performance of the algorithm by using the input/output signal-to-noise ratio (SNR).Five versions of SVD algorithms,namely 1) using all singular values,2) using the top 80% singular values,3) using the top 50% singular values,4) using the top 20% singular values and 5) using singular values s such that s2≥max(s2)/rinSNR were tested.The experimental results indicate that when the singular value threshold is set as smax/(rinSNR)1/2,the super-resolution algorithm provides a good compromise between too much noise and too much bias and has good reconstruction results.

The Multi-Fractal Singularity Value Decomposition and Its Application in Extraction of Gravity Anomaly Associated with Gold Mineralization in Tongshi Gold Field

Because of the complication of geological procedures,the recorded data have the feature of nonlinear.The multi-fractal singularity value decomposition (MSVD) was used to decomposed the gravity data.In this paper,the MSVD was utilized to extract the gravity anomaly associated with the gold mineralization in Tongshi gold field in the southwest of Shandong province.The results showed that the Tongshi complex with negative circular gravity anomaly is an important ore-controlling factor.And the positive ring gravity anomaly distributed

Identification of m = 2 competent mode of complex magneto-hydro-dynamics activities during internal soft disruption based on singular value decomposition and tomography of soft-X-ray emission on the HT-7 tokamak

In this paper,the singular value decomposition（SVD） method as a filter is applied before the tomographic inversion of soft-X-ray emission.Series of ＇filtered＇ signals including specific chronos and topos are obtained.（Here,chronos and topos are the decomposed spatial vectors and the decomposed temporal vectors,respectively）.Given specific magnetic flux function with coupling m = 1 and m = 2 modes,the line-integrated soft-X-ray signals at all chords have been obtained.Then m = 1 and m = 2 modes have been identified by tomography of simulated ＇filtered＇ signals extracted by the SVD method.Finaly,using the experimental line-integrated soft-X-ray signals,m = 2 competent mode of complex magnetohydrodynamics（MHD） activities during internal soft disruption is observed.This result demonstrates that m = 2 mode plays an important role in internal disruption（Here,m is the poloidal mode number）.

DIRECT PERTURBATION METHOD FOR REANALYSIS OF MATRIX SINGULAR VALUE DECOMPOSITION

The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.