Singular Value Decomposition Filtering Research Articles

Dynamic estimation of reflectivity by minimum-delay seismic trace decomposition

Spiking deconvolution corrects for the effect of the seismic wavelet, assumed to be minimum delay, by applying an inverse filter to the seismic trace to get an estimate of reflectivity. To compensate for propagation and absorption effects, one may use time-varying deconvolution, in which a different inverse filter is computed and applied for each output sample position. We modified this procedure by estimating a minimum-delay wavelet for each time-sample position of the seismic trace. This gives a decomposition of the seismic trace as a sum of minimum-delay wavelets, each multiplied by a reflectivity coefficient. The data vector is equal to a lower triangular wavelet matrix, with element 1 on the diagonal, multiplied by the seismic reflectivity vector. Recursive solution of this equation provides an estimate of reflectivity. The reflectivity estimation is a single-trace process that is sensitive to nonwhite noise, and it does not take into account lateral continuity of reflections. Therefore, we have developed a new data processing strategy by combining it with adaptive singular value decomposition (SVD) filtering. The SVD filtering process is applied to the data in two steps: (1) in a sliding spatial window on NMO-corrected CMP or common shot gathers (2) next, after local dip estimation and correction, on local patches in common-offset panels. After the SVD filtering, we applied the new reflectivity estimation procedure. The SVD filtering removes noise and improves lateral continuity, whereas the reflectivity estimation increases the high-frequency content in the data and improves vertical resolution. The new data processing strategy was successfully applied to land seismic data from northeast Brazil. Improvements in data quality are evident in prestack data panels, velocity analysis, and the stacked section.

A positioning method for the feature points of a target board image adopting singular value decomposition

This paper presents a positioning method to extract the feature points employing the singular value decomposition for a target board image as an example. In order to discriminate the feature points from the noises existing in the image, the image matrix is factorized into three matrices. The geometrical implication of the three matrices is interpreted by three transformations which are rotation, scaling, and another rotation. The singular values in the diagonal of the scaling matrix are arranged in descending order, which stands for the significance sequence of the image features. The smaller singular values are corresponding to the noises while the greater ones are considered as primary features. Therefore the new singular values matrix is defined by the modified original scaling matrix in which the smaller singular values are removed by a cutting point. The smoothing image is reconstructed by the two original rotation matrices and the new singular values matrix with the same arrangement. The latter detection procedure of the feature points adopts the Harris corner detection method to evaluate the singular value decomposition filter. The experiments are performed on the target images with the noise densities of 0.005, 0.01 and 0.02 respectively. Comparing with the traditional approach, the feature points on the target which are considered as the primary features are preserved by the filter. Moreover, the noises vanish in the images because they are unimportant details. The experiments prove that the outlined method has the potential to feature points positioning as well as appropriateness for the manufacture engineering and mechanical inspection.

An R-peak detection method that uses an SVD filter and a search back system

In this paper, we present a method for detecting the R-peak of an ECG signal by using an singular value decomposition (SVD) filter and a search back system. The ECG signal was detected in two phases: the pre-processing phase and the decision phase. The pre-processing phase consisted of the stages for the SVD filter, Butterworth High Pass Filter (HPF), moving average (MA), and squaring, whereas the decision phase consisted of a single stage that detected the R-peak. In the pre-processing phase, the SVD filter removed noise while the Butterworth HPF eliminated baseline wander. The MA removed the remaining noise of the signal that had gone through the SVD filter to make the signal smooth, and squaring played a role in strengthening the signal. In the decision phase, the threshold was used to set the interval before detecting the R-peak. When the latest R–R interval (RRI), suggested by Hamilton et al., was greater than 150% of the previous RRI, the method of detecting the R-peak in such an interval was modified to be 150% or greater than the smallest interval of the two most latest RRIs. When the modified search back system was used, the error rate of the peak detection decreased to 0.29%, compared to 1.34% when the modified search back system was not used. Consequently, the sensitivity was 99.47%, the positive predictivity was 99.47%, and the detection error was 1.05%. Furthermore, the quality of the signal in data with a substantial amount of noise was improved, and thus, the R-peak was detected effectively.

Dip‐adaptive singular‐value decomposition filtering for seismic reflection enhancement

ABSTRACTWe present a singular value decomposition (SVD) filtering method for the enhancement of coherent reflections and for attenuation of noise. The method is applied in two steps. First normal move‐out (NMO) correction is applied to shot or CMP records, with the purpose of flattening the reflections. We use a spatial SVD filter with a short sliding window to enhance coherent horizontal events. Then the data are sorted in common‐offset panels and the local dip is estimated for each panel. The next SVD filtering is performed on a small number of traces and a small number of time samples centred around the output sample position. Data in a local window are corrected for linear moveout corresponding to the dips before SVD. At the central time sample position, we sum over the dominant eigenimages of a few traces, corresponding to SVD dip filtering. We illustrate the method using land seismic data from the Tacutu basin, located in the north‐east of Brazil. The results show that the proposed method is effective and is able to reveal reflections masked by ground‐roll and other types of noise.

An SVD analysis of equispaced polynomial interpolation

Runge showed more than a century ago that polynomial interpolation of degree N on a uniform, evenly-spaced grid may diverge as N → ∞ . Here, we present a new angle on polynomial interpolation and the Runge Phenomenon by performing an SVD analysis of the interpolation matrix. We prove that the observed exponential-with- N error growth of the Runge Phenomenon requires that at least the smallest singular value of the interpolation matrix must decay to zero exponentially fast with N, and the matrix condition number κ must grow exponentially with N. Experimentally, four-fifths of the singular values are O ( 1 ) ; the remaining one-fifth of singular values decay exponentially fast with increasing mode number j. The corresponding U modes of the SVD factorization are well approximated by Hermite functions. SVD filtering, which is a common strategy for coping with ill-conditioned matrices by omitting modes of tiny singular value, gave errors no smaller than 7% for Runge's exemplary f ( x ) . Any similar anti-Runge strategy that is based on treating the “good” modes (of O ( 1 ) singular values) differently from the “bad” modes of small singular value is likely to fail because these two groups of modes are strongly coupled. A better strategy for exploiting the SVD factorization remains elusive.

Singular value decomposition filtering for enhanced signal extraction from two-dimensional beam emission spectroscopy measurements

The signal-to-noise ratio (SNR) of extracted turbulence features from beam emission spectroscopy (BES) data is significantly enhanced via application of singular value decomposition (SVD) methods. BES measures two-dimensional localized density fluctuations in DIII-D. The SNR of core turbulence characteristics is typically limited by noise arising from electronic noise, photon noise, and fluctuations in the observed neutral beam. SVD filtering has led to a significant enhancement in the SNR, reducing errors in time-resolved measurements of core turbulence characteristics, including correlation lengths, decorrelation rates, and group velocities. The SVD filtration technique is applied to BES data by combining multiple physically adjacent sampling locations to extract spatially correlated signals while partially removing unwanted incoherent noise. Using approximately half of the singular value weighted modes to reconstruct turbulence signals is found to improve SNR by up to a factor of 4, while maintaining the spatial structure of the turbulence. Unique aspects of application of SVD to broadband turbulence data are discussed.

Fixed-Point Error Analysis of CORDIC Arithmetic for Special-Purpose Signal Processors

CORDIC (COordinate Rotation DIgital Computer) is a well known algorithm using simple adders and shifters to evaluate various elementary functions. Thus, CORDIC is suitable for the design of high performance chips using VLSI technology. In this paper, a complete analysis of the computation error of both the (conventional) CORDIC algorithm and the CORDIC algorithm with expanded convergence range is derived to facilitate the design task. The resulting formulas regarding the relative and absolute approximation errors and the truncation error are summarized in the tabular form. As the numerical accuracy of CORDIC processors is determined by the word length of operands and the number of iterations, three reference tables are constructed for the optimal choice of these numbers. These tables can be used to facilitate the design of cost-effective CORDIC processors in terms of areas and performances. In addition, two design examples: singular value decomposition (SVD) and lattice filter for digital signal processing systems are given to demonstrate the goal and benefit of the derived numerical analysis of CORDIC.

Adaptive noise attenuation of seismic images based on singular value decomposition and texture direction detection

Singular value decomposition (SVD) is an efficient tool for the separation of signal and noise subspaces. When it is used to process seismic images, SVD can enhance the signal-to-noise ratio (SNR) of horizontal events effectively. In this paper, an adaptive SVD filter is proposed to enhance the non-horizontal events by detection of seismic image texture direction and then horizontal alignment of the estimated dip through data rotation. The features derived from the co-occurrence matrix are used to estimate the texture direction. The SVD filter parameter is adapted according to the ratio of the stacking energy along the detected direction and the energy of the image. Coherent noise events are recognized by their directions, which are different from the directions of signal events in general, and are first attenuated by high-rank approximation. Then, the signal events are enhanced by low-rank approximation.

Image restoration using truncated SVD filter bank based on an energy criterion

Image restoration is formulated using a truncated singular-value-decomposition (SVD) filter bank. A pair of known data patterns is used for identifying a small convolution operator. This is achieved by matrix pseudo-inversion based on SVD. Unlike conventional approaches, however, here SVD is performed upon a data-pattern matrix that is much smaller than the image size, leading to an enormous saving in computation. Regularisation is realised by first decomposing the operator into a bank of sub-filters, and then discarding some high-order ones to avoid noise amplification. By estimating the noise spectrum, sub-filters that produce noise energy more than that of useful information are abandoned. Therefore high-order components in the spectrum responsible for noise amplification are rejected. With the obtained small kernel, image restoration is implemented by convolution in the space domain. Numerical results are given to show the effectiveness of the proposed technique.

Reliable scaling exponent estimation of long-range correlated noise in the presence of random spikes

Detrended fluctuation analysis (DFA) has been used widely to determine possible long-range correlations in data obtained from diverse settings. In a recent study [Z. Chen, P.Ch. Ivanov, K. Hu, H.E. Stanley, Effects of nonstationarities on detrended fluctuation analysis, Phys Rev E 65 (2002) 041107], uncorrelated random spikes superimposed on the long-range correlated noise (LR noise) were found to affect DFA scaling exponent estimates. In this brief communication, singular-value decomposition (SVD) filter is proposed to minimize the effect random spikes superimposed on LR noise, thus facilitating reliable estimation of the scaling exponents. The effectiveness of the proposed approach is demonstrated on random spikes sampled from normal and uniform distributions.

Testing for multifractality of the slow solar wind

We analyse a time series of the radial component of the Elsässer variable for the low-speed stream of the solar wind plasma representing Alfvénic fluctuations propagating downstream as measured in situ by the Helios spacecraft in the inner heliosphere. We demonstrate that the influence of noise in the data can be efficiently reduced by moving average and singular-value decomposition filters. We calculate the multifractal spectrum for the flow of the solar wind directly from the cleaned experimental signal. The resulting spectrum of dimensions shows a multifractal structure of the solar wind in the inner heliosphere. The obtained multifractal spectrum is consistent with that for the multifractal measure on the self-similar weighted Cantor set with the degree of multifractality of ∼10 −1.

Generalized dimensions for fluctuations in the solar wind

We analyze time series of velocities of the solar wind plasma including the outward-directed component of Alfvénic turbulence within slow wind observed by the Helios 2 spacecraft in the inner heliosphere. We demonstrate that the influence of noise in the data can be efficiently reduced by a singular-value decomposition filter. The resulting generalized dimensions show a multifractal structure of the solar wind attractor in the inner heliosphere. The obtained multifractal spectrum is consistent with that for the multifractal measure on the self-similar weighted baker's map with two parameters describing uniform compression and natural invariant measure on the attractor of the system.

Dynamic force identification based on enhanced least squares and total least-squares schemes in the frequency domain

Identifying dynamic forces from structural responses is necessary when direct measurement of those dynamic forces is impossible using conventional means. A common approach to address this problem is to determine the frequency response function (FRF) matrix, measure the structural responses, and calculate the dynamic forces based on least-squares (LS) scheme. This approach has been proven to be effective in reducing the random errors that occur in structural response signals. Unfortunately, the accuracy of this approach is often hindered by the inversion of an ill-conditioned FRF matrix at frequencies near the structural resonances. To overcome this inversion instability, two regularization filters, namely the truncated singular value decomposition (TSVD) filter and the Tikhonov filter, are used in conjunction with the conventional LS scheme at specific frequencies. Here a criterion for applying these enhanced LS schemes is proposed to aid in determining when the increase in computational effort is better utilized. Furthermore, a new LS form of the Morozov's discrepancy principle is formulated to aid in selecting the optimum regularization parameter for these filters at each frequency. The accuracy in using conventional LS, TSVD-based LS, and Tikhonov filter-based LS schemes are compared analytically and numerically in this paper. It is found that for small-sized FRF matrices, the Tikhonov filter-based LS scheme tends to work better than the TSVD filter-based LS scheme. Since these approaches can only deal with the random errors in the measured structural responses, a total least-squares (TLS) scheme that can also address errors associated with the FRF matrix is proposed in this research. Numerical simulations demonstrate that under certain conditions, the TLS scheme is more effective in reducing the impact of these errors.

Advanced signal processing tools for dispersive waves

ABSTRACTTwo field examples are presented, showing the advantages of using multicomponent sensors for surface‐wave studies. Multicomponent sensors allow the use of specific signal‐processing tools such as the multicomponent singular value decomposition filter and the multicomponent polarization filter, which are both very efficient at separating surface waves from the other waves that comprise a seismic field record.Firstly, some signal‐processing tools for studying surface waves are described. The various filters range from classical to advanced techniques. For processing single‐component data, the filters are the filter and filters based on singular value decomposition and on spectral matrix decomposition. For processing multicomponent data, the filters are the 4C‐singular value decomposition filter and the classical or high‐order polarization filter.Secondly, processing sequences that can be applied to the field data are described and the single‐component processing sequence and the multicomponent processing sequence are compared. Two field examples are presented. The first data set is a land seismic data recording on 2C sensors. The second data set was obtained from a marine acquisition with OBS (4 components). The results obtained illustrate the advantages of using multicomponent filters. The efficiency of the 4C‐SVD filter and the high‐order statistic polarization filter is demonstrated.

Wavelet filtering of chaotic data

Abstract. Satisfactory method of removing noise from experimental chaotic data is still an open problem. Normally it is necessary to assume certain properties of the noise and dynamics, which one wants to extract, from time series. The wavelet based method of denoising of time series originating from low-dimensional dynamical systems and polluted by the Gaussian white noise is considered. Its efficiency is investigated by comparing the correlation dimension of clean and noisy data generated for some well-known dynamical systems. The wavelet method is contrasted with the singular value decomposition (SVD) and finite impulse response (FIR) filter methods.

A transform domain SVD filter for suppression of muscle noise artefacts in exercise ECG's.

The proposed filter assumes the noisy electrocardiography (ECG) to be modeled as a signal of deterministic nature, corrupted by additive muscle noise artefact. The muscle noise component is treated to be stationary with known second-order characteristics. Since noise-free ECG is shown to possess a narrow-band structure in discrete cosine transform (DCT) domain and the second-order statistical properties of the additive noise component is preserved due to the orthogonality property of DCT, noise abatement is easily accomplished via subspace decomposition in the transform domain. The subspace decomposition is performed using singular value decomposition (SVD). The order of the transform domain SVD filter required to achieve the desired degree of noise abatement is compared to that of a suboptimal Wiener filter using DCT. Since the Wiener filter assumes both the signal and noise structures to be statistical, with a priori known second-order characteristics, it yields a biased estimate of the ECG beat as compared to the SVD filter for a given value of mean-square error (mse). The filter order required for performing the subspace smoothing is shown to exceed a certain minimal value for which the mse profile of the SVD filter follows the minimum-mean-quare error (mmse) performance warranted by the suboptimal Wiener filter. The effective filter order required for reproducing clinically significant features in the noisy ECG is then set by an upper bound derived by means of a finite precision linear perturbation model. A significant advantage resulting from the application of the proposed SVD filter lies in its ability to perform noise suppression independently on a single lead ECG record with only a limited number of data samples.

Stair-Like Behavior in the Correlation Integral of Measured Data

The stair-like structure in the correlation integral of measured data is investigated. It is shown that the stairs are caused by the limited floating-point accuracy of the data in the measurement, and are completely different from the noise in the measured phenomenon. It is found that a modified singular value decomposition filter with very large projection dimension can improve the precision of the data and smooth the stairs in the correlation integral.

Testing for an attractor in the solar wind flow

We analyse a time series of velocity fluctuations of the low-speed stream of the solar wind measured by the Helios spacecraft in the inner heliosphere. We use moving average and singular-value decomposition filters to give a faithful representation of the solar wind nonlinear behaviour. We have supported our previous result that trajectories describing the system in the inertial manifold of phase space can asymptotically approach the attractor of low-dimension. The obtained characteristics of the attractor are significantly different from those of the surrogate data. Thus these results suggest that the inner heliosphere is nonlinear and possibly a deterministic chaotic system.

Reduction of the clutter component in Doppler ultrasound signals based on singular value decomposition: a simulation study.

In pulsed Doppler ultrasound systems, the ultrasound radiofrequency (RF) signals received can be employed to estimate noninvasively the time-dependent blood velocity distribution within and artery. The RF signals are composed of signals originating from clutter (e.g., vessel walls) and scatterers (e.g., red blood cells). The clutter, which is induced by stationary or slowly-moving structure interfaces, must be suppressed to get reliable estimates of the mean blood flow velocities. In conventional pulsed Doppler systems, this is achieved with a static temporal high-pass filter. The static cut-off frequency and the roll-off of these filters cause the culture not always to be optimally suppressed. This paper introduces a clutter removal filter that is based on Singular Value Decomposition (SVD). Unlike conventional high-pass filters, which take into account only the information of the temporal direction, the SVD filter makes use of the information of the temporal and spatial directions. The advantage of this approach is that it does not matter where the clutter is located in the RF signal. The performance of the SVD filter is examined with computer-generated Doppler RF signals. The results are compared with those of standard linear regression (SLR) filter. The performance of the SVD filter is good, especially if a large temporal window (i.e., approximately 100 RF signals) is applied, which improves the performance for low blood flow velocities, A major disadvantage of the SVD filter is its computational complexity, which increases considerably for larger temporal windows.

安定及び不安定な力学のための近似的なデータ同化スキーム

Two suboptimal data assimilation schemes for stable and unstable dynamics are introduced. The first scheme, the partial singular value decomposition filter, is based on the most dominant singular modes of the tangent linear propagator. The second scheme, the partial eigendecomposition filter, is based on the most dominant eigenmodes of the propagated analysis error covariance matrix. Both schemes rely on iterative procedures like the Lanczos algorithm to compute the relevant modes. The performance of these schemes is evaluated for a shallow-water model linearized about an unstable Bickley jet. The results are contrasted against those of a reduced resolution filter, in which the gains used to update the state vector are calculated from a lower-dimensional dynamics than the dynamics that evolve the state itself. The results are also contrasted against the exact results given by the Kalman filter. These schemes are validated for the case of stable dynamics as well. The two new approximate assimilation schemes are shown to perform well with relatively few modes computed. Adaptive tuning of a modeled trailing error covariance for all three of these low-rank approximate schemes enhances performance and compensates for the approximation employed.