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In this paper, a unified algebraic transformation approach is presented for designing parallel recursive and adaptive digital filters and singular value decomposition (SVD) algorithms. The approach is based on the explorations of some algebraic properties of the target algorithms' representations. Several typical modern digital signal processing examples are presented to illustrate the applications of the technique. They include the cascaded orthogonal recursive digital filter, the Givens rotation-based adaptive inverse QR algorithm for channel equalization, and the QR decomposition-based SVD algorithms. All three examples exhibit similar throughput constraints. There exist long feedback loops in the algorithms' signal flow graph representation, and the critical path is proportional to the size of the problem. Applying the proposed algebraic transformation techniques, parallel architectures are obtained for all three examples. For cascade orthogonal recursive filter, retiming transformation and orthogonal matrix decompositions (or pseudo-commutativity) are applied to obtain parallel filter architectures with critical path of five Givens rotations. For adaptive inverse QR algorithm, the commutativity and associativity of the matrix multiplications are applied to obtain parallel architectures with critical path of either four Givens rotations or three Givens rotations plus two multiply-add operations, whichever turns out to be larger. For SVD algorithms, retiming and associativity of the matrix multiplications are applied to derive parallel architectures with critical path of eight Givens rotations. The critical paths of all parallel architectures are independent of the problem size as compared with being proportional to the problem size in the original sequential algorithms. Parallelism is achieved at the expense of slight increase (or the same for the SVD case) in the algorithms' computational complexity.

Singular Value Decomposition (SVD) algorithm is a good method in Collaborative Filtering (CF) algorithm. But the traditional SVD method faces the matrix sparse problem, people solve this problem by Latent Factor Model (LMF). The data imbalances problem also troubles us. The algorithm of this paper combines Slope One and SVD algorithm to solve the data imbalances problem. First, we use Slope One algorithm to deal with the users who have few rating records. Then this kind of user will have more rating records. And we can use new data set to train the SVD model to predict the final results. The experiment results show that our algorithm is better than the traditional Slope One and SVD algorithm.

Change point detection is essential to understand the time-evolving structure of dynamic networks. Recent research shows that a latent semantic indexing (LSI)-based algorithm effectively detects the change points of a dynamic network. The LSI-based method involves a singular value decomposition (SVD) on the data matrix. In a dynamic scenario, recomputing the SVD of a large matrix each time new data arrives is prohibitively expensive and impractical. A more efficient approach is to incrementally update the decomposition. However, in the classical incremental SVD (incSVD) algorithm, the information of the newly added columns is not fully considered in updating the right singular space, resulting in an approximation error which cannot be ignored. This paper proposes an enhanced incSVD (EincSVD) algorithm, in which the right singular matrix is calculated in an alternative way. An adaptive EincSVD (AEincSVD) algorithm is also proposed to further reduce the computational complexity. Theoretical analysis proves that the approximation error of the EincSVD is smaller than that of the incSVD. Simulation results demonstrate that the EincSVD and the AEincSVD perform much better than the incSVD on change point detection, and the performance of the EincSVD is comparable to the batch SVD algorithm.

A combination of a multiresolution (MR) preconditioner and matrix decomposition algorithm – singular value decomposition (MDA-SVD) algorithm is proposed for the analysis of densely discretised electromagnetic (EM) scatterers. In realistic EM scattering problems, the discretisation may be very dense to accurately describe the geometric shape of targets. However, dense discretisations will generate two major drawbacks: (i) the badly conditioned method of moments (MoM) matrix which makes iterative solvers difficult to converge; (ii) the dense near-field MoM matrix for which the computational cost is large, even multilevel fast multipole algorithm (MLFMA) is applied. To remedy the first shortcoming of dense discretisations, an MR preconditioner can be applied to improve the condition of MoM matrices. To reduce the computational cost in solving the MoM matrix equation, an MDA-SVD algorithm can be applied to accelerate the matrix−vector multiplication. Compared with the traditional MLFMA, the MDA-SVD algorithm has no limit on the size of octree boxes which makes the MDA-SVD algorithm suitable for the analysis of dense discretisations. Therefore an MR preconditioner is combined with the MDA-SVD algorithm for fast analysis of dense discretisations. Numerical results indicate that the combination of the MR preconditioner and the MDA-SVD algorithm is efficient.

The technology of low-field nuclear magnetic resonance (LF-NMR) is commonly used in food, agriculture, energy and chemical sectors due to its non-destructive, non-invasive, in situ, green and other advantages. Recently, this technology played an increasingly large role in the field of food-safety supervision especially. In oil product quality testing, conventional T2 spectrum inversion methods such as the non-negative singular value decomposition (SVD) algorithm can only reflect T2 spectrum in a smooth model. However, for a sparse model, the inversion result of non-negative SVD algorithm is expected to be very glossy, leading to low resolution of T2 spectrum and inaccurate analysis of sample property. To solve this problem, we propose a sparse T2 spectrum inversion algorithm based on the L1 norm minimization constraint. In this paper, we establish the sparse model expression of NMR echo curve, and obtain the T2 sparse spectrum inversion results based on the inner truncated Newton-point method. Furthermore, the effectiveness of L1 sparse inversion algorithm is examined by the synthetic data of the smooth model and the spare model which have different peak numbers and signaltonoise ratios (SNRs). Synthetic results show that compared with the non-negative SVD algorithm, the L1 sparse algorithm is appropriate for both the smooth model and the sparse model with higher inversion accuracy. When the number of T2 peaks in a sparse model changes from a single peak to a quad peak, the L1 sparse algorithm can still obtain accurate inversion results, while the SVD algorithm results in a gradual deterioration, and cannot even determine the peak number. Under the sparse model, when the SNR of the measured NMR curve is gradually changed from 5 dB to 50 dB, the L1 sparse algorithm at 20 dB or more can obtain accurate inversion results which have less than 10% peak error and less than 5% peak position error and amplitude average error. However, the non-negative SVD algorithm cannot obtain accurate results at each SNR. Finally, multiple sets of frying oil samples are utilized to prove the accuracy and robustness of L1 sparse inversion algorithm. Inversion results of seven sets of frying oil samples show that the L1 sparse algorithm prefers the non-negative SVD algorithm. The obtained T2 spectrum by the L1 sparse algorithm shows three peaks obviously, and the T21 peak area ratio S21 and the single component relaxation time T2w are higher linear with respect to frying time than the results by non-negative SVD algorithm, which is useful for detecting the frying oil quality change. The inversion results of the T2 spectrum at different SNRs are consistent with the synthetic results, i.e., when the SNR is reduced, the T2 spectrum inversion results from the L1 sparse algorithm are better than those from the non-negative SVD algorithm when SNR is greater than 20 dB.

Purpose:The Bayesian estimation algorithm improves the precision of bolus tracking perfusion imaging. However, this algorithm cannot directly calculate Tmax, the time scale widely used to identify ischemic penumbra, because Tmax is a non-physiological, artificial index that reflects the tracer arrival delay (TD) and other parameters. We calculated Tmax from the TD and mean transit time (MTT) obtained by the Bayesian algorithm and determined its accuracy in comparison with Tmax obtained by singular value decomposition (SVD) algorithms.Methods:The TD and MTT maps were generated by the Bayesian algorithm applied to digital phantoms with time-concentration curves that reflected a range of values for various perfusion metrics using a global arterial input function. Tmax was calculated from the TD and MTT using constants obtained by a linear least-squares fit to Tmax obtained from the two SVD algorithms that showed the best benchmarks in a previous study. Correlations between the Tmax values obtained by the Bayesian and SVD methods were examined.Results:The Bayesian algorithm yielded accurate TD and MTT values relative to the true values of the digital phantom. Tmax calculated from the TD and MTT values with the least-squares fit constants showed excellent correlation (Pearson’s correlation coefficient = 0.99) and agreement (intraclass correlation coefficient = 0.99) with Tmax obtained from SVD algorithms.Conclusions:Quantitative analyses of Tmax values calculated from Bayesian-estimation algorithm-derived TD and MTT from a digital phantom correlated and agreed well with Tmax values determined using SVD algorithms.

This paper concerns accurate computation of the singular value decomposition (SVD) of an $m\times n$ matrix $A$ . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of $A^{\rm T}A$ and take the square roots of the large eigenvalues of it as the large singular values of $A$ ; (b) form the Rayleigh quotient of $A^{\rm T}A$ with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of $A^{\rm T}A$ ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of $A$ . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff $\epsilon$ . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.

With the aim to fit the sin wave curve of a multi-position strap-down north seeking system and to calculate the real north angle accurately, the principles of the system and the algorithm of singular value decomposition(SVD) are introduced. The factors influencing the measuring precision are also analyzed. Then, the algorithm is adopted to fit curves. Experimental results indicate that the singular value decomposition is very precise, for its error sum is 1.1×10−2μV, and the squared error is 2.8mV. Moreover, for phase angle measurement, measured six times in same direction, the squared error is 24″ (1σ). The method based on SVD can satisfy the system precision requirements of curve-fitting north seeking systems.

We propose a new algorithm for the computation of a singular value decomposition (SVD) low-rank approximation of a matrix in the matrix product operator (MPO) format, also called the tensor train matrix format. Our tensor network randomized SVD (TNrSVD) algorithm is an MPO implementation of the randomized SVD algorithm that is able to compute dominant singular values and their corresponding singular vectors. In contrast to the state-of-the-art tensor-based alternating least squares SVD (ALS-SVD) and modified alternating least squares SVD (MALS-SVD) matrix approximation methods, TNrSVD can be up to 13 times faster while achieving better accuracy. In addition, our TNrSVD algorithm also produces accurate approximations in particular cases where both ALS-SVD and MALS-SVD fail to converge. We also propose a new algorithm for the fast conversion of a sparse matrix into its corresponding MPO form, which is up to 509 times faster than the standard tensor train SVD method while achieving machine precision accuracy. The efficiency and accuracy of both algorithms are demonstrated in numerical experiments.

Based on the singular value decomposition (SVD) algorithm, this paper proposes the singular value decomposition optimization algorithm for redundant strapdown inertial measurement unit (SIMU) fault detection. Based on the existing singular value decomposition algorithm, the columns of decoupling matrix are unitized. In addition, the decision function for isolation is corrected. Finally, the fault detection of the redundant SIMU regular dodecahedron configuration is simulated. By comparing the simulation results of the SVD algorithm and the SVD optimization algorithm, it proves that the SVD optimization algorithm is the feasible and effective.

Comments on “Homomorphic image watermarking with a singular value decomposition algorithm”

Combined the improved fast independent component analysis (FastICA) algorithm with the singular value decomposition algorithm, a single-channel fetal electrocardiogram (fECG) extraction method is proposed. First, the improved FastICA algorithm is used to estimate the maternal ECG component from a single-channel abdominal signal of pregnant women using an overrelaxation factor. Then, a preliminary estimate of the fECG signal is obtained by subtracting from the single-channel abdominal signal. Subsequently, the singular value decomposition algorithm is used to denoise the preliminarily estimated fECG signal to obtain a high signal-to-noise ratio. In addition, in the singular value decomposition algorithm for fetal arrhythmia, an improved method for constructing the ECG signal reconstruction matrix is proposed. Finally, the fECG extraction experiments on synthetic abdominal signals and actual abdominal signals (data from 49 abdominal channels sourced from DAISY database and the non-invasive fECG database in PhysioNet) are carried out. The experimental results show that the method in this paper can effectively improve the signal-to-noise ratio and the accuracy of fECG signal extraction, and is suitable for maternal or fetal arrhythmias. Compared with the FastICA algorithm, the signal-to-noise ratio of the fECG signal extracted by the method in this paper is improved by about 5 dB, and the accuracy of fECG extraction in the PhysioNet database can reach 96.54%.

Comments on “A digital watermarking scheme based on singular value decomposition and tiny genetic algorithm”

Singular value decomposition (SVD) is a key step in many algorithms in statistics, machine learning and numerical linear algebra. While classical singular value decomposition has been made efficient in terms of computational complexity, classical algorithms are not able to fully utilise modern computing environments. The goal of this work is to survey various implementations and applications of randomized algorithms for SVD. Algorithms are compared in terms of accuracy and time of execution. On example of robust principal component analysis (RPCA), it is shown that using randomized algorithms can yield a significant speedup for image processing and similar applications.

An optimal filtering recursive algorithm was proposed for multi-channel system with correlated multiplicative noises among observation channels. The system is under the condition that the multiplicative noises are general stochastic matrix and correlated in each observation channel at the same time. Considering the numerical stability of the algorithm, an equivalent algorithm based on singular value decomposition (SVD) was given for the same system. The SVD algorithm, as well as the previous algorithm, is optimal in the sense of linear minimum-variance. Simulation examples to demonstrate the performance of both algorithms show that the algorithm based on SVD has higher computational precision and numerical stability.