An automatic abrupt information extraction method based on singular value decomposition and higher-order statistics

The vibration signals of diesel include excess noise that must be eliminated before extraction of characteristic parameters. Firstly, the effects of vibration-signal de-noising among Fourier transform, wavelet decomposition and wavelet packet decomposition are compared. Secondly, singular value decomposition is applied to de-noising vibration signals. Finally, a new de-noise method integrated with wavelet packet and singular value is presented. In this method, vibration signals are decomposed by wavelet packet, and the wavelet packet coefficient is de-noised by singular value decomposition again. The results indicate that the new de-noising method is the best. The SNR (signal-to-noise ratio) of the vibration signals of a diesel cylinder lid is the highest. The diesel vibration waveforms of combustion and valve become clear and the extracted characteristic parameters become more precise.

38 - The application of higher order singular value decomposition to independent component analysis

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.

Effective clutter filtering is essential for ultrafast ultrasound small blood flow imaging, where weak blood flow signals are often obscured by tissue clutter and noise. While singular value decomposition (SVD)-based methods are widely used, their performance is limited by overlapping singular value distributions between blood flow, clutter, and noise. This study proposes a Cauchy-norm-based sparse SVD (S-SVD) clutter filtering method that integrates a sparsity penalty into the classical SVD framework. By leveraging the structural sparsity of blood flow, the method employs a Cauchy norm regularization to suppress noise while preserving small vessel details. A randomized spatial downsampling strategy is further used to increase computational efficiency. The proposed method is validated through simulated and in vivo rat brain experiments, demonstrating superior performance in power Doppler imaging and functional ultrasound (fUS) imaging compared to conventional SVD and robust principal component analysis (RPCA). Results show significant improvements in signal-tonoise ratio (SNR) by about 10dB and contrast-to-noise ratio (CNR) by about 9dB, with enhanced visualization of small vascular networks and dynamic cerebral blood volume (CBV) responses to whisker stimulation. Future work will focus on accelerating optimization and integrating deconvolution for further resolution enhancement.

Modern deep neural networks (DNNs) often require high memory consumption and large computational loads. In order to deploy DNN algorithms efficiently on edge or mobile devices, a series of DNN compression algorithms have been explored, including factorization methods. Factorization methods approximate the weight matrix of a DNN layer with the multiplication of two or multiple low-rank matrices. However, it is hard to measure the ranks of DNN layers during the training process. Previous works mainly induce low-rank through implicit approximations or via costly singular value decomposition (SVD) process on every training step. The former approach usually induces a high accuracy loss while the latter has a low efficiency. In this work, we propose SVD training, the first method to explicitly achieve low-rank DNNs during training without applying SVD on every step. SVD training first decomposes each layer into the form of its full-rank SVD, then performs training directly on the decomposed weights. We add orthogonality regularization to the singular vectors, which ensure the valid form of SVD and avoid gradient vanishing/exploding. Low-rank is encouraged by applying sparsity-inducing regularizers on the singular values of each layer. Singular value pruning is applied at the end to explicitly reach a low-rank model. We empirically show that SVD training can significantly reduce the rank of DNN layers and achieve higher reduction on computation load under the same accuracy, comparing to not only previous factorization methods but also state-of-the-art filter pruning methods.

As one of the most important components in rotating machinery, gearbox is fragile due to its harsh working environment. A new feature extraction method for gearbox fault diagnosis integrating singular value decomposition (SVD) with multidimensional scaling (MDS) is proposed in this article. SVD is used to extract the singular components (SCs), which reflect the main energy of the vibration signal, and the kurtosis values of SCs are calculated to detect the fault sensitivity of each singular value. The kurtosis-weighted singular values are applied to classify different conditions of gearbox supplemented by the MDS and fuzzy C-means clustering (FCM) method. Five gearbox operating conditions including tooth breakage, tooth pitting, gear eccentricity and normal are simulated to test the performance of proposed feature extraction method in an experiment rig. Partition coefficient (PC) and partition entropy (PE) are used to evaluate the classification effect of weighted singular values. The result suggests that the proposed kurtosis-weighted singular values perform better in distinguishing the different conditions of gearbox than original singular values.

We monitored the occupancy of a functionally important non-coordinated water molecule in the distal heme pocket of sperm whale myoglobin over the pH range 4.3-9.4. Water occupancy was assessed by using time-resolved spectroscopy to detect the perturbation of the heme visible band absorption spectrum caused by water entry after CO photodissociation ( Goldbeck, R. A., Bhaskaran, S., Ortega, C., Mendoza, J. L., Olson, J. S., Soman, J., Kliger, D. S., and Esquerra, R. M. (2006) Proc. Natl. Acad. Sci. U. S. A. 103, 1254-1259 ). We found that the water occupancy observed during the time interval between ligand photolysis and diffusive recombination decreased by nearly 20% as the pH was lowered below 6. This decrease accounted for most of the concomitant increase in the observed CO bimolecular recombination rate constant, as the lower water occupancy presented a smaller kinetic barrier to CO entry into the pocket at lower pH. These results were consistent with a model in which the distal histidine, which stabilizes the water molecule within the distal pocket by accepting a hydrogen bond, tends to swing out of the pocket upon protonation and destabilize the water occupancy at low pH. Extrapolation of this model to lower pH suggests that the additional increase in ligand association rate constant observed previously in stopped-flow studies at pH 3 may also be due in part to reduced distal water occupancy concomitant with further His64 protonation and coupled protein conformational change.

We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\\times N$ matrices whose entries are chosen independently from a Bernoulli distribution with parameter $p$. These matrices represent the adjacency matrices of random Erdős–Rényi digraphs and are sparse when $p\\ll 1$. We prove that in the regime $pN\\gg 1$, the distribution of the least singular value is universal in the sense that it is independent of $p$ and equal to the distribution of the least singular value of a Gaussian matrix ensemble. We also prove the universality of the joint distribution of multiple small singular values. Our methods extend to matrix ensembles whose entries are chosen from arbitrary distributions that may be correlated, complex valued, and have unequal variances.

Vibration signal and shaft orbit are important features that reflect the operating state of rotating machinery. Fault diagnosis and feature extraction are critical to ensure the safety and reliable operation of rotating machinery. A novel method of fault diagnosis based on convolutional neural network (CNN), discrete wavelet transform (DWT), and singular value decomposition (SVD) is proposed in this paper. CNN is used to extract features of shaft orbit images, DWT is used to transform the denoised swing signal of rotating machinery, and the wavelet decomposition coefficients of each branch of the signal are obtained by the transformation. The SVD input matrix is formed after single branch reconstruction of the different branch coefficients, and the singular value is extracted to obtain the feature vector. The features extracted from both methods are combined and then classified by support vector machines (SVMs). The comparison results show that this hybrid method has a higher recognition rate than other methods.

When seismic waves encounter abrupt points of stratum or lithology in the process of propagation, such as fault edges, pinch-out points, or the protrusion of unconformity surfaces, the Snell law will break down and many diffractions will be generated. However, the diffraction information is typically masked by strong reflections; thus, separating diffractions is one key issue for diffraction imaging. The traditional singular value decomposition (SVD) method with the ability for wavefield decomposition and reconstruction is useful in removing strong reflections of large singular values. However, the continuously changing characteristics of the singular value of steep-slope reflections make it difficult to choose a suitable parameter for separating diffractions. To solve the problem of singular value selection in diffractions reconstruction by the SVD method, we have developed a Cook-distance SVD method by mathematically least-squares measuring the contribution of every singular value. The Cook-distance SVD can amplify the difference between continuously changing singular values and easily find the singular values representing diffractions. Synthetic examples and field data applications demonstrate that our Cook-distance SVD method can avoid parameter testing and has good performance in diffraction separation and can highlight reservoir-involved small-scale geologic edges and scatterers.

Cellular automata based color image watermarking using non sub sampled contourlet transform (NSCT) and singular value decomposition (SVD) is presented in this paper. Game of life (GOL) cellular automata is used to scramble binary watermark so that no one can recover the watermark without secret scrambling keys. Because of the salient features like multiscale, multidirection and shift invariance of NSCT, 2-level NSCT is applied to red, green and blue component of RGB cover image. Most of the perceptual content of an image focuses on low frequency subband of NSCT and the singular values taken from low frequency subband have certain stability. So a modest change in the singular values taken from low frequency subband of NSCT is applicable. In this proposed scheme, each singular value of scrambled binary watermark is embedded in fixed location of singular value matrix of low frequency subbands of NSCT of red, green and blue components. The scheme offers security, imperceptibility and robustness against various image processing and geometric attacks like Gaussian white noise, salt and pepper noise, median filtering, JPEG compression, cropping and rotation. The proposed scheme is compared with existing watermarking schemes in terms of Peak Signal to Noise ratio (PSNR), Normalized Correlation (NC)and provides satisfactory performance than other schemes.

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regularization methods. In the last decade nonlinear variational approaches such as ` or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular onehomogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values and vectors involving the subdifferential of the regularization functional, although we shall see that the Rayleigh principle may fail for higher singular values. Using those definitions of singular values and vectors, we shall carry over two main properties from the world of linear regularization. The first one is gaining information about scale, respectively the behavior of regularization techniques at different scales. This also leads to novel estimates at different scales, generalizing the estimates for the coefficients in the linear singular value expansion. The second one is to provide classes of exact solutions for variational regularization methods. We will show that all singular vectors can be reconstructed up to a scalar factor by the standard Tikhonov-type regularization approach even in the presence of (small) noise. Moreover, we will show that they can even be reconstructed without any bias by the recently popularized inverse scale space method.

Matrix factorization plays an important role in scientific computation. The widely used one is singular value decomposition (SVD) which approximates the original data matrix with three lower rank matrices with orthogonality constraints. Recently nonnegative matrix factorization (NMF) considering the nonnegativity of data makes the results more interpretable than those of SVD. However NMF finds only two factor matrices and there is no significant index as singular values of SVD which can be used for sorting learned basis vectors. In this paper we take into account the nonnegativity for SVD and propose nonnegative SVD (NNSVD). The preliminary results on the microarray data of spermatogenesis show that NNSVD has advantages of both SVD and NMF.

Image compression is preferred for less storage space, sending image on the internet and often for medical imaging in health industry, technical drawings or comics. Because of wide popularity of sharing media in internet, compression plays a major role in the watermarking algorithm. This paper presents a compression technique through image watermarking algorithm based on Non-subsampled Contourlet Transform (NSCT) and Singular Value Decomposition (SVD). NSCT and SVD both are explored for image watermarking. For image compression, only Singular Value Decomposition is used. The NSCT can give more accurate coefficients of the edges and contours in image which is shift invariant. After getting the coefficients from NSCT decomposition SVD is applied to break the matrix in a convenient way so that image compression can be done without losing meaningful data. The compressed singular values of NSCT decomposed coefficients of cover image are added with the compressed singular values of NSCT decomposed coefficient of gray key image. This helps to improve the performance of the compression using watermarking method. In terms of PSNR and compression ratio, the performance of this scheme is analyzed. The proposed algorithm is robust and suitable for copyright protection and data transmission against compression applications.

The protection of sensitive and confidential data become a challenging task in the present scenario as more and more digital data is stored and transmitted between the end users. The privacy is vitally necessary in case of medical data, which contains the important information of the patients. In this article, a novel biometric inspired medical encryption technique is proposed based on newly introduced parameterized all phase orthogonal transformation (PR-APBST), singular value, and QR decomposition. The proposed technique utilizes the biometrics of the patient/owner to generate a key management system to obtain the parameters involved in the proposed technique. The medical image is then encrypted employing PR-APBST, QR and singular value decomposition and is ready for secure transmission or storage. Finally, a reliable decryption process is employed to reconstruct the original medical image from the encrypted image. The validity and feasibility of the proposed framework have been demonstrated using an extensive experiments on various medical images and security analysis.