Matrix Singular Value Research Articles

Video fragment processing by Ky Fan norm

In this study, we focused on the formalization of video frame descriptions in the context of solving video segmentation problem. Since native video data can have various sizes, dividing each frame into blocks allows present image frame as a square matrix for a formal description. The frame block is a matrix of arbitrary dimensions. The ability to skip the step of matrix transformation to a square dimension or vectorization using some descriptor allows to reduce computational costs, freeing up computational resources required for this transformation. In our study, we used Ky Fan norm value as image frame block descriptor. The Ky Fan norm is built on top of matrix singular values. A singular decomposition does not impose restrictions on either the dimension orthe character of the elements of the original matrix. We conducted a comparative analysis of the effectiveness of the obtained descriptor for different video data sizesand with different aspect ratios, showing that the change in the descriptor for each block is independent of the video sizeand aspect ratios. Changes in the descriptorsfor each block from frame to frame are identical for video data of varying sizes. This means that as a result of such fragment transform, a square matrix of a fixed size iscreated, regardless of the output video size.This makes it possible to unify further processing of the video, which can be useful for the task of information search in large video databases under the conditions of providing a query "ad exemplum". In thiscase, we can analyze the existing database in offline mode and match each video with a fixed square matrix of descriptors, which will significantly reduce the time and amount of resources when matching with the query. Also, this approach can be effectively used to analyze video data for the motion detection and scene change tracking.

Output feedback control of interval type-2 T-S fuzzy fractional order systems subject to actuator saturation

This paper investigates the stabilization problem of interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy fractional order systems (FOSs). Firstly, IT2 fuzzy model is constructed to approximate the subject to actuator saturation of systems. Then, the sufficient conditions of output feedback control of IT2 fuzzy FOSs in terms of linear matrix inequalities (LMIs) are presented by using the method involving matrix singular value decomposition (SVD). Furthwemore, we solve optimization problems to estimate stable regions according to the proposed stability conditions. Finally, we proposed a numerical example to demonstrate the availability of theoretical analysis results.

Study on the aesthetic evaluation method of applying symmetry analysis in industrial product styling design

Abstract The evaluation of product styling design is transformed into a complex nonlinear solution problem by symmetry analysis in this paper. The equilibrium matrix decomposition evaluation system calculates the stored matrix singular values in product styling design and proves that symmetry can be obtained from the stress matrix row vector. The regression algorithm is used to process multiple industrial product design objectives, and each design element in product styling is coded to obtain the fitness evaluation of individual industrial products. It is found that the satisfaction score in product styling design is 97.4, the weight index of logo novelty element is up to 7, and the regression coefficient of plane shape is 0.269, to improve the scientific and feasibility of aesthetic evaluation of industrial product styling design.

Robust dual color images watermarking scheme with hyperchaotic encryption based on quaternion DFrAT and genetic algorithm

A robust dual color images watermarking algorithm is designed based on quaternion discrete fractional angular transform (QDFrAT) and genetic algorithm. To guarantee the watermark security, the original color watermark image is encrypted with a 4D hyperchaotic system. A pure quaternion matrix is acquired by performing the discrete wavelet transform (DWT), the block division and the discrete cosine transform on the original color cover image. The quaternion matrix is operated by the QDFrAT to improve the robustness and the security of the watermarking scheme with the optimal transform angle and the fractional order. Then the singular value matrix is obtained by the quaternion singular value decomposition (QSVD) to further enhance the scheme’s stability. The encryption watermark is also processed by DWT and QSVD. Afterward, the singular value matrix of the encryption watermark is embedded into the singular value matrix of the host image by the optimal scaling factor. Moreover, the values to balance imperceptibility and robustness are optimized with a genetic algorithm. It is shown that the proposed color image watermarking scheme performs well in imperceptibility, security, robustness and embedding capacity.

Tensor Singular Spectrum Decomposition: Multisensor Denoising Algorithm and Application

Realising multi-sensor signal fusion and weak feature adaptive extraction is a challenging task. Therefore, a new algorithm called tensor singular spectrum decomposition is proposed in this study for the adaptive decomposition of multi-sensor time series. Traditional tensor decomposition algorithms, such as CP, HOSVD and Tucker decomposition, are derived from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -mode product. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -mode product essentially uses the idea of matrices to deal with tensors given that it defines the multiplication between matrix and higher-order tensor, thereby creating problems of non-pseudodiagonal core tensor and nonunique decomposition results in traditional tensor decomposition algorithms. To this end, decomposition of the original tensor signal and reconstruction of multi-sensor component signals are realised in this study by combining the trajectory tensor construction, superposition of the Gaussian function spectral model, adaptive iterative optimisation of embedding dimension and diagonal average method on the basis of the principle of tensor–tensor order-preserving multiplication. The proposed algorithm inherits the perfect mathematical theory and excellent properties of matrix singular value decomposition in processing single-sensor signals whilst retaining the inherent structure and coupling relationship between multi-sensor data and realising the organic fusion and adaptive decomposition of multi-sensor signals. The analysis results of simulation, experimental and engineering signals showed that the proposed method can effectively extract weak fault quantification features hidden in original multi-sensor signals compared with existing methods.

PASSIVITY AND PASSIVATION OF FRACTIONAL-ORDER NONLINEAR SYSTEMS

Although the passivity of integer-order systems has been extensively analyzed, the research outcomes on the passivity of fractional-order nonlinear systems (FONSs) are scarce. This paper presents some theoretical results on passivity and passivation of FONSs. Based on the definition of the passivity of FONSs, and by using the Lyapunov stability theory and the linear matrix inequality (LMI) method, some conditions are derived to assure the FONSs is passive, which enrich the existing theoretical knowledge about the passivity of FONSs. Moreover, an observer-based output passive control is established to ensure that the corresponding closed-loop system is passive by means of LMI technique and matrix singular value decomposition (SVD). Ultimately, the practicality of our yielded results is revealed by two numerical simulations.

The Impact of Intelligent Accounting Information Management on Corporate Governance Information Transparency

In order to improve the transparency of corporate governance information, this paper combines intelligent accounting information technology to analyze the information transparency in the process of corporate governance, analyzes the fast adaptive neighbor clustering algorithm, and realizes the joint spectral embedding of data by creating block diagonal anchor graphs. Moreover, this paper re-selects anchor points from the spectral embedding data and constructs a new block diagonal anchor graph. In addition, using the principle of adaptive nearest neighbor clustering, this paper performs structured graph learning on the block diagonal anchor graph matrix. After derivation, the Laplace eigenvalue decomposition problem of the whole graph in adaptive nearest neighbor clustering can be transformed into a block matrix singular value decomposition problem. The regression analysis shows that there is a linear correlation between the advanced degree of intelligent accounting information management technology and the transparency of corporate governance information.

Two-Stage Contrast Enhancement Method for Liver Tumor Segmentation

Tumor segmentation in Computed Tomography (CT) images is a crucial step in image-guided surgery. However, low-contrast CT images impede the performance of subsequent segmentation tasks. Contrast enhancement is then used as a preprocessing step to highlight the relevant structures, thus facilitating not only medical diagnosis but also image segmentation with higher accuracy. The proposed method is based on two concepts, namely adaptive gamma correction using DWT-SVD and OPTimized Guided Contrast Enhancement (OPTGCE). In the proposed DWT-SVD scheme, the technique decomposes the input medical image into four frequency sub-bands by using DWT and then estimates the singular value matrix of the LL sub-band image. An enhanced LL component is generated using an adequate correction factor and inverse SVD. The proposed OPTimized Guided Contrast Enhancement (OPTGCE) scheme exploits both contextual information from the guidance image and structural information from the input image. Tumor segmentation algorithm is applied on the enhanced images to analyze the performance of the proposed method in facilitating tumor segmentation. The qualitative and quantitative analysis using metrics including entropy, MCCEE, and MIGLCM shows the superiority of the proposed method in comparison with the existing methods that do not include guidance mechanism Keywords—: 2D-DWT, Gamma Correction, SSIM, Gradient.

Adaptive fault-tolerant control for nonlinear high-order fully-actuated systems

In this paper, an adaptive fault-tolerant control (FTC) strategy is proposed for a class of high-order fully-actuated (HOFA) systems. A pre-closed-loop treatment is presented to simplify the observer design for the nonlinear system with actuator faults but it leads to a new problem that faults may not satisfy the observation matching conditions. Thus, an intermediate variable observer is introduced to adaptively estimate system states and actuator faults simultaneously. Then the influence of observation errors on system stability is analyzed by analyzing matrix singular value perturbation. Under the proposed adaptive FTC method, the closed-loop global uniformly ultimately bounded stability is achieved by Lyapunov synthesis. Finally, the effectiveness of the proposed FTC method is verified by simulations.

Image Enhancement and Features Extraction of Electron Microscopic Images Using Sigmoid Function and 2D-DCT

An innovative image enhancement and feature extraction technique that is based on the modified sigmoid function and two-dimensional DCT has been developed. The proposed technique uses a modified sigmoid function that accommodates the original microscopic input image characteristics. A novel block-based input value coupled with the modified sigmoid function is used in this proposed technique to provide good contrast enhancement of an image, resulting in localised contrast enhancement. Singular value decomposition played an important role after DCT because the singular value matrix determines the intensity values of the provided microscopic image. Changes in the singular values have an immediate impact on the intensity of the microscopic input image. The proposed methodology essentially converts the input picture into the SVD-DCT domain, normalises the singular value matrix, and finally reconstructs the enhanced image using inverse DCT. Simulation findings demonstrate that the proposed technique produces significantly superior improved results compared to other current approaches. Various essential characteristics of actinomycetes become evident once electron microscopic images are enhanced, such as long filaments, coils or spirals, rod shapes, and spore patterns. The presented method works successfully and efficiently for various bright and dark microscopic images.

A Tensor SVD-based Classification Algorithm Applied to fMRI Data

To analyze the abundance of multidimensional data, tensor-based frameworks have been developed. Traditionally, the matrix singular value decomposition (SVD) is used to extract the most dominant features from a matrix containing the vectorized data. While the SVD is highly useful for data that can be appropriately represented as a matrix, this step of vectorization causes us to lose the high-dimensional relationships intrinsic to the data. To facilitate efficient multidimensional feature extraction, we utilize a projection-based classification algorithm using the t-SVDM, a tensor analog of the matrix SVD. Our work extends the t-SVDM framework and the classification algorithm, both initially proposed for tensors of order 3, to any number of dimensions. We then apply this algorithm to a classification task using the StarPlus fMRI dataset. Our numerical experiments demonstrate that there exists a superior tensor-based approach to fMRI classification than the best possible equivalent matrix-based approach. Our results illustrate the advantages of our chosen tensor framework, provide insight into beneficial choices of parameters, and could be further developed for classification of more complex imaging data. We provide our Python implementation at https://github.com/elizabethnewman/tensor-fmri.

Spectral Properties of the Normalized Rigidity Matrix for Triangular Formations

This work establishes properties of the normalized rigidity matrix in two- and three-dimensional spaces. The upper bound of the normalized rigidity matrix singular values is derived for minimally and infinitesimally rigid frameworks in two- and three-dimensional spaces. We prove that the transformation of a framework does not affect the normalized rigidity matrix properties. The largest minimum singular value of the normalized rigidity matrix for a rigid framework of three agents in two-dimensional space is given as well as necessary and sufficient conditions to reach that value. These results can be used in stability analysis and control design of a distance-based formation control. The numerical simulation for multi-agent systems in two-dimensional space illustrates the theoretical results. Moreover, a real-time simulation is provided to demonstrate the spectral properties of the normalized rigidity matrix.

Retracted] Research on Digital Image Watermarking Algorithm Based on Scrambling and Singular Value Decomposition

At present, the digital image watermarking algorithm has not embedded the synchronization signal in the image, resulting in the poor performance of the embedded image in terms of, for example, imperceptibility, anti‐attack ability, and robustness. Therefore, a digital image watermarking algorithm based on scrambling and singular value decomposition is proposed. The digital watermark is preprocessed by dimensionality reduction and encryption; the digital image is processed in sections and embedded in the synchronization signal. The digital watermark image is embedded in the digital image watermark by using the low‐frequency energy ratio technology of sound channel. The watermark image extraction step is designed to extract the watermark image. The transmittance of the degraded image is calculated, the transmittance is refined by relying on the soft matting algorithm and guiding filter, the image contrast in the image frequency domain is enhanced, and the results are mapped to the appropriate visual range to optimize the visual brightness of the image. Finally, according to the uniqueness of singular value matrix, the distributed characteristics of image matrix data are described, the visual effect is enhanced, and the research of digital image watermarking algorithm is completed. The experimental results show that the algorithm can extract the watermark information completely, the image contrast and singular value have been significantly improved, and the algorithm has better anti‐attack performance.

Internal Damage Analysis of Braided Composites Embedded in Carbon Nanotube Yarn

Carbon nanotube (CNT) yarn sensors were embedded in 3D braided composites in the form of arrays to detect the internal damage of specimens and study the internal damage monitoring of the 3D braided composites. The signals collected by the sensor array of CNT yarn were preprocessed using the dynamic wavelet threshold algorithm. The exact position of the damage was calculated based on the main features of the resistance signal matrix, which was calculated using the quadratic matrix singular value. The results show that the internal damage localization of the specimens was consistent with the actual damage. The localizations in this study can provide a basis for enhancing the structural health monitoring of smart 3D braided composites.

Wideband Frequency Invariant Array Synthesis Based on Matrix Singular Value Decomposition

In this paper, an analytic method for frequency invariant (FI) array synthesis is proposed based on matrix singular value decomposition. By grouping the elements of FI array into a few subarrays, the FI pattern in the whole frequency band is realized. Using this algorithm, the number of sub arrays is reduced. Simulation results show that the proposed algorithm can synthesize the 64-element broadband FI array in 0.52 s. For the 18-element linear array, the half power beam width (HPBW) changes less than 0.6 degrees in the bandwidth. Moreover, the range of HPBW variation decreases rapidly along with the increase in the number of elements. Furthermore, the effectiveness of the algorithm is verified by synthesizing FI array with low side lobe level (SLL), beam scanning, and notch requirements. The examples in this paper show that the proposed algorithm can achieve better pattern characteristics with fewer elements. Finally, a broadband antenna with 2:1 bandwidth is improved, and two FI arrays of 23 elements and 64 elements are formed by using the antenna. The active pattern of the array element is introduced into the proposed algorithm, and two FI arrays synthesized by the algorithm are simulated by full wave software.

Tensor-tensor algebra for optimal representation and compression of multiway data

With the advent of machine learning and its overarching pervasiveness it is imperative to devise ways to represent large datasets efficiently while distilling intrinsic features necessary for subsequent analysis. The primary workhorse used in data dimensionality reduction and feature extraction has been the matrix singular value decomposition (SVD), which presupposes that data have been arranged in matrix format. A primary goal in this study is to show that high-dimensional datasets are more compressible when treated as tensors (i.e., multiway arrays) and compressed via tensor-SVDs under the tensor-tensor product constructs and its generalizations. We begin by proving Eckart-Young optimality results for families of tensor-SVDs under two different truncation strategies. Since such optimality properties can be proven in both matrix and tensor-based algebras, a fundamental question arises: Does the tensor construct subsume the matrix construct in terms of representation efficiency? The answer is positive, as proven by showing that a tensor-tensor representation of an equal dimensional spanning space can be superior to its matrix counterpart. We then use these optimality results to investigate how the compressed representation provided by the truncated tensor SVD is related both theoretically and empirically to its two closest tensor-based analogs, the truncated high-order SVD and the truncated tensor-train SVD.

Quantum tensor singular value decomposition* *This research is supported in part by Hong Kong Research Grant council (RGC) grants (No. 15208418, No. 15203619, No. 15506619) and Shenzhen Fundamental Research Fund, China under Grant No. JCYJ20190813165207290.

Tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.

Finite-time analysis for networked predictive control systems with induced time delays and data packet dropouts

This paper is concerned with the problem of finite-time stability (FTS) and finite-time boundedness (FTB) issues of networked predictive control systems (NPCs) with induced time delays and data packet dropouts. First, a new networked predictive control (NPC) scheme is established based on the designed observer and prediction strategy, and a model of NPCs is obtained. Second, in order to handle different time delays and data packet dropouts, an improved method of combining matrix singular value decomposition (SVD) with multivariate presupposition is proposed. Third, an observer-based predictive controller is designed to make the closed loop system FTS and FTB via finite-time theory, then the induced time delays and data packet dropouts are disposed actively. In addition, new sufficient conditions are obtained, which can guarantee the desired controller and observer gain matrices be subsequently acquired by solving some linear matrix inequalities (LMIs). Finally, two numerical examples are illustrated to validate the effectiveness of the theoretical findings.

Differential Mean Deviation Factor based Contrast Enhancement Technique Using DWT-SVD for Non-Contrast CT Scan Images

In proper clinical analysis and diagnosis of a patient, accurate visual representation of CT scan images plays a vital role. Due to technical constraints and poor imaging device, sometimes low contrast images can be obtained. To interpret and analyzed the minute details present in the image, the image needs to be enhanced appropriately. This proposed novel low contrast enhancement technique rely on the singular value decomposition (SVD) with discrete wavelet transform (DWT) for improving the minute visual information present in a image as well as retaining the mean brightness. By scaling the values of the singular matrix properly these existing SVD based techniques enhances the contrast of image but the contrast in case of the original CT scan image is too low and hinder proper enhancement result in poor analysis. In this suggested algorithm, the proposed differential mean-deviation factor properly equalized the singular matrix. Introduction of this parameter adjusted the singular value matrix in such a way that proper contrast enhancement is attained. Results of the simulation also reveal that the suggested approach also balanced the brightness of the picture more reliably and increases the contrast comparatively with negligible visual artifacts. It outperforms traditional methodologies of equalization of images such as GHE (Global histogram equalization) and LHE (local histogram equalization). Our proposed algorithm is evaluated and compared using different CT scan images and different algorithms based on SVD and DWT but the inclusion of our proposed factor mean deviation provides a niche over the existing ones in terms of subjective and objective analysis.

Generalized Unitarily Invariant Gauge Regularization for Fast Low-Rank Matrix Recovery.

Spectral regularization is a widely used approach for low-rank matrix recovery (LRMR) by regularizing matrix singular values. Most of the existing LRMR solvers iteratively compute the singular values via applying singular value decomposition (SVD) on a dense matrix, which is computationally expensive and severely limits their applications to large-scale problems. To address this issue, we present a generalized unitarily invariant gauge (GUIG) function for LRMR. The proposed GUIG function does not act on the singular values; however, we show that it generalizes the well-known spectral functions, including the rank function, the Schatten- p quasi-norm, and logsum of singular values. The proposed GUIG regularization model can be formulated as a bilinear variational problem, which can be efficiently solved without computing SVD. Such a property makes it well suited for large-scale LRMR problems. We apply the proposed GUIG model to matrix completion and robust principal component analysis and prove the convergence of the algorithms. Experimental results demonstrate that the proposed GUIG method is not only more accurate but also much faster than the state-of-the-art algorithms, especially on large-scale problems.