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The discovery of community structure is the basis of understanding the topology structure and social function of the network. It is also an important factor for recommendation technology, information dissemination, event prediction, and more. In this paper, we consider the structure and characteristics of the social network and propose an algorithm based on semi-supervised matrix factorization and random walk. The proposed method first calculates the transition probability between nodes through the topology of the network. The random walk model is then used to obtain the final walk probability, and the feature matrix is constructed. At the same time, we combine a priori content information in the network to build a must-link matrix and a cannot-link matrix. We then merge them into the feature matrix of the random walk to form a new feature matrix. Finally, the expectation of the number of edges is defined according to the factorized membership matrix. Results demonstrate the effectiveness and better performance of our method.

This paper proposes a valid and fast algorithm for low-rank matrix factorization. There are multiple applications for low-rank matrix factorization, and numerous algorithms have been developed to solve this problem. However, many algorithms do not use rank directly; instead, they minimize a nuclear norm by using Singular Value Decomposition (SVD), which requires a huge time cost. In addition, these algorithms often fix the dimension of the factorized matrix, meaning that one must first find an optimum dimension for the factorized matrix in order to obtain a solution. Unfortunately, the optimum dimension is unknown in many practical problems, such as matrix completion and recommender systems. Therefore, it is necessary to develop a faster algorithm that can also estimate the optimum dimension. In this paper, we use the Hidden Matrix Factorized Augmented Lagrangian Method to solve low-rank matrix factorizations. We also add a tool to dynamically estimate the optimum dimension and adjust it while simultaneously running the algorithm. Additionally, in the era of Big Data, there will be more and more large, sparse data. In face of such highly sparse data, our algorithm has the potential to be more effective than other algorithms. We applied it to some practical problems, e.g. Low-Rank Representation(LRR), and matrix completion with constraint. In numerical experiments, it has performed well when applied to both synthetic data and real-world data.

This paper aims to extend the conjugate gradient least squares method to solve the least squares problem of the generalized Sylvester-conjugate matrix equation. For any initial values, the proposed iterative method can obtain the least squares Frobenius norm solution within finite iteration steps in the absence of roundoff errors. Finally, two numerical examples are given to show the efficiency of the presented iterative method. And it’s also proved that our proposed iterative algorithm is better than the existing LSQR iterative algorithm.

We consider the matrix completion problem that aims to construct a low rank matrix X that approximates a given large matrix Y from partially known sample data in Y . In this paper we introduce an efficient greedy algorithm for such matrix completions. The greedy algorithm generalizes the orthogonal rank-one matrix pursuit method (OR1MP) by creating s ⩾ 1 candidates per iteration by low-rank matrix approximation. Due to selecting s ⩾ 1 candidates in each iteration step, our approach uses fewer iterations than OR1MP to achieve the same results. Our algorithm is a randomized low-rank approximation method which makes it computationally inexpensive. The algorithm comes in two forms, the standard one which uses the Lanzcos algorithm to find partial SVDs, and another that uses a randomized approach for this part of its work. The storage complexity of this algorithm can be reduced by using an weight updating rule as an economic version algorithm. We prove that all our algorithms are linearly convergent. Numerical experiments on image reconstruction and recommendation problems are included that illustrate the accuracy and efficiency of our algorithms.

Image rectification is an important stage of applying a pair of projective transformations, or homographies, to a pair of stereo images, so that epipolar lines in the original images map to horizontally aligned lines in the rectified images. Considering that for some stereo rigs the intrinsic parameters of the cameras are known but their external parameters are unknown, in this paper, we present a novel method for stereo rectification based on the essential matrix which is derived from the fundamental matrix. Without any optimization process, closed-form analytical solutions of the projective transformations for epipolar rectification can be directly obtained by conducting SVD on the essential matrix. Experimental results show the proposed rectification method not only has higher efficiency and rectification precision, but also its scale invariance and robustness are superior to existing methods.

In this study, a novel Z-bus matrix-building algorithm based on graph theory is proposed as an alternative method for radial distribution systems. The proposed algorithm is based on the branch-path incidence matrix K of a radial distribution network as opposed to a traditional Z-bus building algorithm and inverse Y-bus matrix with lower and upper triangular matrix (LU) method. Advantages of the proposed algorithm include fewer logical judgments, less execution time, and suitability for computing-aided analyses. The algorithm possessed potential for applications in all power system analyses related to a Z-bus impedance matrix including power flow analysis, short circuit fault current analysis, and harmonic power flow analysis. Four standard IEEE test systems and random test systems are selected to demonstrate the superiority of the proposed method with respect to execution times. The results indicate that the proposed algorithm exhibited a performance that exceeded those of other methods, particularly with respect to large-scale unbalanced radial distribution systems with hybrids of single, two-phase, and three-phase elements and nodes.

An algorithm for computing the antitriangular factorization of symmetric matrices, relying only on orthogonal transformations, was recently proposed. The computed antitriangular form straightforwardly reveals the inertia of the matrix. A block version of the latter algorithm was described in a different paper, where it was noticed that the algorithm sometimes fails to compute the correct inertia of the matrix. In this paper we analyze a possible cause of the failure of detecting the inertia and propose a procedure to recover it. Furthermore, we propose a different algorithm to compute the antitriangular factorization of a symmetric matrix that handles most of the singularities of the matrix at the very end of the algorithm. Numerical results are also given showing the reliability of the proposed algorithm.

Although the success of the projective geometry applications in stereo vision, a number of criticisms have been raised in the literature about its use. Most of them concern the performance of the eight-point algorithm that is used to find the fundamental matrix F. And few directly target the application of the projective geometry in computer vision. This paper first, reports on some of these criticisms followed by some flawed derivations of the fundamental matrix equation. Then, in a simple and unquestionable analysis, it demonstrates that the equation of the fundamental matrix mTrFml = 0 does not hold for all image points ml and mr. The matrix F is independent of the scene structure, it depends only on the rotation and translation of the second camera with respect to the first one; (F t [t]xR). The vectors ml and mr are not orthogonal for every point M. And because the dot product (mTrF) · ml) is equal to zero if and only if the two vectors mTrF and ml are orthogonal, the equation is invalid.

Recently, Tang et al. [Numer Algorithms. 2014;66(2):379–397] have offered a cyclic iterative method for determining the unique solution of the coupled matrix equationsAnalogues to the gradient-based algorithm, the proposed algorithm relies on a fixed parameter whereas it has wider convergence region. Nevertheless, the application of the algorithm to find the centro-symmetric solution of the mentioned problem has been left as a project to be investigated and the optimal value for the fixed parameter has not been derived. In this paper, we focus on a more general class of the coupled linear matrix equations that incorporate the mentioned ones in the earlier refereed work. More precisely, we first develop the authors’ propounded algorithm to resolve our considered coupled linear matrix equations over centro-symmetric matrices. Afterwards, we disregard the restriction of the existence of the unique (centro-symmetric) solution and also modify the authors’ algorithm by applying an oblique projection technique w...

Rank-reduction methods have been successfully used for seismic data interpolation and noise attenuation. However, highly intense computation is required for singular value decomposition (SVD) in most rank-reduction methods. In this paper, we propose a simple yet efficient interpolation algorithm, which is based on the Hankel matrix, for randomly missing traces. Following the multichannel singular spectrum analysis (MSSA) technique, we first transform the seismic data into a low-rank block Hankel matrix for each frequency slice. Then, a fast orthogonal rank-one matrix pursuit (OR1MP) algorithm is employed to minimize the low-rank constraint of the block Hankel matrix. In the new algorithm, only the left and right top singular vectors are needed to be computed, thereby, avoiding the complexity of computation required for SVD. Thus, we improve the calculation efficiency significantly. Finally, we anti-average the rank-reduction block Hankel matrix and obtain the reconstructed data in the frequency domain. Numerical experiments on 3D seismic data show that the proposed interpolation algorithm provides much better performance than the traditional MSSA algorithm in computational speed, especially for large-scale data processing.

Matrices are appropriate for representing a wealth of data with complex structures such as images and electroencephalogram data (EEG). To learn a classifier dealing with these matrix data, the structure information of the feature matrix is useful. In this paper, we focus on the regularized matrix classifiers whose input samples and weight parameters are both in the form of a matrix. Some existing approaches assume that the weight matrix has a low-rank structure and then utilize the popular nuclear norm of the weight matrix as a regularization term. However, the optimization methods for these matrix classifiers often involve numbers of expensive singular value decomposition (SVD) operations, which prevents scaling beyond moderate matrix sizes. To reduce the time complexity, we propose a novel learning algorithm called Atom Decomposition Based Subgradient Descent (ADBSD), which solves the optimization problem for the matrix classifier whose objective function is the combination of the Frobenius matrix norm and nuclear norm of the weight matrix along with the hinge loss function. Our ADBSD is an iterative scheme which selects the most informative rank-one matrices from the subgradient of the objective function in each iteration. We consider using the atom decomposition based methods to minimize nuclear norm because they mainly rely on the computation of top singular vector pair which leads to great advantages on efficiency. We empirically evaluate the performance of the proposed algorithm ADBSD on both synthetic and real-world datasets. Results show that our approach is more efficient and robust than the state-of-the-art methods.

We consider the problem of approximately computing saddle-point of a zero-sum matrix game when either the columns of the matrix are revealed incrementally in time or the matrix is too large to apply traditional methods. We leverage the established adaptive multiplicative weights algorithm but introduce a novel simple criterion to determine whether the approximately computed minimizer's best strategy needs to be re-computed when a new column of the matrix is introduced. Our main results are two-fold. First, we show that our proposed incremental approach achieves the same accuracy as applying the adaptive multiplicative weights algorithm on the entire matrix, if known a priori. Second, when the columns of the matrix are generated independently and from the same distribution, we show that the expected number of times the approximate strategy is re-computed grows at most logarithmically with the number of columns of the matrix, thereby being computationally efficient.

In this paper, we propose a two-step proximal gradient algorithm to solve nuclear norm regularized least squares for the purpose of recovering low-rank data matrix from sampling of its entries. Each iteration generated by the proposed algorithm is a combination of the latest three points, namely, the previous point, the current iterate, and its proximal gradient point. This algorithm preserves the computational simplicity of classical proximal gradient algorithm where a singular value decomposition in proximal operator is involved. Global convergence is followed directly in the literature. Numerical results are reported to show the efficiency of the algorithm.

The estimation of the fundamental matrix (F) is to determine the epipolar geometry and to establish a geometrical relation between two images of the same scene or elaborate video frames. In the literature, we find many techniques that have been proposed for robust estimations such as RANSAC (random sample consensus), least squares median (LMeds), and M estimators as exhaustive. This article presents a comparison between the different detectors that are (Harris, FAST, SIFT, and SURF) in terms of detected points number, the number of correct matches and the computation speed of the ‘F’. Our method based first on the extraction of descriptors by the algorithm (SURF) was used in comparison to the other one because of its robustness, then set the threshold of uniqueness to obtain the best points and also normalize these points and rank it according to the weighting function of the different regions at the end of the estimation of the matrix ''F'' by the technique of the M-estimator at eight points, to calculate the average error and the speed of the calculation ''F''. The results of the experimental simulation were applied to the real images with different changes of viewpoints, for example (rotation, lighting and moving object), give a good agreement in terms of the counting speed of the fundamental matrix and the acceptable average error. The results of the simulation it shows this technique of use in real-time applications.

In this paper, we propose a novel method to estimate the relative camera motions of three consecutive images. Given a set of point correspondences in three views, the proposed method determines the fundamental matrix representing the geometrical relationship between the first two views by using the eight-point algorithm. Then, by minimizing the proposed cost function with the fundamental matrix, the relative camera motions over three views are precisely estimated. The experimental results show that the proposed method outperforms the conventional two-view and three-view geometry-based method in terms of the accuracy.

The measurement matrix of compressed sensing has a significant impact for sampling and reconstruction algorithm of the original signal. At present, the majority of the measurement matrix is randomly constructed, and it is difficulty for hardware implementation in the practical applications. In this paper, we use the sparse characteristic of parity- check matrix of LDPC codes, construct measurement matrix based on QC-LDPC (Quasi-cyclic low-density parity-check) matrix, which is a structural and sparse deterministic measurement matrix. The simulation results show that, the measurement matrix is proposed in this paper not only can obtain a better reconstructed image quality, but also it can reduce the complexity of hardware implementation for quasi-cyclic.

In computer vision, it is a challenge to compute the relationship of multiple views from scene images. The view relationship can be obtained from the fundamental matrix. Thus, it is very important to compute an accurate fundamental matrix from unevenly distributed features in complex scene images. This study proposes a robust method to estimate the fundamental matrix from corresponding images. First, the authors introduce how to find matched features from scene images efficiently. The epipolar geometry can restrict the point correspondences to the polar line, but cannot cope with the false points lying on the line. To eliminate such mismatches, the authors present an affine constraint which can also merge the uniform regions produced by mean-shift segmentation. Second, inspired by the success of random sample consensus, the authors moderately improve the weighting function based on M-estimator to increase the accuracy of the fundamental matrix estimation. Experimental results on simulated data and real images show these works are efficient for estimating fundamental matrix. The authors also evaluated the accuracy of their method on computing the external parameters of two cameras. The result shows that this method obtains comparable performance to the more sophisticated calibration method.

In this paper, we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized Sylvester-transpose matrix equation A X B + C X T D = F . The idea is from (Ding and Chen, 2005; Niu et?al., 2011; Wang et?al., 2012) in which some efficient algorithms were developed for solving the Sylvester matrix equation and the Lyapunov matrix equation. On the basis of the information generated in the previous half-step, we further introduce a relaxation factor to obtain the solution of the generalized Sylvester-transpose matrix equation. We show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions. Finally, some numerical examples are given to illustrate that the introduced iterative algorithm is efficient.

In this paper, a novel algorithm, called the general similar sensing matrix pursuit (GSSMP), is proposed to deal with the deterministic sensing matrix with high coherence. First, the columns of the sensing matrix are divided into a number of similar column groups based on the similarity distance. Each similar column group presents a set of coherent columns or a single incoherent column, which provides a unified frame work to construct the similar sensing matrix. The similar sensing matrix is with low coherence provided that the minimum similar distance between any two condensed columns is large. It is proved that under appropriate conditions the GSSMP algorithm can identify the correct subspace quite well, and reconstruct the original K-sparse signal perfectly. Moreover, we have enhanced the proposed GSSMP algorithm to cope with the unknown sparsity level problem, by testing each individual contributing similar column group one by one to find the true vectors spanning the correct subspace. The simulation results show that the modified GSSMP algorithm with the contributing similar column group test process can estimate the sparse vector representing the radar scene with an unknown number of targets successfully. HighlightsA novel GSSMP algorithm is proposed to tackle the deterministic sensing matrix with high coherence.A rigorous proof of the guaranteed reconstruction performance of the GSSMP algorithm is also provided.The proposed GSSMP algorithm is further enhanced to cope with the unknown sparsity problem.

Fundamental matrix estimation has been studied extensively in the area of computer vision and previously proposed techniques include those that only use feature points. In this study, we propose a new technique for calculating the fundamental matrix combined with feature lines, which is based on the epipolar geometry of horizontal and vertical feature lines. First, a method for parameterizing the fundamental matrix is introduced, where the camera orientation elements and relative orientation elements are used as the parameters of the fundamental matrix, and the equivalent relationships are deduced based on the horizontal and vertical feature lines. Next, the feature lines are used as the interior points by the RANSAC algorithm to search for the optimal feature point subset, before determining the weight of each factor using the M-estimators algorithm and building a unified adjustment model to estimate the fundamental matrix. The experimental results obtained using simulated images and real images demonstrate that the proposed approach is feasible in practice and it can greatly reduce the dependency on feature points in the traditional method, while the introduction of feature lines can improve the accuracy and stability of the results to some extent.

We propose a new iterative algorithm to compute the symmetric solution of the matrix equations $AXB=E$ and $CXD=F$ . The greatest advantage of this new algorithm is higher speed and lower computational cost at each step compared with existing numerical algorithms. We state the solutions of these matrix equations as the intersection point of some closed convex sets, and then we use the alternating projection method to solve them. Finally, we use some numerical examples to show that the new algorithm is feasible and effective.

Five-point algorithm is a powerful tool for relative orientation, because it requires no initial assumption of camera position. This algorithm determines an essential matrix from five point correspondences between two calibrated cameras, but results multiple solutions and some selecting process is required. This paper proposes Quasi-Five-Point Algorithm which is non-linear solver with seed solution of 8 point algorithm. The method tries to calculate the appropriate essential matrix without selecting process among multiple solutions. It is one of non-linear approach, but tries to find an appropriate seed before non-linear calculation. Using correspondences of 3 or more additional points, seed values of the solution is calculated. In this paper relationship between traditional parametric relative orientation and essential matrix is discussed, and after that quasi-five-point algorithm is introduced.

This paper considers the complex mixing matrix estimation in the under-determined blind source separation. An effective estimation algorithm through detecting single source points contributed by only one source is proposed. First, the single source points are detected by utilizing the real and the imaginary components of the time---frequency coefficients of mixed signals. The algorithm is suitable for the case in which the mixing matrix is complex, while traditional algorithms usually estimate the real mixing matrix. Then, through modeling and calculating, the mixing matrix of mixed signals can be estimated. Finally, the clustering process is improved in order to get more accurate results. The algorithm can estimate the complex mixing matrix when the number of sensors is less than that of sources. The experimental results validate the efficiency of the estimation algorithm.

In this paper, we propose a new mean value algorithm for the Toeplitz matrix completion based on the singular value thresholding (SVT) algorithm. The completion matrices generated by the new algorithm keep a feasible Toeplitz structure. Meanwhile, we prove the convergence of the new algorithm under some reasonal conditions. Finally, we show the new algorithm is much more effective than the ALM (augmented Lagrange multiplier) algorithm through numerical experiments and image inpainting.

The calibration of projection matrix between cameras and the optimization of the projection matrix based on Levenberg-Marquardt algorithm will be introduced in this article. The optimized parameters of projection matrix can be obtained by the least squares principle and the parameters are the iterative initial values of Levenberg-Marquardt (called L-M for short) to implement the second optimization of projection matrix parameters. This method will further improve the accuracy of measurement system. The results of simulation and experimentation demonstrate that measurement system based on the principle of least squares to obtain the initial optimization parameters and using the L-M algorithm to second times optimization has a higher precision, it is well suited for high accuracy measurement.

When considering non-central imaging devices, the computation of the relative pose requires the estimation of the rotation and translation that transform the 3D lines from one coordinate system to the second. In most of the state-of-the-art methods, this transformation is estimated by the computing a 6i?6 matrix, known as the generalized essential matrix. To allow a better understanding of this matrix, we derive some properties associated with its singular value decomposition.

Rank estimation is an important factor for low-rank based matrix completion, and most works devoted to this problem have considered the minimization of nuclear norm instead of matrix rank. However, when nuclear norm minimization shifts to ‘regularization’ due to noise, it is difficult to estimate original matrix rank, precisely. In present paper, we propose a new fast algorithm to precisely estimate matrix rank and perform completion without using nuclear norm. In our extensive experiments, the proposed algorithm significantly outperformed nuclear-norm based method for accuracy, especially and Incremental OptSpace regarding computational time. Our model selection scheme has many promising extensions for constrained matrix factorizations and tensor decompositions, and these extensions could be useful for wide range of practical applications.

This paper deals with the problem of finding the minimum norm least-squares solution of a quite general class of the coupled linear matrix equations defined over over field of complex numbers. To this end, we examine a gradient-based approach and present the convergence properties of the algorithm. The highlight of the elaborated results in the current work is using a new sight of view for construction of the gradient-based algorithm which turns out that we can ignore some of the limitations assumed by the authors in the recently published works for the application of the algorithm to obtain the solution of the referred problems. To the best of our knowledge, so far, computing the optimal convergence factor of the algorithm to determine the (least-squares) solution of general complex linear matrix equations has left as a project to be investigated. In the current work, we determine the optimal convergence factor of the algorithm. Some numerical experiments are reported to illustrate the validity of the presented results.

In this paper, we propose an image registration algorithm based on improved SIFT (Scale-Invariant Feature Transform) algorithm and essential matrix estimation based on RANSAC (Random Sample Consensus) and AC-RANSAC (A Contrario RANSAC) algorithm. So that in the 3D reconstruction, we can directly restore the parameters of the camera by using the essential matrix model estimated by image registration algorithm. The essential matrix is a 5-parameter model, reflecting the relationship between the representation of the spatial image points in the camera coordinate system under different viewing angles. SIFT algorithm not only maintains the invariance of scale, rotation, brightness and so on, but also maintains a certain degree of stability to the angle change, affine transformation and noise, but the time performance is low and the matching accuracy is not high enough. Therefore, we propose to narrow the dimension of the SIFT feature vector to reduce the time consumption, and increase the similarity measure of the nearest neighbor distance less than 0.3 to calculate the feature point correspondence between images. The experimental results have demonstrated that our method not only can guarantee better time performance, but also can effectively eliminate the wrong match point, greatly improving the matching accuracy.

Recently, many works demonstrated that block-sparse signals recoveries applied model based algorithm could achieve better performance than conventional sense manner. Furthermore, a carefully designed measurement matrix can not only improve the recovery success rate, but also reduce the number of sampling. In this paper, we introduce a novel approach to construct measurement matrix for block-sparse recovery. First, we make the Gram matrix of the measurement matrix close to identity for initialization, and its optimization is demonstrated in our paper. Then, we update the measurement matrix to minimize the weighted sum of whose sub-block coherence and inter-block coherence by using eigenvalues decomposition. The numerical results show that the proposed framework for block-sparse data recoveries significantly improves the ability of block orthogonal matching pursuit algorithm compared with conventional senses.

RPCA (Robust Principal Component Analysis) recovers sparse and low rank components from the original observation data, RPCA commonly uses ADM (Alternate Direction Method) for solving, the efficiency of algorithm depends on the nuclear norm optimization solution, that is SVD. And the application of RPCA in computer vision, image and video and so on, large amounts of data, such as images and video, make it difficult for large-scale data SVD. In this paper, we used the random matrix algorithm to improve the SVD, respectively the Count Sketch algorithm, the Prototype Randomized k-SVD algorithm and the Faster Randomized k-SVD algorithm, The main idea is to reduce the size of the original large-scale data matrix and sample randomly. Using the random projection algorithm to obtain an approximation of the original matrix, and operate QR decomposition of the approximate matrix, get the unitary matrix corresponding to the approximate matrix, and do the corresponding SVD, finally we can get the results which is similar to the original matrix calculation. Obtaining approximation of the original data matrix. But the cost time and space have been greatly optimized. Simulation experiments based on single image and video foreground detection show that the proposed methods can greatly improve the efficiency of RPCA iterative optimization.

This paper deals with the problem of active calibration when channel gain/phase uncertainties and sensor mutual coupling effects are simultaneously present. The numerical algorithms used to compensate for array error matrix, which is formed by the product of mutual coupling matrix and channel gain/phase error matrix, are presented especially tailored to uniform linear array (ULA) and uniform circular array (UCA). First, the array spatial responses corresponding to different azimuths are numerically evaluated using a set of time-disjoint auxiliary sources at known locations. Subsequently, a least-squares (LS) minimization model with respect to array error matrix is established. To solve this LS problem, two novel algorithms, namely algorithm I and algorithm II, are developed. In algorithm I, the array error matrix is considered as a whole matrix parameter to be optimized and an explicit closed-form solution to the error matrix is obtained. Compared with some existing algorithms with similar computation framework, algorithm I is able to utilize all potentially linear characteristics of ULA's and UCA's error matrix, and the calibration accuracy can be increased. Unlike algorithm I, algorithm II decomposes the array error matrix into two matrix parameters (i.e., mutual coupling matrix and channel gain/phase error matrix) to be optimized and all (nonlinear) numerical properties of the error matrix can be exploited. Therefore, algorithm II is able to achieve better calibration precision than algorithm I. However, algorithm II is more computationally demanding as a closed-form solution is no longer available and iteration computation is involved. In addition, the compact Cramer---Rao bound (CRB) expressions for all array error parameters are deduced in the case where auxiliary sources are assumed to be complex circular Gaussian distributed. Finally, the two novel algorithms are appropriately extended to the scenario where non-circular auxiliary sources are used, and the estimation variances of the array error parameters can be further decreased if the non-circularity is properly employed. Simulation experiments show the superiority of the presented algorithms.

This paper proposes a new matrix shrinkage algorithm for matrix rank minimization problems. The proposed algorithm provides a low rank solution by estimating a matrix rank and shrinking non-dominant singular values iteratively. We study the convergence properties of the algorithm, which indicate that the algorithm gives approximate low-rank solutions. Numerical results show that the proposed algorithm works efficiently for hard problems with low computing time.

In this paper we present a novel algorithm, that will never fail, for inverting a general nonsingular k -tridiagonal matrix. The computational cost of the algorithm is given. Some illustrative examples are introduced.

This paper presents a new iterative algorithm as an upgrade to sufficient LMI conditions for the H∞ static-output-feedback (SOF) control of discrete-time systems. Based on an analysis of the structures of the coordinate transformation matrix and the Lyapunov matrix, the open question of how to fix the Lyapunov matrix structure raised by G. I. Bara and M. Boutayeb is replaced with the question of how to choose the coordinate transformation matrix. Then, an iterative algorithm for selecting the optimum coordinate transformation matrix that produces a locally optimal solution is presented. Finally, numerical examples demonstrate the effectiveness and advantages of this method.

The paper applies the matrix calculation in source entropy calculation and propose the algorithm implementation, which can decrease the calculation cost and increase the accuracy significantly significantly.

Deterministic sensing matrices are useful, because in practice, the sampler has to be a deterministic matrix. It is quite challenging to design a deterministic sensing matrix with low coherence. In this paper, we consider a more general condition, when the deterministic sensing matrix has high coherence and does not satisfy the restricted isometry property (RIP). A novel algorithm, called the similar sensing matrix pursuit (SSMP), is proposed to reconstruct a K-sparse signal, based on the original deterministic sensing matrix. The proposed algorithm consists of off-line and online processing. The goal of the off-line processing is to construct a similar compact sensing matrix containing as much information as possible from the original sensing matrix. The similar compact sensing matrix has low coherence, which guarantees a perfect reconstruction of the sparse vector with high probability. The online processing begins when measurements arrive, and consists of rough and refined estimation processes. Results from our simulation show that the proposed algorithm obtains much better performance while coping with a deterministic sensing matrix with high coherence compared with the subspace pursuit (SP) and basis pursuit (BP) algorithms.

The problem of automatic robust estimation of the epipolar geometry for wide-baseline image pair is difficult because the putative correspondences include a low percentage of inlier correspondences, and it could become a severe problem when the veridical data are themselves degenerate or near-degenerate. In this paper, Clustering Pairing Consensus (CPC) algorithm is proposed to estimate the fundamental matrix. The CPC algorithm first produces the Matched Regions Clusters (MRCs) using topological clustering (TC) algorithm given a scale parameter. An estimation is produced from each valid pair of MRCs and is then provided to M-estimation to compute a fundamental matrix. Finally, the best one is chosen as the final model from all the estimation. The proposed CPC algorithm has been demonstrated to be able to effectively estimate fundamental matrix and avoid the degeneracy of the traditional method for some difficult image pairs.

Matrix completion is the extension of compressed sensing. In compressed sensing, we solve the underdetermined equations using sparsity prior of the unknown signals. However, in matrix completion, we solve the underdetermined equations based on sparsity prior in singular values set of the unknown matrix, which also calls low-rank prior of the unknown matrix. This paper firstly introduces basic concept of matrix completion, analyses the matrix suitably used in matrix completion, and shows that such matrix should satisfy two conditions: low rank and incoherence property. Then the paper provides three reconstruction algorithms commonly used in matrix completion: singular value thresholding algorithm, singular value projection, and atomic decomposition for minimum rank approximation, puts forward their shortcoming to know the rank of original matrix. The Projected Gradient Descent based on Soft Thresholding (STPGD), proposed in this paper predicts the rank of unknown matrix using soft thresholding, and iteratives based on projected gradient descent, thus it could estimate the rank of unknown matrix exactly with low computational complexity, this is verified by numerical experiments. We also analyze the convergence and computational complexity of the STPGD algorithm, point out this algorithm is guaranteed to converge, and analyse the number of iterations needed to reach reconstruction error. Compared the computational complexity of the STPGD algorithm to other algorithms, we draw the conclusion that the STPGD algorithm not only reduces the computational complexity, but also improves the precision of the reconstruction solution.

We describe a fraction free version of the Matrix Berlekamp/Massey algorithm. The algorithm computes a minimal matrix generator of linearly generated square matrix sequences in an integral domain. The algorithm performs all operations in the integral domain, so all divisions performed are exact. For scalar sequences, the matrix algorithm specializes to a different algorithm than the algorithm currently in the literature. This new scalar algorithm has smaller intermediate values than the known fraction free Berlekamp/Massey algorithm.

In this work we generalize the Quasi-Minimal Residual Variant of the Bi-CGSTAB algorithm (QMRCGSTAB) to obtain a matrix iterative method for solving the Sylvester-transpose matrix equation ∑ i = 1 l ( A i X B i + C i X T D i ) = E . In order to compare the generalized algorithm with some existing methods, we present two numerical examples.

We analyze the Matrix Berlekamp/Massey algorithm, which generalizes the Berlekamp/Massey algorithm [Massey 1969] for computing linear generators of scalar sequences. The Matrix Berlekamp/Massey algorithm computes a minimal matrix generator of a linearly generated matrix sequence and has been first introduced by Rissanen [1972a], Dickinson et al. [1974], and Coppersmith [1994]. Our version of the algorithm makes no restrictions on the rank and dimensions of the matrix sequence. We also give new proofs of correctness and complexity for the algorithm, which is based on self-contained loop invariants and includes an explicit termination criterion for a given determinantal degree bound of the minimal matrix generator.

In this paper, firstly we try to look for ways to avoid the camera parameters in order to reconstruct 3D model. We attempt to use the parallel stereo visual system and carry out the mathematical derivation of argumentation. Then we use epipolar geometry to solve this problem. And compare the computation algorithms of fundamental matrix. Then for the algorithm, we propose some improvement to compute the fundament matrix more precisely so that the algorithm is more stable and the robustness is stronger.

A camera self-calibration algorithm based on the Kruppas equation has been put forward by decomposing the fundamental matrix on the basis of computing the geometrical static moment. The fundamental matrix could be estimated through the normalized 8 points algorithm in which the centroid of the irregular polygon was calculated by computing the geometrical static moment. The five intrinsic parameters of a camera were calculated in accordance with the optimal target function established from the simplified Kruppas equation and the linear equation about the five internal parameters of a camera. The experiment showed that the five intrinsic parameters of a camera could be realized by this method.