Search

Here, a novel image‐based quantized compressive sensing (QCS) framework based on complex‐valued Vandermonde (Vander) matrix is proposed. In the proposed QCS framework, a discrete wavelet transform (DWT) serves as a sparse basis and a partial complex‐valued Vander matrix serves as a measurement matrix. The theoretical analysis based on mutual coherence metric of compressive sensing (CS) theory shows that the proposed Vander measurement matrix has the best reconstruction performance among other conventional measurement matrices. The simulation results also show that the recovery quality using the proposed measurement matrix can be greatly improved compared with the other existing real‐valued measurement matrices. In particular, the experiment results also show that under the same measurement matrix, the reconstruction performance of Smoothed l0 norm (SL0) algorithm is better than that of Orthogonal Matching Pursuit (OMP) algorithm, sparsity adaptive matching pursuit (SAMP) algorithm and approximate message passing (AMP) algorithm. In addition, a sparse measurement matrix scheme is further proposed to achieve a trade‐off between recovery performance and computational complexity. The theoretical analysis and simulation results both show the proposed image‐based QCS is efficient. [ABSTRACT FROM AUTHOR]

The main research object of this paper is to present a systematic computational procedure for computing the inverse of a generalized Vandermonde matrix. Short and rigorous proofs for the formulas of the determinant and the inverse of a generalized Vandermonde matrix are proposed. The computational cost of this method is O (n 2) . The proposed method can be used efficiently for hand calculation as well as for computer programming. Some examples are given for the sake of illustration. Furthermore, we present a simulation study to compare the time spent to calculate the inverse using the proposed algorithm and the inverse function in Maple. [ABSTRACT FROM AUTHOR]

The spatial smoothing-based and matrix reconstruction-based decorrelation algorithms are widely used to realize the accurate direction-of-arrival (DOA) estimation with the coexistence of both uncorrelated and coherent sources. However, these methods apply only to the traditional uniform linear array (ULA), of which the manifold matrix is a Vandermonde structure. In this paper, to achieve the DOA estimation of mixed signals on coprime array, the coprime array is decomposed into two independent sub-ULAs, and the received data is processed, respectively. An improved spatial differencing technique is exploited to estimate the uncorrelated and coherent sources separately, and the lost degree of freedom (DOF) caused by array decomposition is recovered. The final high-precision DOA estimation is obtained by conducting an ambiguity resolution method to the estimation results of the decomposed subarrays. Furthermore, double-phase partial spectral search scheme is introduced to diminish the computational complexity. The feasibility and superiority of the proposed algorithm are validated through detailed experimental simulations. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Phased-array multibeam RF beamformers require calibration of receivers used in an array of elements before the signals can be applied to an analog beamforming network. This article presents a fast O(n2) algorithm for solving linear systems having a delay Vandermonde matrix (DVM) for the analog beamforming matrix. The structure of the DVM enables to obtain a fast algorithm to efficiently reverse multibeams at the DVM output such that calibration of the input low noise amplifiers can be achieved. The arithmetic complexity of the proposed DVM system solving algorithm is preferable over the standard matrix inversion consuming O(n3) operations. Numerical experiments are presented for forward accuracy of the proposed algorithm computed at the calibration frequency. Moreover, numerical results are shown to compare the order of the arithmetic complexity and the execution time of the proposed algorithm. Signal flow graphs are presented for four- and eight-element multibeam phased arrays. [ABSTRACT FROM AUTHOR]

Multiple images steganography refers to hiding secret messages in multiple natural images to minimize the leakage of secret messages during transmission. Currently, the main multiple images steganography algorithms mainly distribute the payloads as sparsely as possible inmultiple cover images to improve the detection error rate of stego images. In order to enable the payloads to be accurately and efficiently distributed in each cover image, this paper proposes a multiple images steganography for JPEG images based on optimal payload redistribution. Firstly, the algorithm uses the principle of dynamic programming to redistribute the payloads of the cover images to reduce the time required in the process of payloads distribution. Then, by reducing the difference between the features of the cover images and the stego images to increase the detection error rate of the stego images. Secondly, this paper uses a data decomposition mechanism based on Vandermonde matrix. Even if part of the data is lost during the transmission of the secret messages, as long as the data loss rate is less than the data redundancy rate, the original secret messages can be recovered. Experimental results show that the method proposed in this paper improves the efficiency of payloads distribution compared with existing multiple images steganography. At the same time, the algorithm can achieve the optimal payload distribution of multiple images steganography to improve the anti-statistical detection performance of stego images. [ABSTRACT FROM AUTHOR]

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its ${M}+1$ consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is $1/{M}$ and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than $1/{M}$ apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like $\text {SRF}^{2\lambda -2} \times \text {Noise}$ , where the Super-Resolution Factor (SRF) governs the difficulty of the problem and $\lambda $ is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of $M$ in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Signal Processing is the property of Journal of Signal Processing and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Linear prediction methods, based on a Hankel data matrix, suffer from subspace leakage and degraded resolution when applied to data models that do not result in a mode matrix with Vandermonde structure, such as the constant- Q model. In the absence of noise, the Vandermonde structure ensures the equivalence between the number of backscattered signals and the rank of the data matrix. This paper first identifies the origin of subspace leakage residing in subspace-based and linear prediction methods when applied to data of the constant- Q model. Second, it proposes a frequency-distortion technique, based on the extension theorems, for suppressing the leakage and preserving the time resolution performance of these methods. The effectiveness of the distortion technique is then demonstrated on GPR simulated data by extending the damped MUSIC algorithm to the joint parameter estimation of the constant- Q model. [ABSTRACT FROM AUTHOR]

Direction of arrival (DOA) estimation problem for multiple-input multiple-output (MIMO) radar with unknown mutual coupling is studied, and an algorithm for the DOA estimation based on root multiple signal classification (MUSIC) is proposed. Firstly, according to the Toeplitz structure of the mutual coupling matrix, output data of some specified sensors are selected to eliminate the influence of the mutual coupling. Then the reduced-dimension transformation is applied to make the computation burden lower as well as obtain a Vandermonde structure of the direction matrix. Finally, Root-MUSIC can be adopted for the angle estimation. The angle estimation performance of the proposed algorithm is better than that of estimation of signal parameters via rotational invariance techniques (ESPRIT)-like algorithm and MUSIC-like algorithm. Furthermore, the proposed algorithm has lower complexity than them. The simulation results verify the effectiveness of the algorithm, and the theoretical estimation error of the algorithm is also derived. [ABSTRACT FROM AUTHOR]

Abstract: In the article, we propose a recursive algorithm for calculating the inversion of the confluent Vandermonde matrix. Besides, contrary to other works in this field, the article derives an explicit analytic formula for the calculation of the inverse of the confluent Vandermonde matrix. The results of this article do not require any symbolic calculations and therefore can be performed by a numerical algorithm implemented in any general-purpose programming language. Moreover, a ready-to-use C++ full implementation of the recursive algorithm is attached. [Copyright &y& Elsevier]

In a recent paper [3] we have proposed a deterministic stable sparse FFT algorithm for M‐sparse vectors of length N = 2J with runtime 𝒪(M2 log2N) which generalizes the approach in [2]. Our method is based on a divide‐and‐conquer technique and requires to solve an equation system of Vandermonde type at each iteration step. To ensure numerical stability of the algorithm, the number of rows of the employed Vandermonde matrices at each step has been chosen adaptively in [3]. In this paper, we compare the results of [3] with a random sampling approach for determining the rows of the coefficient matrix. [ABSTRACT FROM AUTHOR]

Linear systems of equations Ax=b, where the matrix A has some particular structure, arise frequently in applications. Very often, structured matrices have huge condition numbers and, therefore, standard algorithms fail to compute accurate solutions of Ax=b. We say in this paper that a computed solution is accurate if being the unit roundoff. In this work we introduce a framework that allows many classes of structured linear systems to be solved accurately, independently of the condition number of A and efficiently, that is, with cost For most of these classes no algorithms are known that are both accurate and efficient. The approach in this work relies on first computing an accurate rank-revealing decomposition of A, an idea that has been widely used in the last decades to compute singular value and eigenvalue decompositions of structured matrices with high relative accuracy. In particular, we illustrate the new method by accurately solving Cauchy and Vandermonde linear systems with any distribution of nodes, that is, without requiring A to be totally positive for most right-hand sides b. [ABSTRACT FROM PUBLISHER]

We survey some results related to classical secret sharing schemes defined in Shamir [10] and Blakley [1], and developed in Brickell [2] and Lai and Ding [4]. Using elementary symmetric polynomials, we describe in a unified way which allocations of identities to participants define Shamir's threshold scheme, or its generalization by Lai and Ding, with a secret placed as a fixed coefficient of the scheme polynomial. This characterization enabled proving in Schinzel et al. [8], [9] and Spież et al. [13] some new and non-trivial properties of such schemes. Also a characterization of matrices corresponding to the threshold secret sharing schemes of Blakley and Brickell's type is given. Using Gaussian elimination we provide an algorithm to construct all such matrices which is efficient in the case of relatively small matrices. The algorithm may be useful in constructing systems where dynamics is important (one may generate new identities using it). It can also be used to construct all possible MDS codes. MSC: primary 94A62; secondary 11T71; 11C20 [ABSTRACT FROM AUTHOR]

Vandermonde matrices with real nodes are known to be severely ill-conditioned. We investigate numerically the extent to which the condition number of such matrices can be reduced, either by row-scaling or by optimal configurations of nodes. In the latter case we find empirically the condition of the optimally conditioned n× n Vandermonde matrix to grow exponentially at a rate slightly less than $(1+\sqrt{2})^{n}$. Much slower growth-essentially linear-is observed for optimally conditioned Vandermonde-Jacobi matrices. We also comment on the computational challenges involved in determining condition numbers of highly ill-conditioned matrices. [ABSTRACT FROM AUTHOR]

Abstract: We propose a numerical method (implemented in Matlab) for computing approximate Fekete points on compact multivariate domains. It relies on the search of maximum volume submatrices of Vandermonde matrices computed on suitable discretization meshes, and uses a simple greedy algorithm based on QR factorization with column pivoting. The method gives also automatically an algebraic cubature formula, provided that the moments of the underlying polynomial basis are known. Numerical tests are presented for the interval and the square, which show that approximate Fekete points are well suited for polynomial interpolation and cubature. [Copyright &y& Elsevier]

A fast O(n2) algorithm is derived for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix VR(x) = [rj-1(xi)] with polynomials {rk(x)} defined by a Hessenberg matrix with quasiseparable structure. The result generalizes the well-known Björck- Pereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of Reichel-Opfer for VR(x) involving Chebyshev polynomials of Higham for VR(x) involving real orthogonal polynomials, and a recent algorithm of the authors for VR(x) involving Szegö polynomials. The new algorithm applies to a fairly general new class of (H, k)-quasiseparable polynomials (Hessenberg, order k quasiseparable) that includes (along with the above mentioned classes of real orthogonal and Szegö polynomials) several other important classes of polynomials, e.g., defined by banded Hessenberg matrices. Numerical experiments are presented that coincide with previous experiences with Björck-Pereyra-type algorithms giving better forward error than Gaussian elimination, and this accuracy is consistent with the so-called Chan-Foulser conditioning of the system. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we carry over the Björck-Pereyra algorithm for solving Vandermonde linear systems to what we suggest to call Szegö-Vandermonde systems V Φ(x), i.e., polynomial-Vandermonde systems where the corresponding polynomial system Φ is the Szegö polynomials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n 2) computational procedure. We present numerical experiments that indicate that for ill-conditioned matrices the new algorithm yields better forward accuracy than Gaussian elimination. [Copyright &y& Elsevier]

Abstract: This paper deals with Vandermonde matrices V whose nodes are the equidistant points in [0,1]. We give an analytic factorization and explicit formula for the entries of their inverse, and explore its computational issues. We also give asymptotic estimates of the Frobenius norm of both V and its inverse and show that a new representation of the floating point number system allows one to build an accurate algorithm for the interpolation problem on equidistant nodes in [0,1]. [Copyright &y& Elsevier]

The known deterministic algorithms for loss-resilient encoding/decoding involve computations with Cauchy matrices but only weakly exploit the matrix structure. We propose several modifications with more extensive use of the matrix structure to accelerate the computations substantially. [Copyright &y& Elsevier]

This paper is concerned with the inversion of confluent Vandermonde matrices. A novel and simple recursive algorithm for inverting confluent Vandermonde matrices is presented. The algorithm is suitable for classroom use in both numerical as well as symbolic computation. Examples are included to illustrate the proposed algorithm. [Copyright &y& Elsevier]

In this paper, we first introduce a new concept on matrices which we call annihilator vectors. Then using the shifted form of these vectors, we propose new algorithms for (1) computing the inverse of a Vandermonde matrix, (2) solving some Vandermonde systems of linear equations, both algorithms in O (n 2) arithmetic operations. Our methods are based on a new technique which lead to novel algorithms, need less storage and, from experimental results, are more efficient than the previous usual methods. Finally, comparisons with other methods are broached and some numerical examples are supplied. [ABSTRACT FROM AUTHOR]