The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix ...An of order n is computed by means of their first two and last two columns or rows under the additional assumption that a certain 2×2 matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel matrix An−2 of order n−2 in the center of An is established and sufficient invertibility criteria for both An and An−2 are obtained. Hereby a main tool is to use known inversion formulas involving not only rows or columns of the inverse but also solutions of equations the right hand sides of which depend on entries of the Toeplitz-plus-Hankel matrix itself.
The text may contain information such as personal privacy and national security, and it is vital to protect these sensitive information. The traditional text encryption methods only encrypt text into ...the garbled code. Although the encrypted text appears to be garbled, there may be text length exposure and semantic preservation. Attackers can use these information leakages to analyze and attack. Therefore, in order to confuse the attackers, we propose a technique to encrypt the text into the image in which it can hide the information that may be exposed in the ciphertext such as text length, language structure, frequency distribution, etc. It mainly uses coordinate substitution encryption and image encryption to encrypt text into image. First, the text is encrypted using coordinate substitution to initially disrupt the correlation between characters. Second, the encrypted text is converted to ASCII values and presented as the image. It is then multiplied with the Hankel matrix to adjust its elements values close to the true range of pixel values. Finally, the two-dimensional polynomial chaotic system and the image encryption algorithm are combined to obfuscate and diffusion encrypt images. It is experimentally verified that the scheme has good security and robustness for protecting text data.
Purpose
Quantitative susceptibility mapping (QSM) inevitably suffers from streaking artifacts caused by zeros on the conical surface of the dipole kernel in k‐space. This work proposes a novel and ...accurate QSM reconstruction method based on k‐space low‐rank Hankel matrix constraint, avoiding the over‐smoothing problem and streaking artifacts.
Theory and Methods
Based on the recent theory of annihilating filter‐based low‐rank Hankel matrix approach (ALOHA), QSM is formulated as deconvolution under low‐rank Hankel matrix constraint in the k‐space. The computational complexity and the high memory burden were reduced by successive reconstruction of 2‐D planes along 3 independent axes of the 3‐D phase image in Fourier domain. Feasibility of the proposed method was tested on a simulated phantom and human data and were compared with existing QSM reconstruction methods.
Results
The proposed ALOHA‐QSM effectively reduced streaking artifacts and accurately estimated susceptibility values in deep gray matter structures, compared to the existing QSM methods.
Conclusions
The suggested ALOHA‐QSM algorithm successfully solves the 3‐dimensional QSM dipole inversion problem using k‐space low rank property with no anatomical constraint. ALOHA‐QSM can provide detailed brain structures and accurate susceptibility values with no streaking artifacts.
This paper studies the robust matrix completion (RMC) problem with the objective to recover a low-rank matrix from partial observations that may contain significant errors. If all the observations in ...one column are erroneous, existing RMC methods can locate the corrupted column at best but cannot recover the actual data in that column. Low-rank Hankel matrices characterize the additional correlations among columns besides the low-rankness and exist in power system monitoring, magnetic resonance imaging (MRI) imaging, and array signal processing. Exploiting the low-rank Hankel property, this paper develops an alternating-projection-based fast algorithm to solve the nonconvex RMC problem. The algorithm converges to the ground-truth low-rank matrix with a linear rate even when all the measurements in a constant fraction of columns are corrupted. The required number of observations is significantly less than the existing bounds for the conventional RMC. Numerical results are reported to evaluate the proposed algorithm.
The indeterminate Hamburger moment problem is considered, jointly with all its real axis supported probability density functions. As a consequence of entropy functional concavity, out of such ...densities there is one which has largest entropy and that plays a fundamental role: we call it fhmax. It is proved that the approximate Maximum Entropy (MaxEnt) densities constrained by an increasing number of moments converge in entropy to fhmax where the value of its entropy can be finite or −∞.
Let Hμ be the Hankel matrix with entries μn,k=∫0,1)(n+1)tn+kdμ(t), where μ is a positive Borel measure on the interval 0,1). The matrix acts on the space of all analytic functions in the unit disk by ...multiplication on Taylor coefficients and induces formally the operatorDHμ(f)(z)=∑n=0∞(∑k=0∞μn,kak)zn, where f(z)=∑n=0∞anzn is an analytic function in D. In this paper, we characterize the measures μ for which DHμ is a bounded (resp., compact) operator from the Bergman space Ap (0<p<∞) into the space Aq (q≥p and q>1), or from Ap (0<p≤1) into A1.
Recently, coprime arrays have attracted lots of interest due to their ability of providing enhanced degrees-of-freedom and reduced mutual coupling effect compared to conventional uniform linear ...arrays. Benefitting from these excellent properties, coprime multiple-input multiple-output (MIMO) radar has been recently suggested for improving parameter identifiability and target detection. In this paper, we address the problem of joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation of coherent targets in coprime MIMO radar. First of all, we establish an extended virtual uniform rectangular array (URA) by performing array interpolation on the transmit and receive arrays of coprime MIMO radar. Subsequently, we derive a low-rank block Hankel matrix that is constructed using the correlation information of the virtual URA expected outputs, and utilize the matched filter outputs to recover the block Hankel matrix by solving a low-rank structured matrix completion problem. Finally, we estimate the DODs and DOAs of coherent targets by applying the modified matrix pencil method on the recovered low-rank matrix. We also derive the Cramér-Rao bounds with closed-form expressions for DOD and DOA estimation of coherent targets using coprime MIMO radar. The proposed algorithm can identify multiple coherent targets and achieve parameter automatic pairing. Numerical results demonstrate the superiority of the proposed algorithm over several existing approaches when handling coherent targets.
As for noise reduction adopting singular value decomposition (SVD) algorithm, it is important that how to choose the number of effective singular values. This paper improves SVD algorithm. The ...improved algorithm can automatically calculate the number of effective singular values of the signal with noise, avoiding the misjudgment caused by artificial selection of the number of effective singular values. The relationship between effective singular values of a signal and the dimension of Hankel matrix is confirmed. The relationship of the singular values between the signal with noise, the original signal and the noise signal, as well as the relationship between the magnitude of singular values and the signal energy, are clarified. A hyperparameter ACC to control two data obtained by processing singular values is proposed, so that the number of effective singular values can be identified more accurately. Finally, this algorithm is adopted to process noise successfully in vibration signal of a rotor.
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