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.
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.
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.
A software package is presented that computes locally optimal solutions to low-rank approximation problems with the following features: •mosaic Hankel structure constraint on the approximating ...matrix,•weighted 2-norm approximation criterion,•fixed elements in the approximating matrix,•missing elements in the data matrix, and•linear constraints on an approximating matrix’s left kernel basis. It implements a variable projection type algorithm and allows the user to choose standard local optimization methods for the solution of the parameter optimization problem. For an m×n data matrix, with n>m, the computational complexity of the cost function and derivative evaluation is O(m2n). The package is suitable for applications with n≫m. In statistical estimation and data modeling–the main application areas of the package–n≫m corresponds to modeling of large amount of data by a low-complexity model. Performance results on benchmark system identification problems from the database DAISY and approximate common divisor problems are presented.
An n-by-n real symmetric matrix
$ H=h_{i,j} $
H
=
h
i
,
j
is said to be a Hankel matrix if
$ h_{i,j}=h_{i-1,j+1} $
h
i
,
j
=
h
i
−
1
,
j
+
1
, for each
$ i=2,\ldots,n $
i
=
2
,
...
,
n
and
$ ...j=1,\ldots,n-1 $
j
=
1
,
...
,
n
−
1
. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list
$ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $
Λ
=
{
λ
1
,
...
,
λ
n
}
of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list
$ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $
Λ
=
{
λ
1
,
...
,
λ
n
}
for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if
$ \sum _{i=1}^{n}\lambda _{i}=0 $
∑
i
=
1
n
λ
i
=
0
, we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix.
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.
A spectrally sparse signal of order r is a mixture of r damped or undamped complex sinusoids. This paper investigates the problem of reconstructing spectrally sparse signals from a random subset of n ...regular time domain samples, which can be reformulated as a low rank Hankel matrix completion problem. We introduce an iterative hard thresholding (IHT) algorithm and a fast iterative hard thresholding (FIHT) algorithm for efficient reconstruction of spectrally sparse signals via low rank Hankel matrix completion. Theoretical recovery guarantees have been established for FIHT, showing that O(r2log2(n)) number of samples are sufficient for exact recovery with high probability. Empirical performance comparisons establish significant computational advantages for IHT and FIHT. In particular, numerical simulations on 3D arrays demonstrate the capability of FIHT on handling large and high-dimensional real data.
Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology, and medical imaging. For fast data acquisition or other inevitable reasons, however, ...only a small amount of samples may be acquired, and thus, how to recover the full signal becomes an active research topic, but existing approaches cannot efficiently recover N-dimensional exponential signals with N ≥ 3. In this paper, we study the problem of recovering N-dimensional (particularly N ≥ 3) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC tensor structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.
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