The Experts below are selected from a list of 405 Experts worldwide ranked by ideXlab platform
Wenjing Liao - One of the best experts on this subject based on the ideXlab platform.
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music for single snapshot spectral estimation stability and super resolution
Applied and Computational Harmonic Analysis, 2016Co-Authors: Wenjing Liao, Albert FannjiangAbstract:Abstract This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.
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music for single snapshot spectral estimation stability and super resolution
arXiv: Information Theory, 2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho.
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MUSIC for single-snapshot spectral estimation: Stability and super-resolution,” arXiv preprint arXiv:1404.1484
2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh length, the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies sep-arated above 4RL, MUSIC becomes the best performing one for frequencies separated between 2RL and 3RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to one RL and below when all other methods fail. Indeed, the resolution of MUSIC apparently decreases to zero as noise decreases to zero
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MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution
2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh length, the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies sep-arated above 4RL, MUSIC becomes the best performing one for frequencies separated between 2RL and 3RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to one RL and below when all other methods fail. Indeed, the resolution of MUSIC apparently decreases to zero as noise decreases to zero
Albert Fannjiang - One of the best experts on this subject based on the ideXlab platform.
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music for single snapshot spectral estimation stability and super resolution
Applied and Computational Harmonic Analysis, 2016Co-Authors: Wenjing Liao, Albert FannjiangAbstract:Abstract This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.
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music for single snapshot spectral estimation stability and super resolution
arXiv: Information Theory, 2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho.
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MUSIC for single-snapshot spectral estimation: Stability and super-resolution,” arXiv preprint arXiv:1404.1484
2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh length, the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies sep-arated above 4RL, MUSIC becomes the best performing one for frequencies separated between 2RL and 3RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to one RL and below when all other methods fail. Indeed, the resolution of MUSIC apparently decreases to zero as noise decreases to zero
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MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution
2014Co-Authors: Wenjing Liao, Albert FannjiangAbstract:This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (Nonzero) Singular Value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh length, the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest Nonzero Singular Values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies sep-arated above 4RL, MUSIC becomes the best performing one for frequencies separated between 2RL and 3RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to one RL and below when all other methods fail. Indeed, the resolution of MUSIC apparently decreases to zero as noise decreases to zero
Lynn Mcindoo - One of the best experts on this subject based on the ideXlab platform.
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The Singular Value Decomposition
1998Co-Authors: Jody Hourigan, Lynn McindooAbstract:. We explore the derivation of the SVD and its role in digital image processing. By using MATLAB, we will demonstrate how the SVD is used to minimize the size needed to store an image. Introduction The Singular Value decomposition is a highlight of linear algebra. It plays an interesting, fundamental role in many different applications, namely in digital image processing. The beauty of the SVD within its digital applications is that it provides a robust method of storing large images as smaller, more manageable square ones. This is accomplished by reproducing the original image with each succeeding Nonzero Singular Value. Furthermore, to reduce storage size even further, one may approximate a "good enough" image with using even fewer Singular Values. What is the Singular Value Decomposition? The Singular Value decomposition of a matrix factors an m x n matrix A into the form A = UFV T (1) where U is an m x m orthogonal matrix; V an n x n orthogonal matrix, and F an m x n matrix ..
Jody Hourigan - One of the best experts on this subject based on the ideXlab platform.
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The Singular Value Decomposition
1998Co-Authors: Jody Hourigan, Lynn McindooAbstract:. We explore the derivation of the SVD and its role in digital image processing. By using MATLAB, we will demonstrate how the SVD is used to minimize the size needed to store an image. Introduction The Singular Value decomposition is a highlight of linear algebra. It plays an interesting, fundamental role in many different applications, namely in digital image processing. The beauty of the SVD within its digital applications is that it provides a robust method of storing large images as smaller, more manageable square ones. This is accomplished by reproducing the original image with each succeeding Nonzero Singular Value. Furthermore, to reduce storage size even further, one may approximate a "good enough" image with using even fewer Singular Values. What is the Singular Value Decomposition? The Singular Value decomposition of a matrix factors an m x n matrix A into the form A = UFV T (1) where U is an m x m orthogonal matrix; V an n x n orthogonal matrix, and F an m x n matrix ..
Xu Shu-fang - One of the best experts on this subject based on the ideXlab platform.
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An eigenspace method for computing derivatives of semi-simple eigenValues and corresponding eigenvectors of quadratic eigenValue problems
applied numerical mathematics, 2015Co-Authors: Lu Xin, Xu Shu-fangAbstract:This paper concerns computing derivatives of semi-simple eigenValues and corresponding eigenvectors of the quadratic matrix polynomial Q(p,??)=??2M(p)+??C(p)+K(p) at p=p?. Computing derivatives of eigenvectors usually requires solving a certain Singular linear system by transforming it into a nonSingular one. However, the coefficient matrix of the transformed linear system might be ill-conditioned. In this paper, we propose a new method for computing these derivatives, where the condition number of the coefficient matrix is the ratio of the maximum Singular Value to the minimum Nonzero Singular Value of Q(p?,??(p?)), which is generally smaller than those in current literature and hence leads to higher accuracy. Numerical examples show the feasibility and efficiency of our method. ? 2015 Published by Elsevier B.V. on behalf of IMACS.SCI(E)EI0ARTICLEinbelief@gmail.com; xsf@pku.edu.cn201-2089
