The Experts below are selected from a list of 207 Experts worldwide ranked by ideXlab platform
Huan Li - One of the best experts on this subject based on the ideXlab platform.
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optimized projections for compressed sensing via direct Mutual Coherence minimization
Signal Processing, 2018Co-Authors: Canyi Lu, Huan LiAbstract:Abstract Compressed Sensing (CS) is a new data acquisition theory based on the existence of a sparse representation of a signal and a projected dictionary PD , where P ∈ R m × d is the projection matrix and D ∈ R d × n is the dictionary. To recover the signal from a small number m of measurements, it is expected that the projected dictionary PD is of low Mutual Coherence. Several previous methods attempt to find the projection P such that the Mutual Coherence of PD is low. However, they do not minimize the Mutual Coherence directly and thus they may be far from optimal. Their used solvers lack convergence guarantee and thus the quality of their solutions is not guaranteed. This work aims to address these issues. We propose to find an optimal projection matrix by minimizing the Mutual Coherence of PD directly. This leads to a nonconvex nonsmooth minimization problem. We approximate it by smoothing, solve it by alternating minimization and prove the convergence of our algorithm. To the best of our knowledge, this is the first work which directly minimizes the Mutual Coherence of the projected dictionary and has convergence guarantee. Numerical experiments demonstrate that our method can recover sparse signals better than existing ones.
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optimized projections for compressed sensing via direct Mutual Coherence minimization
arXiv: Information Theory, 2015Co-Authors: Canyi Lu, Huan LiAbstract:Compressed Sensing (CS) is a novel technique for simultaneous signal sampling and compression based on the existence of a sparse representation of signal and a projected dictionary $PD$, where $P\in\mathbb{R}^{m\times d}$ is the projection matrix and $D\in\mathbb{R}^{d\times n}$ is the dictionary. To exactly recover the signal with a small number of measurements $m$, the projected dictionary $PD$ is expected to be of low Mutual Coherence. Several previous methods attempt to find the projection $P$ such that the Mutual Coherence of $PD$ can be as low as possible. However, they do not minimize the Mutual Coherence directly and thus their methods are far from optimal. Also the solvers they used lack of the convergence guarantee and thus there has no guarantee on the quality of their obtained solutions. This work aims to address these issues. We propose to find an optimal projection by minimizing the Mutual Coherence of $PD$ directly. This leads to a nonconvex nonsmooth minimization problem. We then approximate it by smoothing and solve it by alternate minimization. We further prove the convergence of our algorithm. To the best of our knowledge, this is the first work which directly minimizes the Mutual Coherence of the projected dictionary with a convergence guarantee. Numerical experiments demonstrate that the proposed method can recover sparse signals better than existing methods.
E Heyman - One of the best experts on this subject based on the ideXlab platform.
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modal theory for the two frequency Mutual Coherence function in random media beam waves
Waves in Random Media, 1998Co-Authors: Jasmin Oz, E HeymanAbstract:Abstract Pulse propagation in a random medium is mainly determined by the two-frequency Mutual Coherence function which satisfies the parabolic equation. It has recently been shown that this equation can be solved by separation of variables, thereby reducing the solution for any structure function to the solution of ordinary differential equations. In this paper, the method is applied for a beam-wave excitation in a random medium. The exact solution for a quadratic medium is derived. For non-quadratic power-law media an analytical expression at equal positions is presented.
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modal theory for the two frequency Mutual Coherence function in random media general theory and plane wave solution ii
Waves in Random Media, 1997Co-Authors: Jasmin Oz, E HeymanAbstract:Abstract In a previous publication (part I) it has been shown that for an arbitrary statistically isotropic and homogeneous medium the parabolic equation for the two-frequency Mutual Coherence function can be separated and thereby expressed as a superposition of modes. A parameterization based on the longitudinal part of this representation has also been treated. This paper explores the transverse structure and parameterization of the field solution by employing dimensional, variational and the modified WKB procedures for solving the eigenfunction/eigenvalue problem. General expressions are derived first for a general structure function and then specialized for a power-law structure function with emphasis on quadratic and Kolmogorov media.
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modal solution to the plane wave two frequency Mutual Coherence function in random media
Radio Science, 1996Co-Authors: Jasmin Oz, E HeymanAbstract:Pulse propagation in a random medium is mainly determined by the two-frequency Mutual Coherence function which satisfies the parabolic equation. It has been shown recently that this equation can be solved by separation of variables, thereby reducing the solution for any structure function into solutions of ordinary differential equations. Via a proper modal-expansion theorem, this representation may also be applied to any source problem. The modal approach also provides new physical interpretations for relevant physical parameters. This new solution approach is being reviewed here within the simplified framework of plane wave initial conditions. In particular, a general power law structure function is investigated, and the results are compared with the known exact solution for quadratic medium and a numerical solution for a Kolmogorov medium. Using the new modal approach, we present two alternative representations: a “mode series” and a “collective mode solution.” Both representations are suitable for extension into the time domain, giving a series of “wave front arrivals” and “collective resonance” contributions respectively.
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modal theory for the two frequency Mutual Coherence function in random media
IEEE Antennas and Propagation Society International Symposium, 1996Co-Authors: Jasmin Oz, E HeymanAbstract:Short pulse propagation in random media is mainly determined by the two-frequency Mutual Coherence function which is governed in the multiple scattering regime by a parabolic equation. In this paper the modal expansion theory is presented as a new analytical approach for media which are statistically isotropic and homogeneous. By performing a separation of variables, the problem of the 3D partial differential equation is reduced to solving a one-dimensional eigenvalue problem. The full expansion theorem is presented applicable for any initial source configuration. For media characterized by a quadratic structure function, the eigenvalue problem is exactly solvable. The two-frequency Coherence function is obtained as a modal series for the three most important source configurations, namely the plane wave, the point source and the beam wave. By Poisson's theorem, the series is summed up into a closed form expression and is shown to yield the known solutions in the literature. In this paper, we only present the general modal expansion theorem and the exact solution for a beam in a quadratic medium.
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Modal solution to the plane wave two‐frequency Mutual Coherence function in random media
Radio Science, 1996Co-Authors: Jasmin Oz, E HeymanAbstract:Pulse propagation in a random medium is mainly determined by the two-frequency Mutual Coherence function which satisfies the parabolic equation. It has been shown recently that this equation can be solved by separation of variables, thereby reducing the solution for any structure function into solutions of ordinary differential equations. Via a proper modal-expansion theorem, this representation may also be applied to any source problem. The modal approach also provides new physical interpretations for relevant physical parameters. This new solution approach is being reviewed here within the simplified framework of plane wave initial conditions. In particular, a general power law structure function is investigated, and the results are compared with the known exact solution for quadratic medium and a numerical solution for a Kolmogorov medium. Using the new modal approach, we present two alternative representations: a “mode series” and a “collective mode solution.” Both representations are suitable for extension into the time domain, giving a series of “wave front arrivals” and “collective resonance” contributions respectively.
Jasmin Oz - One of the best experts on this subject based on the ideXlab platform.
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modal theory for the two frequency Mutual Coherence function in random media beam waves
Waves in Random Media, 1998Co-Authors: Jasmin Oz, E HeymanAbstract:Abstract Pulse propagation in a random medium is mainly determined by the two-frequency Mutual Coherence function which satisfies the parabolic equation. It has recently been shown that this equation can be solved by separation of variables, thereby reducing the solution for any structure function to the solution of ordinary differential equations. In this paper, the method is applied for a beam-wave excitation in a random medium. The exact solution for a quadratic medium is derived. For non-quadratic power-law media an analytical expression at equal positions is presented.
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modal theory for the two frequency Mutual Coherence function in random media point source
Waves in Random Media, 1997Co-Authors: Jasmin OzAbstract:Abstract The recently introduced modal expansion representation for the two-frequency Mutual Coherence function is applied here to the solution of a point-source field in a random medium. This approach reduces the solution for any structure function to an eigenvalue problem for an ordinary differential equation. For the initial point source it is shown here that the modal expansion yields a result similar to that for the initial plane wave, modified by a spherical free-space phase which contains a weighted coordinate that does not interact with the medium. Having established these general characteristics, special attention is paid to power-law media and, in particular, to a quadratic medium, for which a new exact solution is derived. Via a collective summation of this new modal solution, we rederive the alternative exact solution which exists in the literature. We also discuss the new parameterization implied by the new modal solution.
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modal theory for the two frequency Mutual Coherence function in random media general theory and plane wave solution ii
Waves in Random Media, 1997Co-Authors: Jasmin Oz, E HeymanAbstract:Abstract In a previous publication (part I) it has been shown that for an arbitrary statistically isotropic and homogeneous medium the parabolic equation for the two-frequency Mutual Coherence function can be separated and thereby expressed as a superposition of modes. A parameterization based on the longitudinal part of this representation has also been treated. This paper explores the transverse structure and parameterization of the field solution by employing dimensional, variational and the modified WKB procedures for solving the eigenfunction/eigenvalue problem. General expressions are derived first for a general structure function and then specialized for a power-law structure function with emphasis on quadratic and Kolmogorov media.
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modal solution to the plane wave two frequency Mutual Coherence function in random media
Radio Science, 1996Co-Authors: Jasmin Oz, E HeymanAbstract:Pulse propagation in a random medium is mainly determined by the two-frequency Mutual Coherence function which satisfies the parabolic equation. It has been shown recently that this equation can be solved by separation of variables, thereby reducing the solution for any structure function into solutions of ordinary differential equations. Via a proper modal-expansion theorem, this representation may also be applied to any source problem. The modal approach also provides new physical interpretations for relevant physical parameters. This new solution approach is being reviewed here within the simplified framework of plane wave initial conditions. In particular, a general power law structure function is investigated, and the results are compared with the known exact solution for quadratic medium and a numerical solution for a Kolmogorov medium. Using the new modal approach, we present two alternative representations: a “mode series” and a “collective mode solution.” Both representations are suitable for extension into the time domain, giving a series of “wave front arrivals” and “collective resonance” contributions respectively.
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modal theory for the two frequency Mutual Coherence function in random media
IEEE Antennas and Propagation Society International Symposium, 1996Co-Authors: Jasmin Oz, E HeymanAbstract:Short pulse propagation in random media is mainly determined by the two-frequency Mutual Coherence function which is governed in the multiple scattering regime by a parabolic equation. In this paper the modal expansion theory is presented as a new analytical approach for media which are statistically isotropic and homogeneous. By performing a separation of variables, the problem of the 3D partial differential equation is reduced to solving a one-dimensional eigenvalue problem. The full expansion theorem is presented applicable for any initial source configuration. For media characterized by a quadratic structure function, the eigenvalue problem is exactly solvable. The two-frequency Coherence function is obtained as a modal series for the three most important source configurations, namely the plane wave, the point source and the beam wave. By Poisson's theorem, the series is summed up into a closed form expression and is shown to yield the known solutions in the literature. In this paper, we only present the general modal expansion theorem and the exact solution for a beam in a quadratic medium.
Canyi Lu - One of the best experts on this subject based on the ideXlab platform.
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optimized projections for compressed sensing via direct Mutual Coherence minimization
Signal Processing, 2018Co-Authors: Canyi Lu, Huan LiAbstract:Abstract Compressed Sensing (CS) is a new data acquisition theory based on the existence of a sparse representation of a signal and a projected dictionary PD , where P ∈ R m × d is the projection matrix and D ∈ R d × n is the dictionary. To recover the signal from a small number m of measurements, it is expected that the projected dictionary PD is of low Mutual Coherence. Several previous methods attempt to find the projection P such that the Mutual Coherence of PD is low. However, they do not minimize the Mutual Coherence directly and thus they may be far from optimal. Their used solvers lack convergence guarantee and thus the quality of their solutions is not guaranteed. This work aims to address these issues. We propose to find an optimal projection matrix by minimizing the Mutual Coherence of PD directly. This leads to a nonconvex nonsmooth minimization problem. We approximate it by smoothing, solve it by alternating minimization and prove the convergence of our algorithm. To the best of our knowledge, this is the first work which directly minimizes the Mutual Coherence of the projected dictionary and has convergence guarantee. Numerical experiments demonstrate that our method can recover sparse signals better than existing ones.
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optimized projections for compressed sensing via direct Mutual Coherence minimization
arXiv: Information Theory, 2015Co-Authors: Canyi Lu, Huan LiAbstract:Compressed Sensing (CS) is a novel technique for simultaneous signal sampling and compression based on the existence of a sparse representation of signal and a projected dictionary $PD$, where $P\in\mathbb{R}^{m\times d}$ is the projection matrix and $D\in\mathbb{R}^{d\times n}$ is the dictionary. To exactly recover the signal with a small number of measurements $m$, the projected dictionary $PD$ is expected to be of low Mutual Coherence. Several previous methods attempt to find the projection $P$ such that the Mutual Coherence of $PD$ can be as low as possible. However, they do not minimize the Mutual Coherence directly and thus their methods are far from optimal. Also the solvers they used lack of the convergence guarantee and thus there has no guarantee on the quality of their obtained solutions. This work aims to address these issues. We propose to find an optimal projection by minimizing the Mutual Coherence of $PD$ directly. This leads to a nonconvex nonsmooth minimization problem. We then approximate it by smoothing and solve it by alternate minimization. We further prove the convergence of our algorithm. To the best of our knowledge, this is the first work which directly minimizes the Mutual Coherence of the projected dictionary with a convergence guarantee. Numerical experiments demonstrate that the proposed method can recover sparse signals better than existing methods.
Jigen Peng - One of the best experts on this subject based on the ideXlab platform.
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Influences of preconditioning on the Mutual Coherence and the restricted isometry property of Gaussian/Bernoulli measurement matrices
Linear & Multilinear Algebra, 2020Co-Authors: Yingtong Chen, Jigen PengAbstract:This article discusses the influence of preconditioning on the Mutual Coherence and the restricted isometry property of Gaussian or Bernoulli measurement matrices. The Mutual Coherence can be reduced by preconditioning, although it is fairly small due to the probability estimate of the event that it is less than any given number in (0, 1). This can be extended to a set that contains either of the two types of matrices with a high probability but a subset with Lebesgue measure zero. The numerical results illustrate the reduction in the Mutual Coherence of Gaussian or Bernoulli measurement matrices. However, the first property can be true after preconditioning for a large type of measurement matrices having the property of s-order restricted isometry and being full row rank. This leads to a better estimate of the condition number of the corresponding submatrices and a more accurate error estimate of the conjugate gradient methods for the least squares problems typically used in greedy-like recovery algorithms.
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influences of preconditioning on the Mutual Coherence and the restricted isometry property of gaussian bernoulli measurement matrices
Linear & Multilinear Algebra, 2016Co-Authors: Yingtong Chen, Jigen PengAbstract:This article discusses the influence of preconditioning on the Mutual Coherence and the restricted isometry property of Gaussian or Bernoulli measurement matrices. The Mutual Coherence can be reduced by preconditioning, although it is fairly small due to the probability estimate of the event that it is less than any given number in (0, 1). This can be extended to a set that contains either of the two types of matrices with a high probability but a subset with Lebesgue measure zero. The numerical results illustrate the reduction in the Mutual Coherence of Gaussian or Bernoulli measurement matrices. However, the first property can be true after preconditioning for a large type of measurement matrices having the property of s-order restricted isometry and being full row rank. This leads to a better estimate of the condition number of the corresponding submatrices and a more accurate error estimate of the conjugate gradient methods for the least squares problems typically used in greedy-like recovery algorithms.
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Improved Mutual Coherence of some random overcomplete dictionaries for sparse repsentation
arXiv: Numerical Analysis, 2014Co-Authors: Yingtong Chen, Jigen PengAbstract:The letter presents a method for the reduction in the Mutual Coherence of an overcomplete Gaussian or Bernoulli random matrix, which is fairly small due to the lower bound given here on the probability of the event that the aforesaid Mutual Coherence is less than any given number in (0, 1). The Mutual Coherence of the matrix that belongs to a set which contains the two types of matrices with high probability can be reduced by a similar method but a subset that has Lebesgue measure zero. The numerical results are provided to illustrate the reduction in the Mutual Coherence of an overcomplete Gaussian, Bernoulli or uniform random dictionary. The effect on the third type is better than a former result.
