The Experts below are selected from a list of 45480 Experts worldwide ranked by ideXlab platform
Oded Regev - One of the best experts on this subject based on the ideXlab platform.
-
the restricted isometry property of subsampled Fourier matrices
Symposium on Discrete Algorithms, 2016Co-Authors: Ishay Haviv, Oded RegevAbstract:A Matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant e if it preserves the e2 norm of all k-sparse vectors up to a factor of 1 ± e. We prove that a Matrix A obtained by randomly sampling q = O(k · log2 k · log N) rows from an N ×N Fourier Matrix satisfies the restricted isometry property of order k with a fixed e with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).
-
the restricted isometry property of subsampled Fourier matrices
arXiv: Data Structures and Algorithms, 2015Co-Authors: Ishay Haviv, Oded RegevAbstract:A Matrix $A \in \mathbb{C}^{q \times N}$ satisfies the restricted isometry property of order $k$ with constant $\varepsilon$ if it preserves the $\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\pm \varepsilon$. We prove that a Matrix $A$ obtained by randomly sampling $q = O(k \cdot \log^2 k \cdot \log N)$ rows from an $N \times N$ Fourier Matrix satisfies the restricted isometry property of order $k$ with a fixed $\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).
Ishay Haviv - One of the best experts on this subject based on the ideXlab platform.
-
the restricted isometry property of subsampled Fourier matrices
Symposium on Discrete Algorithms, 2016Co-Authors: Ishay Haviv, Oded RegevAbstract:A Matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant e if it preserves the e2 norm of all k-sparse vectors up to a factor of 1 ± e. We prove that a Matrix A obtained by randomly sampling q = O(k · log2 k · log N) rows from an N ×N Fourier Matrix satisfies the restricted isometry property of order k with a fixed e with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).
-
the restricted isometry property of subsampled Fourier matrices
arXiv: Data Structures and Algorithms, 2015Co-Authors: Ishay Haviv, Oded RegevAbstract:A Matrix $A \in \mathbb{C}^{q \times N}$ satisfies the restricted isometry property of order $k$ with constant $\varepsilon$ if it preserves the $\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\pm \varepsilon$. We prove that a Matrix $A$ obtained by randomly sampling $q = O(k \cdot \log^2 k \cdot \log N)$ rows from an $N \times N$ Fourier Matrix satisfies the restricted isometry property of order $k$ with a fixed $\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).
Jingxin Zhang - One of the best experts on this subject based on the ideXlab platform.
-
multichannel compressive sensing mri using noiselet encoding
PLOS ONE, 2015Co-Authors: Kamlesh Pawar, Gary F Egan, Jingxin ZhangAbstract:The incoherence between measurement and sparsifying transform matrices and the restricted isometry property (RIP) of measurement Matrix are two of the key factors in determining the performance of compressive sensing (CS). In CS-MRI, the randomly under-sampled Fourier Matrix is used as the measurement Matrix and the wavelet transform is usually used as sparsifying transform Matrix. However, the incoherence between the randomly under-sampled Fourier Matrix and the wavelet Matrix is not optimal, which can deteriorate the performance of CS-MRI. Using the mathematical result that noiselets are maximally incoherent with wavelets, this paper introduces the noiselet unitary bases as the measurement Matrix to improve the incoherence and RIP in CS-MRI. Based on an empirical RIP analysis that compares the multichannel noiselet and multichannel Fourier measurement matrices in CS-MRI, we propose a multichannel compressive sensing (MCS) framework to take the advantage of multichannel data acquisition used in MRI scanners. Simulations are presented in the MCS framework to compare the performance of noiselet encoding reconstructions and Fourier encoding reconstructions at different acceleration factors. The comparisons indicate that multichannel noiselet measurement Matrix has better RIP than that of its Fourier counterpart, and that noiselet encoded MCS-MRI outperforms Fourier encoded MCS-MRI in preserving image resolution and can achieve higher acceleration factors. To demonstrate the feasibility of the proposed noiselet encoding scheme, a pulse sequences with tailored spatially selective RF excitation pulses was designed and implemented on a 3T scanner to acquire the data in the noiselet domain from a phantom and a human brain. The results indicate that noislet encoding preserves image resolution better than Fouirer encoding.
Wiegand, Troy M. - One of the best experts on this subject based on the ideXlab platform.
-
Identifying Complex Hadamard Submatrices of the Fourier Matrices via Primitive Sets
'Elsevier BV', 2021Co-Authors: Herr, John E., Wiegand, Troy M.Abstract:For a given selection of rows and columns from a Fourier Matrix, we give a number of tests for whether the resulting subMatrix is Hadamard based on the primitive sets of those rows and columns. In particular, we demonstrate that whether a given selection of rows and columns of a Fourier Matrix forms a Hadamard subMatrix is exactly determined by whether the primitive sets of those rows and columns are compatible with respect to the size of the Fourier Matrix. This allows the partitioning of all submatrices into equivalence classes that will consist entirely of Hadamard or entirely of non-Hadamard submatrices and motivates the creation of compatibility graphs that represent this structure. We conclude with some results that facilitate the construction of these graphs for subMatrix sizes 2 and 3.Comment: 22 pages, 6 figure
-
Identifying Complex Hadamard Submatrices of the Fourier Matrices via Primitive Sets
2020Co-Authors: Herr, John E., Wiegand, Troy M.Abstract:For a given selection of rows and columns from a Fourier Matrix, we give a number of tests for whether the resulting subMatrix is Hadamard based on the primitive sets of those rows and columns. In particular, we demonstrate that whether a given selection of rows and columns of a Fourier Matrix forms a Hadamard subMatrix is exactly determined by whether the primitive sets of those rows and columns are compatible with respect to the size of the Fourier Matrix. This motivates the creation of compatibility graphs for the Fourier matrices. We conclude with some results that facilitate the construction of these graphs for subMatrix sizes 2 and 3.Comment: 22 pages, 6 figure
Grimm Uwe - One of the best experts on this subject based on the ideXlab platform.
-
Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings
2021Co-Authors: Baake Michael, Grimm UweAbstract:Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot--Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier Matrix cocycle in internal space. This cocycle leads to a transfer Matrix equation and thus to a closed expression of Matrix Riesz product type for the Fourier transforms of the windows for the covering model sets. In general, these windows are complicated Rauzy fractals and thus difficult to handle. Equivalently, this approach permits a construction of the (always continuously representable) eigenfunctions for the translation dynamical system induced by the inflation rule. We review and further develop the underlying theory, and illustrate it with the family of Pisa substitutions, with special emphasis on the Tribonacci case.Comment: 31 pages, several figures; revised versio
-
Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings
2020Co-Authors: Baake Michael, Grimm UweAbstract:Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot-Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier Matrix cocycle in internal space. This cocycle leads to a transfer Matrix equation and thus to a closed expression of Matrix Riesz product type for the Fourier transforms of the windows for the covering model sets. In general, these windows are complicated Rauzy fractals and thus difficult to handle. Equivalently, this approach permits a construction of the (always continuously representable) eigenfunctions for the translation dynamical system induced by the inflation rule. We review and further develop the underlying theory, and illustrate it with the family of Pisa substitutions, with special emphasis on the classic Tribonacci case