Abstract
In this paper, a new method of measurement matrix optimization for compressed sensing based on alternating minimization is introduced. The optimal measurement matrix is formulated in terms of minimizing the Frobenius norm of the difference between the Gram matrix of sensing matrix and the target one. The method considers the simultaneous minimization of the mutual coherence indexes including maximum mutual coherence μmax, t-averaged mutual coherence μave and global mutual coherence μall, and solves the problem that minimizing a single index usually results in the deterioration of the others. Firstly, the threshold of the shrinkage function is raised to be higher than the Welch bound and the relaxed Equiangular Tight Frame obtained by applying the new function to the Gram matrix is taken as the initial target Gram matrix, which reduces μave and solves the problem that μmax would be larger caused by the lower threshold in the known shrinkage function. Then a new target Gram matrix is obtained by sequentially applying rank reduction and eigenvalue averaging to the initial one, leading to lower. The analytical solutions of measurement matrix are derived by SVD and an alternating scheme is adopted in the method. Simulation results show that the proposed method simultaneously reduces the above three indexes and outperforms the known algorithms in terms of reconstruction performance.
Highlights
Compressed sensing (CS) [1] can sample the sparse or compressible signals at a subNyquist rate, which brings great convenience for data storage, transmission, and processing.By adopting the reconstruction algorithms, the signal can be exactly reconstructed from the sampled data
For a given dictionary matrix Ψ ∈ R80×120, x ∈ R120×1 has a sparse representation as x = Ψs where s is K-sparse and each non-zero entry is randomly positioned with a Gaussian distribution of i.i.d. zero-mean and unit variance
This paper focused on the optimization of measurement matrix for compressed sensing
Summary
Compressed sensing (CS) [1] can sample the sparse or compressible signals at a subNyquist rate, which brings great convenience for data storage, transmission, and processing. The simulation results carried out in [8] show that the optimized Φ leads to smaller μ ave and a substantially better CS reconstruction performance is obtained. A suitable point between the current solution and the one obtained using a new shrinkage function is chosen to design the Gt in [10] It is of very strong competitiveness in μ ave and μmax. The maximum absolute value of off-diagonal entries in G is almost always greater than μwelch In this case, the optimization usually implies a solution D with low μ ave but high μmax. The simulation results confirm the effectiveness of the proposed method in decreasing the mutual coherence indexes and improving reconstruction performance.
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