A New Method of Measurement Matrix Optimization for Compressed Sensing Based on Alternating Minimization

• Main Authors: Renjie Yi, Chen Cui, Biao Wu, Yang Gong
• Format: Article
• Language: English
• Published: MDPI AG 2021-02-01
• Series: Mathematics
• Subjects: compressed sensing; measurement matrix; Equiangular Tight Frame; mutual coherence
• Online Access: https://www.mdpi.com/2227-7390/9/4/329

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>, <i>t</i>-averaged mutual coherence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>a</mi><mi>v</mi><mi>e</mi></mrow></msub></mrow></semantics></math></inline-formula> and global mutual coherence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>a</mi><mi>l</mi><mi>l</mi></mrow></msub></mrow></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>a</mi><mi>v</mi><mi>e</mi></mrow></msub></mrow></semantics></math></inline-formula> and solves the problem that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> 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.