Measurement Matrix Optimization via Mutual Coherence Minimization for Compressively Sensed Signals Reconstruction

Ziran Wei,Krzysztof Okarma,Zhiyong Xu,Jianlin Zhang,Yong Liu

doi:10.1155/2020/7979606

Abstract

For signals reconstruction based on compressive sensing, to reconstruct signals of higher accuracy with lower compression rates, it is required that there is a smaller mutual coherence between the measurement matrix and the sparsifying matrix. Mutual coherence between the measurement matrix and sparsifying matrix can be expressed indirectly by the property of the Gram matrix. On the basis of the Gram matrix, a new optimization algorithm of acquiring a measurement matrix has been proposed in this paper. Firstly, a new mathematical model is designed and a new method of initializing measurement matrix is adopted to optimize the measurement matrix. Then, the loss function of the new algorithm model is solved by the gradient projection-based method of Gram matrix approximating an identity matrix. Finally, the optimized measurement matrix is generated by minimizing mutual coherence between measurement matrix and sparsifying matrix. Compared with the conventional measurement matrices and the traditional optimization methods, the proposed new algorithm effectively improves the performance of optimized measurement matrices in reconstructing one-dimensional sparse signals and two-dimensional image signals that are not sparse. The superior performance of the proposed method in this paper has been fully tested and verified by a large number of experiments.

Highlights

For signals reconstruction based on compressive sensing, to reconstruct signals of higher accuracy with lower compression rates, it is required that there is a smaller mutual coherence between the measurement matrix and the sparsifying matrix
Owing to some previous research conclusions, we focus on seeking the optimal measurement matrices that are for the binary measurement matrices and the two-dimensional image signals, and for adaptive measurement matrices and one-dimensional sparse signals and the medical image, and our works are further conducted under different signal sparsity levels and different compression rates
The gradient projection based strategy is used to solve our proposed new algorithm model with the new method of initializing measurement matrix for compressively sensed signals reconstruction, which is a new idea for acquiring optimized measurement matrices via minimizing the mutual coherence between measurement matrix and sparsifying basis

Summary

Measurement Matrix Formulation

From the above theoretical introduction of CS, one can know that the performance of the measurement matrix directly influences the results of signals reconstruction, and the measurement matrix is the important intermediate link between signal sparsifying and signal reconstruction. E literature [30] proves that if the measurement matrix and the orthogonal sparsifying basis are as incoherent (orthogonal) as possible, the measurement matrix would have better performance with a large probability. According to the related mathematical theory [47, 57], the incoherence between the measurement matrix and the sparsifying matrix can be indirectly transformed into the problem that Gram matrix approximates an identity matrix as closely as possible. According to the definition of the Gram matrix, minimizing the mutual coherence between Φ and Ψ is equivalent to minimizing the absolute off-diagonal elements in the corresponding Gram matrix and making the Gram matrix as close as possible to the identity matrix [57]. In the literature [47],Elad’s work is just based on the model (6) via the minimization of the average of the off-diagonal entries in the Gram matrix to seek an optimized measurement matrix

Measurement Matrix Optimization

Experiments

Findings

Conclusions