Measurement Matrix Optimization for Compressed Sensing System with Constructed Dictionary via Takenaka-Malmquist Functions.

Qiangrong Xu,Liming Zhang,Yong Fang,Zhichao Sheng

doi:10.3390/s21041229

Qiangrong Xu, Liming Zhang + Show 2 more

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https://doi.org/10.3390/s21041229

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Abstract

Compressed sensing (CS) has been proposed to improve the efficiency of signal processing by simultaneously sampling and compressing the signal of interest under the assumption that the signal is sparse in a certain domain. This paper aims to improve the CS system performance by constructing a novel sparsifying dictionary and optimizing the measurement matrix. Owing to the adaptability and robustness of the Takenaka–Malmquist (TM) functions in system identification, the use of it as the basis function of a sparsifying dictionary makes the represented signal exhibit a sparser structure than the existing sparsifying dictionaries. To reduce the mutual coherence between the dictionary and the measurement matrix, an equiangular tight frame (ETF) based iterative minimization algorithm is proposed. In our approach, we modify the singular values without changing the properties of the corresponding Gram matrix of the sensing matrix to enhance the independence between the column vectors of the Gram matrix. Simulation results demonstrate the promising performance of the proposed algorithm as well as the superiority of the CS system, designed with the constructed sparsifying dictionary and the optimized measurement matrix, over existing ones in terms of signal recovery accuracy.

Highlights

Sparse representation has become a powerful tool, as it can efficiently model signals to facilitate compressing and processing [1,2,3]
The original signal is sparse in a different transform domain, e.g., discrete cosine transform (DCT), discrete wavelet transform (DWT), and so on
The results demonstrate that the constructed TM dictionary of 15 has a better performance in terms of reconstruction quality compared with the12random, DCT, and DWT dictionaries

Summary

Introduction

Sparse representation has become a powerful tool, as it can efficiently model signals to facilitate compressing and processing [1,2,3]. Due to the redundancy in the majority of signals in nature, transforming the original signal to a sparse or compressed version through a certain domain is meaningful for reducing costs in transportation and storage. In this case, the signal can be expressed as a linear combination of a few given basis functions, which indicates that most of the representation’s coefficients are zero or close to zero. If an N-dimensional signal admits K-sparse representation in a dictionary, one can reconstruct exactly the original signal with large probability with M = O(Klog(N/K)) measurements [10,13]. The0 -norm used here denotes the number of the nonzero elements in s.

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