Sparse kronecker pascal measurement matrices for compressive imaging

Journal of the European Optical Society-Rapid Publications, Jun 2017

Background The construction of measurement matrix becomes a focus in compressed sensing (CS) theory. Although random matrices have been theoretically and practically shown to reconstruct signals, it is still necessary to study the more promising deterministic measurement matrix. Methods In this paper, a new method to construct a simple and efficient deterministic measurement matrix, sparse kronecker pascal (SKP) measurement matrix, is proposed, which is based on the kronecker product and the pascal matrix. Results Simulation results show that the reconstruction performance of the SKP measurement matrices is superior to that of the random Gaussian measurement matrices and random Bernoulli measurement matrices. Conclusions The SKP measurement matrix can be applied to reconstruct high-dimensional signals such as natural images. And the reconstruction performance of the SKP measurement matrix with a proper pascal matrix outperforms the random measurement matrices.

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Sparse kronecker pascal measurement matrices for compressive imaging

Jiang et al. Journal of the European Optical Society-Rapid Publications Sparse kronecker pascal measurement matrices for compressive imaging Yilin Jiang 0 Qi Tong 0 Haiyan Wang 0 Qingbo Ji 0 0 College of Information and Communication Engineering, Harbin Engineering University , Harbin 150001 , China Background: The construction of measurement matrix becomes a focus in compressed sensing (CS) theory. Although random matrices have been theoretically and practically shown to reconstruct signals, it is still necessary to study the more promising deterministic measurement matrix. Methods: In this paper, a new method to construct a simple and efficient deterministic measurement matrix, sparse kronecker pascal (SKP) measurement matrix, is proposed, which is based on the kronecker product and the pascal matrix. Results: Simulation results show that the reconstruction performance of the SKP measurement matrices is superior to that of the random Gaussian measurement matrices and random Bernoulli measurement matrices. Conclusions: The SKP measurement matrix can be applied to reconstruct high-dimensional signals such as natural images. And the reconstruction performance of the SKP measurement matrix with a proper pascal matrix outperforms the random measurement matrices. Compressed sensing; Deterministic measurement matrix; Kronecker product; Pascal matrix - Background Compressed sensing (CS) theory is a novel sampling scheme, which indicates that a sparse signal can be recovered from much fewer samples than conventional method [1, 2]. The sampling and the compression procedure are completed by the linear projection in CS. In matrix notation, it can be expressed as where x ∈ ℝN is the original signal, Φ is an M × N(M ≪ N) measurement matrix, y ∈ ℝM is the measurement vector. x is said to be K-sparse if ‖x‖0 ≤ K. CS theory asserts that if the measurement matrix Φ satisfies some conditions, the signal x can be recovered from measurements y without distortion. The emergence of CS provides a new inspiration for optical imaging. Actually most of the nature images are compressible in terms of some sparsity basis, such as Discrete cosine transform (DCT) and Discrete wavelet transform (DWT). The compressibility of the real-word images shows the potential for optical compressive imaging. In the past few years, CS technique has made great progress in many research fields, which include terahertz compressive imaging [3], spectral imaging [4], single pixel imaging [5] and infrared imaging [6]. Some optical imaging applications have been implemented in specific physical experiments. Measurement matrix construction is a crucial problem in CS. The measurements obtained by measurement matrix are related to whether the signal can be accurately reconstructed. If there is enough information within the measurements, the signal can be recovered with high probability. Random measurement matrices are proved to have the merit of universality but suffer from several shortcomings. Firstly, random measurement matrices consume lots of storage resources. Secondly, there is no feasible algorithm to verify whether the random matrix satisfies the requirement as a measurement matrix [7, 8]. The research on deterministic sampling can be tracked back to the binary matrices via polynomials over finite field [9]. The deterministic measurement matrix has the superiority in physical implementation and the advantage of saving storage space. Therefore, many researches on © The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. the deterministic measurement matrix construction have been carried out. Lu introduced a construction of ternary matrices with small coherence [10]. Yao presented a novel simple and efficient measurement matrix named incoherence rotated chaotic matrix [11]. Huang proposed a symmetric Toeplitz measurement matrix [12]. Zhao introduced a deterministic complex measurement matrix to sample the signals in the single pixel imaging [13]. In this paper, we propose a new construction method of deterministic measurement matrix, termed sparse kronecker pascal (SKP) measurement matrix. The SKP measurement matrix combines the properties of the kronecker product and the pascal matrix. It is suitable for the reconstruction of natural images, which are usually high-dimensional signals. Simulations and analyses confirm that the SKP measurement matrices can reconstruct the natural images with a better performance. Methods The SKP measurement matrix construction In mathematics, the kronecker product is an operation on two matrices of arbitrary size resulting in a block matrix [14]. (...truncated)


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Yilin Jiang, Qi Tong, Haiyan Wang, Qingbo Ji. Sparse kronecker pascal measurement matrices for compressive imaging, Journal of the European Optical Society-Rapid Publications, 2017, pp. 17, Volume 13, Issue 1, DOI: 10.1186/s41476-017-0045-9