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  • Sensors & Transducers, Vol. 165, Issue 2, February 2014, pp. 119-124

    119

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    2014 by IFSA Publishing, S. L. http://www.sensorsportal.com

    Compressive Sensing with Tent Chaotic Sequence

    Li Liu, Peng Yang, Jianguo Zhang, Huayu Ja Key Lab of Advanced Transducers and Intelligent Control System, Ministry of Education, Taiyuan

    University of Technology, Taiyuan 030024, China College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China

    E-mail: liu_li82@163.com

    Received: 11 November 2013 /Accepted: 28 January 2014 /Published: 28 February 2014 Abstract: Compressive sensing is a new sampling theory to capture signals at sub-Nyquist rate. To guarantee exact recovery from sparse measurements, specific sensing matrix, which satisfies the Restricted Isometry Property, should be well chosen. Random matrix has been proved to meet the property with high probability; however, the practical implementation is expensive in hardware design. Chaotic matrices which generated by Logistic sequence, Chua and Lorenz dynamical systems have been verified to be Toeplitz-structured and sufficient to satisfy the property. In this paper, we propose that another chaotic sequence - Tent map can also be used to construct the sensing matrix. By numerical performance, we show that, the proposed Tent chaotic sensing matrix has similar performance to random matrix or Logistic chaotic matrix for exact reconstructing compressible signals and images from fewer measurements. Copyright 2014 IFSA Publishing, S. L. Keywords: Compressive Sensing, Reconstruction, Chaos, Tent Map, Logistic Map. 1. Introduction

    Recently, a new sampling theory proposed by

    Cands et al. [1-2] and Donoho [3], called Compressive sensing (CS), has attracted an overwhelming research attention and has been used in different fields, such as video processing [4], medical imaging [5], bio-sensing [6], wireless channel mapping [7], compressive imaging [8-11], radar [12] and so on. The basic idea of CS theory is that when the signal is very sparse or highly compressible in some basis (i.e., most basis coefficients are small or zero-valued), far fewer measurements suffice to exactly reconstruct the signal than needed by Nyquist-Shannon theory. Measurements of the signal are taken using the measurement matrix, which should satisfies the so-called Restricted Isometry Property (RIP), that is the measurement matrix is supposed to be incoherent with the matrix describing the sparse basis. Recent

    Studies show that feasible, judicious selection of the type of the measurement matrix may dramatically improve the ability to extract high-quality images from a limited number of measurements and may reduce the hardware complexity in design [9].

    In CS framework, to find a proper measurement matrix satisfying RIP is one of the central problems. Usually, the measurement matrix is constructed by a random sequence followed by a random distribution, such as Gaussian, uniform and Bernoulli, for these random distributions satisfy RIP with high probability [13]. However, the use of purely random sequences is expensive in hardware design in practice. Since chaotic signals exhibit similar properties to random signals and easily implemented on hardware, some authors have attempted using chaotic signals to construct measurement matrices [14-20]. J. A. Tropp, et al. [14] firstly present compressive sampling with random filter. N. Linh-Trung, et al. [16] examine the use of chaos filters in

    Article number P_1887

    http://www.sensorsportal.com/

  • Sensors & Transducers, Vol. 165, Issue 2, February 2014, pp. 119-124

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    CS with filter taps calculated from the Logistic map and claim that chaos filters generated by the Logistic map outperform random filters. L. Yu, et al. [18-19] construct the measurement matrix with chaotic sequence from Logistic map and prove that this chaotic matrix is Toeplitz-structured and sufficient to satisfy RIP with high probability. V. Kafedziski, et al. [20] use Chua and Lorenz chaotic signals to construct measurement matrices in compressive sampling and show that chaotic signals have similar performance to random Gaussian, Bernoulli and uniformly distributed sequences.

    To the best of our knowledge in the literature on using chaotic sequences in CS, only Logistic map, Chua and Lorenz dynamical systems have been used, thus it is desirable to examine and compare the performance of other chaotic maps. Here, we use Tent map to construct the measurement matrix in CS for reconstructing the 1D signal and 2D high-resolution image. The main contribution of this paper is to propose a new chaotic sequence, which can be effectively used in CS.

    The rest of this paper is organized as follows. In Section 2 we give an overview of compressive sensing. Section 3 describes chaotic sensing matrix using Tent map. Numerical results of compressive sensing with chaotic signals are presented and the performance comparison between chaos-based measurement matrix and random matrix is given in Section 4. In Section 5 we draw some conclusions from the results of our simulation study.

    2. Compressive Sensing Theory Consider an N-dimensional vector Nx is very

    sparse or highly compressible in some basis (i.e. Fourier, Wavelet, Discrete-Cosine basis), then the signal can be represented as x=s (s is K-sparse, meaning it has K significant components). The measurement vector My is captured through x, where M

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    121

    4. Numerical Simulation We provide some numerical examples to verify

    that Tent chaotic sequence is a powerful approach to compressive signal acquisition and reconstruction and the performance of it can be comparable to that of random Gaussian distributed sequence and Logistic chaotic sequence.

    Since this paper does not focus on the reconstruction algorithms, we just briefly use the OMP algorithm for signal reconstruction, whose Matlab implementation can be found in Internet easily.

    4.1. 1D Signal Reconstruction Using Tent Chaotic Sensing Matrix

    Consider a discrete time-sparse signal x with

    N=512 samples and K=20 spikes, as shown in Fig. 3(a), is being reconstruction from M=128 measurements vectors. Here, we use Tent chaotic sensing matrix (TCM), and Logistic chaotic sensing matrix (LCM), Gaussian random matrix (GRM) for comparison, to implement the CS process. Fig. 1 (a)-(c) show the Gaussian random sequence, Logistic map and Tent map respectively, all normalized to have zero mean and unity variance. The obtained measurements, generated from the corresponding random or chaotic sensing matrix, are shown in Fig. 2. It can be seen that the original signal x has been randomly projected on the measurements y.

    Fig. 1. Random and chaotic sequences. The reconstructed signal using OMP is shown in

    Fig. 3(b)-(d). Clearly, when N=512, M=128, K=20, we can obtain exact reconstruction using chaotic sequences or Gaussian sequence. We also give another example for M=64, K=10, as illustrated in Fig. 4. From it, we can see that the performance of

    Tent and Logistic chaotic sensing matrix is better than that of Gaussian matrix (using Gaussian matrix cannot obtain exact reconstruction).

    Fig. 2. The measurements of the signal.

    Fig. 3. Illustrative example showing successful reconstruction using Gaussian and chaotic sensing matrix,

    N=512, M=128, K=20. To father verify the performance of Tent chaotic

    sequence, a comparison between TCM, LCM and GRM is carried out by considering their probability of incorrect reconstruction, that is, the error rate, as follows:

    0 01e x x x Pr / . , (6)

    where x is the reconstructed vector and is the l2 norm.

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    Fig. 4. The performance comparison: Gaussian vs. chaotic sensing matrix, N=512, M=64, K=10.

    In Fig. 5, we plot the error rate varies with the sparsity of the signal for various measurement matrices. We can see that the error rate for TSM is similar with LSM but little smaller than GRM, meaning that the performance of Tent matrix is comparable to Logistic matrix and something better than that of Gaussian random matrix.

    Fig. 5. Error rate as a function of the signal sparsity K for N=512 and M=64.

    4.2. Image Reconstruction Using Tent Chaotic Sensing Matrix

    We simulate reconstruction of the monochrome

    Cameraman image with size 256256 (shown in Fig. 6(a)), which being reconstructed from 32768 samples (sample rate is 32768/2562=0.5) by projection with TCM, LCM and GSM following CS procedure. We use Haar wavelet for our compression basis and OMP as the solver to approximate the solution in 20 iterations. The Peak Signal to Noise Ratio (PSNR) is used as a criterion for the reconstruction performance:

    22

    2 1 10

    n M NPSNR lg

    x x

    , (7)

    where MN is the image size and n means the bit value per sample of the pixel (usually, n=8).

    (a) The original image

    (b) GRM (PSNR=23.9690)

    (c) LCM (PSNR=24.2346)

    (d) TCM (PSNR=24.0190)

    Fig. 6. Reconstruction of CS with GRM, LCM and TCM.

    The results are listed in Fig. 6 and the

    corresponding PSNR values are given. It can be seen that (especially seen from the eye position) the performance of TCM is comparable to that of LCM and litter better than GRM, which is consistent with the results in 4.1.

    To prove the efficiency of TCM, we measure the image with sample rate at 0.3, 0.5, 0.7 and 0.9 respectively, then use OMP algorithm to reconstruct the original

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