compressive sensing with tent chaotic...

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

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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: [email protected]

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 Candès 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

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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 ϒ is very

sparse or highly compressible in some basis Ψ (i.e. Fourier, Wavelet, Discrete-Cosine basis), then the signal can be represented as x=Ψs (s is K-sparse, meaning it has K significant components). The measurement vector My ϒ is captured through Φx, where M<N and M Nϒ is well chosen matrix satisfying RIP. Mathematically, the procedure of CS can be expressed as a linear projection:

y x s , (1)

where Φ is the measurement (sensing) matrix, Ψ is the sparse (compression) basis, and Θ=ΦΨ is the compressive sensing matrix. Given y, Φ and Ψ, the objective is then to faithfully recover s (and hence x) from y with as small M as possible.

The problem of reconstruction can be stated formally ass:

1min s. t. s y s (2)

If Φ is incoherent with Ψ according to the RIP,

s can be recovered from y when M is such that cK N K M N log / , where c is some constant,

using various sparse approximation algorithms, such as l1-optimization based Basic Pursuit (BP), Orthogonal Matching Pursuit (OMP), Gradient Projection for Sparse Reconstruction (GPRS), and so on.

In image reconstruction, the signal x can be represented an N-dimensional high-resolution image and y is its low-resolution image. We wish to reconstruct x from y, then the resulting inverse problem is highly underdetermined and ill-posed for M<N. But in CS, the x can be obtained a unique solution using the sparsity of x.

3. Chaotic Sensing Matrix In this paper, we use chaotic sequence to

construct the measurement matrix, which has been proved to be Toeplitz-structured and sufficient to satisfy RIP with high probability [19]. The process of construction of the measurement matrix M Nϒ is as follows:

1) To generate a chaotic sequence 0 1 1MNc , ,Λc c c of length M×N. The chaotic

sequences are sampled from the output sequence produced by Tent map:

1 0 5 0 1 1

0 0 5 0 5

c n . c n n , , ,MN

c . , .

Λ, (3)

where α is the control parameter. The initial condition c(0) is rather sensitive to the resulting behavior of the chaotic sequence. Note that the value of c(0) and α should be ensuring the signal is chaotic. In this paper we select α=1.6, c(0)=0.3 for Tent map.

For comparison, Logistic map is given by:

1 1

0 1 1 0 4 , 0 0 1

c n c n c n

n , , ,MN , , c ,

Λ

, (4)

where α is also the control parameter, and we choose α=4, c(0)=0.3.

2) To construct an M×N matrix of size column wise (taking M contiguous samples and putting them in a column of Φ), as follows:

0 1

1 1 1 1

1 2 1 1

1

M M N

M M N

M M MN

c c c

c c c

M

c c c

( )

( )

Λ

Λ

Μ Μ Μ Μ

Λ

, (5)

where σ2

is the variance of the sampled signal (or sequence). All elements of the matrix are scaled with

the factor 1 M/ for normalization.


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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 ˆPr / . , (6)

where x̂ 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 256×256 (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

ˆx x

, (7)

where M×N 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 images. The results are shown in Fig. 7. Clearly, the more the samples or measurements are used, the better the performance of reconstruction is obtained. But even if the samples are fewer, the outline of the image can be get due to CS theory. Therefore, compressive sensing with Tent sequence can be applied for supper-resolution image reconstruction.

5. Conclusions In this paper, we demonstrate the use of Tent

chaotic map to construct measurement matrix in compressive sensing. Numerical simulations demonstrate that Tent chaotic matrix has similar performance to Logistic chaotic matrix and litter better than Gaussian random matrix. So far, the


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chaotic signals, such as Logistic, Tent, Chua, Lorenz have been proved to effectively construct measurement matrices for CS theory. Other usable chaotic signal should be verified and the general property of these chaotic signals should be summarized, which is our next work.

(a) sample rate: 0.3, PSNR=18.2382

(b) sample rate: 0.5, PSNR=24.0190

(c) sample rate: 0.7, PSNR=28.5710

(d) sample rate: 0.9, PSNR=31.9698

Fig. 7. Reconstruction of CS with TCM under different sample rate: from top to bottom, 0.3, 0.5, 0.7, 0.9 (only part

of the image is shown for clear visual comparison).

Acknowledgements This work was supported by the National

Natural Science Foundation of China (Grant No. 61240017), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20121402120019), the Natural Science Foundation of Shanxi Province (Grant No. 2012021011-4) and the Youth Foundation of Taiyuan University of Technology (Grant No. 2012L035). References [1]. E. Candès, J. Romberg, T. Tao, Robust uncertainty

principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions Information Theory, 52, 2, 2006, pp. 489-509.

[2]. E. Candès, T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies, IEEE Transactions Information Theory, 52, 12, 2006, pp. 5406-5425.

[3]. D. Donoho, Compressed sensing, IEEE Transactions Information Theory, 52, 4, 2006, pp. 1289-1306.

[4]. M. Shankar, N. P. Pitsianis, D. J. Brady, Compressive video sensors using multichannel imagers, Applied Optics, 49, 10, 2010, pp. B9-B17.

[5]. M. Lustig, D. Donoho, J. M. Pauly, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magnetic Resonance in Medicine, 58, 6, 2007, pp. 1182-1195.

[6]. M. A. Sheikh, O. Milenkovic, R. G. Baraniuk, Designing compressive sensing DNA microarrays, in Proceedings of the IEEE International Workshop on ‘Computational Advances in Multi-sensor Adaptive Processing (CAMP-SAP 2007)’, St. Thomas, U.S. Virgin Islands, 12-14 December 2007, pp. 141-144.

[7]. Y. Mostofi, P. Sen, Compressed mapping of communication signal strength, in Proceedings of the Military Communications Conference (Milcom 2008), San Diego, CA, 17-19 November 2008, pp. 1-7.

[8]. M. F. Duarte, M. A. Davenport, D. Takhar, et al., Single pixel imaging via compressive sampling, IEEE Signal Processing Magazine, 25, 2, 2008, pp. 83-91.

[9]. F. Marcia, R. M. Willett, Compressive coded aperture super resolution image reconstruction, in Proceedings of the IEEE International Conference on ‘Acoustics, Speech and Signal Processing’, Las Vegas, NV, March 31-April 4 2008, pp. 833-836.

[10]. J. C. Yang, J. Wright, T. Huang, et al., Image super-resolution via sparse representation, IEEE Transactions on Image Processing, 19, 11, 2010, pp. 2861-2873.

[11]. J. C. Yang, Z. W. Wang, Z. Lin, et al., Coupled dictionary training for image super-resolution, IEEE Transactions on Image Processing, 21, 8, 2012, pp. 3467-3478.

[12]. R. Baraniuk, P. Steeghs, Compressive radar imaging, in Proceedings of the IEEE Radar Conference, Boston, MA, 17-20 April 2007, pp. 128-133.

[13]. J. Haupt, R. Nowak, Signal reconstruction from noisy random projections, IEEE Transactions on Information Theory, 52, 9, 2006, pp. 4036-4048.


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[14]. J. A. Tropp, W. B. Wakin, M. F. Duarte, et al., Random filters for compressive sampling and reconstruction, in Proceedings of the International Conference on ‘Acoustics, Speech, and Signal Process (ICASSP'06), Toulouse, France, 14-19 May 2006, pp. 872-875.

[15]. J. Romberg, Compressive sensing by random convolution, SIAM Journal on Imaging Sciences, 2, 4, 2009, pp. 1098-1128.

[16]. N. L. Trung, D. V. Phong, Z. M. Hussain, et al., Compressed sensing using chaos filters, in Proceedings of ‘Australasian Telecommunication Networks and Applications Conference (ATNAC'08), Adelaide, SA, 7-10 December 2008, pp. 219-223.

[17]. T. D. Tan, L. Vu-Ha, N. L. Trung, Spread spectrum for chaotic compressed sensing techniques in parallel magnetic resonance imaging, in Proceedings of the 8th International Conference on Information,

Communication and Signal Processing (ICICS'11), Singapore, 13-16 December, 2011, pp. 1-5.

[18]. L. Yu, J. P. Barbot, G. Zheng, H. Sun, Compressive sensing with chaotic sequence, IEEE Signal Processing Letters, 17, 8, 2010, pp. 731-734.

[19]. L. Yu, J. P. Barbot, G. Zheng, H. Sun, Toeplitz-structured chaotic sensing matrix for compressive sensing, in Proceedings of the IEEE, IET International Symposium on ‘Communication Systems Networks and Digital Signal Processing (CSNDSP'10), Newcastle upon Tyne, 21-23 July 2010, pp. 229-233.

[20]. V. Kafedziski, T. Stojanovski, Compressive sampling with chaotic dynamical systems, in Proceedings of the 19th Telecommunications Forum (TELFOR), Belgrade, Yugoslavia, 22-24 November 2011, pp. 695-989.

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