IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 8, AUGUST 2010

731

Compressive Sensing With Chaotic Sequence Lei Yu, Jean Pierre Barbot, Gang Zheng, and Hong Sun

Abstract—Compressive sensing is a new methodology to capture signals at sub-Nyquist rate. To guarantee exact recovery from compressed measurements, one should choose specific matrix, which satisfies the Restricted Isometry Property (RIP), to implement the sensing procedure. In this letter, we propose to construct the sensing matrix with chaotic sequence following a trivial method and prove that with overwhelming probability, the RIP of this kind of matrix is guaranteed. Meanwhile, its experimental comparisons with Gaussian random matrix, Bernoulli random matrix and sparse matrix are carried out and show that the performances among these sensing matrix are almost equal. Index Terms—Chaos, compressive sensing, logistic map.

I. INTRODUCTION

O

VER the recent years, a new sampling theory, called Compressive Sensing [9]–[11] (CS for short), has attracted lots of researchers. The central goal of CS is to capture attributes of a signal using very few measurements: for any -dimensional signal (w.l.g. is -sparse vector), the meais captured through , where surement and is a well chosen matrix satisfying the Restricted Isometry Property (RIP)[8]. satisfies the Restricted Definition 1.1: Matrix Isometry Property of order if there exists a constant such that (1) for all -sparse vectors . In CS framework, finding a proper sensing matrix satisfying RIP is one of the central problems. Candès and Tao have proposed that matrix with elements drawn by Gaussian distribution or Bernoulli distribution satisfies RIP with overwhelming [10]. probability, providing that sparsity And the randomly selected Fourier basis also retains RIP with overwhelming probability with sparsity Manuscript received May 18, 2010 accepted May 29, 2010. Date of publication June 07, 2010; date of current version June 21, 2010. This work was supported by “Bourses Doctorales en Alternance,” PEPS-A2SDC of INSIS-CNRS, NSFC (60872131) and FEDER through CPER 2007–2013 with INRIA LilleNord Europe. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Jared W. Tanner. L.Yu is with Signal Processing Laboratory, Electronic and Information School, Wuhan University, Wuhan 430079 China, and also with with ECS ENSEA and EPI ALIEN, INRIA, 95014 Cergy-Pontoise, France (e-mail: [email protected]). J.-P. Barbot is with ECS ENSEA and EPI ALIEN, INRIA,, 95014 CergyPontoise, France (e-mail: [email protected]). G. Zheng is with INRIA Lille-Nord Europe, 59650 Villeneuve d’Ascq, France (e-mail: [email protected]). H. Sun is with Signal Processing Laboratory, Electronic and Information School, Wuhan University, Wuhan 430079 China (e-mail: [email protected]). Digital Object Identifier 10.1109/LSP.2010.2052243

[10]. On the other hand, many researchers have employed some other techniques to construct deterministic sensing matrix: one group satisfying Statistical Isometry Property (StRIP) [7], such as Chirp Sensing Codes, second order Reed–Muller code, BCH code by R. Calderbank et al. [2], [7], [14]; one group satisfying RIP-1 [5], such as sparse random matrix by P. Indyk et al. [6] and LDPC by D. Baron et. al [4]; one group satisfying deterministic RIP, such as finite fields by R. A. Devore [12], etc. In this paper, we employ chaotic sequence to construct the sensing matrix, called chaotic matrix. Comparing to other techniques, chaotic system generates the “pseudo-random” matrix in deterministic approach and hence verifies the RIP similar to Gaussian or Bernoulli matrix. Moreover, it is easy to be implemented in physical electric circuit and only one initial state is necessary to be memorized. Based on the statistical property of chaotic sequence, it is shown that chaotic matrix satisfies RIP with overwhelming probability, providing that . The main contribution of this paper is to make a connection between chaotic sequence and CS. It is shown by the experiments that the performance of chaotic matrix is somewhat equal to the famous Gaussian random matrix and sparse random matrix. The paper is organized as below. In Section II, one chaotic system is recalled and its statistical property is presented. Section III shows the construction of chaotic matrix and proves its RIP. In Section IV, experiments are carried out to simulate the performance of chaotic matrix. At the end, the conclusion is given. II. CHAOTIC SEQUENCE AND ITS STATISTICAL PROPERTY Let us consider the following quadratic recurrence equation (2) where is a positive constant sometimes known as the “biotic potential” giving the so-called Logistic map. For the special case , the solution for system (2) can be written as follows [17]: (3) where satisfying with the initial condition of (2). It is well known that chaotic system (2) can produce very complex sequences. Even more, it is often used as the random number generator in practice since (2) takes a very simple dynamics [17]. In this section, we will analyze its statistical properties, the distribution, the correlations and the sampling distance which guarantees the statistical independence. Denote

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(4)

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IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 8, AUGUST 2010

obviously, takes the similar statistical property with since and the linear transformation, for instance the fact that are statistically independent, would result in and are statistically independent. A. Distribution Function (4) possesses the following features: zero mean, , invariant denvalues bounded within the interval and . sity given by B. Correlations It can be checked that the if is odd and

-th moment of

satisfies

(5) if

is even.

C. Statistical Independence In [19], it has been proved that sequence generated by (4) is not independent. However, we can measure its independence through the high order correlations, which is determined by the sampling distance. We have the following lemma. the seLemma 2.1: Denote , and quence generated by (4) with initial state integer the sampling distance, then for any positive integer , it has (6) Proof: If there exists at least one odd number in the right side of (6) is equal to 0. For the left side, we have

,

Fig. 1. Probability density (x ) (a) and (x ) (b); (c) and (d) joint prob) for sampling distance d = 5, 15. ability density P (x ; x

Compare it with (5), we have (6). Remark 2.2: Lemma 2.1 implies that and are statis, and this result corresponds tically independent when to that given in [18]. Approximately, if the sampling distance , is chosen large enough, for instance for all , hence and can be considered approximately independent, as illustrated in Fig. 1. III. CHAOTIC SENSING MATRIX Let be the chaotic sequence sampled from the output sequence produced by Logistic map (2) with sampling distance and initial condition , and let denote the regularization of as follows: (7)

where the last equation uses the fact that and , with if otherwise equals to 1. is the summation over all possible configuraand . tions, where and are analysis as follows: All possible cases for is odd: and , 1) hence ; 2) is even: it is possible that , is odd (the assumption at the bewhile since , hence ginning of this part of proof), . Then we can conclude that the left side of (6) is also equal to 0. and are even numbers, after a trivial combinaIf both torial analysis, we get

where just corresponds to (4) and hence fulfils the statistical properties discussed in the previous section. To con, generate sampled Lostruct the sensing matrix with length , then create gistic sequence a matrix column by column with this sequence, written as

.. .

.. .

.. .

(8)

where the scaler is for normalization. By chosen sam, then elements of sequence pling distance are approximately independent and satisfy identical distribution , i.e., a.i.i.d, and hence elements of matrix are a.i.i.d. Theorem 3.1: Chaotic matrix constructed folwith overwhelming lowing (8) satisfies RIP for constant . probability, providing that Remark 3.2: Inherently, this matrix is sub-gaussian with a.i.i.d elements. In [16], A. Pajor et. al have proved that all sub-

YU et al.: COMPRESSIVE SENSING WITH CHAOTIC SEQUENCE

733

gaussian matrix verify the RIP from geometrical point of view. In what follows, a brief proof following R. Baraniuk’s idea [3] connecting Johnson–Lindenstrauss property [1], [15] and RIP, is presented. Moreover, we can see what Lemma 2.1 implies for RIP. Before giving the proof, let us recall a lemma stated in [1]. Lemma 3.3: For ,

where with being any row vector of and being any unit vector. Remark 3.4: In Lemma 3.3, represents approximately less, . which goes to be strictly when sampling distance Proof for Theorem 3.1: The proof contains two parts: first prove the J–L property for any submatrix of , then conclude the RIP using permutation theory. the arbitrary column sub matrix J-L Property: Denote . For any unit vector , from of , with index set Chernoff’s inequality, given some positive value , it has

N = 800 M = 200

Fig. 2. Maximum sparsity for fixed signal size and variable number 2 (left) and for variable signal size 2 of measurements and fixed number of measurements (right).

[300;1000]

and denote i.e.,

M

[100; 500]

N

the union of all possible complementary events, . Then one obtains

where, for a fixed constant , whenever , the bound will only have the exponent with the exponential provided that . Hence we can choose sufficiently small to ensure that . Consequently, the probability for satisfying RIP is at least . where the last inequality is obtained by Taylor expansion and , which is the extremum point, and setting . Similarly, we can calculate the lower bound of its probability as follows:

where the last inequality is obtained by Taylor expansion and setting , which is the extremum point, and . , then one finally gets Choose (9) RIP: For any -sparse vector , denote the set of loca. tions where elements are nonzero, then defined in previous part can be set The column sub matrix one complementary event up and satisfies (9). Let us denote of condition in (1), i.e.,

IV. EXPERIMENTS As presented in Section III, we choose sampling distance , then generate the chaotic matrix following (8). The synthetic sparse signals adopted throughout this section are with only nonzero entries. The locations and signs of the peaks are chosen randomly. The measurement vector is computed by . Then the reconstruction from is solving by Linear Programming, which is accomplished using the SparseLab [13]. . The decision for failure reconstruction is One interest is the maximum sparsity which allows exact reconstruction of the signal. The results are given in Fig. 2 and show that the maximum sparsity in the case of chaotic matrix is similar to that in the case of Sparse matrix [6], Gaussian random matrix and Bernoulli random matrix. The delicate experiment for the maximum sparsity with respect to measurements for chaotic matrix is given in Fig. 3. Also, the probability of successful recovery (recovery rate) and fixed measurement number for fixed signal size is compared among these matrices, shown in Fig. 4. The result shows that chaotic matrix performs similar to the other three matrices. In addition, to evaluate the influence of the initial condition of the chaotic system (2), we set the initial state respectively to , and redo the experiment to test the recovery rate, shown in Fig. 5. The result shows that the initial state takes no influence to the recovery rate.

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sequence also satisfies RIP with overwhelming probability. From the experiments, it shows that chaotic matrix has the similar performance to Sparse matrix, Gaussian random matrix and Bernoulli random matrix. REFERENCES

Fig. 3. Probability of correct recovery for fixed signal size

N = 1000.

Fig. 4. Recovery rate for different sensing matrix.

Fig. 5. Recovery rate for different initial conditions for chaotic sequence.

V. CONCLUSION In this letter, we firstly recall the statistical property of one special chaotic system – Logistic map and prove that the generated sequence is approximately independent with ). Then sampling distance large enough (for instance we prove that matrix constructed with this sampled chaotic

[1] D. Achlioptas, “Database-friendly random projections,” in Proc. 20th ACM SIGMOD-SIGACT-SIGART Symp. Principles of Database Systems (PODS), New York, 2001, pp. 274–281. [2] L. Applebaum, S. Howard, S. Searle, and R. Calderbank, “Chirp sensing codes: Deterministic compressed sensing measurements for fast recovery,” Appl. Comput. Harmon. Anal., vol. 26, no. 2, Mar. 2009. [3] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Construct. Approx., vol. 28, no. 3, pp. 253–263, Dec. 2008. [4] D. Baron, S. Sarvotham, and R. G. Baraniuk, “Bayesian compressive sensing via belief propagation,” IEEE Trans. Signal Process., vol. 58, no. 1, pp. 269–280, Jan. 2010. [5] R. Berinde, A. C. Gilbert, P. Indyk, H. Karloff, and M. J. Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” in 46th Annu. Allerton Conf. Communication, Control, and Computing, Sep. 2008, pp. 798–805. [6] R. Berinde and P. Indyk, Sparse Recovery Using Sparse Random Matrices. Tech. Rep. MIT, 2008. [7] R. Calderbank, S. Howard, and S. Jafarpour, “Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 358–374, Apr. 2010. [8] E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comp. Rendus Acad. des Sci. I, vol. 346, no. 9-10, May 2008. [9] E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inform. Theory, vol. 51, no. 12, pp. 4203–4215, Dec. 2005. [10] E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?,” IEEE Trans. Inform. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006. [11] E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21–30, Mar. 2008. [12] R. A. DeVore, “Deterministic constructions of compressed sensing matrices,” J. Complexity, vol. 23, no. 4-6, Aug. 2007. [13] D. Donoho, I. Drori, V. Stodden, Y. Tsaig, and M. Shahram, Sparselab [Online]. Available: http://sparselab.stanford.edu/ [14] S. Howard, R. Calderbank, and S. Searle, “A fast reconstruction algorithm for deterministic compressive sensing using second order reedmuller codes,” in Conf. Information Sciences and Systems (CISS), Mar. 2008. [15] W. Johnson and J. Lindenstrauss, “Extensions of Lipschitz maps into a Hilbert space,” Contemporary Math., vol. 26, pp. 189–206, 1984. [16] S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, “Uniform uncertainty principle for bernoulli and subgaussian ensembles,” Construct. Approx., vol. 28, no. 3, pp. 277–289, Dec. 2008. [17] Logistic Map Wolfram Research [Online]. Available: http://documents. wolfram.com/ [18] A. Vlad, A. Luca, and M. Frunzete, “Computational measurements of the transient time and of the sampling distance that enables statistical independence in the logistic map,” in Proc. Int. Conf. Computational Science and Its Applications (ICCSA), Berlin, Germany, 2009, pp. 703–718. [19] K. Wang, W. Pei, H. Xia, M. G. Nustes, and J. A. Gonzalez, “Statistical independence in nonlinear maps coupled to non-invertible transformations,” Phys. Lett. A, vol. 372, no. 44, pp. 6593–6601, 2008.

Compressive Sensing With Chaotic Sequence - IEEE Xplore

Index Terms—Chaos, compressive sensing, logistic map. I. INTRODUCTION ... attributes of a signal using very few measurements: for any. -dimensional signal ...

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