The Chaotic Measurement Matrixfor Compressed Sensing

Shihong Yao1, Tao Wang1, Weiming Shen1, Shaoming Pan1,and Yanwen Chong2(&)

1 State Key Laboratory for Information Engineering in Surveying, Mapping andRemote Sensing, Wuhan University, 129 Luoyu Road, Wuhan 430079, China

yao_shi_hong@whu.edu.cn2 Lmars, Wuhan University, Wuhan 430079, China

ywchong@whu.edu.cn

Abstract. How to construct a measurement matrix with good performance andeasy hardware implementation is the core research problem in compressedsensing. In this paper, we present a simple and efficient measurement matrixnamed Incoherence Rotated Chaotic (IRC) matrix. We take advantage of thewell pseudorandom of chaotic sequence, introduce the concept of the incoher-ence factor and rotation, and adopt QR decomposition to obtain the IRC mea-surement matrix which is suited for sparse reconstruction. Simulation resultsdemonstrate IRC matrix has a better performance than Gaussian random matrix,Bernoulli random matrix and other state-of-the-art measurement matrices. Thusit can efficiently work on both natural image and remote sensing image.

Keywords: Measurement matrix � Compressed sensing � Chaotic sequence �Restricted Isometry Property (RIP)

1 Introduction

Compressed sensing (CS) technique brings great convenience for data storage, trans-mission and processing and has already attracted broad attention in the fields of remotesensing and medical imaging and become a research issue in recent years. In CS theory,there are three core issues that are signal sparse representation, measurement matrixdesign and reconstruction algorithm construction [1]. The designing level of mea-surement matrix directly influences signal reconstruction error. In order to evaluate theperformance of measurement matrices, Restricted Isometry Property (RIP) [2] is pro-posed by Candes et al. in 2007, which described three mathematics criterions that themeasurement matrix should satisfy. In practice, it is difficult to prove whether themeasurement matrices satisfy RIP, thus we adopt an equivalent condition of RIPcriterion, which is the incoherence between the measurement matrix and the com-pressible signal, to guide the design of measurement matrix.

Random measurement matrix [3, 4] has high computational complexity and diffi-cult hardware implementation and needs large storage space. Deterministic measure-ment matrix [5–7] can make up the shortfall on hardware implementation of random

© Springer International Publishing Switzerland 2015D.-S. Huang et al. (Eds.): ICIC 2015, Part I, LNCS 9225, pp. 58–64, 2015.DOI: 10.1007/978-3-319-22180-9_6

measurement matrix. On this account the contradiction between determinacy andrandomness come into being.

Chaos theory reveals the unity of determinacy and randomness exactly. Thesequences generated by chaotic system have outstanding pseudo-randomness and areeasy to generate and reproduce. Therefore, chaotic sequence [8, 9] can be used toconstruct measurement matrix in CS. So this paper makes use of strong pseudo-randomness of chaotic system, introduces the concept of the incoherence factors androtation based on the discrete chaotic sequence, and adopts QR decomposition to obtaina measurement matrix suited for sparse reconstruction. This measurement matrix isnamed Incoherence Rotated Chaotic (IRC) matrix. With the IRC matrix, we canreconstruct the sparse signal with a high accuracy.

2 Background of CS Theory

CS is an efficient signal acquisition framework for signals that are sparse or com-pressible in an appropriate domain. Suppose that a signal X is sparse in a certain set oforthogonal basis or a tight frame domain w (such as wavelet domain), where X 2 RN .In other words, there exists a sparse vector a such that X ¼ wa, where the vector a isequivalent or approximate sparse representation of X in the domain w. We use ameasurement basis U which is incoherent with transformation basis w to measuresignal X, where U 2 RM�N and M\\N, hence we obtain the observed vectorY 2 RM�1. Finally we can make use of optimization algorithm to reconstruct the ori-ginal signal X from the observed sets accurately or with a high probability.

It is difficult to directly construct a measurement matrix U to make Uv obey RIPcriterion. v is strictly s-sparse N- dimensional vector. We can only assure the sensingmatrix Uv to obey RIP as much as possible. Only when matrix U is adequately inco-herent with sparse coefficient v, L1 norm optimization method can be used to solve a.

a ¼ min ak k1 s:t Y ¼ Aa ð1Þ

Where A ¼ UW. The problem of L1 norm minimization is equivalent to the problem ofsignal reconstruction. At present, OMP algorithm is an effective in signal reconstruc-tion and chosen as the reconstruction algorithm in this paper.

3 Construction of Incoherence Rotated Chaotic Matrix

Chaotic is a kind of seeming random or irregular movement, which appears in adeterministic nonlinear system. Chaotic system has both determinacy and randomness.Therefore adopting a chaotic system to construct measurement matrix in CS is feasible.

Logistic chaotic system is the most popular used nonlinear dynamics chaotic sys-tems [10]. In this paper, we select Logistic mapping system to create chaotic mea-surement matrix. The Logistic map function is expressed as:

The Chaotic Measurement Matrix for Compressed Sensing 59

xnþ1 ¼ lxnð1� xnÞ ð2Þ

where l 2 0; 4ð Þ; xn 2 0; 1ð Þ. In order to enhance the randomness of chaotic factors,we remove the first 1000 elements, and then perform downsampling on thechaotic factors with sampling interval d. Hence the chaotic factors aremi ¼ 1 - 2x1000þid ; i ¼ 1; 2; . . .ð Þ.

The IRC matrix Uk of the kth compression ratio can be denoted as

U ¼ffiffiffiffiffiffiffi

12kk

r

v1þkn v2þkn � � � vn�1þkn vnþkn

gvnþkn v1þkn v2þkn � � � vn�1þkn

gvn�1þkn g2vnþkn v1þkn � � � vn�2þkn

gvn�2þkn g2vn�1þkn g3vnþkn � � � vn�3þkn

..

. ... ..

. . .. ..

.

gvn�kkþ2þkn g2vn�kkþ3þkn g3vn�kkþ4þkn � � � vn�kkþ1þkn

2

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

5

ð3Þ

where λk (k = 1, 2,…M) is observed frequency of the kth compression ratio. As shownin formula (3), when the elements move to the beginning of a row from the end, theyare multiply by the fixed incoherent coefficient g, where g 2 0:6; 1½ � for each rotation.To construct a measurement matrix suited for variety compression ratios, we only needto determine the first row elements on different compression ratios and adopt the sameconstruction method with formula (3). Then we have Uk ¼ RT

k QTk . The elements of the

main diagonal in Rk are reserved and off-diagonal elements are set to zero, thus a newdiagonal matrix R

0k is generated. The IRC matrix can denoted as U

0k ¼ R

0Tk Q

Tk . The IRC

matrix U0k of variety compression ratios can be obtained by the above steps.

4 Experimental Results

According to the construction procedures of IRC matrix presented in Sect. 3, wechoose sampling distance d ¼ 10, then generate the IRC matrix by formula (3). In thissection, we perform simulations using variety images of size 256 × 256 by Intel(R),Core(TM), dual-core i3 processor, 3.07 GHz, RAM 2.99 G, Matlab 2009a. The imagesparse algorithm is the wavelet transform and the image reconstruction algorithm is theOMP reconstruction algorithm [11].

Figure 1 shows the varied original standard test images including Remote sensingimage, natural scenery, fingerprints, as well classical Lena, Barbara, Peppers, Tiffany,Baboon.

Figure 2 The reconstruction results are different on different compression ratios forone image. Figure 2 shows that the peak signal-to-noise ratios (PSNRs) between ori-ginal test images and reconstruction images on different compression ratios usingGaussian random measurement matrix and IRC matrix. The X-axis shows the serialnumber of images and the Y-axis shows the PSNRs between original test images and

60 S. Yao et al.

reconstruction images. ‘Chaos’ represent the PSNRs using IRC matrix and ‘rand’represent the PSNRs using Gaussian random measurement matrix.

As shown in Fig. 2, on different compression ratios, we can see that the PSNRs ofthe reconstructed images, except the 30th image named testpat, using the IRC matrixare better than Gaussian random measurement and the PSNRs are improved by. 1.5 to2.5 dB.

To further test the reconstruction quality of IRC matrix, we take structure similaritymeasure (SSIM) that takes into account some of the spatial relationships betweenregions as another important measure of image reconstruction performance [12]. Table 1shows that the mean values of SSIM for all the test images on different compressionratios using Gaussian random measurement matrix and IRC matrix and it indicates thatthe SSIM under IRC matrix are higher than the SSIM under Gaussian random mea-surement matrix. From the view of the results of two measurable indicators, the

(1) (2) (3) (4) (5) (6) (7) (8)

(9) (10) (11) (12) (13) (14) (15) (16)

(17) (18) (19) (20) (21) (22) (23) (24)

(25) (26) (27) (28) (29) (30) (31) (32)

Fig. 1. The original standard test images

(a) (b) (c)

(dB)

(dB)

(dB)

Fig. 2. The reconstructed image PSNRs on different compression ratios using Gaussian randommeasurement matrix and IRC matrix. (compression ratios are 0.4, 0.55, 0.7 respectively)

The Chaotic Measurement Matrix for Compressed Sensing 61

reconstruction performance of IRC is significantly better than that of Gaussian randommeasurement matrix.

Figure 3 shows when the compression ratio is 0.4 or 0.8, the reconstruction resultcomparisons of classical images, including Barbara、Lena、Peppers. The originalimages are in the first column, the obvious distinction between reconstruction imagesusing Gaussian random measurement matrix are in the second column and fourthcolumn and the obvious distinction between reconstruction images using IRC matrixare in the third column and fifth column. We can see that the visual quality ofreconstructed images using IRC matrix is much better than Gaussian random mea-surement matrix and the noise points of reconstructed images using IRC matrix are lessthan Gaussian random measurement matrix.

The comparison of the reconstructed image PSNR using different measurementmatrix under different compression ratios are shown in Table 2. The experimentalprocesses refer to the reference [13]. We select the discrete wavelet transform as thesparse algorithm and the OMP algorithm as the image reconstruction algorithm. Theincoherence coefficient η is set to 0.85. The experiment of each measurement matrixunder different compression ratios to reconstruct the 256 × 256 Lena.bmp 2-D imagehas been conducted. As shown in Table 2, the reconstructed image PSNRs of theseeight measurement matrices increase with compression ratio increase and the PSNR of

Table 1. The mean values of SSIM for all the test images on different compression ratios usingGaussian random measurement matrix and IRC matrix. (compression ratios are 0.4, 0.55, 0.7respectively)

M/NThe measurement matrices

0.4 0.55 0.7

SSIM of Gaussian random measurement matrix 0.6277 0.7654 0.8410SSIM of IRC matrix 0.7584 0.8358 0.8880

original random matrix chaotic matrixcompression ratio is 0.4

random matrix chaotic matrixcompression ratio is 0.8

Fig. 3. The reconstruction result comparisons of classical images

62 S. Yao et al.

the reconstructed image using the IRC matrix is higher than the other measurementmatrices. In the same simulation environment and simulation process, Table 3 showsthat IRC matrix performs better than the other measurement matrices in the imagereconstruction effect of PSNR under the fixed compression ratio of 0.6, and IRC matrixhas less time consumed in reconstructing the image.

All the experiment results show that the reconstruction performance of the IRCmatrix is better than the other state-of-the-art measurement matrices. The constructionprocedures of IRC matrix only need a small amount of calculations and storage spacethus it has an easy embedded hardware implementation.

5 Conclusion

With the development and promotion of compressed sensing theory in practicalapplication, data acquisition has significantly overcome the limitations of Nyquistsampling theory. However, the existing random measurement matrices and determin-istic measurement matrices have their own deficiency respectively. In this paper, wetake advantage of the well pseudorandom of chaotic sequence, introduce the concept ofthe incoherence factor and rotation, and adopt QR decomposition to obtain the IRCmeasurement matrix which is suited for sparse reconstruction. Simulated resultsdemonstrate IRC matrix has a better performance than Gaussian random measurementmatrix and the other state-of-the-art measurement matrices. The reconstructed imagePSNRs of IRC matrix are improved at 1.5 dB to 2.5 dB at least and IRC matrix canefficiently work on both natural image and remote sensing image.

Table 2. Each measurement matrix under different compression ratios of reconstructed imagePSNR values

M/N Chao(dB) Gaus(dB) Four(dB) Bern(dB) Ploy(dB) Tope(dB) Rota(dB) Kron(dB)

0.4 26.3461 24.2485 24.0759 23.4901 17.9578 18.3825 18.9907 22.74380.55 29.2168 27.5921 27.3359 26.9858 20.4652 20.5625 20.9537 24.64910.7 31.5148 29.9208 28.7732 29.3375 23.5479 23.6343 23.5233 27.77580.8 33.1058 31.3538 30.1493 30.5902 24.7167 25.0692 25.2591 29.0746

Chao, Chaotic noncorrelation matrix; Gaus, Gaussian random matrix; Four, Fourier randommatrix; Bern, Bernoulli matrix;Poly, polynomial matrix; Tope, Toeplitz matrix; Rota, rotationmatrix; Kron, based on orthogonal basis Kronecker product of matrix.

Table 3. Each measurement matrix reconstruction effect and time consumed.

The measurementmatrices

Chao Gaus Four Bern Ploy Tope Rota Kron

PSNR(dB) 30.1309 28.4385 27.9497 28.0258 22.0542 21.3743 22.4629 26.2274The image reconstructiontime consumed (s)

7.0509 13.7541 13.9357 11.0960 13.0644 7.8477 7.2542 7.5738

The Chaotic Measurement Matrix for Compressed Sensing 63

Acknowledgments. This paper was supported by the Foundation Research Funds for the centralUniversities (204201kf0242, 204201kf0263), National Natural Science Foundation of China(41271398), Shanghai Aerospace Science and Technology Innovation Fund Projects(SAST201425).

References

1. Candès, E.J.: Compressive sampling. In: Proceedings of the International Congress ofMathematicians, vol. 3, pp. 1433 -1452, (2006)

2. Candes, E.J., Tao, T.: Near-optimal signal recovery from random projections: universalencoding strategies? IEEE Trans. Inf. Theor. 52(12), 5406–5425 (2006)

3. Sun, R., Zhao, H., Xu, H.: The application of improved Hadamard measurement matrix incompressed sensing. In: Proceedings of International Conference on Systems andInformatics (ICSAI), pp. 1994–1997 (2012)

4. Zhang, G., Jiao, S., Xu, X., et al.: Compressed sensing and reconstruction with bernoullimatrices. In: Proceedings of International Conference on Information and Automation(ICIA), pp. 455–460 (2010)

5. Yin, W., Morgan, S., Yang, J., et al.: Practical compressive sensing with Toeplitz andcirculant matrices. In: Proceedings of International Society for Optics and Photonics onVisual Communications and Image Processing, pp. 77440K–77440K (2010)

6. DeVore, Ronald A.: Deterministieeon structions of compressed sensing matrices.J. Complex. 23(4–6), 918–925 (2007)

7. Bajwa, W., Haupt, J., Raz, G., Wright, S.J., Nowak, R.D.: Toeplitz-structured compressedsensing matrices. In: Proceedings of the IEEE Workshop on Statistical Signal Processing,Washington D. C., USA, pp. 294–298 (2007)

8. Yu, L., Barbot, J.P., Zheng, G., et al.: Compressive sensing with chaotic sequence. SignalProcess. Lett. 17(8), 731–734 (2010)

9. Linh-Trung, N., Van Phong, D., Hussain, Z.M., et al.: Compressed sensing using chaosfilters. In: Proceedings of IEEE Conference on Telecommunication Networks andApplications (ATNAC), pp. 219–223 (2008)

10. Huang, W., Ye, X.: Topological complexity, return times and weak disjointness. ErgodicTheor. Dyn. Syst. 24(3), 825–846 (2004)

11. Donoho, D.L., Tsaig, Y., Drori, I., et al.: Sparse solution of underdetermined linearequations by stagewise orthogonal matching pursuit. Technique report TR-2006-2,Standford University, Department of Statistics (2006)

12. Wang, H., Maldonado, D., Silwal, S.: A nonparametric-test-based structural similaritymeasure for digital images. Comput. Stat. Data Anal. 55(11), 2925–2936 (2011)

13. Zhang, B., Tong, X., Wang, W., et al.: The research of Kronecker product-basedmeasurement matrix of compressive sensing. EURASIP J. Wirel. Commun. Networking2013(1), 1–5 (2013)

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