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Reconstructing chaotic signals from noised data plays a critical role in many areas of science and engineering. However, the inherent features, such as aperiodic property, wide band spectrum, and extreme sensitivity to initial values, present a big challenge of reducing the noises in the contaminated chaotic signals. To address the above issues, a novel noise reduction algorithm based on the collaborative filtering is investigated in this paper. By exploiting the fractal self-similarity nature of chaotic attractors, the contaminated chaotic signals are reconstructed subsequently in three steps, i.e., grouping, collaborative filtering, and signal reconstruction. Firstly, the fragments of the noised signal are collected and sorted into different groups by mutual similarity. Secondly, each group is tackled with a hard threshold in the two-dimensional (2D) transforming domain to attenuate the noise. Lastly, an inverse transformation is adopted to estimate the noise-free fragments. As the fragments within a group are closely correlated due to their mutual similarity, the 2D transform of the group should be sparser than the one-dimensional transform of the original signal in the first step, leading to much more effective noise attenuation. The fragments collected in the grouping step may overlap each other, meaning that a signal point could be included in more than one fragment and have different collaborative filtering results. Therefore, the noise-free signal is reconstructed by averaging these collaborative filtering results point by point. The parameters of the proposed algorithm are discussed and a set of recommended parameters is given. In the simulation, the chaotic signal is generated by the Lorenz system and contaminated by addictive white Gaussian noise. The signal-to-noise ratio and the root mean square error are introduced to measure the noise reduction performance. As shown in the simulation results, the proposed algorithm has advantages over the existing chaotic signal denoising methods, such as local curve fitting, wavelet thresholding, and empirical mode decomposition iterative interval thresholding methods, in the reconstruction accuracy, improvement of the signal-to-noise ratio, and recovering quality of the phase portraits.
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Keywords:
- chaotic signal /
- collaborative filtering /
- noise reduction
[1] Feng J C 2012 Chaotic Signals and Information Processing (Beijing: Tsinghua University Press) pp32-35 (in Chinese) [冯久超 2012 混沌信号与信息处理 (北京: 清华大学出版社) 第3235页]
[2] Badii R, Broggi G, Derighetti B, Ravani M, Ciliberto S, Politi A, Rubio M A 1988 Phys. Rev. Lett. 60 979
[3] Liu X Y, Qiu S S, Lau C M 2005 J. Syst. Eng. Electron. 16 253
[4] Cawley R, Hsu G H 1992 Phys. Rev. A 46 3057
[5] Leontitsis A, Bountis T, Pange J 2004 Chaos 14 106
[6] Han M, Liu Y H, Xi J H, Guo W 2007 IEEE Signal Proc. Lett. 14 62
[7] Constantine W L B, Reinhall P G 2001 Int. J. Bifurcat. Chaos 11 483
[8] Kopsinis Y, McLaughlin S 2009 IEEE Trans. Signal Proc. 57 1351
[9] Wang X F, Qu J L, Gao F, Zhou Y P, Zhang Y X 2014 Acta Phys. Sin. 63 170203 (in Chinese) [王小飞, 曲建岭, 高峰, 周玉平, 张翔宇 2014 物理学报 63 170203]
[10] Wei X L, Lin R L, Liu S Y, Zhang C H 2016 Shock Vib. 2016 1
[11] Wang W B, Zhang X D, Wang X L 2013 Acta Phys. Sin. 62 050201 (in Chinese) [王文波, 张晓东, 汪祥莉 2013 物理学报 62 050201]
[12] Li G M, L S X 2015 Acta Phys. Sin. 64 160502 (in Chinese) [李广明, 吕善翔 2015 物理学报 64 160502]
[13] Tung W W, Gao J B, Hu J, Yang L 2011 Phys. Rev. E 83 046210
[14] Wang M J, Wu Z T, Feng J C 2015 Acta Phys. Sin. 64 040503 (in Chinese) [王梦蛟, 吴中堂, 冯久超 2015 物理学报 64 040503]
[15] Hu J F, Zhang Y X, Yang M, Li H Y, Xia W, Li J 2016 Nonlinear Dynam. 84 1469
[16] Donoho D L, Johnstone I M 1994 Biometrika 81 425
[17] Dabov K, Foi A, Katkovnik V, Egiazarian K 2007 IEEE Trans. Image Proc. 16 2080
[18] Lebrun M 2012 Image Proc. On Line 2 175
[19] Yu S M 2011 Chaotic Systems and Chaotic Circuits (Xi'an: Xidian University Press) pp10-12 (in Chinese) [禹思敏 2011 混沌系统与混沌电路 (西安: 西安电子科技大学出版社) 第1012页]
[20] He T, Zhou Z O 2007 Acta Phys. Sin. 56 693 (in Chinese) [贺涛, 周正欧 2007 物理学报 56 693]
[21] Tang Y F, Liu S L, Lei N, Jiang R H, Liu Y H 2012 Acta Phys. Sin. 61 170504 (in Chinese) [唐友福, 刘树林, 雷娜, 姜锐红, 刘颖慧 2012 物理学报 61 170504]
[22] Coifman R R, Donoho D L 1995 Lect. Notes Stat. 103 125
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[1] Feng J C 2012 Chaotic Signals and Information Processing (Beijing: Tsinghua University Press) pp32-35 (in Chinese) [冯久超 2012 混沌信号与信息处理 (北京: 清华大学出版社) 第3235页]
[2] Badii R, Broggi G, Derighetti B, Ravani M, Ciliberto S, Politi A, Rubio M A 1988 Phys. Rev. Lett. 60 979
[3] Liu X Y, Qiu S S, Lau C M 2005 J. Syst. Eng. Electron. 16 253
[4] Cawley R, Hsu G H 1992 Phys. Rev. A 46 3057
[5] Leontitsis A, Bountis T, Pange J 2004 Chaos 14 106
[6] Han M, Liu Y H, Xi J H, Guo W 2007 IEEE Signal Proc. Lett. 14 62
[7] Constantine W L B, Reinhall P G 2001 Int. J. Bifurcat. Chaos 11 483
[8] Kopsinis Y, McLaughlin S 2009 IEEE Trans. Signal Proc. 57 1351
[9] Wang X F, Qu J L, Gao F, Zhou Y P, Zhang Y X 2014 Acta Phys. Sin. 63 170203 (in Chinese) [王小飞, 曲建岭, 高峰, 周玉平, 张翔宇 2014 物理学报 63 170203]
[10] Wei X L, Lin R L, Liu S Y, Zhang C H 2016 Shock Vib. 2016 1
[11] Wang W B, Zhang X D, Wang X L 2013 Acta Phys. Sin. 62 050201 (in Chinese) [王文波, 张晓东, 汪祥莉 2013 物理学报 62 050201]
[12] Li G M, L S X 2015 Acta Phys. Sin. 64 160502 (in Chinese) [李广明, 吕善翔 2015 物理学报 64 160502]
[13] Tung W W, Gao J B, Hu J, Yang L 2011 Phys. Rev. E 83 046210
[14] Wang M J, Wu Z T, Feng J C 2015 Acta Phys. Sin. 64 040503 (in Chinese) [王梦蛟, 吴中堂, 冯久超 2015 物理学报 64 040503]
[15] Hu J F, Zhang Y X, Yang M, Li H Y, Xia W, Li J 2016 Nonlinear Dynam. 84 1469
[16] Donoho D L, Johnstone I M 1994 Biometrika 81 425
[17] Dabov K, Foi A, Katkovnik V, Egiazarian K 2007 IEEE Trans. Image Proc. 16 2080
[18] Lebrun M 2012 Image Proc. On Line 2 175
[19] Yu S M 2011 Chaotic Systems and Chaotic Circuits (Xi'an: Xidian University Press) pp10-12 (in Chinese) [禹思敏 2011 混沌系统与混沌电路 (西安: 西安电子科技大学出版社) 第1012页]
[20] He T, Zhou Z O 2007 Acta Phys. Sin. 56 693 (in Chinese) [贺涛, 周正欧 2007 物理学报 56 693]
[21] Tang Y F, Liu S L, Lei N, Jiang R H, Liu Y H 2012 Acta Phys. Sin. 61 170504 (in Chinese) [唐友福, 刘树林, 雷娜, 姜锐红, 刘颖慧 2012 物理学报 61 170504]
[22] Coifman R R, Donoho D L 1995 Lect. Notes Stat. 103 125
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