Denoising of contaminated chaotic signals based on collaborative filtering

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 signalto- 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.