In this paper, we consider sparse decomposition (SD) of twodimensional (2D) signals on overcomplete dictionaries with separable atoms. Although, this problem can be solved by converting it to the SD of one-dimensional (1D) signals, this approach requires a tremendous amount of memory and computational cost. Moreover, the uniqueness constraint obtained by this approach is too restricted. Then in the paper, we present an algorithm to be used directly for sparse decomposition of 2D signals on dictionaries with separable atoms. Moreover, we will state another uniqueness constraint for this class of decomposition. Our algorithm is obtained by modifying the Smoothed L0 (SL0) algorithm, and hence we call it two-dimensional SL0 (2D-SL0).

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In this paper, we propose an image compression-encryption method based on two-dimensional (2D) sparse representation and chaotic system. In the first step of this method, the input image is extended in a transform domain to obtain a sparse representation. To achieve better performance of image compression by 2D sparse recovery, the sparse representation is scrambled via a chaotic confusion. This step helps the satisfaction of the uniqueness conditions for sparse recovery, and the security level of encryption is increased. Then, two orthogonal measurement matrices are generated using the chaotic time series. The singular value decomposition is used to compress the sparse scrambled representation in two dimensions. Finally, to reduce the correlation between adjacent pixels in the compressed matrix, and obtain a uniform distribution in the encrypted image, a compressed scrambling matrix based on chaotic confusion is used. Then, XOR operation is applied for final encryption. In the decryption process, to improve the compression efficiency, the total variation constraint is added to the 2D sparse recovery problem based on the smoothed norm. The simulation results demonstrate the satisfying performance of the proposed method for different compression ratios. Security analysis describes the effectiveness of the proposed encryption approach.

Objective. The accurate decomposition of a mother’s abdominal electrocardiogram (AECG) to extract the fetal ECG (FECG) is a primary step in evaluating the fetus’s health. However, the AECG is often affected by different noises and interferences, such as the maternal ECG (MECG), making it hard to evaluate the FECG signal. In this paper, we propose a deep-learning-based framework, namely ‘AECG-DecompNet’, to efficiently extract both MECG and FECG from a single-channel abdominal electrode recording. Approach. AECG-DecompNet is based on two series networks to decompose AECG, one for MECG estimation and the other to eliminate interference and noise. Both networks are based on an encoder-decoder architecture with internal and external skip connections to reconstruct the signals better. Main results. Experimental results show that the proposed framework performs much better than utilizing one network for direct FECG extraction. In addition, the comparison of the proposed framework with popular single-channel extraction techniques shows superior results in terms of QRS detection while indicating its ability to preserve morphological information. AECG-DecompNet achieves exceptional accuracy in the precision metric (97.4%), higher accuracy in recall and F 1 metrics (93.52% and 95.42% respectively), and outperforms other state-of-the-art approaches. Significance. The proposed method shows a notable performance in preserving the morphological information when the FECG within the AECG signal is weak.

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.