Gradient Immune-based Sparse Signal Reconstruction Algorithm for Compressive Sensing

Abstract

The reconstruction aspect is the main core of the compressive sensing theory, in which the sparse signal is reconstructed from an incomplete set of random measurements. The constraint of spare signal reconstruction is the minimization of l0-norm, especially under noise condition. Thus, this paper proposes a new method called Gradient Immune-based Sparse Signal Reconstruction Algorithm for Compressive Sensing (GISSRA-CS) to optimize the trade-off between the reconstruction error and the sparsity requirements. The principle of the GISSRA-CS method is embedding the Gradient Local Search (GLS) method in the evolutionary process of the Immune Algorithm (IA) for solving the sparsity problem. Here, the sparsity problem is formulated as a multi-objective problem (MOP) by combining l0 and l1-norms of a solution and l2-norm of a residual error in the same criterion to optimize the trade-off between the sparsity requirements and the error. This MOP problem is solved in a several subproblems manner by assigning different weights for each subproblem to increase the population diversity. For a long-term sparse signal, the window method is used to divide it into multiple short signals to improve the performance and computational complexity of the proposed method. Mathematical analysis and simulation experiments are presented to validate the performance and complexity of the GISSRA-CS method. Results of different simulation scenarios based on the benchmark and simulated signals show that the GISSRA-CS method outperforms the other methods in recovering the sparse signals with a small reconstruction error from noiseless and noisy measurements. Furthermore, the convergence of GISSRA-CS is faster than the other evolutionary recovery methods, but it is slower than the traditional recovery methods.