Nonconvex Compressed Sensing by Nature-Inspired Optimization Algorithms

The l₀ regularized problem in compressed sensing reconstruction is nonconvex with NP-hard computational complexity. Methods available for such problems fall into one of two types: greedy pursuit methods and thresholding methods, which are characterized by suboptimal fast search strategies. Nature-inspired algorithms for combinatorial optimization are famous for their efficient global search strategies and superior performance for nonconvex and nonlinear problems. In this paper, we study and propose nonconvex compressed sensing for natural images by nature-inspired optimization algorithms. We get measurements by the block-based compressed sampling and introduce an over-complete dictionary of Ridgelet for image blocks. An atom of this dictionary is identified by the parameters of direction, scale and shift. Of them, direction parameter is important for adapting to directional regularity. So we propose a two-stage reconstruction scheme (TS_RS) of nature-inspired optimization algorithms. In the first reconstruction stage, we design a genetic algorithm for a class of image blocks to acquire the estimation of atomic combinations in all directions; and in the second reconstruction stage, we adopt clonal selection algorithm to search better atomic combinations in the sub-dictionary resulted by the first stage for each image block further on scale and shift parameters. In TS_RS, to reduce the uncertainty and instability of the reconstruction problems, we adopt novel and flexible heuristic searching strategies, which include delicately designing the initialization, operators, evaluating methods, and so on. The experimental results show the efficiency and stability of the proposed TS_RS of nature-inspired algorithms, which outperforms classic greedy and thresholding methods.