The Improvement Of Sparse Decomposition Algorithm Based On Smooth L₀ Norm

The representation of signal(image)has always been one of the most important research topics in the field of signal(image)analysis and processing,because the more concise representation ways are,the more efficient and convenient subsequent signal(image)analysis and processing would be.In 1993,Mallat and Zhang proposed a complete dictionary to represent the signal based on wavelet analysis and introduced the idea of sparse representation.In 2006,Donoho et al.put forward the theory of compressive sensing,which was realized with much lower sampling rate to represent and compress the signal,compared to the Nyquist sampling rate.Since then,sparse representation theory has gained a lot of attention in the field of signal(image).At the same time,sparse representation theory has been widely extended to data processing,machine learning,pattern classification,blind source separation,computer vision and robotics due to its solid mathematical theory.The theory of sparse representation shows that if a signal is compressible in a transform domain,it can be sparsely represented by an over-complete dictionary with the uncorrelated transform basis,and the signal after sparse representation can be reconstructed with high precision.Based on the characteristics of the signal,it is possible to adaptively select suitable over-complete dictionary.Over the years,people have made a lot of explorations on the two major tasks:sparse decomposition of signals and construction of over-complete dictionary,and have obtained remarkable achievements.In the sparse decomposition task,many algorithms for solving the sparse representation model have been put forward from various aspects.The typical one is the sparse representation method based on the smooth l0 norm.The basic idea is to introduce a smoothing function to approximate the to norm,so the problem of minimizing the norm is transformed into an optimization problem aiming at the smoothing function so as to avoid the NP difficulty caused by minimizing the norm of l0.The most representative of these methods is the SLO algorithm proposed by Mohimani et al.(Smoothed l0 Norm),which chooses a Gaussian function with parameters to form a smoothing function.The steepest descent method and the gradient projection principle are used to approximate the l0 norm of the vector.With its high efficiency,effectiveness and no need for priori sparseness,the SLO algorithm has become a typical algorithm in sparse representation.In this paper,the SLO algorithm is further discussed.Based on the original algorithm,two improvements are made:the ONReSLO algorithm is proposed and the performance with different parameters is compared;the two-dimensional threshold SLO algorithm is proposed to reduce the algorithm computational complexity.Finally,in-depth analysis is done regarding the algorithm's good robustness against large noises.The effectiveness of the improved algorithm is also verified by the simulation results.