Research Of Nonconvex Convolutional Dictionary Learning Problem Based On Proximal Gradient Algorithm

Convolutional dictionary learning(CDL)aims to learn a structured local convolutional dictionary and sparse coefficient maps from signals of interest,achieving better results than traditional patch-based dictionary learning methods on many image processing tasks.Current CDL methods using the patch-based,Fourier-based,or slice-based convolutional sparse representation are often limited in l1-constrained convex CDL problem,which may lead to non sparse coefficient maps,thus degrading the performance of applications.Recent studies have focused on nonconvex optimization algorithms for l0-constrained nonconvex CDL problem.Firstly,two classical algorithms for solving the CDL problem are reviewed in this paper.One is based on slice dictionary learning(SBDL algorithm)to solve convex CDL problem,and the other is based on Fourier domain dictionary learning algorithm(FB algorithm)to solve nonconvex CDL problem.Through the analysis,it can be found that the two algorithms have their advantages in solving the CDL problem,but also have limitations and can not solve the CDL problem well.Secondly,a new proximal gradient nonconvex optimization algorithm(PGNOA algorithm)is proposed for the nonconvex CDL problem.PGNOA is a method to solve nonconvex CDL problem based on slicing,PGNOA is also a nonconvex generalization of the slice-based l1 constrained convex CDL method.The convergence and complexity of the PGNOA algorithm are analyzed,and a large number of experiments are carried out on the benchmark data set.The results show that PGNOA is superior to the existing slice-based convex CDL method and Fourier-based non-convex CDL method in terms of sparsity and objective function values.Thirdly,the practical application effect of the PGNOA algorithm is analyzed,and it is applied to the field of image processing.The dictionary learned from PGNOA are applied to image inpainting and image separation tasks.The related algorithms are deduced and the experimental results show that the algorithm has better performance than the most advanced methods at present.Finally,based on the PGNOA algorithm and adding inertia term,an inertial proximal gradient algorithm(IPGM algorithm)is proposed.With the addition of the inertia term,the convergence speed of the algorithm is improved,then the complexity of the algorithm is analyzed,and the convergence of the algorithm is proved.Finally,the performance and effect of the IPGM algorithm and other algorithms are compared through experiments.The results show that the IPGM algorithm has a lower objective function value,lower sparse value,and higher peak signal-to-noise ratio of reconstruction.To sum up,the IPGM algorithm has many good theoretical properties,and the experimental results are efficient and intuitive.