A Survey On Convolutional Dictionary Learning Based On Proximal Gradient Algorithm

Convolutional dictionary learning is an important research direction of convolutional sparse representation.It learns dictionaries and corresponding sparse coefficient maps from signals.At present,convolutional dictionary learning plays an important role in many image processing tasks.In order to facilitate convolution calculation,most of the current convolutional dictionary learning algorithms are based on the Fourier domain,but the algorithms based on the Fourier domain tend to cause undesired boundary ambiguity and lack of local processing power.In view of this,this paper studies the problem of convex convolutional dictionary learning based on time domain computation from a mathematical point of view.Firstly,since the traditional block-based model loses the relation between image blocks,the slice-based convolutional dictionary learning problem is used,and the proximal gradient algorithm is used to solve the problem.In order to accelerate the convergence rate of the traditional proximal gradient algorithm,this paper introduces the inertia term into the proximal gradient algorithm,constructs the inertial proximal gradient method,namely,the IPGM algorithm,and applies it to the slice-based convolutional dictionary learning problem.Experimental results show that the introduction of the inertia term speeds up the convergence rate of the algorithm.Secondly,on the basis of the heavy ball system,this paper introduces the dry friction damping in physics into the heavy ball system to obtain the HBDF system,and deduces a new inertial proximal gradient method based on the dry friction and Moreau envelope,namely the IPGM-DM algorithm.The IPGM-DM algorithm is used to solve the optimization problem of single convex function by combining the relationship of proximal mapping and Moreau envelope.IPGM-DM algorithm transforms the optimization problem of the objective function to be optimized into the optimization problem of its Moreua envelope.In this paper,the convergence of IPGM-DM algorithm is proved theoretically,and the experimental results show that IPGM-DM algorithm has better performance in image reconstruction,image denoising and image fusion.Finally,considering that the introduction of Moreau envelope may bring about the approximation of the results,this paper extends the HBDF system to the additive HBDF system to solve the optimization problem of two convex functions,and deduces the inertial proximal gradient method with dry friction,namely the IPGM-DF algorithm.In the IPGM-DF algorithm,the solutions of proximal mapping of the dictionary and sparse coefficient are presented,and the convergence of the algorithm is proved and the computational complexity is analyzed.On the image reconstruction task,the objective function value obtained by IPGM-DF algorithm is the minimum,which effectively jumps out of the local minimum.