Tensor Completion Methods Based On TV Regularization And TT Low Rank Nonconvex Optimization

There is a large amount of high-dimensional data in the field of modern engineering(such as color images,hyperspectral images,video and Internet tra c data streams),which play an important role in face recognition,medical imaging,tra c flow monitoring and other fields.These data bring convenience to us,but also bring many headaches.In real life,the data we observe are often incomplete or contaminated due to various reasons(such as damage,detector failure,etc).Therefore,how to use incomplete data information to estimate or recovery the complete data has become the focus of attention.Tensor completion aims to recover missing entries from partial observations for high-dimensional data.In the traditional tensor completion algorithm,the tensor is always unfolded into a matrix along the mode-k,which can help us better recover highdimensional data.However,this unfold way cannot fully capture the correlation between high-dimensional data.Therefore,how to use tensor decomposition to accurately characterize the intrinsic characteristics of high-dimensional data has become an important research direction.Traditional tensor decomposition includes Tucker decomposition,CP decomposition,t-product decomposition and Tensor train decomposition,which correspond to Tucker rank,CP rank,tube rank and TT rank.Previous studies have shown that TT rank has the ability to capture the hidden correlation between different modes of tensor,and can better reveal the internal structure of data in many applications.Based on this facts,we considers using the tensor TT rank to describe the low rank of data and realize the excavate of the intrinsic structure of the data.In chapter 3,we proposes a low rank tensor completion(LRTC)model based on TT rank non-convex optimization.In this model,a simple concave function is introduced to replace the rank function of the matrix obtained by mode-(1,2,· · ·,k)unfold,which will characterize the low-rank of tensor data.Based on this,we propose an implementable alternating minimization algorithm to solve the underlying optimization model.Experiments on color image restoration demonstrate that the proposed method has higher peak signal to noise ratio(PSNR)value,especially in the 10%-40% sampling rate case.Visual data not only has global low rank,but also has local smoothness.The low rank of a tensor is usually not su cient to recover the relevant data.Therefore,in Chapter4,we propose a new TV regularization based tensor completion model for color image inpainting.The model not only preserves the low-rank structure of the image,but also uses the TV regularization term to enhance the smoothness of the tensor,and explore the local smoothness of the image in the spatial domain.Numerical results show that our method has higher recovery accuracy than many existing state-of-the-art matricization and tensorization approaches.