A Research Of Low-rank Tensor Completion Model And Algorithm

In recent years,tensor analysis has attracted the increasing attention.Tensor,as a high-dimensional array,is able to represent most realistic data more naturally than ”flat”matrices,especially for digital images,including real color images,video,multispectral(MSI),hyperspectral(HSI),etc.However,during the acquisition and transmission of images,some elements of the observed tensor may be lost due to the interference of various factors such as atmosphere,sensors,transmission lines,etc.This problem seriously affects image quality and poses great difficulties for subsequent image processing applications.Tensor completion is designed to recover lost elements in the observation tensor and has been widely used in the fields of image and video inpainting,multispectral and hyperspectral data recovery,high-dimensional network data analysis and network personalization recommendations.Over the years,the low-rankness of matrices as a prior describing the intrinsic structure of real data has had more successful applications in matrix completion problems.But unlike in the case of matrices,the tensor rank is not yet uniquely defined.So how to more accurately describe the low order of the tensor is a question worth studying.Recently,tensor singular value decomposition(t-SVD)based on discrete Fourier transform(DFT)has achieved good performance on low-rank tensor completion.This paper is based on t-SVD a d study in two main ways.Firstly,we consider boundary conditions in t-SVD.The periodic boundary conditions that take into account the discrete Fourier transform assumptions on which the tproduct in TNN are not realistic.We consider reflective boundary conditions and study the corresponding discrete cosine transform.Based on the property that the t-product of two tensors can be calculated by the product of the block Toplitz plus Hankel matrix and the block unfolding matrix of the tensor obtained by the discrete cosine transform,we can quickly calculate the t-product.Based on this,we present a tensor singular value decomposition based on the discrete cosine transform and a new tensor nuclear norm.With the new tensor nuclear norm,we propose a new low-rank tensor completion model and develop the alternating direction multiplier method to solve this problem.Numerical experiments proved that our method performs better in video and multispectral image recovery.Secondly,we consider the deep image prior.The low-rank term is not sufficient for effective tensor completion,and we chose to add additional deep learning-based image a prior to enhance the performance.We use convolutional neural networks to denoise forward slices in the spatial domain of multidimensional images,and the regularization term based on deep learning are able to preserve the details of the images well.At the same time,for low sampling rates,this data-driven regularization term also refines the previous flaw of low rank tensor completion.In order to efficiently solve the proposed model,we develop the alternating direction multiplier method in a highly flexible plugand-play(PnP)framework,which allows us to insert a deep denoiser.The numerical experiments have shown that our proposed method is able to recover better detail and has a huge advantage over traditional methods.