Research On Tensor Robust Principal Component Analysis Based On Truncated Nuclear Norm

With the popularization and development of mobile Internet and multimedia technology,people are not only users of pictures and videos,but also their creators.Under the limitation of subjective and objective conditions,there will be a lot of noise in the process of shooting or transmission.The use of some advanced optical hardware equipment can reduce the noise,but many advanced optical hardware is rather expensive and difficult to popularize,so the method of image denoising with software is becoming more and more important.In this paper,the image with noise is taken as the research object to study the image data expressed by tensor.Based on the robust principal component analysis,the image denoising model has been widely studied,and good performance has been achieved under certain conditions.A noise matrix is decomposed into the sum of a low-rank matrix and a sparse matrix,the nuclear norm is used to describe the low-rank matrix,and the 1? norm is used to describe the sparsity of the noise matrix.However,the model has two major disadvantages: 1)The nuclear norm can not effectively describe the matrix rank function,since the penalty for the large singular value is too heavy,and the incoherent condition of the matrix is difficult to satisfy.Therefore,this method can only obtain the suboptimal solution.2)The robust principal component analysis model can only deal with two-dimensional matrix data.In real life,most of the data exist in the form of high-dimensional tensors.When dealing with data expressed by tensors,tensors are usually converted into vectors or matrices,while the description of the potential multidimensional constraint relations of tensors is ignored.In view of the shortcomings of robust principal component analysis,tensor data with multiple linear structures are studied.Based on the singular value decomposition of tensors,the truncated nuclear norm of tensors is defined to make up for the deficiency of the nuclear norm,and a new model,tensor robust principal component analysis model based on truncated nuclear norm,is established.The augmented Lagrangian function is constructed and solved by alternate direction multiplier when the incoherent tensor condition is satisfied.The validity and feasibility of the robust principal component analysis model with tensor truncated nuclear norm for high-dimensional data recovery are testified by the experiments of composite data and multi-group color images on MATLAB.The content of the paper is arranged as follows:Firstly,the research background and significance of image denoising are introduced,the concepts of compressed sensing,matrix reconstruction and low rank tensor recovery are given,and the research contents are summarized.Secondly,the preliminary knowledge of matrix and tensor related mathematics is enumerated,and the tensor robust principal component analysis algorithm and tensor completion algorithm based on truncated nuclear norm are emphasized.The augmented Lagrange function of the algorithm is constructed and the iterative process of the model is given by using the alternate direction multiplier method.Thirdly,a tensor robust principal component analysis model based on truncated nuclear norm is proposed.The two-step iterative method is adopted in the solution process,and the effectiveness of the algorithm is proved by numerical experiments.Finally,summarize the work done.