Researches On Tensor Sparse Models Based On Circular Convolution And Applications

Recently,tensor sparse representations of multi-dimensional images have received widely attention and are successfully applied to various areas,including image recognition,video tracking,image restoration,since they can preserve the spatial structures of images,prune the redundant information,dig deeply intrinsic features of images.Among them,due to similar computational rules as matrices,tensor models based on circular convolution have become hot topics in the area of the tensor research,which can explore the correlations among the dimensions of images.This thesis focuses on building novel circular convolution-based tensor sparse models,including tensor sparse coding models and low rank tensor completion models.The proposed models analyze the special properties under the circular convolution,combine the classical matrix analysis,and try to give some physical explanations of the proposed models.The effectiveness and efficiency of the proposed models have been demonstrated in various applications,including hyperspectral image denoising,color video reconstruction,image clustering,color image completion.The main contributions of this thesis are given as follows:1.A novel tensor sparse coding model using t-linear combination is proposed.We explore the small-size dictionary,shifting invariance,and richer physical explanations hidden in the t-linear combination,and adopt more concise dictionaries for multi-dimensional data analysis,which significantly reduce the storage and computational complexity.Based on different sparsity measurements,we proposed two tensor sparse coding models.Meanwhile,we also propose an efficient alternating minimization optimization algorithm.For tensor coefficient learning,we design a tensor-based fast iterative thresholding algorithm,and the coefficient update only includes circular convolution,which can be efficient and parallel computed.For tensor dictionary learning, we first divide the original problems into several nearly-independent subproblems to reduce the computational complexity,and then adopt Lagrange dual algorithms to reduce the optimal variables.Finally,we demonstrate the small-size dictionary property and the efficiency of the proposed algorithms in hyperspectral image denoising and color video reconstruction.2.A graph regularized tensor sparse coding model is proposed for image clustering.The proposed model can preserve the spatial structures of images by tensor representation,and extract the intrinsic features by sparse coding models.Moreover,to enhance the clustering performance,we incorporate the graph Laplacian to preserve the local invariant in the projected space.The objective function of the proposed model includes matrices and tensors,which is efficiently solved by introducing a tensor representation of graph Laplacian.We design a novel tensor-based fast iterative thresholding algorithm for tensor coefficient learning.For image clustering,due to the small-size dictionary,we utilize the original image sizes without dimension reduction.We demonstrate the advantages of tensor representations and graph Laplacian regularization on the clustering results of four public datasets.3.A novel sparse orthogonal tensor dictionary learning model is proposed.Considering that similar samples may not contain so many patterns and do not require overcomplete dictionary,we propose a sparse orthogonal tensor dictionary learning model.The dictionary under the orthogonal constraint can simultaneously reduce the correla-tions among atoms and improve the representation power of the dictionary.Moreover,we propose a fast iterative algorithm.Both tensor coefficient learning and tensor dictionary learning problems have closed-form solutions,without iteratively updating, which significantly reduce the running time.In the hyperspectral image denoising problems,we demonstrate the efficiency of the proposed model by recovery results and running time.4.An anisotropic total variation regularized low-rank tensor completion model is proposed for color image completion.First,the model adopts tensor nuclear norm to preserve the low-rank properties of color images,which considers the correlations among the dimensions.Secondly,an anisotropic total variation regularization is proposed to preserve the smooth properties of the spatial dimensions in the recovered color images.Thirdly,we propose an efficient algorithm for solving the proposed model.Last,we demonstrate the efficiencies of the tensor nuclear norm and anisotropic total variation regularization in the color image restoration experiment.