Research On Tensor Modeling?Efficient Algorithms And Theory For Inverse Problems In High-Dimensional Image Processing

With the rapid development of data collection technology,it is easy to obtain highdimensional images,such as videos,remote sensing images,and magnetic resonance images.However,due to the limitation of environment or equipment,high-dimensional images may be degraded by missing entries,incomplete structure,and low-resolution.These degradations will seriously affect the subsequent research,thereby reducing the practical value of high-dimensional images.Estimating the original data from the observed one is a typical inverse problem with the ill-posed property,while the regularization method can effectively overcome the ill-posedness by utilizing the prior knowledge of high-dimensional images.In this dissertation,we mainly focus on tensor completion,hyperspectral unmixing,and hyperspectral super-resolution problems.For a specific problem,we exploit the global,local and nonlocal structure priors of high-dimensional images to design the reasonable regularizers,build the tensor-based regularization models,develop the effective and efficient algorithms,and then analyze the theoretical properties of proposed models and algorithms.The main contents are summarized as four parts:1.TMac-TT,a tensor completion method based on tensor train rank,suffers from serious block-artifacts when restoring low-order tensor images.To tackle this issue,by exploring the global low-rankness and the local piecewise smoothness,we build a tensor completion model combining low-rank matrix factorization based on tensor train rank and the total variation regularization.We apply the low-rank matrix factorization on the canonical matricizations of the tensor to characterize the correlation between different modes of high-order tensors and use the total variation to enhance the spatial local smoothness of high-dimensional images.Meanwhile,based on the block coordinate descent algorithm,we introduce the proximal operator to alternately solve the proposed model and theoretically prove that the proposed algorithm converges to the coordinatewise minimizers.Extensive numerical experiments demonstrate that the proposed method can effectively reduce the block-artifacts and keep the piecewise smoothness of tensor data in spatial dimensions.2.Based on the first part,we exploit the nonlocal self-similarity of high-dimensional images and propose a tensor completion model based on the nonlocal redundancy.By block-matching,we stack the similar -th order cubes into a( + 1)-th order data and apply the tensor train low-rank constraint on the grouped tensor,which can simultaneously learn the correlations along the spatial and temporal/spectral modes.Moreover,we establish a perturbation analysis for the tensor train low-rankness of groups consisting of similar cubes.An efficient alternating direction method of multipliers-based algorithm is developed to solve the proposed model.Extensive experiments show that the proposed method can handle various completion problems of high-dimensional images under different missing cases,and the results are better than these of compared methods.3.We propose a block-term tensor decomposition and the total variation-based hyperspectral unmixing model,which simultaneously explores the global low-rankness and piecewise smoothness of hyperspectral images.We design a two-block parameterization strategy to avoid solving the large-scale subproblems.In specific,we recast the block-term tensor decomposition model as two parameters,i.e.,spectral signatures and abundance maps,that is natural to impose the structural priors(nonnegativity and smoothness),thereby admiting lower per-iteration complexity and high efficiency.On the other hand,we design a fast orthogonal projection solver for computing one subproblem with low-rank and probability simplex structures.Simulations verify the efficiency and effectiveness of the proposed method.4.Based on the third part,we propose a coupled block-term tensor decompositionbased hyperspectral super-resolution model.Hyperspectral super-resolution aims at fusing a pair of hyperspectral and multispectral images to recover a super-resolution image.In the proposed model,the spectral images are modeled as tensors with low-rank blockterm decomposition representations.The recoverability of the super-resolution image is guaranteed under mild conditions.A salient feature of the proposed framework is that the latent factors under the block-term tensor decomposition admit physical meaning.Therefore,structural constraints and regularization terms that reflect prior information about the latent factors can be flexibly incorporated in our hyperspectral super-resolution framework.We consider the nonnegativity of the latent factors and the spatial smoothness of abundance maps.We propose an accelerated alternating gradient projection algorithm to solve our model with the guaranteed convergence.Extensive simulated and real experiments show that the proposed method achieves the best recovered results.