Low-Rank Tensor Models For Image Restoration

With the rapid growth of big data applications in the world,visual signals such as images and videos have become the main carriers of multimedia data,and the quality of digital images plays a key role in data applications and processing.However,the observed images or videos may suffer from some distortion factors such as noise pollution,camera shake,resolution limitation,motion blur and packet loss during the process of imaging,compression and transmission,all of which will lead to image degradation.Therefore,image restoration algorithms that use distorted images to reconstruct original real images play an important role in the applications of medical image analysis,remote sensing,public security monitoring and digital entertainment.It is a topic of great concern in the field of image processing.Visual signals are usually high-dimensional,and the task of efficiently recovering them is challenging.Fortunately,these high-dimensional signals are not unstructured,but have specific characteristics.In the past decade,people have conducted much research on signals with specific properties,especially sparse and low-rank data.Studies have shown that these signals can be recovered from the data sampled at a rate much lower than the Nyquist rate.Therefore,using prior knowledge of low-rank structure to improve image restoration performance is a general and effective method.The purpose of tensor completion is to fill in or estimate missing or unobserved entries by using the low-rank properties of tensors,which means it recovers a complete tensor from a part of known elements.The work of this dissertation is based on low-rank tensor models,with the goal of image restoration based on the constraint of tensor rank minimization,and mainly focuses on the research topics on the following restoration problems: image denoising,image completion,and image super-resolution reconstruction.The research objects include two-dimensional image data and threedimensional image data(such as color images,videos and MRI images).The research content is mainly divided into the following four parts:(1)An improved weighted kernel norm minimization algorithm for image denoising is proposed.When searching for similar blocks,we convert each block from the spatial domain to the frequency domain,shrink the transform coefficients using a hard threshold operation,and match blocks with truncated transformation coefficients to improve the accuracy of similar block matching.During the denoising process using the weighted nuclear norm minimization,we use a strategy to adaptively adjust the residual image feedback ratio according to the image noise variance,so as to make full use of the residual image information to denoise an image.Finally,we judge whether to terminate the iteration by comparing the changes of the correlation coefficient between the active regions of the denoised image and its residual image in two consecutive iterations,so as to obtain the best denoising performance when the iteration is finished.Experimental results show that compared with existing methods,the proposed method can maintain clearer edges and structure.(2)An image completion algorithm combining low-rank tensor and total variation(TV)regularization is proposed.For incomplete tensors(such as color images and videos with missing partial information),based on the tensor-train rank-1(TTr1)decomposition,an iterative soft threshold method is proposed to solve the tensor rank and reduce the influence of missing entries on singular values.Considering the algorithmic complexity,we adopt the RGB channels instead of utilizing the non-local similarity of an image to find the similar block groups to construct a tensor.Although the low-rank constraint can capture the global structure of data well,it cannot capture the local correlation of images or videos.In order to exploit the local smoothness of visual data,we integrate TV into the proposed model as a regularization term.Finally,we use a primal-dual splitting method to achieve the optimization.Experimental results on the restoration of color images,videos and MRI data have shown that compared with the state-of-the-art tensor completion techniques,the proposed method can better preserve the clarity of image and video structure.(3)An image completion algorithm based on the low-rank tensor of sub-images is proposed.In color image restoration,a common way is to treat each color image as a third-order tensor with the mode(row × column × RGB),and then apply the tensor nuclear norm to approximate it.The current tensor presentation of color image data has a certain low-rank feature,but does not have a strong lowrank feature.In order to enhance the low-rank feature of an image,the algorithm is based on the tensor singular value decomposition(t-SVD),utilizing the local similarity to sub-sample an image to obtain a sub-image set which has a strong low-rank property.We use this sub-image set to recover the low-rank tensor from the corrupted observation image.In addition,because the tensor nuclear norm is direction-dependent,the value of the tensor nuclear norm may be different if a tensor is rotated or its mode is permuted.In our completion method,the mode(row × column × RGB)of a low-rank third-order tensor is permuted to the mode with RGB in the middle(row × RGB × column),and then the low-rank optimization completion is performed on the permuted tensor.This permutation operation can make a better restoration.Finally,the alternating direction method of multipliers is used to solve the problem.Experimental results on color images and videos show that the proposed algorithm can recover the texture structure of color images and videos well.(4)An image super-resolution algorithm based on the tensor rank minimization theory is proposed.The innovation of this algorithm is to integrate the low-rank tensor and TV regularization and apply it to the image super-resolution reconstruction.The TV regularization term is used to keep the reconstructed image smooth locally,and to avoid the ringing effect while reconstructing image edges.In solving the minimization problem of the tensor nuclear norm,the proximal operator based on the TTr1 decomposition is applied to approximate the tensor nuclear norm,which maintains the inherent correlation of three RGB channels of the reconstructed color image and makes the color of the reconstructed image more realistic.Experimental results show that,compared with several classical super-resolution reconstruction algorithms,the proposed algorithm can obtain richer image details and clearer image contours while maintaining the global image structure.