# Low Tensor Rank Constrained Image Inpainting Using a Novel Arrangement Scheme
## 1 Introduction  
Image inpainting refers to the process of completing missing entries or restoring damaged regions of an image. It is a typical ill-posed inverse problem, generally solved by exploiting the image priors [1, 2], such as smoothness, sparsity, and low rankness. In recent years, tensor analysis, including tensor low-rank decomposition and tensor completion, has attracted increasing attention [3, 4, 5, 6]. A color image itself is an order-3 tensor, or it can be used to construct a high-order (greater than 3) tensor; then, the image inpainting problem becomes a tensor completion problem. A tensor is more challenging to analyze than a matrix due to the complicated nature of higher-order arrays [7]. Low tensor rank can be constrained to recover the missing pixels. The effectiveness relies on the tensor rank. The lower the tensor rank, the better the recovery results. Thus, finding ways to decrease the tensor rank is essential in the tensor completion problem. Unlike the matrix rank, the definition of tensor rank is not unique and is related to the tensor decomposition scheme. Low tensor rank completion methods can be categorized according to the tensor decomposition frameworks they use [8]. The traditional tensor decomposition tools include CANDECOMP/PARAFAC (CP) and Tucker decomposition [8, 9]. Recently proposed decomposition frameworks include tensor singular value decomposition (t-SVD) [10, 11, 12], tensor train (TT) decomposition [13, 14], tensor tree (TTR) decomposition [6, 15], tensor ring (TR) decomposition [16], fully connected tensor network (FCTN) decomposition [17], etc. The CP rank is hard to estimate. Tucker rank is multi-rank, whose elements are the ranks ofmode-n matrices that are highly unbalanced. TT rank is also multi-rank, whose elements are the ranks of TT matrices. For a high-order tensor, TT matrices are more balanced than the mode-n matrices. As matrix rank minimization is only efficient when the matrix is balanced, TT decomposition is more suitable for describing global information of high-order tensors than Tucker decomposition. T-SVD defines the tubal rank of the high-order tensor, which can be easily estimated according to a fast Fourier-based method. The tubal rank has been shown to be more efficient than the matrix rank and Tucker multi-rank in video applications [18; 19; 20]. TTR rank is essentially equivalent to Tucker multi-rank. It has to be mentioned that both TT and TR decompose the \(N\)th-order tensor into a list of third-order core tensors. Among them, TR decomposition is an improvement of TT decomposition, and TR rank is also a type of multi-rank. TR connects multiple core tensors in a circular manner, but TR rank only constrains the factor relationship between two adjacent core tensors. FCTN decomposition can flexibly characterize the correlation of multiple core tensor factors. Many popular tensor completion methods have applied the traditional CP or Tucker decomposition on color image inpainting. Some recent works have exploited the sparse Tucker core tensor and nonnegative Tucker factor matrices for image restoration [7; 21; 22]. Some works have constrained the low rankness of the mode-n matrix caused by the decomposition of a color image for inpainting [23; 24]. As low tensor rank constraints cannot fully capture the local smooth and global sparsity priors of tensors, some works have combined Tucker and total variation (TV). The SPCTV (smooth PARAFAC tensor completion and total variation) method [25] uses the PD (PARAFAC decomposition, a derivation of Tucker decomposition) framework and constrains the TV (total variation) on every factor matrix of PD. Some works have combined the constraints of the low rankness of every mode-n matrix and the TV regularization on every mode-n matrix for color image inpainting [26; 27]. Some works have proposed data restoration methods based on Bayesian tensor completion [28; 29; 30; 31].