# Image Completion with Hybrid Interpolation in Tensor Representation
## 1 Introduction
Due to the orthogonality condition, low-rank representations yielded by SVD contain negative values, which is not profitable for representing a large class of images. Other low-rank models (not necessarily restricted to SVD) can be used to tackle this problem [27]. One of the commonly used methods for extracting low-rank part-based representations from nonnegative matrices is nonnegative matrix factorization (NMF) [28]. It has already found many relevant applications in image analysis, and can also be used for solving image completion problems [29; 30]. In this approach, the missing regions are sequentially updated with an NMF-based low-rank approximation of an observed image, which resembles the phenomena of propagating the neighboring information towards the missing regions in the PDE-based inpainting methods. However, due to the non-convexity of alternating optimization, NMF is sensitive to its initialization, especially for insufficiently sparse data. The ambiguity effects can be considerably relaxed if tensor decomposition models are used [31]. Moreover, multi-linear decomposition models prevent cross-modal interactions, which is particularly useful for image representations. Such a low-rank image completion methodology is mostly based on the concept of tensor completion issues that have been extensively studied [32; 33; 34; 35; 36]. There are many tensor decomposition models that are used for image completion tasks, including the fundamental ones, such as CANDECOMP/PARAFAC (CP) [37; 38] and Tucker decomposition [39; 40; 41], as well as tensor networks, such as tensor ring [42], tensor train [43; 44], hierarchical Tucker decomposition [45], and other tensor decomposition models [46]. Tensor completion methods are intrinsically addressed for processing color images (3D multi-way arrays), but they can also be applied to gray-scale images using various tensorization or scanning operators, e.g., the ket augmentation [47]. They are also very flexible in incorporating penalty terms or constraints; however, their efficiency strongly depends on the image to be completed. In practice, a low-rank representation is always a certain approximation of the underlying image, controlled by the rank of a tensor decomposition model. The problem of rank selection is regarded in terms of a trade-off between under- and over-fitting, and many approaches exist to tackle it. For example, [PERSON] et al. [38] proposed to increase the rank with recursive updates. In the early recursive steps, a low-rank structure is recovered, and then it is gradually updated to a higher-rank structure that contains more details. The rank can also be controlled by the decreasing rank procedure [46] or using the singular value thresholding strategy [43]. Thus, one of the drawbacks of this kind of image completion method is the problem of selecting the optimal rank of decomposition, and it is usually resolved by heuristic procedures. Moreover, the tensor decomposition-based image-completion methods usually involve a high computational cost if all factor matrices or core tensors are updated in each iterative step. To relax the problem with the right rank selection while keeping computational costs at a very low level, we propose a very simple alternative approach to low-rank image completion. Our strategy is based on the Tucker decomposition model in which the full-rank factor matrices are previously estimated with a hybrid connection of two interpolation methods. Since the factor matrices are precomputed, only the core tensor must be estimated. Despite the full-rank assumption, it involves a relatively low computational cost because the core tensor is sparse with non-zero entries determined by the available pixels.