# Image Completion with Hybrid Interpolation in Tensor Representation
## 1 Introduction  
Image completion aims to synthesize missing regions in an incomplete image or a video sequence with the content-aware information captured from accessible or unperturbed regions. The missing regions may result from removing unwanted sub-areas, occlusion, or unobserved or considerably damaged random pixels. Image completion is one of the fundamental research topics in the area of computer vision and graphics technology, motivated by widespread applications in various applied sciences [1]. It is used for the restoration of old photographs, paintings, and films by removing scratches, dust spots, occlusions, or other user-marked objects, such as annotations, subtitles, stamps, logos, etc. In telecommunications, image completion techniques may fix error concealment problems or recover missing data-blocks lost during transmission or video compression [2, 3]. The recent works also emphasize their usefulness in remote sensing imaging to remove occlusions, such as clouds and \"dead\" pixels [4, 5]. The topic of image completion has been extensively studied for at least two decades, resulting in the development of several computational approaches to address this problem. The pioneering work in fully automated digital image inpainting dates back to 2000 when [PERSON] et al. [6] introduced a milestone algorithm that was able to automatically fill in missing regions from their neighboring information without providing user-defined sample images. This algorithm is based on partial differential equations (PDEs) that determine isophotes (brightness-level lines) along which the information on the surrounding structure is propagated upwards. It is a fully automatic algorithm, but it inpaints efficiently only in narrow lines, does not recover textures, and produces blurred results. Another approach to image inpainting is to synthesize the texture. This concept was introducedby [PERSON] and [PERSON] [7] in 1999, but their algorithm requires a sample texture image to recover the texture in the missing region. [PERSON] et al. [8] also developed a hybrid version of both inpainting techniques, in which an incomplete image is decomposed into its texture and structure components, one reconstructed by texture synthesis, and another by PDE-based image inpainting. When a region to be completed is relatively large, the exemplar-based image synthesis algorithms [9; 10; 11] or their hybrid versions [12] seem to be more appropriate. The hybrid strategies have been studied in many other research papers [13; 14; 15; 16], and currently, image completion based on simultaneous fill-in of texture and structure is a fundamental approach, especially for recovering large missing regions. Various neural network architectures, e.g., convolutional neural network and generative adversarial network, can also perform hybrid image completion [17; 18; 19; 20]. A survey of image completion methods can be found in [1; 5; 21; 22]. The above-mentioned image completion methods are efficient even for very large missing regions; however, they cannot be applied for incomplete images with many uniformly distributed missing pixels (e.g., about 90 %). No texture information can be learned from any subregions of such an image. There is also no continuous boundary of missing regions, and hence the neighboring information cannot be propagated towards the centers of missing regions. In such cases, different methods must be applied. Assuming that an incomplete image has a low-rank structure and satisfies the incoherence condition [23], usually represented by clusters of similar patches, the image completion boils down to a rank minimization problem. Since it is an NP-hard problem, many computational strategies have been proposed to approximate it by a convex relaxation, usually involving matrix or tensor decomposition methods. One of them assumes a convex approximation of a rank minimization problem with a nuclear-norm minimization problem, which can be solved easily using singular value decomposition (SVD) [24; 25; 26].