Multi-Dimensional Visual Data Completion via Low-Rank Tensor Representation under Coupled Transform

This paper addresses the tensor completion problem, which aims to recover missing information of multi-dimensional images. How to represent a low-rank structure embedded in the underlying data is the key issue in tensor completion. In this work, we suggest a novel low-rank tensor representation based on coupled transform, which fully exploits the spatial multi-scale nature and redundancy in spatial and spectral/temporal dimensions, leading to a better low tensor multi-rank approximation. More precisely, this representation is achieved by using two-dimensional framelet transform for the two spatial dimensions, one/two-dimensional Fourier transform for the temporal/spectral dimension, and then Karhunen-Loéve transform (via singular value decomposition) for the transformed tensor. Based on this low-rank tensor representation, we formulate a novel low-rank tensor completion model for recovering missing information in multi-dimensional visual data, which leads to a convex optimization problem. To tackle the proposed model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the structured optimization problem. Numerical examples on color images, multispectral images, and videos illustrate that the proposed method outperforms many state-of-the-art methods in qualitative and quantitative aspects.