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–Loeve 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. To tackle the proposed model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the structured optimization problem.