The ability of capturing fine spectral discriminative information enables hyperspectral images (HSIs) to observe, detect and identify objects with subtle spectral discrepancy. However, the captured HSIs may not represent true distribution of ground objects and the received reflectance at imaging instruments may be degraded, owing to environmental disturbances, atmospheric effects and sensors' hardware limitations. These degradations include but are not limited to: complex noise (i.e., Gaussian noise, impulse noise, sparse stripes, and their mixtures), heavy stripes, deadlines, cloud and shadow occlusion, blurring and spatial-resolution degradation and spectral absorption, etc. These degradations dramatically reduce the quality and usefulness of HSIs. Low-rank tensor approximation (LRTA) is such an emerging technique, having gained much attention in HSI restoration community, with ever-growing theoretical foundation and pivotal technological innovation. Compared to low-rank matrix approximation (LRMA), LRTA is capable of characterizing more complex intrinsic structure of high-order data and owns more efficient learning abilities, being established to address convex and non-convex inverse optimization problems induced by HSI restoration. This survey mainly attempts to present a sophisticated, cutting-edge, and comprehensive technical survey of LRTA toward HSI restoration, specifically focusing on the following six topics: Denoising, Destriping, Inpainting, Deblurring, Super-resolution and Fusion. The theoretical development and variants of LRTA techniques are also elaborated. For each topic, the state-of-the-art restoration methods are compared by assessing their performance both quantitatively and visually. Open issues and challenges are also presented, including model formulation, algorithm design, prior exploration and application concerning the interpretation requirements.
