Haar Nuclear Norms With Applications to Remote Sensing Imagery Restoration

Remote sensing image restoration, which aims to reconstruct corrupted or missing regions, heavily relies on low-rank models. A recent trend in this field is to jointly model low-rank and local smoothness priors using a single regularization term, in order to better recover fine textures. However, due to the entanglement of low- and high-frequency components in an image, existing methods often struggle to simultaneously capture both coarse-grained structures and fine-grained textures, while also suffering from high computational complexity. To address these issues, this paper proposes a novel regularization, the Haar Nuclear Norm (HNN), for efficient and effective remote sensing image restoration. HNN transforms images into wavelet coefficients that separate low-frequency (coarse-grained) and high-frequency (fine-grained) components, and enforces low-rankness via nuclear norms on the mode-3 unfolding matrices of these wavelet coefficients. Experimental evaluations conducted on hyperspectral image inpainting, multi-temporal image cloud removal, and hyperspectral image denoising have revealed the HNN's potential. Typically, HNN achieves a performance improvement of 1-4 dB and a speedup of 10-28x compared to some state-of-the-art methods (e.g., tensor correlated total variation, and fully-connected tensor network) for inpainting tasks. The code is available at https://github.com/isyuchang/HNN.