Image Restoration Learning via Noisy Supervision in Fourier Domain

Noisy supervision refers to supervising network learning with targets corrupted by noise, encompassing both weakly supervised learning with noisy targets and fully unsupervised denoising using unpaired noisy images. It alleviates the data collection burden and enhances the practical applicability of deep learning techniques. Existing methods face two main limitations: they are ineffective at handling noise with long-range correlations, commonly found in real-world scenarios such as low-light imaging and remote sensing, and rely on pixel-wise loss functions that offer limited supervision for image deblurring and super-resolution. This work addresses these challenges by leveraging the Fourier domain, where spatially correlated noise exhibits sparsity and independence, and Fourier coefficients capture global information that enables stronger supervision. We prove that Fourier coefficients of a wide range of noise converge in distribution to the Gaussian distribution and establish a statistical equivalence between learning with clean and noisy targets in the Fourier domain. Based on these insights, we develop a weakly supervised framework for image restoration learning with noisy targets, and construct a fully unsupervised denoising method tailored to stripe-wise noise. Extensive experiments show that our approaches achieve superior performance in both quantitative metrics and perceptual quality.