Self-Supervised Image Prior Learning with GMM from a Single Noisy Image

The lack of clean images undermines the practicability of supervised image prior learning methods, of which the training schemes require a large number of clean images. To free image prior learning from the image collection burden, a novel Self-Supervised learning method for Gaussian Mixture Model (SS-GMM) is proposed in this paper. It can simultaneously achieve the noise level estimation and the image prior learning directly from only a single noisy image. This work is derived from our study on eigenvalues of the GMM's covariance matrix. Through statistical experiments and theoretical analysis, we conclude that (1) covariance eigenvalues for clean images hold the sparsity; and that (2) those for noisy images contain sufficient information for noise estimation. The first conclusion inspires us to impose a sparsity constraint on covariance eigenvalues during the learning process to suppress the influence of noise. The second conclusion leads to a self-contained noise estimation module of high accuracy in our proposed method. This module serves to estimate the noise level and automatically determine the specific level of the sparsity constraint. Our final derived method requires only minor modifications to the standard expectation-maximization algorithm. This makes it easy to implement. Very interestingly, the GMM learned via our proposed self-supervised learning method can even achieve better image denoising performance than its supervised counterpart, i.e., the EPLL. Also, it is on par with the state-of-the-art self-supervised deep learning method, i.e., the Self2Self.