Robust estimation for image noise based on eigenvalue distributions of large sample covariance matrices

Abstract

In this paper, we propose a novel algorithm to estimate Gaussian noise levels for captured natural images by rigorously analyzing the limiting distributions of the eigenvalue spectrum of a large covariance matrix with Gaussian samples. In order to select a relatively homogeneous region that best represents the noise, the corresponding image patches are first rearranged to construct a high-dimensional noise covariance matrix. And then, an optimal criterion for classifying homogeneous regions is derived based on the statistical relationship between the largest and the second largest eigenvalues of a sample covariance matrix. Moreover, we further explore the reasons for the bias of the maximum likelihood estimator of the noise variance both in high-dimensional settings and finite samples. According to random matrix theory, we clarify the asymptotic properties of the trace of a sample covariance matrix to measure the error bounds of estimation and then propose a new bias-corrected estimator. To this end, an effective estimation method for the noise level is devised based on the boundness and asymptotic behavior of pure noise eigenvalues of the selected patches. The estimation performance of our method has been guaranteed both theoretically and empirically. Experimental results have demonstrated that our approach can reliably infer true noise variance and is superior to the competing methods in terms of both estimation accuracy and robustness.