Separating Signal from Noise Using Patch Recurrence across Scales

Recurrence of small clean image patches across different scales of a natural image has been successfully used for solving ill-posed problems in clean images (e.g., super-resolution from a single image). In this paper we show how this multi-scale property can be extended to solve ill-posed problems under noisy conditions, such as image denoising. While clean patches are obscured by severe noise in the original scale of a noisy image, noise levels drop dramatically at coarser image scales. This allows for the unknown hidden clean patches to "naturally emerge" in some coarser scale of the noisy image. We further show that patch recurrence across scales is strengthened when using directional pyramids (that blur and sub sample only in one direction). Our statistical experiments show that for almost any noisy image patch (more than 99%), there exists a "good" clean version of itself at the same relative image coordinates in some coarser scale of the image. This is a strong phenomenon of noise-contaminated natural images, which can serve as a strong prior for separating the signal from the noise. Finally, incorporating this multi-scale prior into a simple denoising algorithm yields state-of-the-art denoising results.