Projection and Proximal Operators in Image Denoising

Image denoising is a classical inverse problem in which the recovery of a clean image from its noisy observation is complicated by ill-posedness. This paper develops a functional-analytic framework in which a broad class of denoising methods is expressed as the iteration of firmly nonexpansive or quasi-nonexpansive operators on a Hilbert space. Total variation regularization, wavelet shrinkage, projection onto convex sets (POCS), projection onto epigraph sets (PESC), and regularization by denoising via fixed-point projection (RED-PRO) are each shown to fit this structure, and their convergence is obtained from a single Krasnosel'skii–Mann argument. The proximal nature of wavelet shrinkage is distinguished from metric projection onto the \(\ell^1\)-ball, and the demicontractive hypothesis underlying RED-PRO is stated explicitly. Numerical experiments on standard test images show that RED-PRO with a BM3D denoiser attains the highest PSNR at every noise level tested, that Chambolle's TV projection follows within \(1.2\)–\(1.7\) dB, and that POCS and wavelet shrinkage trade accuracy for speed. The framework clarifies the operator-theoretic content of each method and provides a common language for their analysis and comparison.