Local Neighborhood Operators: Stability, Edge Preservation, and Diffusion

We develop a unified framework for image denoising based on the Local Neighborhood Product N, an algebraic operator that aggregates information over closed neighborhoods through a signal-dependent weighting mechanism U. The associated Neighborhood Diffusion Equation (NDE) provides a common formulation for three classes of denoising filters: when U = 1 it recovers Gaussian smoothing; when U decreases with the local gradient it implements anisotropic diffusion (Perona--Malik); when U is learned from data it yields a trainable adaptive filter. We introduce the Neighborhood Product Filter (NPF) family—NP-Gaussian, NP-EdgePreserving, NP-Anisotropic, and NP-Adaptive—and establish rigorous stability bounds: for \|U\|_∞ < 1/5, the NDE contracts exponentially to zero. Moreover, we prove that NP-EdgePreserving generalizes both the Perona--Malik model and the bilateral filter, with explicit edge-preserving guarantees. Extensive experiments on standard datasets (Set12, BSD68, Kodak, LIVE1) show that NP-EdgePreserving outperforms Gaussian filtering by up to $1.7$ dB and bilateral filtering by $0.9$ dB in PSNR, while NP-Adaptive achieves performance comparable to BM3D with 5× higher speed and only $1.2$K parameters.