A Local Neighborhood Product on 2D and 3D Matrix Spaces

We introduce a novel Local Neighborhood Product $\star_{\mathcal N}$ defined on two-dimensional and three-dimensional matrix spaces over an arbitrary field. The proposed operator departs from classical matrix multiplication by replacing global interactions with localized neighborhood-based algebraic couplings on discrete grids, thereby inducing a new grid-algebra structure. We establish several fundamental algebraic properties of the operator, including commutativity, distributivity, and non-associativity, highlighting its inherently non-classical behavior. The construction provides a natural framework for modeling discrete spatial interactions and neighborhood-driven dynamics. To demonstrate its computational relevance, we present an illustrative application in image processing, where grayscale images are represented as matrices and processed via the proposed operator under additive Gaussian noise.Experimental results on standard benchmark images (including the Cameraman dataset) show that the proposed operator achieves effective noise attenuation while preserving dominant structural features. Quantitative evaluation using Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR) indicates competitive performance compared to classical Gaussian filtering. The results suggest that the proposed framework offers a unified algebraic approach to neighborhood-based operators, with potential applications in image processing, lattice systems, graph-based models, and discrete computational structures.