OCTREE NEUTROSOPHIC SETS USING ROUGH MATRIX

The octree is a hierarchical tree data structure commonly employed in computer graphics, spatial indexing, and three-dimensional data representation. Traditional octree methodologies generally categorise regions as black (completely shaded), white (unshaded), or grey (partially shaded), aligning with a binary (0–1) logic, with the intermediate state managed in an ad hoc manner. This article extends binary logic to a three-valued logic octree, facilitating a more nuanced and systematic depiction of spatial uncertainty. We present a neutrosophic quadtree framework that extends the traditional quadtree by explicitly representing the components of truth, indeterminacy, and falsity for each node, in accordance with neutrosophic set theory. We propose a consolidated methodology for expressing imprecise spatial data by integrating these two approaches, wherein each node conveys either three-valued (true, false, intermediate) or neutrosophic (T, I, F) information. Additionally, we illustrate how both structures can be represented and examined using rough matrices, employing the concepts of lower, higher, and boundary approximations to measure certainty and indeterminacy inside hierarchical divisions. This comprehensive perspective amplifies the descriptive capability of tree-based spatial models and paves the way for advancements in image reconstruction, feature extraction, and uncertainty management in multidimensional datasets.