Threshold Topologies and Separation Polynomials of Digital Images

A digital image is modeled as a finite metric space by equipping its pixel set with a color distance induced by a norm on ${\mathbb{R}}^3$. For a threshold $\delta$, we study the threshold topology $T_{\delta }$, generated by $\delta$-chain connectedness, and the separation polynomial $C_{\Psi }(z;\delta )$, which enumerates $\delta$-separated pixel sets. We prove that each $T_{\delta }$ is a partition topology and classify these spaces up to homeomorphism by component cardinalities. The filtration $T_{\delta }$ is shown to be equivalent to the maximal subdominant pseudo-ultrametric of the color metric, with $\delta$-components precisely its closed balls. Its change points coincide with the distinct nonzero weights of a minimum spanning tree; we state explicitly which of these facts are reformulations of classical results on single-linkage clustering, and we treat the spatially constrained variant through a graph-restricted ultrametric. We establish a sharp Lipschitz stability estimate and derive filtration interleavings under uniform color perturbations, together with exact invariance under suitable margin conditions. Algebraically, $C_{\Psi }(z;\delta )$ is identified with the independence polynomial of the similarity graph and the face enumerator of its independence complex. A packing bound yields an efficient computation method. Finally, explicit greyscale examples establish the logical independence of the topological and algebraic invariants. The framework is illustrated on a $4\times 4$ digital image through exact computations of filtrations, entropy, separation polynomials, and Euler characteristics.