Threshold Topologies and Separation Polynomials of Digital Images

A digital image is modelled as a finite metric space by equipping its pixel set with a colour 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 colour metric, with \( \delta \)-components precisely its closed balls. Its change points coincide with the distinct nonzero weights of a minimum spanning tree. We establish a sharp Lipschitz stability estimate and derive filtration interleavings under uniform colour 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 \( 4\times4 \) digital image through exact computations of filtrations, entropy, separation polynomials, and Euler characteristics.