Rendering Separoid Information: Rate-Distortion Reconstruction of Convex Apartness Scenes

Abstract

A convex scene communicates more than shape: the pattern of which groups of objects are mutually apart and which cross is a discrete relational payload. We treat the apartness table of a finite family of convex bodies, a separoid, as a source signal; a renderable convex scene as its encoder; and the rendered image as a noisy visual channel from which the apartness structure is decoded. For disjoint index sets $A,B$, the source bit records whether $\operatorname{conv}(\bigcup_{a\in A} C_a)$ and $\operatorname{conv}(\bigcup_{b\in B} C_b)$ are disjoint. Within this view, apartness-preserving rendering becomes a rate--distortion problem: the rate is a differentiable geometric code length for the carrier scene, while the distortion is closure-aware and weights maximal separations and minimal Radon partitions by the number of consequences they control. A differentiable support-function realization turns separability into a soft directional margin and represents each separation by a distribution over witnessing directions, yielding a variational lower bound on apartness mutual information $I(Σ;Y)$ and an information-theoretic account of view selection. Experiments on planar convex scenes show that scenes are recovered from the apartness table alone at 99.9% bit accuracy, with the certificate skeleton already determining the full table; coordinate quantization gives a clean operational rate--distortion frontier where certificate distortion is more stringent than Hamming error; and rendered $48\times48$ images transmit about 0.72 of the apartness-graph entropy under mild noise. Increasing the viewpoint-robustness term widens separating cones with only a modest geometry-rate surcharge. The result is a certificate-aware rendering objective for scenes whose purpose is to make relational convex structure recoverable rather than merely pixel-faithful.