Topology-Aware Differentiable Triangle-Soup Reconstruction via Persistent Homology

Abstract

Differentiable triangle-soup reconstruction inherits a limitation from its objective: photometric and geometric losses cannot measure topology, so a reconstruction with a collapsed loop or a punctured enclosed void can score exactly as well as a correct one (on Chamfer-equal probes the diagrams differ 35-40x in bottleneck distance). The standard implicit remedy -- steer *where* the resampler spends its budget -- does not repair this: in a controlled study, a topology-informed prior is largely matched by an equally wide random one, and no prior shape repairs loops. We therefore move topology into the objective: a differentiable persistence term compares the evolving surface's diagram, measured on live surface samples, to a fixed target; gradients flow through a pair-frozen backward re-expressing matched birth/death simplices as closed-form circumradii, plus a recruitment term restoring the gradient optimal matching provably lacks when a feature is missing; one ratio knob calibrates the loss against the photometric gradient, no curriculum needed. Every claim passes a channel-controlled verdict: the loss must beat a norm-matched *non-topological* control through the identical gradient channel, at Chamfer parity. Under that rule the loss is topology-specific for enclosed voids (4.0-7.9x lower error) and -- the class every allocation prior failed -- for loops (2.3x, zero phantom handles, while the control collapses one); loss and prior compose; component counts (H0) are a null result. The verdicts replicate without per-shape tuning on eight external genus-known meshes in two pre-registered groups (group means: loops 1.52x, voids 4.87x; the one non-pass is a no-headroom null), degrading gracefully under noise. All evidence is synthetic and single-machine, with the target diagram known in advance; real scans are future work. Prescription: correct topology in the loss, allocate wide, combine.