基于持续同调的拓扑感知可微三角 soup 重建
Topology-Aware Differentiable Triangle-Soup Reconstruction via Persistent Homology
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中文总结 AI 辅助
该研究针对可微三角 soup 重建的拓扑缺陷，提出将持续同调纳入损失函数的方法，在环和空洞的拓扑修复上取得显著效果，且方法可组合、鲁棒性好。
中文摘要 AI 辅助
可微三角 soup 重建继承了其目标函数的一个局限：光度损失和几何损失无法测量拓扑，因此带有塌陷环或穿孔封闭空洞的重建结果，在 Chamfer 距离相等的探测点上，其瓶颈距离的图差异达 35-40 倍，却能获得与正确重建完全相同的评分。标准的隐含补救措施——引导重采样器分配预算的位置——并不能解决该问题：在一项对照研究中，拓扑先验的效果在很大程度上与同等宽度的随机先验相当，且没有任何先验形状能修复环。因此，我们将拓扑纳入目标函数：一个可微的持续项将演化表面的图（通过活动表面样本测量）与固定目标进行比较；梯度通过一对冻结的反向传播流动，将匹配的出生/死亡单形重新表示为闭式形式的外接圆半径，再加上一个招募项，用于恢复特征缺失时梯度最优匹配所缺失的部分；一个比率旋钮可校准损失与光度梯度的平衡，无需课程学习。所有主张均通过通道控制的判定：损失必须通过相同的梯度通道击败范数匹配的非拓扑对照，且在 Chamfer 距离相当的情况下满足要求。根据该规则，该损失对封闭空洞具有拓扑特异性（误差降低 4.0-7.9 倍），且对环（每一种分配先验都失败的类别）也具有拓扑特异性（误差降低 2.3 倍，无幻影柄，而对照会塌陷一个）；损失与先验可组合；分量数（H0）为无效结果。判定在 8 个外部属已知的网格上进行了无逐形调整的重复，分为两个预注册组（组均值：环 1.52 倍，空洞 4.87 倍；唯一未通过的是无余量的无效结果），在噪声下表现平稳。所有证据均为合成且单机生成，目标图预先已知；真实扫描是未来的工作。方案：在损失中校正拓扑，分配宽范围，进行组合。
英文摘要
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.
