Fence Theorem: Towards Dual-Objective Semantic-Structure Isolation in Preprocessing Phase for 3D Anomaly Detection

• URL: http://arxiv.org/abs/2503.01100v2
• Date: Tue, 04 Mar 2025 04:33:21 GMT
• Title: Fence Theorem: Towards Dual-Objective Semantic-Structure Isolation in Preprocessing Phase for 3D Anomaly Detection
• Authors: Hanzhe Liang, Jie Zhou, Xuanxin Chen, Tao Dai, Jinbao Wang, Can Gao,
• Abstract summary: Fence Theorem formalizing preprocessing as a dual-objective semantic isolator.<n>Patch3D, consisting of Patch-Cutting and Patch-Matching modules, to segment semantic spaces and consolidate similar ones.<n>Experiments on Anomaly-ShapeNet and Real3D-AD with different settings demonstrate that progressively finer-grained semantic alignment in preprocessing directly enhances point-level AD accuracy.
• Score: 32.44179060918441
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 3D anomaly detection (AD) is prominent but difficult due to lacking a unified theoretical foundation for preprocessing design. We establish the Fence Theorem, formalizing preprocessing as a dual-objective semantic isolator: (1) mitigating cross-semantic interference to the greatest extent feasible and (2) confining anomaly judgments to aligned semantic spaces wherever viable, thereby establishing intra-semantic comparability. Any preprocessing approach achieves this goal through a two-stage process of Emantic-Division and Spatial-Constraints stage. Through systematic deconstruction, we theoretically and experimentally subsume existing preprocessing methods under this theorem via tripartite evidence: qualitative analyses, quantitative studies, and mathematical proofs. Guided by the Fence Theorem, we implement Patch3D, consisting of Patch-Cutting and Patch-Matching modules, to segment semantic spaces and consolidate similar ones while independently modeling normal features within each space. Experiments on Anomaly-ShapeNet and Real3D-AD with different settings demonstrate that progressively finer-grained semantic alignment in preprocessing directly enhances point-level AD accuracy, providing inverse validation of the theorem's causal logic.

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