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

TL;DR

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 Fence Theorem's causal logic.

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.

Figures (10)

• Figure 1: Visualisation of interference between structures. (a) details an anomaly in the Cup wall characterized by a significant curvature. In contrast, (b) presents a normal, smooth curve on the Cup wall. (c) showcases a normal curve with a large curvature on the Cup handle, resembling the anomaly depicted in (a). Utilizing a memory bank to model the entire point cloud could result in incorrectly identifying the anomaly in (a) as normal due to the similarities with (c). Registration in (d) ensures structural similarity, while (e) ensures feature comparability via rotation-invariant embedding.
• Figure 2: Visualisation of the Fence Theorem.
• Figure 3: Pipeline of Patch3D.
• Figure 4: Experimental results of the registration approach. There is a significant positive correlation between the improved point-level detection performance and the registration accuracy. This is attributed to the increased orthogonality of the individual semantic spaces in the presence of increased registration accuracy.
• Figure 5: Experimental results of the Patch3D. There is a significant positive correlation between the improvement in point-level detection performance and the number of semantic spaces. This is consistent with the interpretation of the Fence Theorem.