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

arXiv: https://arxiv.org/abs/2503.01100 · HTML: https://mlsanity.com/paper/2503.01100/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 28 references that are in the map.
- First posted on arXiv 3 Mar 2025.

## Builds on (most cited first, 20 of 20)

- [Masked Autoencoders for Point Cloud Self-supervised Learning](https://mlsanity.com/paper/2203.06604/) (2022)
- [The MVTec 3D-AD Dataset for Unsupervised 3D Anomaly Detection and Localization](https://mlsanity.com/paper/2112.09045/) (2021)
- [Deep Industrial Image Anomaly Detection: A Survey](https://mlsanity.com/paper/2301.11514/) (2023)
- [Multimodal Industrial Anomaly Detection via Hybrid Fusion](https://mlsanity.com/paper/2303.00601/) (2023)
- [Asymmetric Student-Teacher Networks for Industrial Anomaly Detection](https://mlsanity.com/paper/2210.07829/) (2022)
- [Back to the Feature: Classical 3D Features are (Almost) All You Need for 3D Anomaly Detection](https://mlsanity.com/paper/2203.05550/) (2022)
- [Complementary Pseudo Multimodal Feature for Point Cloud Anomaly Detection](https://mlsanity.com/paper/2303.13194/) (2023)
- [Towards Scalable 3D Anomaly Detection and Localization: A Benchmark via 3D Anomaly Synthesis and A Self-Supervised Learning Network](https://mlsanity.com/paper/2311.14897/) (2023)
- [R3D-AD: Reconstruction via Diffusion for 3D Anomaly Detection](https://mlsanity.com/paper/2407.10862/) (2024)
- [PO3AD: Predicting Point Offsets toward Better 3D Point Cloud Anomaly Detection](https://mlsanity.com/paper/2412.12617/) (2024)
- [Look Inside for More: Internal Spatial Modality Perception for 3D Anomaly Detection](https://mlsanity.com/paper/2412.13461/) (2024)
- [PointAD: Comprehending 3D Anomalies from Points and Pixels for Zero-shot 3D Anomaly Detection](https://mlsanity.com/paper/2410.00320/) (2024)
- [SplatPose & Detect: Pose-Agnostic 3D Anomaly Detection](https://mlsanity.com/paper/2404.06832/) (2024)
- [Towards High-resolution 3D Anomaly Detection via Group-Level Feature Contrastive Learning](https://mlsanity.com/paper/2408.04604/) (2024)
- [Exploring the Potential of Encoder-free Architectures in 3D LMMs](https://mlsanity.com/paper/2502.09620/) (2025)
- [Teacher-Student Network for 3D Point Cloud Anomaly Detection with Few Normal Samples](https://mlsanity.com/paper/2210.17258/) (2022)
- [Revisiting Multimodal Fusion for 3D Anomaly Detection from an Architectural Perspective](https://mlsanity.com/paper/2412.17297/) (2024)
- [SplatPose+: Real-time Image-Based Pose-Agnostic 3D Anomaly Detection](https://mlsanity.com/paper/2410.12080/) (2024)
- [Towards Zero-shot 3D Anomaly Localization](https://mlsanity.com/paper/2412.04304/) (2024)
- [3D-CSAD: Untrained 3D Anomaly Detection for Complex Manufacturing Surfaces](https://mlsanity.com/paper/2404.07748/) (2024)

## Built on by (most cited first, 4 of 4)

- [Examining the Source of Defects from a Mechanical Perspective for 3D Anomaly Detection](https://mlsanity.com/paper/2505.05901/) (2025)
- [A Comprehensive Survey for Real-World Industrial Defect Detection: Challenges, Approaches, and Prospects](https://mlsanity.com/paper/2507.13378/) (2025)
- [The 1st International Workshop on Disentangled Representation Learning for Controllable Generation (DRL4Real): Methods and Results](https://mlsanity.com/paper/2509.10463/) (2025)
- [Open-Set Supervised 3D Anomaly Detection: An Industrial Dataset and a Generalisable Framework for Unknown Defects](https://mlsanity.com/paper/2604.01171/) (2026)

## Abstract (arXiv)

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.