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High Dimensional Data Decomposition for Anomaly Detection of Textured Images

Ji Song, Xing Wang, Jianguo Wu, Xiaowei Yue

TL;DR

This work tackles texture-driven anomaly detection in high-dimensional images by introducing TBSD, a decomposition framework that explicitly models textures via quasi-periodicity and couples this with a smooth background prior. The method learns texture basis functions from a small set of defect-free images and uses them as priors in an optimization that separates background, textures, and anomalies, thereby reducing texture misidentification. TBSD demonstrates superior anomaly detection performance on both simulated and real industrial textures, with competitive computational complexity and robustness to non-ideal training data. The approach offers practical benefits for smart manufacturing where textured surfaces pose distinct challenges for conventional anomaly detectors.

Abstract

In the realm of diverse high-dimensional data, images play a significant role across various processes of manufacturing systems where efficient image anomaly detection has emerged as a core technology of utmost importance. However, when applied to textured defect images, conventional anomaly detection methods have limitations including non-negligible misidentification, low robustness, and excessive reliance on large-scale and structured datasets. This paper proposes a texture basis integrated smooth decomposition (TBSD) approach, which is targeted at efficient anomaly detection in textured images with smooth backgrounds and sparse anomalies. Mathematical formulation of quasi-periodicity and its theoretical properties are investigated for image texture estimation. TBSD method consists of two principal processes: the first process learns the texture basis functions to effectively extract quasi-periodic texture patterns; the subsequent anomaly detection process utilizes that texture basis as prior knowledge to prevent texture misidentification and capture potential anomalies with high accuracy.The proposed method surpasses benchmarks with less misidentification, smaller training dataset requirement, and superior anomaly detection performance on both simulation and real-world datasets.

High Dimensional Data Decomposition for Anomaly Detection of Textured Images

TL;DR

This work tackles texture-driven anomaly detection in high-dimensional images by introducing TBSD, a decomposition framework that explicitly models textures via quasi-periodicity and couples this with a smooth background prior. The method learns texture basis functions from a small set of defect-free images and uses them as priors in an optimization that separates background, textures, and anomalies, thereby reducing texture misidentification. TBSD demonstrates superior anomaly detection performance on both simulated and real industrial textures, with competitive computational complexity and robustness to non-ideal training data. The approach offers practical benefits for smart manufacturing where textured surfaces pose distinct challenges for conventional anomaly detectors.

Abstract

In the realm of diverse high-dimensional data, images play a significant role across various processes of manufacturing systems where efficient image anomaly detection has emerged as a core technology of utmost importance. However, when applied to textured defect images, conventional anomaly detection methods have limitations including non-negligible misidentification, low robustness, and excessive reliance on large-scale and structured datasets. This paper proposes a texture basis integrated smooth decomposition (TBSD) approach, which is targeted at efficient anomaly detection in textured images with smooth backgrounds and sparse anomalies. Mathematical formulation of quasi-periodicity and its theoretical properties are investigated for image texture estimation. TBSD method consists of two principal processes: the first process learns the texture basis functions to effectively extract quasi-periodic texture patterns; the subsequent anomaly detection process utilizes that texture basis as prior knowledge to prevent texture misidentification and capture potential anomalies with high accuracy.The proposed method surpasses benchmarks with less misidentification, smaller training dataset requirement, and superior anomaly detection performance on both simulation and real-world datasets.
Paper Structure (32 sections, 3 theorems, 25 equations, 17 figures, 5 tables, 6 algorithms)

This paper contains 32 sections, 3 theorems, 25 equations, 17 figures, 5 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Related Methods for Textured Image Anomaly Detection
  4. RPCA and SSD
  5. Proposed TBSD Method
  6. Quasi-periodicity of Texture Component
  7. Decomposition and Examples of Source Image
  8. One-dimensional Periodicity
  9. Two-dimensional Periodicity
  10. One-dimensional Quasi-periodicity
  11. Two-dimensional Quasi-periodicity
  12. Section Conclusion
  13. Texture Basis Integrated Smooth Decomposition (TBSD) Method
  14. Proposed Optimization Model
  15. Estimation of Low-rank Smooth Background
  16. ...and 17 more sections

Key Result

Theorem 3.1

A composite 1D signal set $S$, which is a linear combination of periodic 1D signal sets $\{S^{(k)}\vert S^{(k)}=do(S_{T^{(k)}}, \mathbb{F}_{n^{(k)}}), 1\le k \le K, k\in \mathbb{Z}^+\}$, is also periodic if and only if all ratios of any two periodic lengths $\{T^{(k_1)}/T^{(k_2)}\vert 1\le k_1, k_2\

Figures (17)

  • Figure 1.1: Real-world textured images in manufacturing: (a) Wood plate surfacebergmann2021mvtec, (b) Rolling steel inspection (from OG Technology), (c) Surface non-destructive testing (from Boeing), (d) 3D printing
  • Figure 1.2: Schematic diagram of relationships among Core mathematical assumptions, optimization models, and technical algorithms of the proposed TBSD method.
  • Figure 3.1: Background, textures, and anomalies in simulation/real-world image examples
  • Figure 3.2: Examples of textured images: (a) Manually constructed image with mainly linear textures for Simulation Study; (b) Manually constructed image with theoretically quasi-periodic textures; (c) Real-world image of wooden surfaces for Case Study
  • Figure 3.3: How quasi-periodicity derives from periodicity: an originally periodic 1D signal set $S$, with uncertainty attached onto lengths and elemental sets of its periodic segments, could be transformed into a 1D quasi-periodic signal set. (For the convenience of illustration, discrete signal sets are connected as continuous curves in this figure).
  • ...and 12 more figures

Theorems & Definitions (8)

  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1
  • Proposition 3.1
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.2