Noise & pattern: identity-anchored Tikhonov regularization for robust structural anomaly detection

TL;DR

The paper tackles robust structural anomaly detection by training a self-supervised Filtering Autoencoder (FAE) to repair artificially corrupted images. It introduces a structured corruption model that spans shape, texture, and opacity, and augments training with isotropic Gaussian noise to realize identity-anchored Tikhonov regularization, stabilizing reconstructions and improving localization. The authors provide a theoretical expansion showing how the noise term yields a Jacobian penalty that anchors the mapping toward the identity on the data manifold, and they validate the method on the MVTec AD benchmark where they achieve state-of-the-art segmentation. The combination of a versatile corruption model and Gaussian-regularized restoration yields robust, domain-agnostic anomaly detection with practical impact for automated industrial inspection.

Abstract

Anomaly detection plays a pivotal role in automated industrial inspection, aiming to identify subtle or rare defects in otherwise uniform visual patterns. As collecting representative examples of all possible anomalies is infeasible, we tackle structural anomaly detection using a self-supervised autoencoder that learns to repair corrupted inputs. To this end, we introduce a corruption model that injects artificial disruptions into training images to mimic structural defects. While reminiscent of denoising autoencoders, our approach differs in two key aspects. First, instead of unstructured i.i.d.\ noise, we apply structured, spatially coherent perturbations that make the task a hybrid of segmentation and inpainting. Second, and counterintuitively, we add and preserve Gaussian noise on top of the occlusions, which acts as a Tikhonov regularizer anchoring the Jacobian of the reconstruction function toward identity. This identity-anchored regularization stabilizes reconstruction and further improves both detection and segmentation accuracy. On the MVTec AD benchmark, our method achieves state-of-the-art results (I/P-AUROC: 99.9/99.4), supporting our theoretical framework and demonstrating its practical relevance for automatic inspection.

Figures (11)

• Figure 1: Robust Structural Anomaly Detection in a Nutshell. Our qualitative results on MVTec AD (hazelnut): input images, predicted heatmaps, and ground-truth masks.
• Figure 2: Illustration of our AD process. Given input $\hat{\boldsymbol{x}}$, our model produces an output $f_{\boldsymbol{\theta}}(\hat{\boldsymbol{x}})$, which replicates normal regions and replaces irregularities with locally consistent patterns. Then we compute a spatial difference map $\Delta(\hat{\boldsymbol{x}}, f_{\boldsymbol{\theta}}(\hat{\boldsymbol{x}})) \in \mathbb{R}^{h \times w}$. In the last step we apply a series of averaging convolutions $G_k$ to the difference map to produce final anomaly heatmap $\text{anomap}_{f_{\boldsymbol{\theta}}}^{n,k}(\hat{\boldsymbol{x}})$.
• Figure 3: Comparison of reconstruction quality between our models trained with and without Gaussian noise. The first row shows an input image $\boldsymbol{x}$ and its reconstructions $f(\boldsymbol{x})$ from both models. The second row provides zoom-ins of the region marked by a red square. Reconstructions in normal regions are visibly improved when the model is regularized with Gaussian noise.
• Figure 4: Comparison of reconstruction quality between our models trained with and without Gaussian noise. Each example shows a defective input $\hat{\boldsymbol{x}}$ (top row), its reconstructions $f(\hat{\boldsymbol{x}})$ from both models, and zoom-ins (middle row) of the region marked by a red square for detailed comparison. The bottom row shows the ground-truth anomaly mask and corresponding heatmaps. Reconstructions in corrupted regions and the corresponding anomaly heatmaps are visibly improved when the model is regularized with Gaussian noise.
• Figure 5: Illustration of a combined realization of our design principles for simulating structural anomalies, covering variations in shape, texture, and occlusion opacity. The first row shows partially corrupted images, and the second row displays the corresponding anomaly masks, where mask intensity reflects the opacity of the occlusions. In the third example, Gaussian noise is applied across the entire image as part of our regularization strategy.

Theorems & Definitions (6)

• Theorem 1: Identity-Anchored Tikhonov Regularization
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Proposition 1