This paper studies the integration problem in differential fields that may involve quantities reminiscent of the classical Weierstrass ℘ function, which are defined by a first-order nonlinear differential equation. We extend the classical notion of special polynomials to elements of Weierstrass-like extensions and present algorithms for reduction in such extensions. As an application of these results, we derive some new formulae for integrals of powers of ℘.

Humans detect real-world object anomalies by perceiving, interacting, and reasoning based on object-conditioned physical knowledge. The long-term goal of Industrial Anomaly Detection (IAD) is to enable machines to autonomously replicate this skill. However, current IAD algorithms are largely developed and tested on static, semantically simple datasets, which diverge from real-world scenarios where physical understanding and reasoning are essential. To bridge this gap, we introduce the Physics Anomaly Detection (Phys-AD) dataset, the first large-scale, real-world, physics-grounded video dataset for industrial anomaly detection. Collected using a real robot arm and motor, Phys-AD provides a diverse set of dynamic, semantically rich scenarios. The dataset includes more than 6400 videos across 22 real-world object categories, interacting with robot arms and motors, and exhibits 47 types of anomalies. Anomaly detection in Phys-AD requires visual reasoning, combining both physical knowledge and video content to determine object abnormality. We benchmark state-of-the-art anomaly detection methods under three settings: unsupervised AD, weakly-supervised AD, and video-understanding AD, highlighting their limitations in handling physics-grounded anomalies. Additionally, we introduce the Physics Anomaly Explanation (PAEval) metric, designed to assess the ability of visual-language foundation models to not only detect anomalies but also provide accurate explanations for their underlying physical causes. Our project is available at https://guyao2023.github.io/Phys-AD/.

Object anomaly detection is essential for industrial quality inspection, yet traditional single-sensor methods face critical limitations. They fail to capture the wide range of anomaly types, as single sensors are often constrained to either external appearance, geometric structure, or internal properties. To overcome these challenges, we introduce MulSen-AD, the first high-resolution, multi-sensor anomaly detection dataset tailored for industrial applications. MulSen-AD unifies data from RGB cameras, laser scanners, and lock-in infrared thermography, effectively capturing external appearance, geometric deformations, and internal defects. The dataset spans 15 industrial products with diverse, real-world anomalies. We also present MulSen-AD Bench, a benchmark designed to evaluate multi-sensor methods, and propose MulSen-TripleAD, a decision-level fusion algorithm that integrates these three modalities for robust, unsupervised object anomaly detection. Our experiments demonstrate that multi-sensor fusion substantially outperforms single-sensor approaches, achieving 96.1% AUROC in object-level detection accuracy. These results highlight the importance of integrating multi-sensor data for comprehensive industrial anomaly detection. The dataset and code are available at https://github.com/ZZZBBBZZZ/MulSen-ADtosupportfurtherresearch.

3D point cloud anomaly detection is essential for robust vision systems but is challenged by pose variations and complex geometric anomalies. Existing patch-based methods often suffer from geometric fidelity issues due to discrete voxelization or projection-based representations, limiting fine-grained anomaly localization. We introduce Pose-Aware Signed Distance Field (PASDF), a novel framework that integrates 3D anomaly detection and repair by learning a continuous, pose-invariant shape representation. PASDF leverages a Pose Alignment Module for canonicalization and a SDF Network to dynamically incorporate pose, enabling implicit learning of high-fidelity anomaly repair templates from the continuous SDF. This facilitates precise pixel-level anomaly localization through an Anomaly-Aware Scoring Module. Crucially, the continuous 3D representation in PASDF extends beyond detection, facilitating in-situ anomaly repair. Experiments on Real3D-AD and Anomaly-ShapeNet demonstrate state-of-the-art performance, achieving high object-level AUROC scores of 80.2% and 90.0%, respectively. These results highlight the effectiveness of continuous geometric representations in advancing 3D anomaly detection and facilitating practical anomaly region repair. The code is available at https://github.com/ZZZBBBZZZ/PASDF to support further research.

A complete reduction ϕ for derivatives in a differential field is a linear operator on the field over its constant subfield. The reduction enables us to decompose an element f as the sum of a derivative and the remainder ϕ(f). A direct application of ϕ is that f is in-field integrable if and only if ϕ(f) = 0.

Recently, 3D anomaly detection, a crucial problem in-volving fine-grained geometry discrimination, is getting more attention. However, the lack of abundant real 3D anomaly data limits the scalability of current models. To enable scalable anomaly data collection, we propose a 3D anomaly synthesis pipeline to adapt existing large-scale 3D models for 3D anomaly detection. Specifically, we construct a synthetic dataset, i.e., Anomaly-ShapeNet, based on ShapeNet. Anomaly-ShapeNet consists of 1600 point cloud samples under 40 categories, which provides a rich and varied collection of data, enabling efficient training and enhancing adaptability to industrial scenarios. Meanwhile, to enable scalable representation learning for 3D anomaly localization, we propose a self-supervised method, i.e., It-erative Mask Reconstruction Network (IMRNet). During training, we propose a geometry-aware sample module to preserve potentially anomalous local regions during point cloud down-sampling. Then, we randomly mask out point patches and sent the visible patches to a trans-former for reconstruction-based self-supervision. During testing, the point cloud repeatedly goes through the Mask Reconstruction Network, with each iteration's output be-coming the next input. By merging and contrasting the final reconstructed point cloud with the initial input, our method successfully locates anomalies. Experiments show that IMRNet outperforms previous state-of-the-art methods, achieving 66.1% in I-AUC on our Anomaly-ShapeNet dataset and 72.5% in I-AUC on ReaI3D-AD dataset. Our benchmark will be released at https://github.com/Chopper-233/Anomaly-ShapeNet.