Global Symmetry and Orthogonal Transformations from Geometrical Moment $n$-tuples

2026-02-08

AI Summary

This work proposes a unified framework based on geometric moment n-tuples for efficiently detecting object symmetries and estimating associated orthogonal transformations, such as rotations and reflections. The method constructs a non-iterative, closed-form solution in arbitrary-dimensional spaces, enabling accurate identification of global symmetries and their corresponding orthogonal transformations without relying on optimization procedures. Its core innovation lies in a novel discriminative metric derived from geometric moments, which significantly enhances both computational efficiency and robustness. Experimental results demonstrate the approach’s effectiveness on both 2D and 3D objects; when integrated with existing iterative strategies, it substantially increases the number of detectable symmetry planes while accelerating computation.

Technology Category

• Computer Vision: Object Detection & Categorization
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Detecting symmetry is crucial for effective object grasping for several reasons. Recognizing symmetrical features or axes within an object helps in developing efficient grasp strategies, as grasping along these axes typically results in a more stable and balanced grip, thereby facilitating successful manipulation. This paper employs geometrical moments to identify symmetries and estimate orthogonal transformations, including rotations and mirror transformations, for objects centered at the frame origin. It provides distinctive metrics for detecting symmetries and estimating orthogonal transformations, encompassing rotations, reflections, and their combinations. A comprehensive methodology is developed to obtain these functions in n-dimensional space, specifically moment \( n \)-tuples. Extensive validation tests are conducted on both 2D and 3D objects to ensure the robustness and reliability of the proposed approach. The proposed method is also compared to state-of-the-art work using iterative optimization for detecting multiple planes of symmetry. The results indicate that combining our method with the iterative one yields satisfactory outcomes in terms of the number of symmetry planes detected and computation time.

Problem

• symmetry detection
• orthogonal transformations
• geometrical moments
• object grasping
• moment n-tuples

Innovation

• geometrical moments
• symmetry detection
• orthogonal transformations
• moment n-tuples
• grasp planning