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

TL;DR

This paper introduces a novel method using geometrical moments to detect symmetries and estimate orthogonal transformations in objects, enhancing object grasping strategies by providing robust symmetry detection in 2D and 3D.

Contribution

It develops a comprehensive approach based on geometrical moment n-tuples for symmetry detection and orthogonal transformation estimation in multi-dimensional objects, validated through extensive tests.

Findings

Effective symmetry detection in 2D and 3D objects.

Combining the proposed method with iterative optimization improves results.

The approach is robust and compares favorably with state-of-the-art techniques.

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.