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

Omar Tahri

TL;DR

This work introduces moments $\mathbf{n}$-tuples, a closed-form, training-free framework that detects symmetry and estimates orthogonal transformations in $\mathbb{R}^n$. By constructing $n$-tuples from higher-order geometrical moments and leveraging their equivariance under rotations and reflections, the method recovers rotations and reflections in both 2D and 3D without data-driven training. Key contributions include a concrete derivation of triplet descriptors in 3D, a linear constraint-based approach for estimating $\boldsymbol{\alpha}$ vectors, and demonstrations on 2D and 3D objects showing accurate symmetry detection and transform estimation with favorable computation times. The approach holds significant potential for robotics and computer vision tasks such as grasping, pose estimation, and symmetry-aware processing, and it is extensible to affine transformations in future work.

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.

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

TL;DR

This work introduces moments -tuples, a closed-form, training-free framework that detects symmetry and estimates orthogonal transformations in . By constructing -tuples from higher-order geometrical moments and leveraging their equivariance under rotations and reflections, the method recovers rotations and reflections in both 2D and 3D without data-driven training. Key contributions include a concrete derivation of triplet descriptors in 3D, a linear constraint-based approach for estimating vectors, and demonstrations on 2D and 3D objects showing accurate symmetry detection and transform estimation with favorable computation times. The approach holds significant potential for robotics and computer vision tasks such as grasping, pose estimation, and symmetry-aware processing, and it is extensible to affine transformations in future work.

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 -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.
Paper Structure (17 sections, 36 equations, 8 figures, 1 table)

This paper contains 17 sections, 36 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Geometrical representation using moments
  3. Geometrical p-th order moments
  4. Orthogonal transformation and rotational speed
  5. Moment time variation and rotational speeds
  6. N-tuple descriptors definition
  7. Orthogonal transformation estimation using n-tuples
  8. Orthogonal transformations and n-tuples
  9. N-tuple derivation: 3D space case
  10. Reflection transformation and symmetry
  11. Experiments in 2D space
  12. Experiments in 3D space
  13. Symmetry with respect to a plane
  14. Symmetry with respect to a radial axis
  15. Boosting b-plane symmetry by axial symmetry
  16. ...and 2 more sections

Figures (8)

  • Figure 1: Case study of symmetry: (a) object center, (b) x-axis (c) y-axis
  • Figure 2: Case study of a non-symmetrical object: (a) the object, (b) $n$-tuples from ${\bf v}_{(2,3)}^{(1,1)}$, (c) $n$-tuples from ${\bf v}_{(4,5)}^{(1,1)}$, (d) $n$-tuples from ${\bf v}_{(6,3)}^{(1,1)}$
  • Figure 3: Example of mirror symmetry
  • Figure 4: Mug, a symmetrical object case of study: (a) Plane of symmetry of the object with initial conditions, $\mathbf{R}=\mathbb{I}$. (b) Plane of symmetry recovered after being applied to a random $\mathbf{R}$ rigid transformation. (c) Representation of moment's $n$-tuples in Cartesian space.
  • Figure 5: Axial symmetry:(a) bottle model, (b) the rotated bottle , (c) the $n$-tuples computed from moments
  • ...and 3 more figures