# Detecting Approximate Reflection Symmetry in a Point Set using Optimization on Manifold

## TL;DR

This paper introduces an optimization-based algorithm on manifolds for detecting approximate reflection symmetry in point sets, demonstrating state-of-the-art accuracy and robustness in 2D and 3D cases.

## Contribution

It formulates symmetry detection as a combined correspondence and transformation optimization on a Riemannian manifold, independent of descriptors.

## Key findings

- Achieves state-of-the-art performance on benchmark datasets.
- Robustly detects symmetry under varying distortions.
- Effective in both 2D and 3D symmetry detection.

## Abstract

We propose an algorithm to detect approximate reflection symmetry present in a set of volumetrically distributed points belonging to $\mathbb{R}^d$ containing a distorted reflection symmetry pattern. We pose the problem of detecting approximate reflection symmetry as the problem of establishing correspondences between the points which are reflections of each other and we determine the reflection symmetry transformation. We formulate an optimization framework in which the problem of establishing the correspondences amounts to solving a linear assignment problem and the problem of determining the reflection symmetry transformation amounts to solving an optimization problem on a smooth Riemannian product manifold. The proposed approach estimates the symmetry from the geometry of the points and is descriptor independent. We evaluate the performance of the proposed approach on the standard benchmark dataset and achieve the state-of-the-art performance. We further show the robustness of our approach by varying the amount of distortion in a perfect reflection symmetry pattern where we perturb each point by a different amount of perturbation. We demonstrate the effectiveness of the method by applying it to the problem of 2-D and 3-D reflection symmetry detection along with comparisons.