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

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 the correspondences between the points which are reflections of each other and determining 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 an optimization problem on a smooth Riemannian product manifold. The proposed approach estimates the symmetry from the distribution of the points and is descriptor independent. We evaluate 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 reflection symmetry detection along with relevant comparisons.