  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.
