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

TL;DR: The robustness of the approach is shown by varying the amount of distortion in a perfect reflection symmetry pattern where the authors perturb each point by a different amount of perturbation, and the effectiveness of the method is demonstrated by applying it to the problem of 2-D and 3-D reflection 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 (two-dimensional) and 3-D reflection symmetry detection along with comparisons.

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### "Detecting Approximate Reflection Sy..." refers background in this paper

...proposed approach, we first find the set of corner points [58]....

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### "Detecting Approximate Reflection Sy..." refers methods in this paper

...Our algorithm is similar to Iterative Closest Point (ICP) algorithm ([48], [49]) only in the sense that we also follow the alternation between the optimization of reflection transformation (rotation and translation in ICP) and the correspondences between the mirror symmetric points (correspondences between the points of two different shapes in ICP)....

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### "Detecting Approximate Reflection Sy..." refers background in this paper

...the optimization algorithms are well studied [50]....

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...In order to introduce the essential differential geometry of the set M, we follow [50]....

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