Reflection Invariant and Symmetry Detection

Symmetry detection and discrimination are of fundamental meaning in science, technology, and engineering. This paper introduces reflection invariants and defines the directional moment to detect symmetry for shape analysis and object recognition. And it demonstrates that detection of reflection symmetry can be done in a simple way by solving a trigonometric system derived from the directional moment, and discrimination of reflection symmetry can be achieved by application of the reflection invariants in 2D and 3D. Rotation symmetry can also be determined based on that.The experiments in 2D and 3D, including the regular triangle, the square, and the five Platonic objects, show that all the reflection lines or planes can be deterministically found using directional moments up to order six.This result can be used to simplify the efforts of symmetry detection in research areas,such as protein structure, model retrieval, reverse engineering, and machine vision etc.