An energy of asymmetry for accurate detection of global reflection axes

Abstract

We define in this paper two energy terms, symmetric energy term and asymmetric energy term, which respectively correspond to the symmetric and asymmetric components of an object. The asymmetric energy term must be zero, if the studied object is invariant under a reflection about the x-axis. Accordingly, we formulate the problem of detecting reflectional symmetries as a problem of minimising the asymmetric energy term. From the local minima of the asymmetric energy term, we can detect all the symmetric axes of any object. Since the asymmetric energy term is expressed as a summation of a set of generalised complex (GC) moments computed for an object, the proposed symmetry detection method is robust against both noise and slight deformation. We use the steepest descent technique to calculate the local minima of the asymmetric energy term, whose initialisation is calculated from the most dominant GC moment. Experiments on typical logo images and human brain image have shown the effectiveness and the robustness of the proposed method. To our knowledge, this is the first theory on energy functions that describe the symmetric and asymmetric components of a 2D pattern.

Introduction

Reflectional symmetry is a very important type of symmetries in planar image and the presence of such cue helps in performing a wide variety of tasks such as guiding shape matching, model-based object matching and object recognition [1], [2].

A straight line is an axis of symmetry of a planar object if the object is invariant under reflection about that line. Mirror symmetry is usually referred to an object with only one axis of symmetry. Generally, objects may have two or more axes of symmetries, denoted as reflection-symmetric axes in this paper. Various methods have been proposed to detect the reflection-symmetric axes of an object. However, the review given below shows that none of the previous methods are complete in the sense that they are unable to detect all reflection-symmetric axes of any object.

Previous methods were mainly focused on detecting one single reflection-symmetric axis. Labonte et al. [3] studied the problem of detecting global bilateral symmetry in images consisting of dense arrangements of local features, such as dots or oriented segments. Zielke et al. [4] only looked at vertical or near-vertical symmetry axes in an image for car-following.

Other methods attempted to detect two or more reflection-symmetric axes. Atallah [5] detected the axes of reflectional symmetries by first determining the centroid position and then using a string pattern matching technique, which considered all possible lines passing through the centroid. However, the main drawback is that this method is only applicable to planar figures made up of (possibly intersecting) segments, circles, points, etc. Marloa [6] presented an algorithm for finding the number and positions of the symmetry axes of a symmetric or almost-symmetric planar object, while this technique required the evaluation of some rational functions. Masuda et al. [7] presented a method to detect reflectional symmetry by performing correlation with the reflected images. But the disadvantage with this method is high computational cost and memory requirement, since all possible transformations have to be tested. Sun and Sherrah [8] formulated the symmetry detection problem as a correlation of the Gaussian image. As stated in the paper, their technique determines the symmetric axis assuming a priori that the input image is symmetric. However, this technique cannot be used to determine whether the input image is symmetric or not.

A spectral-based method has been designed for detecting the centre of a quasi-symmetrical object [9]. The disadvantage of this method is that it can only be applied to quasi-symmetrical objects, not any type of symmetrical objects.

In this paper, we present a new formulation of the symmetry detection problem in terms of energy minimisation. The advantage of this formulation is that all the global reflection-symmetric axes can be detected accurately. We define a novel energy function based on a set of Generalised Complex (GC) moments [10], [11] computed for an object. This novel energy function can be separated into two terms, symmetric energy term and asymmetric energy term, which, respectively, correspond to the symmetric and the asymmetric components of the object. We proved that the asymmetric energy term must be zero, if the studied object is invariant under a reflection about the x-axis. That means, we can compute the orientations of the reflectional axes by searching the orientations that make the asymmetric energy term zero. However, in real application the asymmetric energy term might not be exactly zero at those orientations, since the studied symmetric object might be corrupted by noise or even imperfectly symmetric. This way, we calculate a set of orientations that make the asymmetric energy term minimal, and from this set of orientations we screen out the false orientations and obtain the reflection-symmetric orientations. The local minima of the asymmetric energy term are calculated by the steepest descent technique, whose initialisation is determined by a GC moment with the largest magnitude. The two constraints are proposed to screen out the false orientations.

This paper is organised as follows. The properties of GC moments of the reflection-symmetric object are described in Section 2. In particular, we show that the asymmetric energy term must be zero for the object invariant under a reflection about the x-axis. Based on these properties, Section 3 proposes a method of detecting reflectional symmetries. Two constraints for screening out false symmetry angles are also given in the same section. Experiments on typical logo images and human brain image are provided in Section 4 for demonstrating the performance of our method. Section 5 gives the conclusions.