**Introduction**

Symmetry is a prolific phenomenon in the world. Many objects around us are strongly constrained. For instance, not only cultural artifacts but also many natural objects are bilateral symmetric (or with mirror symmetry). There are a lot of symmetries in human anatomy, and the symmetry of the ribs is one example. Ideas of symmetry arose among the ancient Greek philosophers and mathematicians in connection with their study of the harmony of the world. The ancient sculptors, artist, and architects created their masterpieces in accordance with the canons of harmony. The method of symmetry has become a powerful and effective instrument of theoretical research in modern science. Many physical laws are formulated as some kind of variational principle: a physically possible state is the state which minimizes (or maximizes) some quantity. A state is said to be homogeneous if it has translation as its symmetry, and isotropic if it has rotation as its symmetry. The goal of an image understanding system is to identify and locate a specified object in the scene. In such cases, the system must have some knowledge of the shape of the desired object. Symmetries are good candidates for describing shape. It is a powerful concept that facilitates object detection and recognition in many situations. These representations can be used in robotics for recognition, inspection, grasping, and reasoning. Symmetry in an image allows it to be described economically. For example, if one half of an object is the mirror image of the other half, then one half need not be described. Symmetry may be defined in terms of three linear transformations in n-dimensional Euclidean space: reflection, rotation and translation. Formally, a subset S of is symmetric with respect to a linear transformation T if T(S)=S. A reflectional symmetry has a reflection line, for which the left half space is a mirror image of the right half. The ribcage in a single CT slice, for instance, is symmetrical with respect to a vertical line that passes through the spine. The ribs in a chest X-ray image also appear to be symmetrical with respect to the mediastinum.

We investigate the use of gradient information for symmetry detection in a grey scale image by analyzing the shape of the orientation histogram. For 2D images, the symmetry information is obtained by using the gradient orientation histogram which is a one-dimensional signal. The key observations are that 2D reflectional image symmetry is related to the reflectional symmetry of this one-dimensional signal and the 2D rotational symmetry is related to the translational property of this one-dimensional signal. When processing this signal, fast Fourier transform is used for symmetry detection. For 3D object the process is much more complicated. First of all one need to find a way to represent orientation histograms for 3D objects. The extended Gaussian image is used to present the 3D orientation histograms. For detailed description, please refer to the following publications.

- C. Sun and D. Si, “Fast Reflectional Symmetry Detection Using Fourier Transform“, Journal of Real-Time Imaging, vol.5, no.1, pp.63-74, February 1999.
- C. Sun, “Fast Recovery of Rotational Symmetry Parameters Using Gradient Orientation“, SPIE Journal of Optical Engineering, vol.36, no.4, pp.1073-1077, April 1997.
- C. Sun and J. Sherrah, “3-D Symmetry Detection Using the Extended Gaussian Image“, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol.19, no.2, pp.164-168, February 1997.
- C. Sun, “Symmetry Detection Using Gradient Information“, Pattern Recognition Letters, vol.16, no.9, pp.987-996, 1995.