Detecting grey level symmetry: The frequency domain approach

Abstract

Applying symmetry detection algorithms to images after edge detection or segmentation is computationally attractive but potentially inadequate in demanding applications. In this paper a theoretical frame-work for reflectional symmetry analysis in grey level images is developed. The symmetry of a one dimensional function can be measured in the frequency domain as the fraction of its energy that resides in the symmetric Fourier basis functions. This intuitively appealing approach is extended to two dimensions. It is shown that the Radon transform is a convenient theoretical framework for grey level reflectional symmetry analysis. In particular, the projection-slice theorem unifies symmetry measurement in one and two dimensions and allows to measure reflectional symmetry in a 2-D function directly from its Fourier transform. Finite support computation requires windowing that can be interpreted in terms of the Gabor decomposition. Symmetry measurement in a natural image is demonstrated.