An energy of asymmetry for accurate detection of global reflection axes

We de®ne 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 re ̄ection about the x-axis. Accordingly, we formulate the problem of detecting re ̄ectional 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 ®rst theory on energy functions that describe the symmetric and asymmetric components of a 2D pattern. q 2001 Elsevier Science B.V. All rights reserved.