Scale space vector fields for symmetry detection

Andrew D. J. Cross, Edwin R. Hancock

Abstract: This paper describes a vectorial representation that can be used to assess the symmetry of objects in 2D images. The method exploits the calculus of vector fields. Commencing from the gradient field extracted from filtered grey-scale images we construct a vector potential. Our image representation is based on the distribution of tangential gradient vectors residing on the image plane. By embedding the image plane in an augmented 3-dimensional space, we compute the vector potential by performing volume integration over the distribution of edge tangents. The associated vector field is computed by taking the curl of the vector potential. The auxiliary spatial dimension provides a natural scale-space sampling of the generating edge-tangent distribution; as the height above the image plane is increased, so the volume over which averaging is effected also increases. We extract edge and symmetry lines through a topographic analysis of the vector-field at various heights above the image plane. Symmetry axes are lines of where the curl of the vector potential vanishes; at edges the divergence of the vector potential vanishes.