Scale space vector fields for symmetry detection Andrew D.J. Cross, Edwin R. Hancock* Department of Computer Science, University of York, York, Y01 5DD, UK Received 10 July 1997; received in revised form 10 March 1998; accepted 5 May 1998 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.  1999 Elsevier Science B.V. All rights reserved. Keywords: Vector fields; Symmetry detection; Continuous symmetry; Canny edge-map 1. Introduction Symmetry is an important way of assessing the shape of both 2D and 3D objects. Several alternative ways of locating axes and points of symmetry have been suggested in the literature. Broadly speaking, these can be divided into those that aim to analyse the properties of pre-segmented shapes [1–5] and those that aim to characterize shape as a continuous attribute [6–10]. Examples of the segmental analysis of symmetry include Friedberg’s moment-based method [11], the use of spline-based representations by Cham and Cipolla [2], Ponce’s idea of exploiting ribbons [3], and the skeletal analysis of binary shapes by Wright et al. [5]. The basic practical limitation of such methods is the availability of reliable segmental entities prior to symmetry analysis. It is for this reason that the early analysis of con- tinuous symmetry proves an attractive alternative. For instance Zabrodsky et al. have characterized the folding symmetries of 2D shapes using point-sets [8–10]. Reisfeld et al. generate a continuous symmetry measure from the orientation distribution for edgels and textons [6,7,12]. We make two observations concerning the existing work on continuous symmetry characterization. Firstly, the input image representation is a sparse pre-segmentation, e.g. points of curvature, edgels or textons. Secondly, existing work shares the common feature of using only a scalar representation to assess symmetry structure. Our standpoint in this paper is that a vectorial representation of the unseg- mented image data offers a more elegant way of assessing the axial symmetries of planar shapes. This is a natural way in which to proceed since the edge tangent vectors asso- ciated with the boundaries of regular 2D shapes exhibit chiral symmetry. At points of symmetry there is a cancella- tion of diametrically placed vectors. To exploit this prop- erty, we appeal to the mathematical concept of vector potential. The potential is found by volume averaging the edge tangent vectors. Individual vectors are weighted by their inverse distance from the sampling position. Asso- ciated with the potential is a vector field which is computed by taking its curl. The motivation underpinning this paper is that although vector calculus has been widely exploited in the analysis of scalar image representations, there has been little effort devoted to the analysis of vector-field represen- tations of early visual feature formation. Our aim in this paper is to develop vectorial theory of contour symmetry in 2D images. Although several authors have exploited electrostatic field analogies in the analysis of shape, these are less ambitious than the work reported here since they are electrostatic in origin [13,14]. For instance, 0262-8856/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved PII: S0262-8856(98)00133-4 * Corresponding author. E-mail address: erh@minster.york.ac.uk (E.R. Hancock) Image and Vision Computing 17 (1999) 337–345