An algebraic approach to symmetry detection Yosi Keller Yale University Mathematics Department New-Haven, Connecticut, USA yosi.keller@yale.edu Abstract We present an algorithm for detecting cyclic and dihe- dral symmetries of an object. Both synwnetry types can be detected by the special patterns they generate in the object’s Fourier transfom These pattern are effectively detected and analyzed using the “angular diflerence $mc- tion” (ADF), which measures the diflerence in the angular content of images. The ADF is accurately computed by us- ing the pseudo-polar Fourier tramfomz, which rapidly com- putes the Fourier tramfom of an object on a near-polar grid. The algorithm detects all the axes of centered and non-centered symmetries. The proposed algorithm is alge- braically accurate and uses m interpolations. 1 Introduction The two most common types of symmetries are rotational and reflectional symmetries. An object is said to have a rota- tional symmetry of order N if it is invariant under rotations ofann= . . . N - 1. An object is said to have a reflec- tion% symmetry if it is invariant under a reflection transfor- mation about some line. Most existing algorithms usually detect either rotational or reflectional symmetry. The algo- rithm presented in this paper is based on the angular differ- ence function (ADF), which measures the difference of two objects in a given angular direction. For symmetric objects the value of this function is shown to be zero in points that correspond to the symmetry axes. The zeros of the ADF identify both rotational and reflectional symmetries. The algorithm characterizes rotational symmetries by the set of rotation angles that keep the object unchanged. Similarly, it characterizes reflectional symmetries by the set of reflection axes. Our idea is related to the work presented in [3]. Both algorithms detect the patterns that symmetries induce in the frequency domain. However, the algorithm we present in this paper uses an algebraically exact method for detecting these patterns. Specifically, it computes the ADF using the pseudo-polar Fourier transform and then uses the zeros of Yoel Shkolnisky Tel-Aviv University Computer Science Department Tel-Aviv, Isreal yoel@post.tau.ac.il the ADF to detect minima ridges in the Fourier domain. II also uses a simpler scheme to infer the reflectional symme- try from the rotational symmetry. The paper is organized as follows. In section 2 we de- scribe previous work related to symmetry detection. In section 3 we describe the pseudo-polar Fourier transform, which evaluates the Fourier transform of an object on a near-polar grid. This transform is the basis for our sym- metry detection algorithm. In section 4 we introduce the Angular difference function (ADF) as a tool for analyzing polar properties of images and utilize it to detect and an- alyze rotational and reflectional symmetries. In sections 6 we present experimental results. 2 Previous work Symmetry is thoroughly studied in the literature from both theoretical, algorithmic and applicative perspectives. The- oretical treatment of symmetry can be found in [4]. The algorithmic approach to symmetry detection can be divided into several categories based on its characteristics. The first characteristic of a symmetry detection algorithm is whether it considers symmetry as a binary or continuous feature that measures the amount of symmetry. A second characteristic is the type of symmetry detected by the algorithm. Mosl algorithms detect either rotational or reflectional symmetry but not both. A third characteristic is the assumptions on the image. For example, whether the algorithm assumesthat the image is symmetric or detects it itself, or whether the algo- rithm assumes that the symmetric feature is located at the center of the image. A Fourth characteristic is whether the algorithm operates in the image domain or transforms the problem into a different domain, like the Fourier domain. A fifth characteristic is the robustness of the algorithm to noise and its ability to operate on real-life non-synthetic im- ages. The last characteristic of an algorithm is its complex- ity. This characteristic is important for symmetry detection algorithms since most algorithms typically require an ex- haustive search over all potential symmetry axes. Such a 0-7695-2128-2/04 $20.00 (C) 2004 IEEE