text: Humans take advantage of real world symmetries for various tasks, yet capturing their superb symmetry perception mechanism into a computational model remains elusive. Encouraged by a new discovery (CVPR 2016) demonstrating extremely high inter-person accuracy of human perceived symmetries in the wild, we have created the first deep-learning neural network for reflection and rotation symmetry detection (Sym-NET), trained on photos from MS-COCO (Common Object in COntext) dataset with nearly 11K symmetry-labels from more than 400 human observers. We employ novel methods to convert discrete human labels into symmetry heatmaps, capture symmetry densely in an image and quantitatively evaluate Sym-NET against multiple existing computer vision algorithms. Using the symmetry competition testsets from CVPR 2013 and unseen MS-COCO photos, Sym-NET comes out as the winner with significantly superior performance over all other competitors. Beyond mathematically well-defined symmetries on a plane, Sym-NET demonstrates abilities to identify viewpoint-varied 3D symmetries, partially occluded symmetrical objects and symmetries at a semantic level.
  @cite_65: We present a computational model for periodic pattern perception based on the mathematical theory of crystallographic groups. In each N-dimensional Euclidean space, a finite number of symmetry groups can characterize the structures of an infinite variety of periodic patterns. In 2D space, there are seven frieze groups describing monochrome patterns that repeat along one direction and 17 wallpaper groups for patterns that repeat along two linearly independent directions to tile the plane. We develop a set of computer algorithms that "understand" a given periodic pattern by automatically finding its underlying lattice, identifying its symmetry group, and extracting its representative motifs. We also extend this computational model for near-periodic patterns using geometric AIC. Applications of such a computational model include pattern indexing, texture synthesis, image compression, and gait analysis.
  @cite_0: Abstract Automatic detection of the principal axes of given shapes with known symmetry properties is studied in this paper. The inapplicability of a well-known equation, which is used to compute the direction of the principal axis of a given shape using moment functions, to a class of so-called degenerate shapes is first pointed out. Rotationally symmetric shapes often encountered in real applications are shown to belong to this class. This problem is solved by extending the notion of principal axis to higher order ones in terms of higher order moment functions. Analytic equations for computing the direction of high-order principal axes are derived. They include the well-known equation for computing the direction of the (second-order) principal axis as a special case. Some experimental results are included finally to show the effectiveness of the derived analytic equations.
  @cite_45: 
  @cite_49: Abstract A template matching procedure based on symmetry detection of planar images is presented. The method works directly in conjunction with unscaled intensity images, independently of position and orientation, and may be adapted to the case of overlapping objects by introducing the concept of local symmetry. Moreover, owing to the moderate amount of needed local processing, the algorithm is particularly noise insensitive.
  @cite_46: 
  @cite_51: A novel and efficient method is presented for grouping feature points on the basis of their underlying symmetry and characterising the symmetries present in an image. We show how symmetric pairs of features can be efficiently detected, how the symmetry bonding each pair is extracted and evaluated, and how these can be grouped into symmetric constellations that specify the dominant symmetries present in the image. Symmetries over all orientations and radii are considered simultaneously, and the method is able to detect local or global symmetries, locate symmetric figures in complex backgrounds, detect bilateral or rotational symmetry, and detect multiple incidences of symmetry.
summary: Earlier work on rotation symmetry detection includes the use of autocorrelation @cite_46 @cite_65 and image moments @cite_45 @cite_49 @cite_0 . Loy and Eklundh @cite_51 use a variation on their SIFT feature-based reflection symmetry approach to find rotation symmetry as well. The orientations of matched SIFT feature pairs are used to find a rotation symmetry center. The detected rotation symmetry centers emerge as maxima in the voting space. This algorithm stands out from all others since the authors have made their code publicly available, and the symmetry competition workshops in CVPR 2011 2013 have used it as the baseline algorithm for both reflection and rotation symmetry detection. Thus far, this algorithm is considered to be the best baseline algorithm for reflection and rotation symmetry detection.