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_58: We present a novel and effective algorithm for affinely skewed rotation symmetry group detection from real-world images. We define a complete skewed rotation symmetry detection problem as discovering five independent properties of a skewed rotation symmetry group: 1) the center of rotation, 2) the affine deformation, 3) the type of the symmetry group, 4) the cardinality of the symmetry group, and 5) the supporting region of the symmetry group in the image. We propose a frieze-expansion (FE) method that transforms rotation symmetry group detection into a simple, 1D translation symmetry detection problem. We define and construct a pair of rotational symmetry saliency maps, complemented by a local feature method. Frequency analysis, using Discrete Fourier Transform (DFT), is applied to the frieze-expansion patterns (FEPs) to uncover the types (cyclic, dihedral, and O(2)), the cardinalities, and the corresponding supporting regions, concentric or otherwise, of multiple rotation symmetry groups in an image. The phase information of the FEP is used to rectify affinely skewed rotation symmetry groups. Our result advances the state of the art in symmetry detection by offering a unique combination of region-based, feature-based, and frequency-based approaches. Experimental results on 170 synthetic and natural images demonstrate superior performance of our rotation symmetry detection algorithm over existing methods.
  @cite_32: We present a novel and effective algorithm for rotation symmetry group detection from real-world images. We propose a frieze-expansion method that transforms rotation symmetry group detection into a simple translation symmetry detection problem. We define and construct a dense symmetry strength map from a given image, and search for potential rotational symmetry centers automatically. Frequency analysis, using discrete Fourier transform (DFT), is applied to the frieze-expansion patterns to uncover the types and the cardinality of multiple rotation symmetry groups in an image, concentric or otherwise. Furthermore, our detection algorithm can discriminate discrete versus continuous and cyclic versus dihedral symmetry groups, and identify the corresponding supporting regions in the image. Experimental results on over 80 synthetic and natural images demonstrate superior performance of our rotation detection algorithm in accuracy and in speed over the state of the art rotation detection algorithms.
  @cite_11: 
summary: Lee and Liu @cite_11 @cite_32 @cite_58 have developed an algorithm to detect (1) the center of rotation, (2) the number of folds, (3) type of symmetry group (dihedral cyclic O(2)), and (4) the region of support. The first step of their algorithm is rotation symmetry center detection where they use a novel frieze expansion to transform the image at each pixel location into polar coordinates and then search for translation symmetry. The second step applies a Discrete Fourier Transform (DFT) on the frieze expansion to determine (2)-(4) listed above. In our work, for rotation symmetries we only focus on detecting rotation symmetry centers.