A deep learning model, EscherNet 101, is constructed to categorize images of 2D periodic patterns into their respective 17 wallpaper groups. Beyond evaluating EscherNet 101 performance by classification rates, at a micro-level we investigate the filters learned at different layers in the network, capable of capturing second-order invariants beyond edge and curvature.

Humans take advantage of real world symmetries for various tasks, yet capturing their superb symmetry perception mechanism with a computational model remains elusive. Motivated by a new study demonstrating the extremely high inter-person accuracy of human perceived symmetries in the wild, we have constructed the first deep-learning neural network for reflection and rotation symmetry detection (Sym-NET), trained on photos from MS-COCO (Microsoft-Common Object in COntext) dataset with nearly 11K consistent 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. On CVPR 2013 symmetry competition testsets and unseen MS-COCO photos, Sym-NET significantly outperforms 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.

High-level robotic assembly planning is concerned with how bodies fit together and how spatial relationships among bodies are established over time. To generate an assembly task specification for robots, it is necessary to represent the geometric shapes of the assembly components in a computational form. One of the principal aspects of shape representation that is relevant for assembly tasks is the symmetry of the shape. Group theory is the standard mathematical tool for describing symmetry.

High-level robotic assembly planning is concerned with how bodies fit together and how spatial relationships among bodies are established over time. To generate an assembly task specification for robots, it is necessary to represent the geometric shapes of the assembly components in a computational form. One of the principal aspects of shape representation that is relevant for assembly tasks is the symmetry of the shape. Group theory is the standard mathematical tool for describing symmetry. The interaction between algebra and geometry within a group theoretic framework has provided us with a unified computational treatment of reasoning about how parts with multiple contacting features fit together.