Yanxi Liu and Robert Collins,
"Frieze and Wallpaper Symmetry Group Classification under Affine
and Perspective Distortion",  Technical Report CMU RI-98-37, 
Robotics Institute, Carnegie Mellon University, December 1998.


In this paper we study classification of 2D repeated patterns in terms of their respective symmetry groups -- the well-known seven Frieze groups and 17 wallpaper groups. Computer algorithms for Frieze and wallpaper symmetry group classification are developed in Euclidean as well as affine spaces. Several symmetry invariants of these groups under affine transformations are analyzed in detail and used to extend the Euclidean group classification algorithm for patterns that are distorted under affine transformations. Experimental results on computer generated images and photos of natural scenes are presented. Precise classification of 2D repeated patterns in terms of their symmetry groups provides a computational means for image indexing, image matching, object recognition, and motion recovery. This is a report of an on-going research effort. Existing problems and future work are discussed.

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