1. A Computational Model for Repeated Pattern Perception Using Frieze and Wallpaper Groups Yanxi Liu and Robert T. Collins, Robotics Institute, Carnegie Mellon University ABSTRACT The theory of Frieze and wallpaper groups is used to extract visually meaningful building blocks (motifs) from a repeated pattern. We show that knowledge of the interplay between translation, rotation, reflection and glide-reflection in the symmetry group of a pattern leads to a small finite set of candidate motifs that exhibit local symmetry consistent with the global symmetry of the entire pattern. The resulting pattern motifs conform well with human perception of the pattern. General idea: find lattice of peaks in an autocorrelation image Problem: many patterns have self-similar structure at multiples of the true lattice frequency, causing spurious candidate peaks to form in the autocorrelation surface Observation: height (magnitude) of a peak value does not imply salience! Our approach: judge salience of a candidate peak by the size of its Region of Dominance, defined as the largest hypersphere, centered on the peak, within which no higher peak can be found. 2) Translational Lattice Extraction Oriental RugAutocorrelation Global Thresholding Lin et.al. (a competing algorithm) Highest 32 from Lin et.al 32 Most- Dominant Peaks An Example: 1) Symmetry Group Theory Main Point: A finite set of symmetry groups completely characterize the structural symmetry of any repeated pattern. Wallpaper Lattice Units VII From a web page by David Joyce, Clark Univ. p1p2pmpgcm pmmpmgpggcmmp4 p4mp4gp3p3m1 p6p6m http://www.clarku.edu/~djoyce/wallpaper/ p31m The 17 Wallpaper Groups The 7 Frieze Groups Frieze Lattice Units IIIIIIIVVVIVII formed by the two shortest vectors parallelogram rectangle square hexagonal rhombic Possible Lattice Types Crystallographic restriction: the order of rotation symmetry in a wallpaper pattern can only be 2 (180 degrees), 3 (120 deg), 4 (90 deg) or 6 (60 deg). Original patternAuto-correlation image Generating region t1 t2 SSD correlation with… Lowest value is match score Rot 180Rot 120Rot 90Rot 60 Ref t1Ref t2Ref t1+t2Ref t1-t2 0.0680.3180.2870.323 0.0850.0620.3050.300 PMM Here 2,3,4, or 6 denotes an n-fold rotational symmetry T n or D n denotes a reflectional symmetry about one of the unit lattice edges or diagonals Y(g) indicates the existence of glide-reflection symmetry 3) Wallpaper Group Classification (for Euclidean, monochrome patterns) An Example: t2 t1 Rot 180 Rot 120Rot 90Rot 60 Ref t1 Ref t2Ref t1+t2Ref t1-t2 Tabular form Lattice unit 5) Some Applications Regular texture replacement: Replace one regular scene texture with another, in an image, while maintaining the same sense of scene occlusions, shading and surface geometry. Pattern Analysis Gait Analysis Graphics original recovered cross correlation(frameI,frameJ) background subtraction (This sequence from R.Cutler at U.Maryland) 4) Motif Selection General idea: for each wallpaper class, the stabilizer subgroups (centers of rotational symmetry) with the highest order belong to a finite number of orbits. Choose a set of candidate motifs centered on each independent point of the highest rotational symmetry. 4) Motif Selection p3 p4 p4m p4g p6 p3m1 p31m p6m pmm cmm pgg pmg pm p1 p2 pg cm More Examples: CMM Orbits of 2-fold rotation centers Poor motif Good candidate motifs An Example Regions of Dominance