A Computational Model for Repeated Pattern Perception Using Frieze and Wallpaper Groups

1. 1) Symmetry Group Theory Main Point: A finite set of symmetry groups completely characterize the structural symmetry of any repeated pattern. The 17 Wallpaper Groups p1 p2 pm pg cm The 7 Frieze Groups pmm pmg pgg cmm p4 p4m p4g p3 p31m p3m1 p6 p6m VII From a web page by Frieze Lattice Units Regions of Dominance David Joyce, Clark Univ. http://www.clarku.edu/~djoyce/wallpaper/ I II III IV V VI VII Wallpaper Lattice Units Possible Lattice Types formed by the two shortest vectors parallelogram rhombic rectangle square hexagonal 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). Orbits of 2-fold rotation centers An Example CMM Poor motif Good candidate motifs p1 p4 p2 Rot 180 Rot 120 Rot 90 Rot 60 p4m pm t2 p4g Ref t1 Ref t2 Ref t1+t2 Ref t1-t2 t1 pg p3 cm p31m Graphics 5) Some Applications Pattern Analysis pmm Gait Analysis p3m1 background subtraction pmg 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. p6 recovered original cross correlation(frameI,frameJ) pgg (This sequence from R.Cutler at U.Maryland) p6m cmm 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. 2) Translational Lattice Extraction 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. An Example: Lin et.al.(a competing algorithm) Global Thresholding Highest 32 from Lin et.al 32 Most- Dominant Peaks Oriental Rug Autocorrelation 3) Wallpaper Group Classification(for Euclidean, monochrome patterns) 4) Motif Selection 4) Motif Selection Tabular form 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. Here 2,3,4, or 6 denotes an n-fold rotational symmetry Tn or Dn denotes a reflectional symmetry about one of the unit lattice edges or diagonals Y(g) indicates the existence of glide-reflection symmetry More Examples: Generating region t2 An Example: t1 Original pattern Auto-correlation image SSD correlation with… PMM Lowest value is match score Rot 180 Rot 120 Rot 90 Rot 60 Lattice unit 0.068 0.318 0.287 0.323 Ref t1 Ref t2 Ref t1+t2 Ref t1-t2 0.085 0.062 0.305 0.300