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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. - PowerPoint PPT Presentation
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A Computational Model for Repeated Pattern Perception Using Frieze and Wallpaper GroupsYanxi Liu and Robert T. Collins, Robotics Institute, Carnegie Mellon University
ABSTRACTThe 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 imageProblem: 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 Rug AutocorrelationGlobal
ThresholdingLin et.al.
(a competingalgorithm)
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.
p1 p2 pm pg cm
pmm pmg pgg cmm p4
p4m p4g p3 p3m1
p6 p6m
http://www.clarku.edu/~djoyce/wallpaper/
p31m
The 17 Wallpaper Groups
The 7 Frieze Groups
Frieze Lattice UnitsI II III IV V VI VII
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).
parallelogram
rectangle
rhombic
hexagonal 6-fold
3-fold T1-ref D1-ref
D1 and D2-ref
T1-ref
T1 and T2-ref
2-fold
Y
N
Y
N
Y
NN
Y
Y
N
N
Y
N
Y
p6m
cm
p3
p3m1p31m
p6
pmm
p2pm
pg
p1
pmg
pgg
2 refs
1 ref 1 glide
2 glides
1 glide
1 ref
1 ref
EuclideanAlgorithm
Latticetype
square 4-fold T1-ref
Y
NY
NY
cmm
p4g
p4
p4m
glide
N
Original pattern Auto-correlation image
Generating region
t1
t2
SSD correlation with…
Lowest valueis match score
Rot 180 Rot 120 Rot 90 Rot 60
Ref t1 Ref t2 Ref t1+t2 Ref t1-t2
0.068 0.318 0.287 0.323
0.085 0.062 0.305 0.300
PMM
Here 2,3,4, or 6 denotes an n-fold rotational symmetryTn or Dn denotes a reflectional symmetry about one of the unit lattice edges or diagonalsY(g) indicates the existence of glide-reflection symmetry
3) Wallpaper Group Classification (for Euclidean, monochrome patterns)
An Example:
t2
t1
Rot 180 Rot 120 Rot 90 Rot 60
Ref t1 Ref t2 Ref t1+t2 Ref 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.
Symmetry of Running Dog
Autocorrelation peaks Lattice Unit tile
0.05580.06130.13110.13110.12410.12840.13010.0513
flipD2flipD1flipT2flipT1rot60rot90rot120rot180
cmm
Symmetry of Walking Human
Autocorrelation peaks Lattice
Unit tile
0.08350.08910.08920.05670.11430.09240.11100.0484
flipD2flipD1flipT2flipT1rot60rot90rot120rot180
p4m
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
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