Transcript Document 7325744

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
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
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
p1
p2
pm
pg
cm
pmm
pmg
pgg
cmm
p4
p4m
p4g
p3
p31m
p3m1
The 7 Frieze Groups
VIIp6
p6m
From a web page by
David Joyce, Clark Univ.
Frieze Lattice Units
I
II
III
IV
V
http://www.clarku.edu/~djoyce/wallpaper/
VI
VII
Wallpaper Lattice Units
Regions of Dominance
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.
Possible Lattice Types
formed by the two shortest vectors
An Example:
parallelogram
rhombic
rectangle
square
hexagonal
Oriental Rug
Autocorrelation
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).
3) Wallpaper Group Classification
parallelogram
2-fold
2 refs
p1
N
p2
Y
rectangle
rhombic
D1 and D2-ref
1 ref
p4m
Y
N
square
Y
4-fold
pgg
1 glide
Y
T1-ref
pg
cm
N
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
p4
glide
p4g
hexagonal
Y
6-fold
N
3-fold
N
Y
Y
T1-ref
N
p6m
N
D1-ref
N
p3
Y
p3m1
Generating region
t2
Orbits of 2-fold
rotation centers
CMM
Poor
motif
Good candidate motifs
More Examples:
p1
p4
Rot 180 Rot 120
Rot 90
Rot 60
p2
p4m
An Example:
Original pattern
32 MostDominant
Peaks
p6
T1-ref
Y
p31m
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.
pmg
2 glides
cmm
N
Lattice
type
Y
T1 and T2-ref
Tabular form
pm
1 ref 1 glide
N
Highest 32
from Lin et.al
An Example
pmm
1 ref
Lin et.al.
(a competing
algorithm)
4) Motif Selection
(for Euclidean, monochrome patterns)
Euclidean
Algorithm
Global
Thresholding
t1
Auto-correlation image
Ref t1
t2
Ref t2 Ref t1+t2 Ref t1-t2
SSD correlation with…
t1
PMM
Lowest value
is match score
Lattice unit
pm
Rot 180 Rot 120
0.068
0.318
Ref t1
0.085
Ref t2
0.062
p4g
Rot 90
Rot 60
0.287
0.323
pg
p3
Ref t1+t2 Ref t1-t2
0.305
0.300
cm
5) Some Applications
Graphics
Pattern Analysis
recovered
original
p31m
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.
pmm
Gait Analysis
p3m1
background subtraction
pmg
p6
cross correlation(frameI,frameJ)
Symmetry of Walking Human
Symmetry of Running Dog
pgg
(This sequence from R.Cutler at U.Maryland)
p4m
cmm
p6m
Unit tile
Autocorrelation peaks
rot180
rot120
0.0484 0.1110
rot90
Lattice
rot60
0.0924 0.1143
flipT1
Unit tile
Autocorrelation peaks
flipT2
0.0567 0.0892
flipD1
flipD2
0.0891 0.0835
rot180
rot120
0.0513 0.1301
rot90
Lattice
rot60
0.1284 0.1241
flipT1
flipT2
0.1311 0.1311
flipD1
flipD2
0.0613 0.0558
cmm