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