member: 洪健超 n26984232 謝豐任 n26984258 謝明德 ieee transactions on pattern analysis and...

Skewed Rotation Symmetry Group Detection

Member: 洪健超 N26984232 謝豐任 N26984258 謝明德

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE

AUG 2009

Outline

IntroductionAlgorithms and procedures Rotation Symmetry Strength (RSS) Symmetry Shape Density (SSD) Affine deformation

DiscussionDemo

Introduction

What is affinely skewed rotation symmetry?

To present algorithm for affinely skewed rotation symmetry group detection from images.

Affine transformation

x’=Ax+b

Introduction

There are three types of Euclidean rotation symmetry groups for 2D objects:

Cyclic(Cn) n:folds

Dihedral(Dn)

A disk O(2)

Algorithms and procedures

Five independent properties of a skew rotation symmetry group:

1) center of the rotation symmetry

2) affine deformation (Orientation & Aspect ratio)

3) type of the symmetry group(Cn, Dn, O(2))

4) cardinality of the symmetry group (number of folds)

5) supporting regions for the symmetry group (annulus)

Algorithms and proceduresInput

RSS SSD

Candidate rotation center

Local Affine Rectification

Input

RSS SSD

Rot

atio

n S

ymm

etry

C

ente

r D

etec

tion

Symmetry Center& Rectified Input

Frieze-expansion

Discrete Fourier Transform

Frequency Analysis

Merging & Elimination

FEP

Output

Output

Rot

atio

n S

ymm

etry

G

roup

Ana

lysi

s

Algorithms and proceduresInput

RSS SSD

Candidate rotation center

Local Affine Rectification

Input

RSS SSD

Rot

atio

n S

ymm

etry

C

ente

r D

etec

tion

Symmetry Center& Rectified Input

Frieze-expansion

Discrete Fourier Transform

Frequency Analysis

Merging & Elimination

FEP

Output

Output

Rot

atio

n S

ymm

etry

G

roup

Ana

lysi

s

Rotation Symmetry Stregth (RSS)

To propose a frieze-expansion method that transform rotation symmetry group into one dimensional translation symmetry detection problem.

(x,y)

Rpx,y(r, n) : frieze expansion pattern

Rotation Symmetry Stregth (RSS)

Frieze expansion

Rotation Symmetry Stregth (RSS)

One dimensional horizontal discrete fourier transform (DFT)

energy spectral density

2( 1)( 1)

, , , ,1

( , ) ( , ) ( , ) = ( , )N i n k

Nx y x y x y x y

n

P r k a r k ib r k p r n e

2 2, , ,( , ) ( , ) ( , )x y x y x yS r k a r k b r k

, ( , )x yS r k : energy spectral density

Rotation Symmetry Stregth (RSS)

Define RSS at an image position (x,y):FEP DFT

, , ,( , ) { ( , )} 2 { ( , )}

2,3,4,...,2

x y peak x y x yS r k mean S r k std S r k

Nk

k

r

kpeak

k

r

Rotation Symmetry Stregth (RSS)

,

1 ,

( ( , ( )))( , )

( ( , ))

Rx y peak

rr x y

mean S r k rRSS x y

mean S r k

1, ( ( ), ( ( ))) 0,

0,

peak peak

r

if Mod k r min k rwhere

otherwise

0 10 20 30 40 50 60 70 80 900

5

10

15x 10

6

k

S

0 10 20 30 40 50 60 70 80 900

5

10x 10

6

k

S

0 10 20 30 40 50 60 70 80 900

1

2

3x 10

6

k

S

Rotation Symmetry Stregth (RSS)

Symmetry Shape Density (SSD)Bidirectional Flow

ci = (xi , yi)

,

,

( ( , 2))( )

( ( , 2))i i

i i

x yi

x y

Re P rr arctan

Im P r

so the slope of ci with slope -1/tan(Фi)

( ( ))i imedian r

The bidirectional flow line is defined as:

tan 11

tan tani

i i i i i i

y xx y x y

The Bidirectional Flow EquationBidirectional flow line

Symmetry Shape Density (SSD)

Bidirectional Flow

Symmetry Shape Density (SSD)

Bidirectional Flow

ci = (xi, yi)

Bidirectional Flow

Consider another point cj = (xj, yj),

X cj = (xj, yj)X

Bidirectional Flow

C

The potential rotation symmetry center C by intersecting the two bidirectional flows detected:

( )

i i j j i

i j

i j i j i j j i

i j

s x s xj y y

s sxC

y s s x x s y s y

s s

where1 1

tan tani j

i j

s s

Symmetry Shape Density (SSD)

Symmetry Shape Density (SSD)

SSD(x,y)=D(x,y) ． G(l,l)D(x, y) corresponds to the cumulative number of intersecting points C at the location (x, y)G(l,l) : Gaussian kernel

Symmetry Shape Density (SSD)

RSS VS SSD

D3 O(2)

Algorithms and proceduresInput

RSS SSD

Candidate rotation center

Local Affine Rectification

Input

RSS SSD

Rot

atio

n S

ymm

etry

C

ente

r D

etec

tion

Symmetry Center& Rectified Input

Frieze-expansion

Discrete Fourier Transform

Frequency Analysis

Merging & Elimination

FEP

Output

Output

Rot

atio

n S

ymm

etry

G

roup

Ana

lysi

s

Affine deformation

Orentation Aspect ratio

Affine deformation

Orientation sine wave pattern repeated twice

Φ1 = median(Φ(r))

,

,

( ( ,31

2

))( )

,3( )

( ( ))x y

x y

Re P rr actan

Im P r

Φ2 = Φ1 + π/2

Affine deformation

Aspect ratioTo compare RSS values for each different aspect ratio of the aligned images by changing the length of the x-axis, between 1:1 and 1:σ(0 < σ < 1)

0 10 20 30 40 50 60 70 80 90 100 1100

100

200

300

400

500

600

700

800

RS

SRSS

Affine deformation

Algorithms and proceduresInput

RSS SSD

Candidate rotation center

Local Affine Rectification

Input

RSS SSD

Rot

atio

n S

ymm

etry

C

ente

r D

etec

tion

Symmetry Center& Rectified Input

Frieze-expansion

Discrete Fourier Transform

Frequency Analysis

Merging & Elimination

FEP

Output

Output

Rot

atio

n S

ymm

etry

G

roup

Ana

lysi

s

Rotation Symmetry Group Analysis

Cardinality of the symmetry group

DFT coefficient -1

The dominant coefficient

Rotation Symmetry Group Analysis

Symmetry Group ClassificationNo vertical reflection

vertical reflection

To flip the cell and slide it over FEP with horizontally while computing correlation. If the periodic match corresponding the principle frequency from DFT, then we conclude it is dihedral. Otherwise , it is cyclic.

And must be local maximum

Discussion

, , ,( , ) { ( , )} 2 { ( , )} 2,3,4,...,2x y peak x y x y

NS r k mean S r k std S r k k

,

1 ,

( ( , ( )))( , )

( ( , ))

Rx y peak

rr x y

mean S r k rRSS x y

mean S r k

( ( ), (1, ,

))

,

(

0

) 0peak pea

r

kif Mod k r mwhere

otherw

i

i

n k

s

r

e

, ( , )x y peakS r k

KpeakKpeak

And must be local maximum

Discussion

, , ,( , ) { ( , )} 2 { ( , )} 2,3,4,...,2x y peak x y x y

NS r k mean S r k std S r k k

,

1 ,

( ( , ( )))( , )

( ( , ))

Rx y peak

rr x y

mean S r k rRSS x y

mean S r k

( ( ), (1, ,

))

,

(

0

) 0peak pea

r

kif Mod k r mwhere

otherw

i

i

n k

s

r

e

, ( , )x y peakS r k

RSS map

Demo