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