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Generic Fourier Descriptor for
Shape-based Image Retrieval
Dengsheng Zhang, Guojun Lu
Gippsland School of Comp. & Info Tech
Monash University
Churchill, VIC 3842
Australia
[email protected]
http://www.gscit.monash.edu.au/~dengs/
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Outline
• Motivations
• Problems
• Generic Fourier Descriptor (GFD)
• Experimental Results
• Conclusions
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Motivations
• Content-based Image Retrieval
– Image description is important for image searching
– Image description constitutes one of the key part of MPEG-7
– Shape is an important image feature along with color and
texture
• Effective and Efficient Shape Descriptor
– good retrieval accuracy, compact features, general
application, low computation complexity, robust retrieval
performance and hierarchical coarse to fine representation
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Fourier Descriptor
• Obtained by applying Fourier transform
on a shape signature, such as the central
distance function r(t).
1
an 
N
N 1
 r (t ) exp(  j 2nt / N ), n  0, 1, , N -1
t 0
No Contour
Shape Signature
Same Contour and
Different content
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Zernike Moments
• Acquired by applying Zernike moment transform on a
shape region in polar space.
–
–
–
–
Complex form
Does not allow finer resolution in radial direction
create a number of repetitions in each order of moment
Shape must be normalized into an unit disk
Z nm 
n 1
 f ( x, y)  V
*
nm
( x, y )
 x y
n 1

 f (r cos  , r sin  )  Rnm (r )  exp( jm ),
 r 
r 1
Vnm ( x, y)  Vnm (r cos  , r sin  )  Rnm (r )  exp( jm )
Rnm (r ) 
( n |m|) / 2

s 0
(1) s
(n  s)!
r n2 s
n | m |
n | m |
s! (
 s)! (
 s)!
2
2
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Generic Fourier Descriptor
• Polar Transform
– For an input image f(x, y), it is first
transformed into polar image f(r,  ):
r  ( x  xc ) 2  ( y  yc ) 2 ,   arctan
1
where xc 
M
N 1
1
x and yc 

N
x 0
y  yc
x  xc
M 1
y
y 0
– Find R = max{ r( ) }
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Generic Fourier Descriptor-II
• Polar Raster Sampling
Polar Grid
Polar image
Polar raster sampled image in Cartesian space
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Generic Fourier Descriptor-III
• Binary polar raster sampled shape
images
Polar raster sampling
Polar raster sampling
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Generic Fourier Descriptor-IIV
• 2-D Fourier transform on polar
raster sampled image f(r,  ):
r
2i
PF (  ,  )   f (r , i ) exp[  j 2 (  
 )]
R
T
r
i
where 0r<R and i = i(2/T) (0 i<T); 0<R, 0<T.
R and T are the radial frequency resolution and
angular frequency resolution respectively.
• The normalized Fourier coefficients
are the GFD.
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Generic Fourier Descriptor-V
• Rotation invariant
Fourier
Polar raster sampled
PF
Fourier
Polar raster sampled
PF
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Generic Fourier Descriptor-VI
• Translation invariant due to using shape
centroid as origin.
• Scale normalization:
| PF (0,0) | | PF (0,1) |
| PF (0, n) |
| PF (m,0) |
| PF (m, n) |
GFD  {
,
,...,
,...,
,...,
}
area
| PF (0,0) |
| PF (0,0) |
| PF (0,0) |
| PF (0,0) |
• Due to f(x, y) is real, only a quarter of the
transformed coefficients are distinct. The first
36 coefficients are selected as shape
descriptor.
• The similarity between two shapes are
measured by the city block distance between
the two set of GFDs.
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Experiment
• Datasets
– MPEG-7 region shape database (CE-2) has been tested. CE-2 has
been organized by MPEG-7 into six datasets to test a shape
descriptor’s behaviors under different distortions.
– Set A1 is for test of scale invariance. 100 shapes in Set A1 has
been classified into 20 groups which are designated as queries.
– Set A2 is for test of rotation invariance. 140 shapes in Set A2 has
been classified into 20 groups which are designated as queries
– Set A3 is for test of rotation/scaling invariance.
– Set A4 is for test of robustness to perspective transform. 330
shapes in Set A4 has been classified into 30 groups which are
designated as queries.
– Set B consists of 2811 shapes from the whole database, it is for
subjective test. 682 shapes in Set B have been manually classified
into 10 groups by MPEG-7.
– For the whole database, 651 shapes have been classified into 31
groups which can be used as queries.
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Performance Measurement
• Precision-Recall
r
number of relevant retrieved images
R 
n1 total number of relevant images in DB
r number of relevant retrieved images
P

n2
number of retrieved images
• For each query, the precision of the retrieval at
each level of the recall is obtained. The result
precision of retrieval is the average precision of
all the query retrievals.
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Results
• Average Precision-Recall on Set A1 and A2
Rotation Invariance Test
100
90
80
70
60
50
40
30
20
10
0
Precision
Precision
Scale Invariance Test
GFD
ZMD
10
20
30
40
50
60
Recall
70
80
90
100
100
90
80
70
60
50
40
30
20
10
0
GFD
ZMD
10
20
30
40
50
60
70
80
90
100
Recall
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Results
• Average Precision-Recall on Set A4 and CE-2
General Invariance Test
100
90
80
70
60
50
40
30
20
10
0
Precision
Precision
Perspective Invariance Test
GFD
ZMD
0
10
20
30
40
50
60
Recall
70
80
90
100
100
90
80
70
60
50
40
30
20
10
0
GFD
ZMD
0
10
20
30
40
50
60
70
80
90
100
Recall
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• Average Precision-Recall on Set B
Subjective Test
80
70
Precision
60
50
GFD
40
ZMD
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Recall
Class
1
2
3
4
5
6
7
8
9
10
Average
No. of shapes
68.0
248
22
28
17
22
45
145
45
42
GFD (%)
47.0
66.4
55.6
50.0
50.0
24.8
30.4
50.8
55.6
29.0
46.0
ZMD (%)
37.0
58.0
55.0
41.2
42.6
22.6
33.6
52.0
41.4
34.0
41.7
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Results
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CE-2
Set B
19
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Conclusions
• A new shape descriptor, generic Fourier descriptor
(GFD) has been proposed.
• It has been tested on MPEG-7 region shape database
• Comparisons have been made between GFD and
MPEG-7 shape descriptor ZMD.
• Compared with ZMD, GFD has four advantages:
– it captures spectral features in both radial and circular
directions;
– it is simpler to compute;
– it is more robust and perceptually meaningful;
– the physical meaning of each feature is clearer.
• The proposed GFD satisfies all the six requirements
set by MPEG-7 for shape representation .
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