ABSTRACT A novel summation invariant of curves under transformation group action is proposed. This new invariant is less sensitive to noise than the differential invariant and does not require an analytical expression for the curve as the integral invariant does. We exploit this summation invariant to define a shape descriptor called a semi-local summation invariant and use it as a new feature for shape recognition. Tested on a database of noisy shapes of fishes, it was observed that the summation invariant feature exhibited superior discriminating power than that of wavelet-based invariant features

Nigel Boston

Wei-Yang Lin

Yu Hen Hu

CiteSeerX

SUMMATION INVARIANT AND ITS APPLICATIONS TO SHAPE RECOGNITION  
 
Wei-Yang Lin, Nigel Boston, and Yu Hen Hu 
 
Department of Electrical and Computer Engineering  
University of Wisconsin, Madison WI, 53706 USA 
weiyanglin@wisc.edu, boston@ece.wisc.edu, hu@engr.wisc.edu  
 
ABSTRACT 
A novel summation invariant of curves under 
transformation group action is proposed. This new 
invariant is less sensitive to noise than the differential 
invariant and does not require an analytical expression 
for the curve as the integral invariant does. We exploit 
this summation invariant to define a shape descriptor 
called a semi-local summation invariant and use it as a 
new feature for shape recognition. Tested on a database of 
noisy shapes of fishes, it was observed that the 
summation invariant feature exhibited superior 
discriminating power than that of wavelet-based invariant 
features. 
1. INTRODUCTION 
 
Invariants for transformation groups play an important 
role in computer vision. The idea that one can compute 
functions of images that do not change under various 
viewing conditions is appealing. It holds potential for 
many applications. Hence the study of invariants for 
certain transformation groups (Euclidean, affine and 
projective)  has flourished.  
Toward the end of the last millennium, algorithms 
based on invariants did not meet our expectations. 
Differential invariants depend on derivatives so are very 
sensitive to noise [1, 2]. There have been several attempts 
to decrease sensitivity to noise. To avoid high-order 
derivatives, a semi-differential invariant was introduced 
in [3]. Affine invariant multiscale analysis was 
investigated in [4]. Potentials were used as coordinates to 
prolong group actions, so that the resulting invariants 
would depend on integrals rather than derivatives and not 
be sensitive to noise [5]. Also another type of integral 
invariant was formulated by integrating with respect to 
affine quasi-invariant arc-length [6]. 
In this paper, we introduce a general method to 
generate invariants that are weighted summations of 
discrete data, as analogues to integral ones. We use the 
summation invariants to measure the similarity between 
shapes and illustrate the potential of our method in real 
world applications. The rest of this paper is organized as 
follows. Section 2 describes the summation invariant. In 
section 3, we use the summation invariant to define a 
novel shape descriptor, which is called the semi-local 
summation invariant.  In section 4, we apply the proposed 
method to the problem of fish recognition. Finally, 
section 5 summarizes the contribution and provides an 
overview of future directions.  
 
2. SUMMATION INVARIANT 
 
The transformation groups acting on 2° , such as 
Euclidean, affine and projective groups are of particular 
importance in planar shape recognition. In this section, 
we describe a systematic method to find invariant 
functions for certain transformation groups.  
 
2.1. Extending group action to potentials 
 
The boundary of an object is extracted and parameterized 
as [ ]x n and [ ]y n . Consider a transformation group G of 
dimension r acting on 2° defined by 
 
( [ ], [ ]) ( [ ], [ ]),g x n y n x n y n g G= ∈o  
 
We prolong the group action to the jet space 
nJ consisting of potentials up to the n-th order. The 
definition of potential and jet space are shown below. 
 
Definition 1. The potential ,i jP of order k is given by 
 
1
,
0
[ ] [ ]
N
i j i j
n
P x n y n
−
=
= ⋅∑ , where i j k+ =  
 
Definition 2. The jet space nJ is the Euclidean space with 
coordinates  
 
( )( [0], [0], [ 1], [ 1], )nx y x N y N P− −  
 
PDF created with pdfFactory Pro trial version www.pdffactory.com
where ( )nP consist of potentials up to n-th order.   
For example, 
 
1 1
1
0 0
( [0], [0], [ 1], [ 1], [ ], [ ])
N N
n n
J x y x N y N x n y n
− −
= =
= − − ∑ ∑  
 
Then, we can find an invariant function of the 
transformation group G by solving the normalization 
equations [7].  
 
2.2. An example: invariant of affine transformation 
 
In this section, we use affine transformation as an 
example to illustrate the proposed method. Consider the 
affine transformation group acting on 2° given by, 
 
( , ) ( , )g x y ax by c dx ey f= + + + +o  
, where det( ) 0
a b
c d
≠
 
  
 
 
After applying the group action, 1,0P  is given by 
  
1
1,0 1,0 0 ,1
0
[ ] [ ]
N
n
P ax n by n c aP bP cN
−
=
= + + = + +∑  
 
Similarly, 
 
1
0 ,1 1,0 0,1
0
[ ] [ ]
N
n
P dx n ey n f dP eP fN
−
=
= + + = + +∑  
 
We can solve for {a, b, c, d, e, f} by setting 
 
1,0 0,1( [0], [0], [ 1], [ 1], , )
(0, 0,1,1, 0, 0)
x y x N y N P P− −
=
 
 
Then, Substituting {a, b, c, d, e, f} into 2,0P  gives us the 
invariant function 
 
2,0 0,1 2 0,2 1,0 2
1,1 1,0 0,1
1,0 0,1 2
2
1,0 0,1
( [0] ) ( [0] )
2 ( [0] )( [0] )
( [0] [0])
[ 1] [0] [0] [ 1]
/
( [ 1] [0]) ( [ 1] [0])
P Ny P P Nx P
P Nx P Ny P
N P y P x
Nx N y Nx y N
P y N y P x N x
α
− + −
= − − −
− −
− − − +
− − − − −
 
 
 
 
 
 
 
 
 
 
3. SEMI-LOCAL SUMMATION INVARIANTS 
 
Since the summation invariant is a map from N° to ° , 
the dimension of the feature vector is one. In some 
applications, such as recognition of similar objects, it 
won’t give us accurate recognition results. Instead of 
doing global summation, we define a summation 
invariant locally to extract local features of the contour 
and also expand the dimension of the feature vector. It’s 
called a semi-local summation invariant. The definition 
of the semi-local summation invariant is given by 
 
( )21 0 0 1 1 0 1 0[ ] ( ) ( ) ( )x ym M x y x y P y y P x xβ = − + − − −  
 
 where 
  
1
[mod( , )]
m M
x
n m
P x n N
+ −
=
= ∑       
1
[mod( , )]
m M
y
n m
P y n N
+ −
=
= ∑  
0 [ ],x x m=    0 [ ],y y m=    1 [mod( 1, )],x x m M N= + −    
1 [mod( 1, )].y y m M N= + −   
 
Here, we can use only the denominator of α to define a 
semi-local summation invariant.  Since it’s transformed 
by 
 
g o denominator(α) = 2( )ae bd−  denominator(α) 
 
for all (2)g A∈ . The scaling factor will be canceled when 
similarity is measured by normalized cross-correlation as 
follows:  
1
1 2
0
1 1
2 2
1 2
0 0
[ ] [ ]
[ ] [ ]
N
n
N N
n m
n n
n m
β β
ρ
β β
−
=
− −
= =
=
∑
∑ ∑
 
 
4. APPLICATIONS TO SHAPE RECOGNITION 
 
In this section, we apply semi-local summation invariant 
to recognize 2D fish contours under affine 
transformations.  
 
4.1. Fish recognition 
 
We randomly selected 100 distinct fish contours  from the 
SQUID database [8], and re-sampled each 2D contour 
curve such that the total number of points is 512. Some of 
these fish contours are shown in Figure 1(a). For each 
curve, we generate 20 variations by applying affine 
transformations with randomly generated parameters. In 
PDF created with pdfFactory Pro trial version www.pdffactory.com
Figure 1(b), we illustrate the 20 variations of the same 
fish contour.  In all, there are 100 distinct types of fish 
contours and 20 variations for each type. For each type of 
fish, its 20 variations are partitioned equally and 
randomly into five disjoint sets of 4 samples each. 
Combining the corresponding disjoint sets of each fish 
type, we have five sets of data, with each consisting of 
exactly four samples of each of the 100 types of fishes.  
Fish recognition is performed by choosing SET 1 as 
the training set and the others as the test set. We can also 
perform 4 other recognition experiments by choosing 
SET 2, SET 3, SET 4 and SET 5 as the training set 
respectively. Specifically, the same pattern classifier will 
be applied to 5 different partitions of the available data 
into training and testing data sets. In the kth partition, the 
kth data set will be the training data set, and the 
remaining four data sets combined will form the testing 
data set. This way, each data set will be used as the 
training data set exactly once in the 5 partitions. 
The semi-local summation invariant is calculated for 
each fish contour with M =51 with cyclic extension at the 
boundary. The feature vector has a dimension of 512. 
A nearest neighbor pattern classifier is used for this 
experiment. For each testing feature vector, its cross-
correlation with each of the training feature vectors is 
computed according to the following formula: 
1
0
1 1
2 2
0 0
[ ] [ ]
[ ] [ ]
N
training test
n
N N
training test
n m
n n
n m
ρ
η η
η η
−
=
− −
= =
=
⋅∑
∑ ∑
 
When the semi-local summation invariant is used for 
matching fish contours, the total number of mismatches is 
177 out of 8000 or 2.21%. We repeat the same procedure 
but replacing the summation invariant with an integral 
invariant. The total number of mismatches is 1165/8000 
= 14.6%.  It is quite clear that the summation invariant 
based feature performed much better than that of the 
integral invariant based features. 
 
4.2. Sensitivity to noise 
 
Next, the sensitivity of the semi-local summation 
invariant based feature will be tested experimentally and 
compared to that of a wavelet invariant feature proposed 
by Khalil and Bayoumi [9]. For this purpose, we add 
Gaussian-distributed noise into the fish contour. Again, 
the database contains 100 distinct types of fish and each 
type has 20 variations. An example of a fish contour and 
its noisy version are depicted in Figure 2.  Two different 
noise levels are used:  σ = 1, and σ = 2.  
It has been shown [9] that the wavelet-based invariant 
function  shows superior discriminating power over other 
traditional invariant features such as the moment 
invariant [10] or Fourier descriptor method [11]. In this 
work, the wavelet affine invariant function , , , , , ( )a b c d e f tη  is 
used.  Within the 8 scale-levels we compute 3,4,5,6,7 ,8 ( )tη . 
During preliminary experiments, it is found that scale 
levels 1 and 2 are too sensitive to noise and are excluded 
from the wavelet invariant features.  
The results in terms of probability of 
misclassification are summarized in the following table: 
Table 1 Sensitivity comparison using probability of 
misclassification 
 Semi-local wavelet invariant 
σ = 1 367/8000 = 4.6% 2253/8000 = 28.2% 
σ = 2 514/8000 = 6.4% 3476/8000 = 43.5%  
 
0 50 100
0
100
200
300
0 50 100 150
0
200
400
0 100 200
0
200
400
600
0 50 100 150
0
200
400
0 50 100 150
0
200
400
0 50 100 150
0
200
400
600
0 50 100 150
0
200
400
0 50 100 150
0
200
400
0 50 100
0
100
200
300
0 50 100
0
200
400
0 50 100 150
0
100
200
300
0 50 100 150
0
100
200
300
0 200 400
0
50
100
0 200 400
0
50
100
150
0 50 100 150
0
100
200
300
0 200 400
0
50
100
150
0 200 400
0
50
100
150
0 50 100 150
0
100
200
300
0 50 100 150
0
100
200
300
0 200 400
0
100
200
 
(a) 
0 50 100 150
-200
0
200
400
-80 -60 -40 -20
-400
-200
0
200
0 100 200 300
-100
0
100
200
-300 -200 -100 0
-100
-50
0
50
-200 -100 0 100
0
50
100
150
-100 0 100 200
0
100
200
300
0 100 200 300
-50
0
50
100
-100 0 100 200
-300
-200
-100
0
-200 -100 0
-50
0
50
-400 -200 0 200
0
100
200
-150 -100 -50 0
-400
-200
0
200
-200 -100 0
-100
0
100
0 100 200 300
-50
0
50
100
-200 0 200 400
-200
-100
0
-400 -200 0 200
0
50
100
150
-100 -50 0 50
0
100
200
300
-300 -200 -100 0
-20
0
20
40
-100 -50 0 50
-300
-200
-100
0
-100 -50 0 50
-200
-100
0
-150 -100 -50 0
-400
-200
0
200
 
(b) 
 
Figure 1. (a) 20 distinct types of fish from the 
database are shown here.  (b) 20 variations of fish 
contour for the same type of fish. 
 
PDF created with pdfFactory Pro trial version www.pdffactory.com
-35 0 -300 -250 -200 -150 -100 -50 0
-180
-160
-140
-120
-100
-80
-60
-40
-20
 
-350 -300 -250 -2 00 -150 -100 -50 0
-180
-160
-140
-120
-100
-80
-60
-40
-20
 
 
(a) (b) 
 
Figure 2. Fish contours after adding Gaussian 
distributed noise, (a) σ = 1, (b) σ = 2. (σ denotes the 
standard deviation of noise) 
 
As we can see, the summation invariant can 
successfully recognize fish in spite of a high level of 
noise. Compared with wavelet-based techniques, it also 
exhibits stronger immunity to noise. 
 
5. CONCLUSION 
 
In this work, we provide a new solution to the 
equivalence problem of planar contours under 
transformation group action. A summation invariant for 
the affine group acting on 2° is explicitly derived and 
applied to the problem of shape recognition. A database 
of marine animals was used to test the proposed method. 
Compared with some traditional methods, experimental 
results show that our method has superior discriminating 
power and better noise immunity. Among possible future 
directions, the major work is to further improve 
recognition performance by using higher order 
summation invariants.  
 
6. REFERENCES 
 
[1]  M. Fels and P. J. Olver, "Moving coframes: I. A 
practical algorithm," Acta Applicandae 
Mathematicae, vol. 51, pp. 161-213, 1998. 
[2]  M. Fels and P. J. Olver, "Moving coframes: II. 
Regularization and theoretical foundations," Acta 
Applicandae Mathematicae, vol. 55, pp. 127-208, 
1999. 
[3]  L. Van Gool, P. Kempenaers, and A. Oosterlinck, 
"Recognition and semi-differential invariants," 
Proc. CVPR '91, 1991. 
[4]  T. Cohignac, C. Lopez, and J. M. Morel, "Integral 
and local affine invariant parameter and application 
to shape recognition," presented at Pattern 
Recognition, 1994. Vol. 1 - Conference A: 
Computer Vision & Image Processing., 1994. 
[5]  C. |Hann, C.E., "Projective curvature and integral 
invariants," in Acta applicandae mathematicae, vol. 
74, 2002, pp. 177-193. 
[6]  J. Sato and R. Cipolla, "Affine integral invariants 
and matching of curves," Proceedings of 13th 
International Conference on Pattern Recognition, 
25-29 Aug. 1996, Vienna, Austria, 1996. 
[7]  P. J. Olver, "Moving frames," Journal of Symbolic 
Computation, vol. 36, pp. 501-12, 2003. 
[8]  F. Mokhtarian, S. Abbasi, and J. Kittler, "Indexing 
an image database by shape content using curvature 
scale space," IEE Colloquium on Intelligent Image 
Databases (Ref. No.1996/119), 22 May 1996, 
London, UK, 1996. 
[9]  M. I. Khalil and M. M. Bayoumi, "A dyadic wavelet 
affine invariant function for 2D shape recognition," 
IEEE Trans. Pattern Analysis and Machine 
Intelligence, vol. 23, pp. 1152-1164, 2001. 
[10]  T. H. Reiss, "The revised fundamental theorem of 
moment invariants," IEEE Trans. Pattern Analysis 
and Machine Intelligence, vol. 13, pp. 830-834, 
1991. 
[11]  K. Arbter, W. E. Snyder, H. Burkhardt, and G. 
Hirzinger, "Application of affine-invariant Fourier 
descriptors to recognition of 3-D objects," IEEE 
Trans. Pattern Analysis and Machine Intelligence, 
vol. 12, pp. 640-647, 1990. 
 
 
PDF created with pdfFactory Pro trial version www.pdffactory.com


Summation invariant and its application to shape recognition

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DB31D3F101C341A0BB6B015AFA604264?doi=10.1.1.1054.321&rep=rep1&type=pdf

Summation invariant and its application to shape recognition

Abstract

Similar works

Full text

Available Versions

CiteSeerX