| figures) 
onsidera- 
itivity is 
he calcu- 
this pro- 
(b) 
(c) 
3.1.3 Affine-invariant Fourier descriptors 
The points of the closed contour may be viewed 
as being in a complex image plane. Starting at 
an arbitrary point on the contour, and tracking 
once around it, yield a sequence of complex num- 
bers. The discrete Fourier transform of this se- 
quence defines the so-called Fourier descriptors 
of the contour. In this coefficients the frequencies 
are represented. Experimentally we found that 16 
coefficients are sufficient for a geometrically satis- 
fying description of the simple geometric features. 
To get affine-invariant features the complex Fou- 
rier descriptors are aggregated in pairs. Each pair 
defines a 2 x 2 matrix of which the determinant 
is calculated. This yields scalar quantities which 
are also called A-invariants (Arbter, 1989). The 
A-invariants or more precisely the absolute A- 
invariants found by a further normalization are 
the primary quantities for the backprojection into 
the spatial domain. By calculating the inverse 
LE a CA 
ame S 
k=0.604 K-0.785 k=1.0 
Figure 4: Starting situation (upper row), the nor- 
med figures and the corresponding form factors 
(lower row) 
Fourier transform the normed figures plotted in 
figure 4 are obtained. From these figures the form 
factor k — 47 F/U? is determined which defines 
the compactness of the object. The calculated 
form factors (see figure 4) are a simple feature 
for the characterization and identification of the 
unknown object. 
3.1.4 Affine-invariant moments 
Given a two-dimensional function g(z, y) the cen- 
tral moments of order p -- q can be expressed as 
Hog = / Fi -— £)"(y — y)*g(z, y) dz dy 
—OO —oO 
where z,j are the weighted mean values. From 
this moments up to order 3 four affine-invariant 
features ],,152,13,14 can be derived. The main 
part of the derivation is the evaluation of a qua- 
dratic form. The resulting formula, for example 
of I, reads as 
I — (p2opoz — p31) / pao - 
The formulas for the other quantities are more 
extensive. The interested reader should consult 
the paper by Hu (1962). 
In contrast to the procedures described before the 
determination of affine-invariant features using 
the moments relies on the area within the border 
line. Some experimental results for this invari- 
ants are listed in table 1. In this case the binary 
image of the region of interest defines the func- 
tion g(z, y), i.e., all pixels within the border line 
of the corresponding figure are equal to 1. The 
  
I I, I; I, 
  
  
triangle 
t: 9.25.1079 -3.25-10-7 -5.49-10-5 4.06-10-° 
e: 9.26.107? -3.23-10-7 -5.47-10-5 4.06-10-° 
$s... 7.82038. ..2.1-10-8 2.0-107% 1.8107 
rectangle 
t: 6.94-10-3 0 0 0 
e: 6.94.10? 0 0 0 
s: 6.8.1078 0 0 0 
circle 
t: 6.25.1073 0 0 0 
e: 6.34.107?  .82.10-16  .2.1.107?  1.0.10-4 
s 17110/* 73-10-16 1.0.107?  5.2.10-!! 
  
Table 1: Invariants I;, I2, I3, I4 for triangle, rec- 
tangle and circle. (t: theoretical, e: experimental, 
s: standard deviation) 
theoretical values are derived analytically. The 
experimentally found mean values refer to a cer- 
tain number of calculations with differently affine 
distorted figures. The empirical values and the 
theoretical values fit well together. The largest 
differences can be observed for the circle. In ad- 
dition to the mean values the corresponding stan- 
dard deviations are listed. The calculated values 
467