International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
2n f 
a(AT= X ADS Y À; (10) 
i=] i=) 
where 0<a<l will govern how much of the variation seen in 
the training set can be represented by a small number of ¢ 
modes. Any shape in the training set can be approximated using 
the mean shape and a weighted sum of the principal 
components from the / modes. 
x=u +P,b, (11) 
where 
P,=(P1P2--P#) (12) 
is the matrix of the first / cigenvectors and 
ii T 
b, —(bjb»...b;) (13) 
is a vector of weights for each eigenvector. The eigenvectors 
are orthogonal so equation (11) can be written as: 
bj=P" (x-ux) (14) 
Since the variance of b; over the training set will be the 
associated eigenvalue À; we might expect that the limits should 
-34A; zb; 3 A; (15) 
where we see that most of the population is in the order of 36 
from the mean. This allows us to generate plausible shapes that 
are not part of the training set. Summarising, the PCA analysis 
has given us the original shapes in terms of their differences 
and similarities. In other words it has identified the statistical 
patterns in the data. Since the variations can be performed with 
the most significant eigenvectors we can reduce the 
dimensionality of the data and describe the variations with 
fewer variables (Hamarneh, 1998). 
be in the order of: 
3.3.4  Back-projection. In order to get the original data 
back we need to add the mean of the original data. So, the new 
generated back-projected data will be given by: 
b'=uy+(P/P(x-u,)) (16) 
The result of this process is shown in Figure 4. 
  
  
  
Lr i i ; 
na 100 ) 100 200 300 400 5001 Bod 
752 
Figure 4. Back-projection of the original data annotated with 
the landmark points 
4. RESULTS 
In our experiments we used 20 training shapes with 70 
landmark points (hence 140 parameters) and found that the first 
four modes accounted for 9296 of the variance of the training 
data. The variance of each mode is as follows: mode | (m=1) 
with variance (v=1.34), mode 2 (m=2) with variance (v=1.03), 
mode 3 (m=3) with variance (v=0.6), and mode 4 with variance 
(v=0.43). Some of the significant modes of variation together 
with how the training data are arranged in the PCA space are 
shown in Figure 5. 
  
nere eno RAT 
  
  
  
  
  
Figure 5. The first four modes of variation of the automatically 
generated model of the hand outlines. 
S. CONCLUSIONS 
In this article we have presented a new method for automatic 
landmark detection on the contour of hand shapes. The method 
is based on Freeman chain code for changes in code direction, 
which indicates a corner in the boundary and on Minimum 
Perimeter Polygon approximation for defining curvatures where 
a change of the slope occurs with the control points 
approximately uniformly spaced along the curvatures. Success 
of the whole procedure is suggested by distinct modes that 
generate eligible shape variations. However, the above method 
considered only hands represented by closed boundaries and 
non-occluding boundaries. These problems are part of our 
current research together with extending the method to high- 
level 3D hand shape variations. 
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