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We now have a way of converting each vector in the dataset into a set of 60
scalars which will form a 60dimensional key coding of the image. The
key is composed of the scalar coefficients that determine the linear
combination of eigenvectors to approximate the original image vector. The
computation of the 60coefficient key
(c_{0}, c_{1},..., c_{59}) is performed
using Equation . In Figure , we
display the 60 scalar code representing one face from our training set. This encoding
is performed for each face x
in the database giving us
a total of N 60element vectors of the form
(c_{x0}, c_{x1}, ...,
c_{x59}).
With KL decomposition, there is no correlation between the coefficients in the
key (i.e., each dimension in the 60 dimensional space populated by facepoints
is fully uncorrelated)[17]. Consequently, the dataset appears as a
multivariate random Gaussian distribution. The corresponding 60 dimensional
probability density function is approximated in the L_{2} sense by
Equation [17]:

(4.35) 
The envelope of this Gaussian distribution is a hyperellipsoid [17]
whose axis along each dimension is proportional to the eigenvalue of the
dimension. In other words, the hyperellipsoid is ``thin'' in the higherorder
dimensions and relatively wide in the lowerorder ones. Although it is
impossible to visualize the distribution in 60 dimensions, an idea of this
arrangement can be seen in Figure which shows the
distribution of the data set along the 3 firstorder coefficients (associated
with the 3 firstorder eigenvectors).
Figure 4.26:
The distribution of the dataset in the first three coefficient dimensions.

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Tony Jebara
20000623