During the construction of our tree, the MDF's are computed locally. For
each subregion 
, we obtain DKL projection matrices
and 
and mean vector 
based on the training
samples within , where is the projection matrix to the
MEF space and is the projection matrix to the MDF space
as defined previously. The leaves of the partition tree correspond to
the regions which contain
the training samples from a single class.
The approximator uses the following decision rule to classify the query
fovea vector 
to the
class of a leaf cell.
Since each local cell has its own DKL projection, in order to logically
compare between two different cells, we use a measurement called
Mixture Distance ( 
).
Intuitively, what is being measured can be seen in Fig. 10. In
Fig. 10, the original image space is a 3D space, the MEF space
is a 2D
subspace, and the MDF space is 1D subspace since two classes are well separated
along the first MDF vector.
The first term under the radical indicates the
distance of the original vector from the population which indicates
how well the MEF subspace represents the query vector . This
term is necessary since it is entirely possible that a query vector
that is miles away from a particular subregion's MEF subspace would
project very near to the region's center. The second term indicates the
distance between the MDF components of the query vector and
the MDF components of the center vector in the original image
space.
Figure 10:
Illustration of components in the Mixture Distance in a 3D original space.