Parent Topic: <A href="http://www.pcigeomatics.com/cgi-bin/pcihlp/CLWORKS|Classify">Classify</a><hr>
<H4>Supervised Algorithms</H4>
There are five algorithms that may be used for a supervised 
classification. They are:
<P>
<UL>
<LI> Parallelepiped
<LI> Parallelepiped with Maximum Likelihood as tie breaker
<LI> Minimum Distance
<LI> Maximum Likelihood
<LI> Maximum Likelihood with NULL class
</UL>
<H5>Parallelepiped Classifiers</H5>
The parallelepiped classifier uses the threshold of 
each class signature to determine if a given
pixel falls within the class or not.  The thresholds specifies
the dimensions (in standard deviation units) of each side of a
parallelepiped surrounding the mean of the class in feature 
space. If the pixel falls inside the parallelepiped, it is
assigned to the class. However, if the pixel falls within more
than one class, it is put in the overlap class (code 255). If
the pixel does not fall inside any class, it is assigned to the
null class (code 0). 
 
The parallelepiped classifier is typically used when speed is
required. Unfortunately, in many cases this results in poor accuracy and a
large number of pixels classified as ties (or overlap, class 255).<p>
The parallelepiped classifier with maximum likelihood as a tie breaker is a
cross between the parallelepiped
classifier and the full maximum likelihood classifier.   The
basic concept is to use parallelepiped classification unless
we have a tie (overlap), in which case the tie is resolved by
using full maximum likelihood classification.
 
This type of classification is an attempt to gain the speed of
the parallelepiped classifier while eliminating the large
number of pixels classed as ties (overlap).
 
Typically, the Ties approach is used as a preliminary step to the full
maximum likelihood classification.
<P>
<H5>Maximum Likelihood</H5>
The full maximum likelihood classifier uses the Gaussian
threshold (THRS) stored in each class signature to determine if
a given pixel falls within the class or not.  The threshold is
the radius (in standard deviation units) of a hyperellipse
surrounding the mean of the class in feature space.  If the
pixel falls inside the hyperellipse, it is assigned to the class.
The class bias (BIAS) is used to resolve overlap between
classes, and weights one class in favour of another.  If the
pixel does not fall inside any class, it is assigned to the 
null class (code 0).
 
The maximum likelihood classifier is considered to give more
`accurate' results than parallelepiped classification. However, it
is much slower due to extra computations. We use the word
`accurate' in quotes because this assumes that classes in the
input data have a Gaussian distribution and that signatures were
well selected. This is not always a safe assumption.
<P>
<H5>Minimum Distance</H5>
The distances between the pixel to be classified and each class
centre are compared. The pixel is assigned to the class whose
centre is the closest to the pixel.<p>
The minimum Euclidian distance 
classifier is defined by the following equation:
<P>
<PRE>
                T
  Gi(X) = (X-Ui) * (X-Ui)

        = SUM[(xj-uj)**2]  for j = 1 to d.

Gi(X)          is the result for class i on pixel X
T              indicates transposition of the elements in brackets
d              is the number of channels in the classification
X=(x1,...,xd)  is the (d by 1) pixel vector of grey-levels
Ui=(u1,...,ud) is the (d by 1) mean vector for class i
j              is the subscript of jth element of a vector
SUM[]          is the total of elements inside brackets


If for all i not equal j, Gj(X) &lt; Gi(X),
then X is classified as j.
</PRE>
The diagram below illustrates the classification of 4 pixels into 3
classes. Pixels are classified to the nearest class centre.
<P>
<PRE>
|             Class A                    *   class centre
|  .    a                      .        ...  class boundary  
|     .           *          .
|        .  b             .               Pixel      Class
|          .          .                  
|              .    .                       a           A
|          c     .                          b           A
|                .                          c           B
|          *      .     *  Class C          d           C
|                 .  d                                          
|       Class B   .                        
|                .                             
|                              
+----------------------------------
</PRE>
Reference:<p>
Hodgson, M.E., 1988. &quot;Reducing the computational requirements
of the minimum-distance classifier.&quot;  Remote Sensing of 
Environment, Vol. 24.
<P>
<hr>
Parent Topic: <A href="http://www.pcigeomatics.com/cgi-bin/pcihlp/CLWORKS|Classify">Classify</a><br>
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