Parent Topic: <A href="http://www.pcigeomatics.com/cgi-bin/pcihlp/MLC">MLC</a><hr>
<H3>ALGORITHM</H3>
This section is included for reference only and gives a brief
description of the algorithm and equations used by MLC.<p>
The maximum likelihood equation used in MLC is the Mahalanobis
minimum distance classifier defined by the following equation:
<P>
<PRE>
                    t  -1
   Gi(X)= -1/2(X-Ui) Ci  (X-Ui) - (d/2)log(2TT) - (1/2)log(|Ci|)
         + log(Pi)

 Gi(X)         is the result for class i on pixel X
 d             is the number of channels in the classification
 X=(x1,...,xd) is the (d by 1) pixel vector of grey-levels
 Ui            is the (d by 1) mean vector for class i  
 Ci            is the (d by d) covariance matrix for class i
 TT            is pi = 3.1415...
 |Ci|          is the determinant of the covariance matrix
 Pi=Bi/SUM(Bi) is the a priori probability for class i
 Bi            is the BIAS for class i
 SUM(Bi)       is the sum of BIASes for all classes used
 t             as a superscript denotes transpose
 -1            as a superscript denotes inverse
 Ti            is the threshold value THRS for class i

                  -1
 d, Ui, Bi, Ti, Ci   and |Ci| are obtained from the signature
 segment (usually generated by CSG).

                          -1
In general, the matrix  Ci  defines the shape and orientation
characteristics of the hyperellipsoid in feature space for
class i.  The Ui vector determines its position and Ti (which
is selected by the user in CSG or CSE) determines its size.
</PRE>
The classification process for a single pixel X is as follows:
<P>
<PRE>
 1)  For each class (i=1,...,n), determine if X lies within the
     hyperellipsoid for the class.

                    t   -1            2
        i.e,  (X-Ui)  Ci  (X-Ui) &lt;= Ti    must be true

 2.  If X is not in any hyperellipsoid, then
        assign the pixel to the NULL class
     else
        compute Gi(X) for each class which passed step (1) and
        assign the pixel to the class where Gi(X) is a maximum
     endif

 3.  The &quot;a posteriori&quot; is calculated using Bayes Rule.

         P(i|X) =  P(X|i)P(i)/P(X)

     where P(i|X) is the &quot;a posteriori&quot; probability of each pixel
           Pi=Bi/SUM(Bi) is the &quot;a priori&quot; probability for class i
</PRE>
<PRE>
                  n  
          P(X) = SUM P(X|i)P(i)    , n = no. of training classes
                 i=1

          Gi(X) = log(P(X|i)) + log(P(i))
</PRE>
For more information see the following publications:
<P>
<PRE>
 Duda and Hart, 1973.  Pattern Classification and Scene Analysis,
 John Wiley and Sons, chapter 2.

 Robert A. Schowengerdt, 1983.  Techniques for Image Processing and 
 Classification in Remote Sensing.  Academic Press.
</PRE>
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