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<H3>Signature Separability</H3>
The ``Signature Separability Report panel'' contains
a scrolled text block with a report of signature separability like that 
produced by SIGSEP program.<p>
The row of tabs selects the Separability Measure to use in 
calculations:
<P>
<UL>
<LI> Bhattacharrya (or Jeffries-Mastusuta) Distance (default) 
<LI> Transformed Divergence 
</UL>
Transformed Divergence is a popular empirical
measure which is computationally simpler than the Bhattacharrya
Distance. However, the Bhattacharrya Distance is more theoretically
sound because it is directly related to the upper bound of the
probabilities of classification errors.<p>
The row of radio buttons selects the Report Format:
<P>
<UL>
<LI> Matrix (default) 
<LI> Sorted List 
</UL>
If Matrix is selected, the report prints a lower triangular matrix of 
separability measures for all signature pairs.
If Sorted List is selected, then signature pairs are sorted from minimum 
to maximum separability. Pressing the column headers on the Sorted List
will sort that column in ascending order.<p>
The ``Save Report'' button allows you to append the separability
report to any text file.<p>
Both the Transformed Divergence and Bhattacharrya Distance measures
are real values between 0 and 2,
where `0' indicates complete overlap between the signatures of two
classes and `2' indicates a complete separation between the two
classes. Both measures are monotonically related to
classification accuracies. The larger the separability values are,
the better the final classification results will be.<p>
The following rules are suggested for possible ranges of separability values:
<P>
<PRE>
   0.0 to 1.0   (very poor separability)
   1.0 to 1.9   (poor separability)
   1.9 to 2.0   (good separability)
</PRE>
Very poor separability (0.0 to 1.0) indicates that the two
signatures are statistically very close to each other. The user has
two options. The first option is that one signature can be arbitrarily 
discarded (which is
suggested when the separability is closer to 0). The second option is 
that the two signatures can be merged using the ``Merge'' function in 
the ``Signature Editing'' panel (which is suggested when the 
separability is closer to 1).<p>
Poor separability (1.0 to 1.9) indicates that the two signatures
are separable to some extent. However, it is desirable to improve
separability if possible. Low signature separability is usually
caused by improper combinations of image bands and/or training
sites which have large internal variability within each class.<p>
The user is encouraged to use this function to examine the quality of
training sites and class signatures before performing
classification. In order to improve class separabilities,
consider the following options:
<P>
<UL>
<LI> Choose a different combination of multispectral image bands
which are better suited for separating certain classes when
creating signature segments. 
<LI> Create a new set of classes which are more separable given the
available image data.
<LI> Modify the existing training sites for classes which have the
largest internal variance. 
<LI> Merge those pairs of classes whose signature separabilities are too low. 
</UL>
Reference:
<P>
<PRE>
 Richards, J. A., 1986, &quot;Remote Sensing Digital Image Analysis&quot;.
 Springer-Verlag, Berlin, Heidelburg, New York, London,
 Paris, Tokyo, pp. 206-225.
</PRE>
Note: The formula to calculate the Bhattacharrya Distance measure
given in the above book is incorrect. Check the reference in the
book to find the right formula.
<P>

See Also: <A HREF = "http://www.pcigeomatics.com/cgi-bin/pcihlp/SIGSEP">SIGSEP</A><P>
<H4>Transformed Divergence</H4>
The Transformed Divergence is given by the following equations:
<P>
<PRE>
  TD(i,j) = 2*[1-exp(-D(i,j)/8)]

where 

  TD(i,j) = Transformed Divergence between classes i and j
   D(i,j) = divergence between classes i and j

   D(i,j) = 0.5*T [M(i)-M(j)]*[InvS(i)+InvS(j)]*[M(i)-M(j)]
            + 0.5*Trace[InvS(i)*S(j)  +InvS(j)*S(i)  -2*I ]

where 

  M(i)    = mean vector of class i, where the vector has Nchannel
            elements (Nchannel is the number of channels used)
  S(i)    = covariance matrix for class i, which has Nchannel by
            Nchannel elements
  InvS(i) = inverse of matrix S(i)
  Trace[] = trace of matrix (sum of diagonal elements)
  T[]     = transpose of matrix
  I       = identity matrix
</PRE>
<H4>Bhattacharrya Distance</H4>
The Bhattacharrya (or Jeffries-Mastusuta) Distance is calculated
using the following formula:
<P>
<PRE>
  BD(i,j) = 2*[1-exp(-a(i,j))]

where 

  BD(i,j) = Bhattacharrya Distance between class i and j
   a(i,j) = 0.125*T[M(i)-M(j)]*Inv[A(i,j)]*[M(i)-M(j)]
            + 0.5  *ln{det(A(i,j))/SQRT[det(S(i))*det(S(j))]}

where 

  M(i)    = mean vector of class i, where the vector has Nchannel
            elements (Nchannel is the number of channels used)
  S(i)    = covariance matrix for class i, which has Nchannel by
            Nchannel elements
  Inv[]   = inverse of matrix
  T[]     = transpose of matrix
  A(i,j)  = 0.5*[S(i)+S(j)]
  det()   = determinant of a matrix
  ln{}    = natural logarithm of scalar value
  SQRT[]  = square root of scalar value
</PRE>
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