as the principal components transformation. The KLT is applied in 
remote sensing [11], generally in the spectral dimension. It has 
been widely used by remote sensing people [3-4] mainly to reduce the 
feature space for better olassification in land use studies. The 
name Principal Components is derived from the fact that most of the 
energy in the original data is compacted into fewer components which 
lends itself to better classfication. 
The KLT is an optimal transform in a statistical sense under a 
variety of criteria. The basis vectors of the KLT are the 
eigenvectors of the input covariance matrix. The corresponding 
eigenvalues is an indication of the energy packed into each one of 
the components. Also most of the energy or variance is packed into 
the fewest number of coefficients possible by any transform. Hence 
it is the optimal transform. The KLT completely decorrelates the 
input data in the transform domain. The KLT simply diagonalizes the 
covariance matrix. The diagonal non-zero values of this diagonal 
matrix are the eigenvalues. Each eigenvalue is also the variance of 
that particular coefficient. The KLT is derived from the statistics 
of the image data. Hence the KLT varies if the image statistics 
vary. This can be considered as its primary disadvantage. One 
needs to compute the transformation every time the statistics 
change. At the same time, if the statistics do not change, the 
transformation matrix (or the eigenvector matrix) does not change. 
As an example, the same transformation can be used for various 
images if all the images to be transformed are assumed to obey the 
Markov-I process. Another important disadvantage about the KLT is 
that it has no fast algorithms in general. However fast algorithms 
have been proposed [12-13] for the KLT for a class of process such 
as a stationary Guass-Markov process. Additionally  signficant 
computational effort is involved in computing the basis vectors of 
KLT. 
In general the eigenvectors are ordered as per the decreasing order 
in corresponding eigenvalues. Now most of the energy is packed into 
the first few transform coefficients. Also the transform components 
are uncorrelated. Since each eigenvalue is the variance of the 
corresponding principal component, it is an indication of the energy 
packed into the particular principal component. The variance 
distribution curve in the KLT domain drops down steeply from the 
first component onwards. Since the KLT is an orthogonal transform 
[14], the total data variance is preserved in the transform domain. 
The very same property guarantees that the inverse transform matrix 
is the same as the transpose of the forward transform matrix. In 
other words if [T] denotes the forward transform matrix, then 
«1 
[T)[IT] s [T] [T3'.s [1] — n 
where [T]' denotes the transpose of [T] and [I] denotes the identity 
matrix. The KL transformation on a data vector [D] is generally 
obtained as [11] 
[Pi =: (TI [:.[D1:- [M1 ] a vn 2) 
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