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karhunen loeve transform is simply a tool for data compression and pattern recognition.
let's have an example.
if you have a image and the application is data compression first you should rearrange it in a vector and calculate sample covariance matrix related to the image vector. then use eigenvalue decomposition. usually the main energy of the image is in first e.g. 20% of eigenvalue. so simply get rid of other eigenvalues and corresponding eigenvectors. project your image vector on remaining eigenvectors. save eigenvectors and projection value as the compressed image.
This application has been briefly explained in Gonzales book on "Image Processing".
There are some other applications such as Pattern Recognition, and Principle Components Analysis.
I want especially to pay your attention on the fact, that K-L transform - is an adaptive technique of Signal Processing. It means, that you obtain (construct) the basis for signal or image decomposition just from the data set (signal or image) without using any other functions. It's a very important property, because it helps take into account local properties and internal structure of the process.
Also, among the other adaptive technique, it's worth mentioning the following:
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