Karhunen-Loeve Transform (KLT ) (continued)

From the normality or unitary transform condition i.e., we get with the variances of along the main diagonal of the matrix. Thus, in terms of column vectors of , the above equation becomes

We see that the basis vectors of the KLT are the eigenvectors of , orthonormalised to satisfy

[Note that whereas the other image transforms like DFT, DCT were independent of data, the KLT transformation depends on 2^nd order satisfies of the data.]

The variances of the KLT coefficients are the eigenvalues of and since is symmetric and positive definite, eigenvalues are real and positive. The KLT basic vectors and transform coefficients are also real. Besides decorrelating transform coefficients, the KLT has another useful property: it maximizes the number of transform coefficients that are small enough so that they are insignificant. For example, suppose the KLT coefficients are ordered according to decreasing variance ie

Also suppose that for reasons of economy, we transmit only the 1^st pN coefficients where . The receiver then uses the truncated column vector to form the reconstructed values as .

The mean squared error (MSE) between the original and the reconstructed pels is then

It can be shown that KLT minimizes the MSE due to truncation.

(continued in the next slide)