Karhunen-Loeve Transform (KLT )

So far no transformation has been found for images that completely removes statistical dependence between transformation coefficients. Suppose a reversible transformation is carried out on N-dimensional vector to produce N-dimensional vector of transform coefficients, denoted by,

	(4.4.1)

where is a linear transformation matrix that is orthonormal or unitary;

ie		(4.4.2)

(where "1" denotes transpose)

Denote the m th column of matrix by column vector , then eqn (4.4.2 ) is equivalent to

The vectors are called orthonormal basis vectors for linear transform .

From eqn (4.4.1) we can express as,

where are transform coefficients, and is represented as a weighted sum of basis vectors.

KLT transform is an orthogonal linear transformation that can remove pairwise statistical correlation between the transform coefficients; ie the KLT transform coefficients satisfy

where summation is over all possible values and and P( ) represents the probability.

This is usually written using the statistical averaging operator E as,

or

where is the variance of .

Note that statistical independence implies uncorrelation, but the reverse is not generally true (except for jointly Gaussian r v s )

The KLT can be derived by assuming

The N x N correlation matrix of then becomes

(continued in the next slide)