Properties of Unitary transforms :(contd.)

2) Energy Compaction Property:

Most unitary transforms have a tendency to pack a large fraction of average energy of an image into relatively few transform coefficients. Since total energy is preserved this implies that many transform coefficients will contain very little energy. If and denote the mean and covariance of vector then corresponding quantities for are :

= =

And	=

	=

	=

Variances of the transform coefficients are given by the diagonal elements of

i.e. =

=

Since is unitary , it implies:

= = =

and

	= =

	=

The average energy of transform coefficients tends to be unevenly distributed, although it may be evenly distributed for input sequence .For a 2D random field , with mean and covariance , its transform coefficients satisfy the properties ,

=

and

= =

If covariance of is separable i.e

=

Then variances of transform coefficients can be written as a separable product,

=

where

= ;