Image and Video Compression

28 Image and Video Compression

where

xy

uv

(,)

,

*

U

represents the inverse transformation kernel. Thus, we can rep-

resent an image as a series combination of the unitary matrices. Equations (3.4)

and (3.5) are called a transform pair. Like the 1D signals, if the image is recon-

structed using the nite coefcients then the reconstruction error is as follows:

IxyIxy{(,) (,)}

xy

N

xy

N

2

,0

1

,0

1

2



∑∑

εε

=−

=

−

=

−

(3.6)

This error can be minimized if

xy

uv

{(,)}

,

U

is complete, and it represents the

set of orthonormal basis functions. As seen earlier, one of the important

properties of orthonormal transform is that the inverse of the transform

matrix is simply its conjugate transpose. Orthonormal transforms are

energy-preserving ones; energy in the native domain i ...