28 Image and Video Compression
where
uv
,
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 coefcients then the reconstruction error is as follows:
IxyIxy{(,) (,)}
2
2
∑∑
εε
−
−
(3.6)
This error can be minimized if
uv
,
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 ...