Databases Reference
In-Depth Information
The forward and inverse transform matrices
A
and
B
are inverses of each other; that is,
AB
I
, where
I
is the identity matrix.
Equations (
9
) and (
10
) deal with the transform coding of one-dimensional sequences, such
as sampled speech and audio sequences. However, transform coding is one of the most popular
methods used for image compression. In order to take advantage of the two-dimensional nature
of dependencies in images, we need to look at two-dimensional transforms.
Let
X
i
,
j
be the
=
BA
=
(
i
,
j
)
th pixel in an image. A general linear, two-dimensional transform for
a block of size
N
×
N
is given as
N
−
1
N
−
1
k
,
l
=
X
i
,
j
a
i
,
j
,
k
,
l
(15)
i
=
0
j
=
0
All two-dimensional transforms in use today are
separable
transforms; that is, we can transform
a two-dimensional block by first taking the transform along one dimension, then repeating the
operation along the other direction. In terms of matrices, this involves first taking the (one-
dimensional) transform of the rows and then taking the column-by-column transform of the
resulting matrix. We can also reverse the order of the operations, first taking the transform of
the columns, and then taking the row-by-row transform of the resulting matrix. The transform
operation can be represented as
N
−
1
N
−
1
k
,
l
=
a
k
,
i
X
i
,
j
a
j
,
l
(16)
i
=
0
j
=
0
which in matrix terminology would be given by
AXA
T
=
(17)
The inverse transform is given as
B
T
(18)
All of the transforms we deal with will be
orthonormal transforms
. An orthonormal
transform has the property that the inverse of the transform matrix is simply its transpose
because the rows of the transform matrix form an orthonormal basis set:
X
=
B
A
−
1
A
T
B
=
=
(19)
For an orthonormal transform, the inverse transform will be given as
A
T
X
=
A
(20)
Orthonormal transforms are energy preserving; that is, the sum of the squares of the
transformed sequence is the same as the sum of the squares of the original sequence. We can
see this most easily in the case of the one-dimensional transform:
N
−
1
i
=
0
θ
2
i
T
=
θ
θ
(21)
T
Ax
=
(
Ax
)
(22)
x
T
A
T
Ax
=
(23)
