Databases Reference
In-Depth Information
For the transform of Example 13.3.1, the outer products are
11
11
1
1
2
1
2
−
1
α
0
,
0
=
α
0
,
1
=
(34)
−
1
1
11
−
1
1
2
1
2
−
1
α
1
,
0
=
α
1
,
1
=
(35)
1
−
1
−
11
From (
20
), the inverse transform is given by
x
01
x
01
x
10
x
11
11
1
θ
00
θ
01
θ
10
θ
11
11
1
1
2
=
(36)
−
1
−
1
θ
00
+
θ
01
+
θ
10
+
θ
11
θ
00
−
θ
01
+
θ
10
−
θ
11
θ
00
+
θ
01
−
θ
10
−
θ
11
θ
00
−
θ
01
−
θ
10
+
θ
11
1
2
=
(37)
=
θ
00
α
0
,
0
+
θ
01
α
0
,
1
+
θ
10
α
1
,
0
+
θ
11
α
1
,
1
(38)
The transform values
θ
ij
can be viewed as the coefficients of the expansion of
x
in terms of
the matrices
α
i
,
j
are known as the
basis
matrices.
For historical reasons, the coefficient
α
i
,
j
. The matrices
α
0
,
0
, is called
the DC coefficient, while the coefficients corresponding to the other basis matrices are called
AC coefficients. DC stands for direct current, which is current that does not change with time.
AC stands for alternating current, which does change with time. Notice that all of the elements
of the basis matrix
θ
00
, corresponding to the basis matrix
α
0
,
0
are the same, hence the DC designation.
In the following section, we will look at some of the variety of transforms available to us,
then at some of the issues involved in quantization and coding. Finally, we will describe in
detail two applications, one for image coding and one for audio coding.
13.4 Transforms of Interest
In Example 13.2.1, we construct a transform that is specific to the data. In practice, it is
generally not feasible to construct a transform for the specific situation for several reasons.
Unless the characteristics of the source output are stationary over a long interval, the transform
needs to be recomputed often, and it is generally burdensome to compute a transform for every
different set of data. Furthermore, the overhead required to transmit the transform itself might
negate any compression gains. Both of these problems become especially acute when the size
of the transform is large. However, there are times when we want to find out the best we can
do with transform coding. In these situations, we can use data-dependent transforms to obtain
an idea of the best performance available. The best-known data-dependent transform is the
discrete Karhunen-Loéve transform (KLT). We will describe this transform in the next section.
13.4.1 Karhunen-Loéve Transform
The rows of the discrete Karhunen-Loéve transform [
189
], also known as the Hotelling trans-
form, consist of the eigenvectors of the autocorrelation matrix. The autocorrelation matrix for
