Graphics Reference
In-Depth Information
6.2.4
Basis Vectors of the HEVC Core Transforms
The left half of the 32
32 matrix specifying the 32-point forward transform is
shown in Fig.
6.2
. The right half can be derived by using the (anti-) symmetry
properties of the basis vectors (property 6 of Sect.
6.2.1
). The inverse transform
matrix of HEVC is defined as the transpose of the matrix resulting from the figure.
The 32
32 matrix contains up to 31 unique numbers as follows.
n
90; 90; 90; 89; 88; 87; 85; 83; 82; 80; 78; 75; 73; 70; 67; 64;
61; 57; 54; 50; 46; 43; 38; 36; 31; 25; 22; 18; 13; 9; 4
o
(6.3)
d
32
i;0
;i
D
1;:::;31
D
These unique numbers are elements 1-31 of the first column of the forward
transform matrix. Note that although the number 90 occurs three times, this is by
accident and not generally true. The unique numbers property was used in [
26
]to
enable 25 % area reduction for hardware designs with practical throughput.
Furthermore, the coefficients
d
ij
of the smaller transform matrices (
N
D
4, 8, 16)
can be derived from the coefficients
d
32
ij
of the 32
32 transform matrix as:
d
ij
D
d
32
i .32=N /;j
;i;j
D
0;:::;N
1
(6.4)
Let
D
4
denote the 4
4 transform matrix. By using (
6.4
) and Fig.
6.2
,
D
4
can be
obtained as:
2
4
3
5
2
4
3
5
d
32
0;0
d
32
0;1
d
32
0;2
d
32
0;3
64
64
64
64
d
32
8;0
d
32
8;1
d
32
8;2
d
32
8;3
83
36
36
83
D
4
D
D
d
32
16;0
d
32
d
32
16;2
d
32
64
64
64
64
16;1
16;3
d
32
24;0
d
32
24;1
d
32
24;2
d
32
24;3
36
83
83
36
The 8
8 transform matrix
D
8
and the 16
16 transform matrix
D
16
can be
similarly obtained from the 32
32 transform matrix as shown in Fig.
6.2
where
different colors are used to highlight the embedded 16
16, 8
8and4
4
forward transform matrices. This property allows for different transform sizes to be
implemented using the same architecture thereby facilitating hardware sharing [
6
].
Note that from the unique numbers property of (
6.3
) and the (anti-)symmetry
properties,
D
4
is also equal to:
2
4
3
5
d
32
16;0
d
32
16;0
d
32
16;0
d
32
16;0
d
32
8;0
d
32
24;0
d
32
24;0
d
32
8;0
D
4
D
(6.5)
d
32
16;0
d
32
16;0
d
32
16;0
d
32
16;0
d
32
24;0
d
32
8;0
d
32
8;0
d
32
24;0
