Graphics Reference
In-Depth Information
Tabl e 6. 1
Comparison of transform design methods
HEVC core transforms
Scaling and rounding
Orthogonality
o
ij
< 0.0029
o
ij
< 0.0037
Closeness to DCT
m
ij
< 0.0213
m
ij
< 0.0077
Norm measure
n
i
< 0.0014
n
i
< 0.0109
To measure the degree of approximation for properties 1-3 of Sect.
6.2.1
,the
following measures are defined for an integer
N
-point DCT approximation with
scaled matrix elements equal to
d
ij
and basis vectors equal to
d
i
D
[
d
i
0
, :::,
d
i
(
N
1)
]
T
where
i
D
0, :::,
N
1.
1. Orthogonality measure:
o
ij
D
d
i
d
j
/
d
0
d
0
,
i
¤
j
2. Closeness to DCT measure:
m
ij
Dj
˛
c
ij
d
ij
j
/
d
00
3. Norm measure:
n
i
Dj
1
d
i
d
i
/
d
0
d
0
j
where
i
,
j
D
0, :::,
N
1,
c
ij
are the DCT matrix elements of (
6.2
), and the scale
factor ˛ is defined as
d
00
N
1/2
.
As a result of careful investigation, it was decided to represent each matrix
coefficient with 8 bit (including sign bit), and to choose the elements of the first
basis vector to be equal to 64 (i.e.
d
0
j
D
64,
j
D
0, :::,
N
1). Note that this results
in a scale factor of 2
6 C
M
/2
for the HEVC transform matrix when compared to the
orthonormal DCT. The remaining matrix elements were hand-tuned (within the
constraints of properties 4-6 of Sect.
6.2.1
) to achieve a good balance between
properties 1-3 of Sect.
6.2.1
. The hand-tuning was performed as follows. First,
the real-valued scaled DCT matrix elements, ˛
c
ij
, were derived. Next, for each
unique number in the resulting matrices, each integer value in the interval [
1.5, 1.5]
around ˛
c
ij
was examined and the resulting values of
o
ij
,
m
ij
,and
n
i
were calculated.
Since there are only 31 unique numbers in the transform matrices (see Sect.
6.2.4
),
various permutations can be examined systematically (although not exhaustively).
The final integer matrix elements were chosen to give a good compromise between
all measures
o
ij
,
m
ij
,and
n
i
. The resulting worst case values of
o
ij
,
m
ij
,and
n
i
are shown in the second column of Table
6.1
. The norm was considered to be
sufficiently close to 1 (i.e. the norm measure
n
i
is sufficiently close to 0) to justify
not using a non-flat default de-quantization matrix in HEVC (i.e. all transform
coefficients scaled equally).
For comparison purposes, the resulting measures when multiplying the real-
valued DCT matrix elements with 2
6 C
M
/2
and rounding to the closest integer are
listed in the third column of Table
6.1
. As can be seen from the table, although the
matrix elements of the HEVC transforms are farther from the scaled DCT matrix
elements, they have better orthogonality and norm properties.
Finally, by using only 8 bit representation, property 7 of Sect.
6.2.1
(trigonomet-
ric relationship between matrix elements) was not easily preserved. The authors are
not aware of any trigonometric property of the HEVC core transforms that can be
utilized to reduce the number of arithmetic operations below those required when
using the (anti-) symmetry properties.
