Graphics Reference
In-Depth Information
Almost equal norm of all basis vectors
Same symmetry properties as the IDCT basis vectors
Smaller transform matrices are embedded in larger transform matrices
Eight-bit representation of transform matrix elements
Sixteen-bit transpose buffer
Multipliers can be represented using 16 bits or less with no cascaded multiplica-
tions or intermediate rounding
Accumulators can be implemented using less than 32 bits
6.2.1
Discrete Cosine Transform
The
N
transform coefficients
v
i
of an
N
-point 1D DCT applied to the input samples
u
i
can be expressed as
N
1
X
v
i
D
u
j
c
ij
(6.1)
j D0
where
i
D
0, :::,
N
1. Elements
c
ij
of the DCT transform matrix
C
are defined as
cos
N
j
C
1
2
i
p
N
c
ij
D
(6.2)
p
2 for
i
D
0and
i
> 0, respectively. Furthermore, the basis vectors
c
i
oftheDCTaredefinedas
c
i
D
[
c
i
0
, :::,
c
i
(
N
1)
]
T
where
i
D
0, :::,
N
1.
The DCT has several properties that are considered useful both for compression
efficiency and for efficient implementation [
22
].
1. The basis vectors are orthogonal, i.e.
c
i
c
j
D
0for
i
¤
j
. This property is desir-
able for compression efficiency by achieving transform coefficients that are
uncorrelated.
2. The basis vectors of the DCT have been shown to provide good energy
compaction which is also desirable for compression efficiency.
3. The basis vectors of the DCT have equal norm, i.e.
c
i
c
i
D
1for
i
D
0, :::,
N
1.
This property is desirable for simplifying the quantization/de-quantization pro-
cess. Assuming that equal frequency-weighting of the quantization error is
desired, equal norm of the basis vectors eliminates the need for quantization/de-
quantization matrices.
4. Let
N
D
2
M
. The elements of a DCT matrix of size 2
M
2
M
is a subset of the
elements of a DCT matrix of size 2
M
C 1
2
M
C 1
. More specifically, the basis
vectors of the smaller matrix is equal to the first half of the even basis vectors of
where
i
,
j
D
0, :::,
N
1andwhere
P
is equal to 1 and
