DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

Eq 38.1
Owing to the eight-line-high vertical transform, eight
lines of image memory are required in the DCT
subsystem of the encoder, and in the IDCT (DCT -1 )
subsystem of the decoder. When the DCT is implemented
in separable form, as is almost always the case,
this is called transpose memory.
Level-shifting
The DCT formulation in JPEG is intended for signed
sample values. In ordinary hardware or firmware, the
DCT is implemented in fixed-point, two’s complement
arithmetic. Standard video interfaces use offset binary
representation, so each luma or color difference sample
is level shifted prior to DCT by subtracting 2k-1 , where k
is the number of bits in use.
Discrete cosine transform (DCT)
The 8×8 forward DCT (FDCT) takes an 8×8 array of 64
sample values (denoted f, whose elements are fi,j ), and
produces an 8×8 array of 64 transform coefficients
(denoted F, whose elements are Fu,v ). The FDCT is
expressed by this equation:
( )
( )
7 7
1
⎡ 2i+ 1uπ⎤⎡2j
+ 1vπ⎤
Fuv , = C( u) C( v) fi,
jcos
⎢ ⎥cos
⎢ ⎥;
4 Σ Σ ⎢ 16 ⎥ ⎢ 16 ⎥
i=
0 j=
0 ⎣ ⎦ ⎣ ⎦
⎧ 1
⎪ ; w = 0
C( w)
= ⎨ 2
⎪
⎩1;
w = 1, 2, …,
7
The cosine terms need not be computed on-the-fly;
they can be precomputed and stored in tables.
The inverse transform – the IDCT, or DCT -1 – is this:
7 7
1
( 2i+ 1)
uπ ( 2j + 1)
vπ
Eq 38.2 fij , = Σ Σ C u C v Fuv
, cos cos
4
16
16
u=
0 v=
0
The forward and inverse transforms involve nearly identical
arithmetic: The complexity of encoding and
decoding is very similar. The DCT is its own inverse
(within a scale factor), so performing the DCT on the
transform coefficients would perfectly reconstruct the
original samples, subject only to the roundoff error in
the DCT and IDCT.
CHAPTER 38 JPEG AND MOTION-JPEG (M-JPEG) COMPRESSION 451