Cryptography Reference
In-Depth Information
the HVS is less sensitive to high frequencies thus they can be coarsely quan-
tized or even discarded to achieve data reduction. There are many types of
transform coding, such as Karhunen-Loeve Transform (KLT), Discrete Fourier
Transform (DFT), Discrete Cosine Transform (DCT) and Walsh-Hadamard
Transform (WHT). Compared to the others, DCT provides the good com-
promise between data reduction, computational complexity and minimizing
blocking artifacts. DCT was first applied to image compression by Chen and
Pratt [8], and is now widely used in most of image and video coding standards.
The two dimensional DCT is defined as follows,
F (u, v)=
2C(u)C(v)
n
,
n−1
n−1
(2j +1)uπ
2n
(2k +1)vπ
2n
f (j, k)cos
cos
(2.6)
j=0
k=0
n−1
n−1
f (j, k)=
2
n
C(u)C(v)
j=0
k=0
;
(2j +1)uπ
2n
(2k +1)vπ
2n
F (u, v)cos
cos
(2.7)
1
√
2
if u, v =0,
C(u),C(v)=
1
otherwise;
where f (i, j),i,j =0, ..., n−1 are the pixel values in the nn image block,
and F (u, v),u,v =0, ..., n−1 are its corresponding DCT coe
cients. Note
that Equation (2.6) is the forward transform and Equation (2.7) is the inverse
transform. In most of image and video coding applications, n = 8. Note also
that at u =0,v = 0, Equation (2.6) becomes F (0, 0) =
7
j=0
7
k=0
f (j, k).
It is kind of the average of the pixel values in the 88 image block, and is
thus called the DC coe
cient. The rest of F (u, v) are usually referred to as
AC coe
cients.
It can be seen from Fig. 2.6(b) that most of the DCT coe
cients are
very small (dark in the image block) and only the lower frequency coe
cients
are having big amplitudes. The lattice of bright dots are exactly where the
DC coe
cients and low frequency coe
cients locate for each of the 88
image blocks. It is also noted that in the areas where the original image have
strong edges, bright dots become brighter, indicating that there is higher
entropy in those blocks, or more none zero AC coe
cients. Fig. 2.6(c) shows
an numerical example of one block of DCT coe
cients, whose image block
locates at the nose tip of Fig. 2.6(a) (the white block). We can observe that
only the top left corner of the block has large values, and the rest of them are
all very small.
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