082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

24.14 The discrete cosine transform 799
f(n)
8
6
4
2
2 4 6 8
Figure24.20
Lossycompression usingthe d.c.t. andthe d.f.t. Markers:circle = original data; square =
d.c.t. derived values; diamond =d.f.t. derived values.
n
Thisisanexampleof lossycompression,whichisusedextensivelyinaudio,image
and video compression. Because some of the samples are known to be zero they do
not need to be stored and hence the file or data storage requirement is much lower. An
approximation of the original data can be recovered even though not all of the original
d.c.t. samples are provided. It can be shown that the more samples we retain, the better
thequalityofthereproduction.Itiswellknownthatthed.c.t.performswellonstraight
line data and is used extensively in compression standards for images such as JPEG. In
image data which is viewed qualitatively a certain amount of data loss is tolerable, and
thiswill be illustratedinthe following example.
Engineeringapplication24.3
Two-dimensionald.c.t.andimagecompression
Sofarwehaveonlyseenthed.c.t.appliedtoaone-dimensional dataset.Thiscould
be used to handle the compression of audio data, for example. When applying the
d.c.t. to a two-dimensional problem such as an image a slightly different form is
required:
F[k,l] = √ 1 ∑N−1
NM
n=0
{ M−1 ∑
[ ( π
f[n,m]cos m + 1 )
l] } [ ( π
cos n + 1 ) ]
k
M 2 N 2
m=0
forn=0,1,2, ... ,N−1
m=0,1,2, ... ,M−1
This formula can be applied to a 2D data set. We take the greyscale image given
in Figure 24.21 as the source data. Each pixel in the image has assigned to it a
numerical brightness value and itistothese values that weapply the d.c.t.
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