Chapter 8 Data Compression c Outline Transform Coding Discrete Cosine Transform Transform Coding x k ID: 674388
Download Presentation The PPT/PDF document "CSI-447 : Multimedia Systems" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
CSI-447
:
Multimedia
Systems
Chapter
8
:
Data Compression
(
c
)Slide2
Outline
Transform
Coding
–
Discrete Cosine
TransformSlide3
Transform
Coding
⎢
⎥
⎢
.
⎥
⎢
⎣
x
k
⎥
⎦
good chance that
a
substantial amount of correlation is inherent among neighboring samples
x
i
. The rational behind transform coding is that if
Y
is the result of
a
linear transform
T
of X in such a way that the components of Y are much less correlated, then Y can be coded more efficiently than X
⎡
x
1
⎤
⎢
⎥
⎢
x
2
⎥
Let
X
⎢
.
⎥
be
a
vector of samples. There is
aSlide4
Transform
Coding
In
dimensions higher than 3, if most information is accurately described in the first few components of
a
transformed vector, the remaining components can be coarsely quantized or even set to zero with little signal
distortion.
The less effect one dimension has on another,
the more
chance we have of dealing differently with axes that store relatively minor amounts of information without affecting reasonably accurate reconstruction of the signal from its quantized or truncated transform coefficients.
Therefore, compression comes from the quantization of
the
components of
Y
.Slide5
Discrete Cosine Transform
2D
DCT: Given
a
function
f
(
i
,
j
)
over two integer variables
i
and
j
(e.g.
a
piece of an image), the 2D DCT transforms it into
a
new function
F
(
u
,
v
), with integers
u
and
v
running over the same range as
i and j such thatwhere i,u = 0, ...,
M
–
1
and
j
,
v
=
0,
...,
N
– 1
and
M
1
N
1
i0 j 0
2M 2N
MN
F (u, v) 2C(u)C(v) cos (2i 1)u cos (2 j 1)v f (i, j)
⎪
⎪
otherwise
⎩
1
if
x
0
2
⎧ 1
C
(
x
)
⎨Slide6
Discrete Cosine Transform
For
N
=
M
= 8
(used for JPEG Standard), the 2D DCT is
...
The inverse DCT (2D-IDCT)
is
7 7
4 16 16
u
0
v
0
f
~
(
i
,
j
)
C
(
u
)
C
(
v
)
cos
(2
i
1)
u
cos
(2
j 1)v F
(u, v)Slide7
Discrete Cosine Transform
1D
DCT: Given
a
function
f
(
i
) over integer variable
i
,
the 1D DCT transforms it into
a
new function
F
(
u
), with
integer
u
running over the same range as
i
such that
(
M
=8)
7
16
2
where i,u = 0, ..., 7 andi0F (u
)
C
(
u
)
cos
(2
i
1)
u
f
(
i
)
⎪
⎪
otherwise
⎩
1if
x 0
2⎧ 1C(x) ⎨Slide8
Discrete Cosine Transform
The inverse 1D-DCT is defined
by
where
i
,
u
=
0, ...,
7
and
7
2 16
u
0
f
~
(
i
)
C
(
u
)
cos
(2
i
1)
u
F
(
u
)
⎪
⎪
otherwise
⎩
1
if
x
0
2
⎧ 1C(x) ⎨Slide9
Discrete Cosine Transform
An electrical signal with constant magnitude is known as
a
DC signal (Direct
Current)
For
example,
a
9-volts
battery.
An electrical signal that changes its magnitude periodically at
a
certain frequency is known as an AC signal (alternating
Current)
For example,
household
electric power circuit (110 volts, 60Hz
vs.
220 volts
50Hz)
Although most signals are complex, any signal can be expressed as
a
sum of multiple signals that are sine or cosine waveforms at various amplitudes and
frequencies.
This is known as Fourier AnalysisSlide10
Discrete Cosine Transform
If a cosine function is used, the process of determining the amplitudes of the AC and DC components of the signal is called a Cosine Transform, and the integer indices make it a Discrete Cosine
Transform.
When
u
=0,
F
(
u
) yields the DC
coefficient
When
u
=
1, 2, ..., 7,
F
(
u
) yields the first, second, ..., seventh AC
coefficient.Slide11
Discrete Cosine Transform
The inverse transform uses a sum of the products of the DC or AC coefficients and the cosine functions to reconstruct the function
f
(
i
), now known as
f
~
(
i
).
Both DCT and IDCT use the same set of cosine functions, known as basis
functions.
The idea behind Transform Coding is to use only a few coefficients that result in the highest
energySlide12
Examples
(1D-DCT)Slide13
Examples
(1D-DCT)Slide14
Examples
(1D-IDCT)Slide15
Discrete Cosine Transform
Approximation of the ramp function using a 3-term DCT approximation vs. a 3-term DFT
approximation.