<s>
A	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
DCT	B-General_Concept
)	O
expresses	O
a	O
finite	O
sequence	O
of	O
data	B-Application
points	I-Application
in	O
terms	O
of	O
a	O
sum	O
of	O
cosine	O
functions	O
oscillating	O
at	O
different	O
frequencies	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
,	O
first	O
proposed	O
by	O
Nasir	O
Ahmed	O
in	O
1972	O
,	O
is	O
a	O
widely	O
used	O
transformation	O
technique	O
in	O
signal	O
processing	O
and	O
data	B-General_Concept
compression	I-General_Concept
.	O
</s>
<s>
It	O
is	O
used	O
in	O
most	O
digital	O
media	O
,	O
including	O
digital	B-Algorithm
images	I-Algorithm
(	O
such	O
as	O
JPEG	O
and	O
HEIF	O
)	O
,	O
digital	O
video	O
(	O
such	O
as	O
MPEG	O
and	O
H.26x	B-Application
)	O
,	O
digital	O
audio	O
(	O
such	O
as	O
Dolby	O
Digital	O
,	O
MP3	B-Application
and	O
AAC	B-Application
)	O
,	O
digital	B-Application
television	I-Application
(	O
such	O
as	O
SDTV	B-Device
,	O
HDTV	B-Device
and	O
VOD	B-Application
)	O
,	O
digital	O
radio	O
(	O
such	O
as	O
AAC+	O
and	O
DAB+	O
)	O
,	O
and	O
speech	B-Application
coding	I-Application
(	O
such	O
as	O
AAC-LD	B-Algorithm
,	O
Siren	B-Algorithm
and	O
Opus	B-Algorithm
)	O
.	O
</s>
<s>
DCTs	B-General_Concept
are	O
also	O
important	O
to	O
numerous	O
other	O
applications	O
in	O
science	O
and	O
engineering	O
,	O
such	O
as	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
,	O
telecommunication	O
devices	O
,	O
reducing	O
network	O
bandwidth	O
usage	O
,	O
and	O
spectral	B-Algorithm
methods	I-Algorithm
for	O
the	O
numerical	O
solution	O
of	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
DCT	B-General_Concept
is	O
a	O
Fourier-related	B-Algorithm
transform	I-Algorithm
similar	O
to	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	B-Algorithm
)	O
,	O
but	O
using	O
only	O
real	O
numbers	O
.	O
</s>
<s>
The	O
DCTs	B-General_Concept
are	O
generally	O
related	O
to	O
Fourier	O
series	O
coefficients	O
of	O
a	O
periodically	O
and	O
symmetrically	O
extended	O
sequence	O
whereas	O
DFTs	O
are	O
related	O
to	O
Fourier	O
series	O
coefficients	O
of	O
only	O
periodically	O
extended	O
sequences	O
.	O
</s>
<s>
DCTs	B-General_Concept
are	O
equivalent	O
to	O
DFTs	O
of	O
roughly	O
twice	O
the	O
length	O
,	O
operating	O
on	O
real	O
data	O
with	O
even	O
symmetry	O
(	O
since	O
the	O
Fourier	O
transform	O
of	O
a	O
real	O
and	O
even	O
function	O
is	O
real	O
and	O
even	O
)	O
,	O
whereas	O
in	O
some	O
variants	O
the	O
input	O
or	O
output	O
data	O
are	O
shifted	O
by	O
half	O
a	O
sample	O
.	O
</s>
<s>
There	O
are	O
eight	O
standard	O
DCT	B-General_Concept
variants	O
,	O
of	O
which	O
four	O
are	O
common	O
.	O
</s>
<s>
The	O
most	O
common	O
variant	O
of	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
is	O
the	O
type-II	O
DCT	B-General_Concept
,	O
which	O
is	O
often	O
called	O
simply	O
the	O
DCT	B-General_Concept
.	O
</s>
<s>
This	O
was	O
the	O
original	O
DCT	B-General_Concept
as	O
first	O
proposed	O
by	O
Ahmed	O
.	O
</s>
<s>
Its	O
inverse	O
,	O
the	O
type-III	O
DCT	B-General_Concept
,	O
is	O
correspondingly	O
often	O
called	O
simply	O
the	O
inverse	O
DCT	B-General_Concept
or	O
the	O
IDCT	B-General_Concept
.	O
</s>
<s>
Two	O
related	O
transforms	O
are	O
the	O
discrete	O
sine	O
transform	O
(	O
DST	O
)	O
,	O
which	O
is	O
equivalent	O
to	O
a	O
DFT	B-Algorithm
of	O
real	O
and	O
odd	O
functions	O
,	O
and	O
the	O
modified	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
MDCT	O
)	O
,	O
which	O
is	O
based	O
on	O
a	O
DCT	B-General_Concept
of	O
overlapping	O
data	O
.	O
</s>
<s>
Multidimensional	O
DCTs	B-General_Concept
(	O
MD	O
DCTs	B-General_Concept
)	O
are	O
developed	O
to	O
extend	O
the	O
concept	O
of	O
DCT	B-General_Concept
to	O
multidimensional	O
signals	O
.	O
</s>
<s>
A	O
variety	O
of	O
fast	O
algorithms	O
have	O
been	O
developed	O
to	O
reduce	O
the	O
computational	O
complexity	O
of	O
implementing	O
DCT	B-General_Concept
.	O
</s>
<s>
One	O
of	O
these	O
is	O
the	O
integer	O
DCT	B-General_Concept
(	O
IntDCT	O
)	O
,	O
an	O
integer	O
approximation	O
of	O
the	O
standard	O
DCT	B-General_Concept
,	O
used	O
in	O
several	O
ISO/IEC	O
and	O
ITU-T	O
international	O
standards	O
.	O
</s>
<s>
DCT	B-General_Concept
compression	O
,	O
also	O
known	O
as	O
block	B-General_Concept
compression	I-General_Concept
,	O
compresses	O
data	O
in	O
sets	O
of	O
discrete	O
DCT	B-General_Concept
blocks	O
.	O
</s>
<s>
DCT	B-General_Concept
blocks	O
sizes	O
including	O
8x8	O
pixels	B-Algorithm
for	O
the	O
standard	O
DCT	B-General_Concept
,	O
and	O
varied	O
integer	O
DCT	B-General_Concept
sizes	O
between	O
4x4	O
and	O
32x32	O
pixels	B-Algorithm
.	O
</s>
<s>
The	O
DCT	B-General_Concept
has	O
a	O
strong	O
energy	O
compaction	O
property	O
,	O
capable	O
of	O
achieving	O
high	O
quality	O
at	O
high	O
data	B-General_Concept
compression	I-General_Concept
ratios	I-General_Concept
.	O
</s>
<s>
However	O
,	O
blocky	O
compression	B-Algorithm
artifacts	I-Algorithm
can	O
appear	O
when	O
heavy	O
DCT	B-General_Concept
compression	O
is	O
applied	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
was	O
first	O
conceived	O
by	O
Nasir	O
Ahmed	O
,	O
T	O
.	O
Natarajan	O
and	O
K	O
.	O
R	O
.	O
Rao	O
while	O
working	O
at	O
Kansas	O
State	O
University	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
was	O
originally	O
intended	O
for	O
image	B-General_Concept
compression	I-General_Concept
.	O
</s>
<s>
Ahmed	O
developed	O
a	O
practical	O
DCT	B-General_Concept
algorithm	O
with	O
his	O
PhD	O
students	O
T	O
.	O
Raj	O
Natarajan	O
,	O
Wills	O
Dietrich	O
,	O
and	O
Jeremy	O
Fries	O
,	O
and	O
his	O
friend	O
Dr.	O
K	O
.	O
R	O
.	O
Rao	O
at	O
the	O
University	O
of	O
Texas	O
at	O
Arlington	O
in	O
1973	O
.	O
</s>
<s>
They	O
presented	O
their	O
results	O
in	O
a	O
January	O
1974	O
paper	O
,	O
titled	O
Discrete	B-General_Concept
Cosine	I-General_Concept
Transform	I-General_Concept
.	O
</s>
<s>
It	O
described	O
what	O
is	O
now	O
called	O
the	O
type-II	O
DCT	B-General_Concept
(	O
DCT-II	O
)	O
,	O
as	O
well	O
as	O
the	O
type-III	O
inverse	O
DCT	B-General_Concept
(	O
IDCT	B-General_Concept
)	O
.	O
</s>
<s>
Since	O
its	O
introduction	O
in	O
1974	O
,	O
there	O
has	O
been	O
significant	O
research	O
on	O
the	O
DCT	B-General_Concept
.	O
</s>
<s>
In	O
1977	O
,	O
Wen-Hsiung	O
Chen	O
published	O
a	O
paper	O
with	O
C	O
.	O
Harrison	O
Smith	O
and	O
Stanley	O
C	O
.	O
Fralick	O
presenting	O
a	O
fast	O
DCT	B-General_Concept
algorithm	O
.	O
</s>
<s>
These	O
research	O
papers	O
,	O
along	O
with	O
the	O
original	O
1974	O
Ahmed	O
paper	O
and	O
the	O
1977	O
Chen	O
paper	O
,	O
were	O
cited	O
by	O
the	O
Joint	O
Photographic	O
Experts	O
Group	O
as	O
the	O
basis	O
for	O
JPEG	O
's	O
lossy	B-Algorithm
image	B-General_Concept
compression	I-General_Concept
algorithm	O
in	O
1992	O
.	O
</s>
<s>
In	O
1975	O
,	O
John	O
A	O
.	O
Roese	O
and	O
Guner	O
S	O
.	O
Robinson	O
adapted	O
the	O
DCT	B-General_Concept
for	O
inter-frame	B-Algorithm
motion-compensated	B-Algorithm
video	B-Algorithm
coding	I-Algorithm
.	O
</s>
<s>
They	O
experimented	O
with	O
the	O
DCT	B-General_Concept
and	O
the	O
fast	O
Fourier	O
transform	O
(	O
FFT	O
)	O
,	O
developing	O
inter-frame	B-Algorithm
hybrid	O
coders	O
for	O
both	O
,	O
and	O
found	O
that	O
the	O
DCT	B-General_Concept
is	O
the	O
most	O
efficient	O
due	O
to	O
its	O
reduced	O
complexity	O
,	O
capable	O
of	O
compressing	O
image	O
data	O
down	O
to	O
0.25-bit	O
per	O
pixel	B-Algorithm
for	O
a	O
videotelephone	B-Application
scene	O
with	O
image	B-Algorithm
quality	I-Algorithm
comparable	O
to	O
an	O
intra-frame	O
coder	O
requiring	O
2-bit	O
per	O
pixel	B-Algorithm
.	O
</s>
<s>
In	O
1979	O
,	O
Anil	O
K	O
.	O
Jain	O
and	O
Jaswant	O
R	O
.	O
Jain	O
further	O
developed	O
motion-compensated	B-Algorithm
DCT	B-General_Concept
video	O
compression	O
,	O
also	O
called	O
block	O
motion	B-Algorithm
compensation	I-Algorithm
.	O
</s>
<s>
This	O
led	O
to	O
Chen	O
developing	O
a	O
practical	O
video	O
compression	B-General_Concept
algorithm	I-General_Concept
,	O
called	O
motion-compensated	B-Algorithm
DCT	B-General_Concept
or	O
adaptive	O
scene	O
coding	O
,	O
in	O
1981	O
.	O
</s>
<s>
Motion-compensated	B-Algorithm
DCT	B-General_Concept
later	O
became	O
the	O
standard	O
coding	O
technique	O
for	O
video	O
compression	O
from	O
the	O
late	O
1980s	O
onwards	O
.	O
</s>
<s>
The	O
integer	O
DCT	B-General_Concept
is	O
used	O
in	O
Advanced	B-Application
Video	I-Application
Coding	I-Application
(	O
AVC	O
)	O
,	O
introduced	O
in	O
2003	O
,	O
and	O
High	B-Algorithm
Efficiency	I-Algorithm
Video	I-Algorithm
Coding	I-Algorithm
(	O
HEVC	B-Algorithm
)	O
,	O
introduced	O
in	O
2013	O
.	O
</s>
<s>
The	O
integer	O
DCT	B-General_Concept
is	O
also	O
used	O
in	O
the	O
High	O
Efficiency	O
Image	O
Format	O
(	O
HEIF	O
)	O
,	O
which	O
uses	O
a	O
subset	O
of	O
the	O
HEVC	B-Algorithm
video	B-Algorithm
coding	I-Algorithm
format	I-Algorithm
for	O
coding	O
still	O
images	O
.	O
</s>
<s>
A	O
DCT	B-General_Concept
variant	O
,	O
the	O
modified	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
MDCT	O
)	O
,	O
was	O
developed	O
by	O
John	O
P	O
.	O
Princen	O
,	O
A.W.	O
</s>
<s>
The	O
MDCT	O
is	O
used	O
in	O
most	O
modern	O
audio	O
compression	O
formats	O
,	O
such	O
as	O
Dolby	O
Digital	O
(	O
AC-3	O
)	O
,	O
MP3	B-Application
(	O
which	O
uses	O
a	O
hybrid	O
DCT-FFT	O
algorithm	O
)	O
,	O
Advanced	B-Application
Audio	I-Application
Coding	I-Application
(	O
AAC	B-Application
)	O
,	O
and	O
Vorbis	B-Algorithm
(	O
Ogg	O
)	O
.	O
</s>
<s>
The	O
discrete	O
sine	O
transform	O
(	O
DST	O
)	O
was	O
derived	O
from	O
the	O
DCT	B-General_Concept
,	O
by	O
replacing	O
the	O
Neumann	O
condition	O
at	O
x	O
=	O
0	O
with	O
a	O
Dirichlet	O
condition	O
.	O
</s>
<s>
The	O
DST	O
was	O
described	O
in	O
the	O
1974	O
DCT	B-General_Concept
paper	O
by	O
Ahmed	O
,	O
Natarajan	O
and	O
Rao	O
.	O
</s>
<s>
Nasir	O
Ahmed	O
also	O
developed	O
a	O
lossless	B-Algorithm
DCT	B-General_Concept
algorithm	O
with	O
Giridhar	O
Mandyam	O
and	O
Neeraj	O
Magotra	O
at	O
the	O
University	O
of	O
New	O
Mexico	O
in	O
1995	O
.	O
</s>
<s>
This	O
allows	O
the	O
DCT	B-General_Concept
technique	O
to	O
be	O
used	O
for	O
lossless	B-Algorithm
compression	I-Algorithm
of	O
images	O
.	O
</s>
<s>
It	O
is	O
a	O
modification	O
of	O
the	O
original	O
DCT	B-General_Concept
algorithm	O
,	O
and	O
incorporates	O
elements	O
of	O
inverse	O
DCT	B-General_Concept
and	O
delta	B-General_Concept
modulation	I-General_Concept
.	O
</s>
<s>
It	O
is	O
a	O
more	O
effective	O
lossless	B-Algorithm
compression	I-Algorithm
algorithm	O
than	O
entropy	B-Algorithm
coding	I-Algorithm
.	O
</s>
<s>
Lossless	B-Algorithm
DCT	B-General_Concept
is	O
also	O
known	O
as	O
LDCT	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
is	O
the	O
most	O
widely	O
used	O
transformation	O
technique	O
in	O
signal	O
processing	O
,	O
and	O
by	O
far	O
the	O
most	O
widely	O
used	O
linear	O
transform	O
in	O
data	B-General_Concept
compression	I-General_Concept
.	O
</s>
<s>
Uncompressed	B-General_Concept
digital	O
media	O
as	O
well	O
as	O
lossless	B-Algorithm
compression	I-Algorithm
had	O
impractically	O
high	O
memory	B-General_Concept
and	O
bandwidth	O
requirements	O
,	O
which	O
was	O
significantly	O
reduced	O
by	O
the	O
highly	O
efficient	O
DCT	B-General_Concept
lossy	B-Algorithm
compression	I-Algorithm
technique	O
,	O
capable	O
of	O
achieving	O
data	B-General_Concept
compression	I-General_Concept
ratios	I-General_Concept
from	O
8:1	O
to	O
14:1	O
for	O
near-studio-quality	O
,	O
up	O
to	O
100:1	O
for	O
acceptable-quality	O
content	O
.	O
</s>
<s>
DCT	B-General_Concept
compression	O
standards	O
are	O
used	O
in	O
digital	O
media	O
technologies	O
,	O
such	O
as	O
digital	B-Algorithm
images	I-Algorithm
,	O
digital	B-Application
photos	I-Application
,	O
digital	O
video	O
,	O
streaming	O
media	O
,	O
digital	B-Application
television	I-Application
,	O
streaming	O
television	O
,	O
video	B-Application
on	I-Application
demand	I-Application
(	O
VOD	B-Application
)	O
,	O
digital	O
cinema	O
,	O
high-definition	O
video	O
(	O
HD	O
video	O
)	O
,	O
and	O
high-definition	B-Device
television	I-Device
(	O
HDTV	B-Device
)	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
,	O
and	O
in	O
particular	O
the	O
DCT-II	O
,	O
is	O
often	O
used	O
in	O
signal	O
and	O
image	B-Algorithm
processing	I-Algorithm
,	O
especially	O
for	O
lossy	B-Algorithm
compression	I-Algorithm
,	O
because	O
it	O
has	O
a	O
strong	O
"	O
energy	O
compaction	O
"	O
property	O
:	O
in	O
typical	O
applications	O
,	O
most	O
of	O
the	O
signal	O
information	O
tends	O
to	O
be	O
concentrated	O
in	O
a	O
few	O
low-frequency	O
components	O
of	O
the	O
DCT	B-General_Concept
.	O
</s>
<s>
For	O
strongly	O
correlated	O
Markov	O
processes	O
,	O
the	O
DCT	B-General_Concept
can	O
approach	O
the	O
compaction	O
efficiency	O
of	O
the	O
Karhunen-Loève	O
transform	O
(	O
which	O
is	O
optimal	O
in	O
the	O
decorrelation	O
sense	O
)	O
.	O
</s>
<s>
DCTs	B-General_Concept
are	O
also	O
widely	O
employed	O
in	O
solving	O
partial	O
differential	O
equations	O
by	O
spectral	B-Algorithm
methods	I-Algorithm
,	O
where	O
the	O
different	O
variants	O
of	O
the	O
DCT	B-General_Concept
correspond	O
to	O
slightly	O
different	O
even/odd	O
boundary	O
conditions	O
at	O
the	O
two	O
ends	O
of	O
the	O
array	O
.	O
</s>
<s>
DCTs	B-General_Concept
are	O
also	O
closely	O
related	O
to	O
Chebyshev	O
polynomials	O
,	O
and	O
fast	O
DCT	B-General_Concept
algorithms	O
(	O
below	O
)	O
are	O
used	O
in	O
Chebyshev	O
approximation	O
of	O
arbitrary	O
functions	O
by	O
series	O
of	O
Chebyshev	O
polynomials	O
,	O
for	O
example	O
in	O
Clenshaw	B-Algorithm
–	I-Algorithm
Curtis	I-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
The	O
DCT	B-General_Concept
is	O
the	O
coding	O
standard	O
for	O
multimedia	O
telecommunication	O
devices	O
.	O
</s>
<s>
DCT	B-General_Concept
compression	O
significantly	O
reduces	O
the	O
amount	O
of	O
memory	B-General_Concept
and	O
bandwidth	O
required	O
for	O
digital	O
signals	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
is	O
widely	O
used	O
in	O
many	O
applications	O
,	O
which	O
include	O
the	O
following	O
.	O
</s>
<s>
The	O
DCT-II	O
,	O
also	O
known	O
as	O
simply	O
the	O
DCT	B-General_Concept
,	O
is	O
the	O
most	O
important	O
image	B-General_Concept
compression	I-General_Concept
technique	O
.	O
</s>
<s>
It	O
is	O
used	O
in	O
image	B-General_Concept
compression	I-General_Concept
standards	O
such	O
as	O
JPEG	O
,	O
and	O
video	B-Algorithm
compression	I-Algorithm
standards	I-Algorithm
such	O
as	O
H.26x	B-Application
,	O
MJPEG	B-Application
,	O
MPEG	O
,	O
DV	O
,	O
Theora	B-Algorithm
and	O
Daala	B-Application
.	O
</s>
<s>
There	O
,	O
the	O
two-dimensional	O
DCT-II	O
of	O
blocks	O
are	O
computed	O
and	O
the	O
results	O
are	O
quantized	B-Algorithm
and	O
entropy	B-Algorithm
coded	I-Algorithm
.	O
</s>
<s>
In	O
this	O
case	O
,	O
is	O
typically	O
8	O
and	O
the	O
DCT-II	O
formula	O
is	O
applied	O
to	O
each	O
row	O
and	O
column	O
of	O
the	O
block	O
.	O
</s>
<s>
Advanced	B-Application
Video	I-Application
Coding	I-Application
(	O
AVC	O
)	O
uses	O
the	O
integer	O
DCT	B-General_Concept
(	O
IntDCT	O
)	O
,	O
an	O
integer	O
approximation	O
of	O
the	O
DCT	B-General_Concept
.	O
</s>
<s>
It	O
uses	O
4x4	O
and	O
8x8	O
integer	O
DCT	B-General_Concept
blocks	O
.	O
</s>
<s>
High	B-Algorithm
Efficiency	I-Algorithm
Video	I-Algorithm
Coding	I-Algorithm
(	O
HEVC	B-Algorithm
)	O
and	O
the	O
High	O
Efficiency	O
Image	O
Format	O
(	O
HEIF	O
)	O
use	O
varied	O
integer	O
DCT	B-General_Concept
block	O
sizes	O
between	O
4x4	O
and	O
32x32	O
pixels	B-Algorithm
.	O
</s>
<s>
,	O
AVC	O
is	O
by	O
far	O
the	O
most	O
commonly	O
used	O
format	O
for	O
the	O
recording	O
,	O
compression	O
and	O
distribution	O
of	O
video	O
content	O
,	O
used	O
by	O
91%	O
of	O
video	O
developers	O
,	O
followed	O
by	O
HEVC	B-Algorithm
which	O
is	O
used	O
by	O
43%	O
of	O
developers	O
.	O
</s>
<s>
Image	B-General_Concept
compression	I-General_Concept
standard	O
Year	O
Common	O
applications	O
JPEG	O
1992	O
The	O
most	O
widely	O
used	O
image	B-General_Concept
compression	I-General_Concept
standard	O
and	O
digital	B-Algorithm
image	I-Algorithm
format	O
,	O
JPEG	B-Device
XR	I-Device
2009	O
Open	B-Device
XML	I-Device
Paper	I-Device
Specification	I-Device
WebP	O
2010	O
A	O
graphic	O
format	O
that	O
supports	O
the	O
lossy	B-Algorithm
compression	I-Algorithm
of	O
digital	B-Algorithm
images	I-Algorithm
.	O
</s>
<s>
Developed	O
by	O
Google	B-Application
.	I-Application
</s>
<s>
High	O
Efficiency	O
Image	O
Format	O
(	O
HEIF	O
)	O
2013	O
Image	O
file	O
format	O
based	O
on	O
HEVC	B-Algorithm
compression	O
.	O
</s>
<s>
BPG	O
2014Based	O
on	O
HEVC	B-Algorithm
compression	O
JPEG	O
XL	O
2020	O
A	O
royalty-free	O
raster-graphics	O
file	O
format	O
that	O
supports	O
both	O
lossy	B-Algorithm
and	O
lossless	B-Algorithm
compression	I-Algorithm
.	O
</s>
<s>
Video	B-Algorithm
coding	I-Algorithm
standard	I-Algorithm
Year	O
Common	O
applications	O
H.261	B-Algorithm
1988	O
First	O
of	O
a	O
family	O
of	O
video	B-Algorithm
coding	I-Algorithm
standards	I-Algorithm
.	O
</s>
<s>
Used	O
primarily	O
in	O
older	O
video	B-Application
conferencing	I-Application
and	O
video	B-Application
telephone	I-Application
products	O
.	O
</s>
<s>
Motion	B-Application
JPEG	I-Application
(	O
MJPEG	B-Application
)	O
1992	O
QuickTime	B-Application
,	O
video	O
editing	O
,	O
non-linear	O
editing	O
,	O
digital	B-Device
cameras	I-Device
MPEG-1	B-Algorithm
Video	O
1993	O
Digital	O
video	O
distribution	O
on	O
CD	B-Device
or	O
Internet	O
video	O
MPEG-2	B-Algorithm
Video	I-Algorithm
(	O
H.262	B-Algorithm
)	O
1995	O
Storage	O
and	O
handling	O
of	O
digital	B-Algorithm
images	I-Algorithm
in	O
broadcast	O
applications	O
,	O
digital	B-Application
television	I-Application
,	O
HDTV	B-Device
,	O
cable	O
,	O
satellite	O
,	O
high-speed	O
Internet	O
,	O
DVD	O
video	O
distribution	O
DV	O
1995	O
Camcorders	O
,	O
digital	O
cassettes	O
H.263	B-Algorithm
(	O
MPEG-4	B-Application
Part	I-Application
2	I-Application
)	O
1996	O
Video	B-Application
telephony	I-Application
over	O
public	O
switched	O
telephone	O
network	O
(	O
PSTN	O
)	O
,	O
H.320	B-Application
,	O
Integrated	B-Protocol
Services	I-Protocol
Digital	I-Protocol
Network	I-Protocol
(	O
ISDN	B-Protocol
)	O
Advanced	B-Application
Video	I-Application
Coding	I-Application
(	O
AVC	O
/	O
H.264	B-Application
/	O
MPEG-4	B-Algorithm
)	O
2003	O
Most	O
common	O
HD	O
video	O
recording/compression/distribution	O
format	O
,	O
Internet	O
video	O
,	O
YouTube	B-General_Concept
,	O
Blu-ray	O
Discs	O
,	O
HDTV	B-Device
broadcasts	O
,	O
web	B-Application
browsers	I-Application
,	O
streaming	O
television	O
,	O
mobile	B-Application
devices	I-Application
,	O
consumer	O
devices	O
,	O
Netflix	O
,	O
video	B-Application
telephony	I-Application
,	O
FaceTime	B-Operating_System
Theora	B-Algorithm
2004	O
Internet	O
video	O
,	O
web	B-Application
browsers	I-Application
VC-1	B-Operating_System
2006	O
Windows	B-Device
media	I-Device
,	O
Blu-ray	O
Discs	O
Apple	B-Algorithm
ProRes	I-Algorithm
2007	O
Professional	O
video	O
production	O
.	O
</s>
<s>
WebM	B-Application
Video	O
2010	O
A	O
multimedia	O
open	O
source	O
format	O
developed	O
by	O
Google	B-Application
intended	O
to	O
be	O
used	O
with	O
HTML5	B-Application
.	O
</s>
<s>
High	B-Algorithm
Efficiency	I-Algorithm
Video	I-Algorithm
Coding	I-Algorithm
(	O
HEVC	B-Algorithm
/	O
H.265	B-Algorithm
)	O
2013	O
The	O
emerging	O
successor	O
to	O
the	O
H.264/MPEG	B-Application
-4	I-Application
AVC	I-Application
standard	O
,	O
having	O
substantially	O
improved	O
compression	O
capability	O
.	O
</s>
<s>
Daala	B-Application
2013	O
Research	O
video	O
format	O
by	O
Xiph.org	B-Application
.	O
</s>
<s>
AV1	B-Application
2018	O
An	O
open	O
source	O
format	O
based	O
on	O
VP10	O
(	O
VP9	B-Algorithm
's	O
internal	O
successor	O
)	O
,	O
Daala	B-Application
and	O
Thor	B-Algorithm
;	O
used	O
by	O
content	O
providers	O
such	O
as	O
YouTube	B-General_Concept
and	O
Netflix	O
.	O
</s>
<s>
Multidimensional	O
DCTs	B-General_Concept
(	O
MD	O
DCTs	B-General_Concept
)	O
have	O
several	O
applications	O
,	O
mainly	O
3-D	O
DCTs	B-General_Concept
such	O
as	O
the	O
3-D	O
DCT-II	O
,	O
which	O
has	O
several	O
new	O
applications	O
like	O
Hyperspectral	O
Imaging	O
coding	O
systems	O
,	O
variable	O
temporal	O
length	O
3-D	O
DCT	B-General_Concept
coding	O
,	O
video	B-Algorithm
coding	I-Algorithm
algorithms	O
,	O
adaptive	O
video	B-Algorithm
coding	I-Algorithm
and	O
3-D	O
Compression	O
.	O
</s>
<s>
Due	O
to	O
enhancement	O
in	O
the	O
hardware	O
,	O
software	O
and	O
introduction	O
of	O
several	O
fast	O
algorithms	O
,	O
the	O
necessity	O
of	O
using	O
M-D	O
DCTs	B-General_Concept
is	O
rapidly	O
increasing	O
.	O
</s>
<s>
DCT-IV	O
has	O
gained	O
popularity	O
for	O
its	O
applications	O
in	O
fast	O
implementation	O
of	O
real-valued	O
polyphase	O
filtering	O
banks	O
,	O
lapped	O
orthogonal	B-Algorithm
transform	I-Algorithm
and	O
cosine-modulated	O
wavelet	O
bases	O
.	O
</s>
<s>
DCT	B-General_Concept
plays	O
a	O
very	O
important	O
role	O
in	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
.	O
</s>
<s>
By	O
using	O
the	O
DCT	B-General_Concept
,	O
the	O
signals	O
can	O
be	O
compressed	O
.	O
</s>
<s>
DCT	B-General_Concept
can	O
be	O
used	O
in	O
electrocardiography	B-Application
for	O
the	O
compression	O
of	O
ECG	B-Application
signals	O
.	O
</s>
<s>
DCT2	O
provides	O
a	O
better	O
compression	O
ratio	O
than	O
DCT	B-General_Concept
.	O
</s>
<s>
The	O
DCT	B-General_Concept
is	O
widely	O
implemented	O
in	O
digital	B-Architecture
signal	I-Architecture
processors	I-Architecture
(	O
DSP	O
)	O
,	O
as	O
well	O
as	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
software	O
.	O
</s>
<s>
Many	O
companies	O
have	O
developed	O
DSPs	O
based	O
on	O
DCT	B-General_Concept
technology	O
.	O
</s>
<s>
DCTs	B-General_Concept
are	O
widely	O
used	O
for	O
applications	O
such	O
as	O
encoding	B-Algorithm
,	O
decoding	B-Algorithm
,	O
video	O
,	O
audio	O
,	O
multiplexing	B-Architecture
,	O
control	O
signals	O
,	O
signaling	O
,	O
and	O
analog-to-digital	O
conversion	O
.	O
</s>
<s>
DCTs	B-General_Concept
are	O
also	O
commonly	O
used	O
for	O
high-definition	B-Device
television	I-Device
(	O
HDTV	B-Device
)	O
encoder/decoder	O
chips	O
.	O
</s>
<s>
A	O
common	O
issue	O
with	O
DCT	B-General_Concept
compression	O
in	O
digital	O
media	O
are	O
blocky	O
compression	B-Algorithm
artifacts	I-Algorithm
,	O
caused	O
by	O
DCT	B-General_Concept
blocks	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
algorithm	O
can	O
cause	O
block-based	O
artifacts	B-Algorithm
when	O
heavy	O
compression	O
is	O
applied	O
.	O
</s>
<s>
Due	O
to	O
the	O
DCT	B-General_Concept
being	O
used	O
in	O
the	O
majority	O
of	O
digital	B-Algorithm
image	I-Algorithm
and	O
video	B-Algorithm
coding	I-Algorithm
standards	I-Algorithm
(	O
such	O
as	O
the	O
JPEG	O
,	O
H.26x	B-Application
and	O
MPEG	O
formats	O
)	O
,	O
DCT-based	O
blocky	O
compression	B-Algorithm
artifacts	I-Algorithm
are	O
widespread	O
in	O
digital	O
media	O
.	O
</s>
<s>
In	O
a	O
DCT	B-General_Concept
algorithm	O
,	O
an	O
image	O
(	O
or	O
frame	O
in	O
an	O
image	O
sequence	O
)	O
is	O
divided	O
into	O
square	O
blocks	O
which	O
are	O
processed	O
independently	O
from	O
each	O
other	O
,	O
then	O
the	O
DCT	B-General_Concept
of	O
these	O
blocks	O
is	O
taken	O
,	O
and	O
the	O
resulting	O
DCT	B-General_Concept
coefficients	O
are	O
quantized	B-Algorithm
.	O
</s>
<s>
This	O
process	O
can	O
cause	O
blocking	O
artifacts	B-Algorithm
,	O
primarily	O
at	O
high	O
data	B-General_Concept
compression	I-General_Concept
ratios	I-General_Concept
.	O
</s>
<s>
DCT	B-General_Concept
blocks	O
are	O
often	O
used	O
in	O
glitch	O
art	O
.	O
</s>
<s>
The	O
artist	O
Rosa	O
Menkman	O
makes	O
use	O
of	O
DCT-based	O
compression	B-Algorithm
artifacts	I-Algorithm
in	O
her	O
glitch	O
art	O
,	O
particularly	O
the	O
DCT	B-General_Concept
blocks	O
found	O
in	O
most	O
digital	O
media	O
formats	O
such	O
as	O
JPEG	O
digital	B-Algorithm
images	I-Algorithm
and	O
MP3	B-Application
digital	O
audio	O
.	O
</s>
<s>
Another	O
example	O
is	O
Jpegs	O
by	O
German	O
photographer	O
Thomas	O
Ruff	O
,	O
which	O
uses	O
intentional	O
JPEG	B-Algorithm
artifacts	I-Algorithm
as	O
the	O
basis	O
of	O
the	O
picture	O
's	O
style	O
.	O
</s>
<s>
Like	O
any	O
Fourier-related	B-Algorithm
transform	I-Algorithm
,	O
discrete	B-General_Concept
cosine	I-General_Concept
transforms	I-General_Concept
(	O
DCTs	B-General_Concept
)	O
express	O
a	O
function	O
or	O
a	O
signal	O
in	O
terms	O
of	O
a	O
sum	O
of	O
sinusoids	O
with	O
different	O
frequencies	O
and	O
amplitudes	B-Application
.	O
</s>
<s>
Like	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
DFT	B-Algorithm
)	O
,	O
a	O
DCT	B-General_Concept
operates	O
on	O
a	O
function	O
at	O
a	O
finite	O
number	O
of	O
discrete	O
data	B-Application
points	I-Application
.	O
</s>
<s>
The	O
obvious	O
distinction	O
between	O
a	O
DCT	B-General_Concept
and	O
a	O
DFT	B-Algorithm
is	O
that	O
the	O
former	O
uses	O
only	O
cosine	O
functions	O
,	O
while	O
the	O
latter	O
uses	O
both	O
cosines	O
and	O
sines	O
(	O
in	O
the	O
form	O
of	O
complex	O
exponentials	O
)	O
.	O
</s>
<s>
However	O
,	O
this	O
visible	O
difference	O
is	O
merely	O
a	O
consequence	O
of	O
a	O
deeper	O
distinction	O
:	O
a	O
DCT	B-General_Concept
implies	O
different	O
boundary	O
conditions	O
from	O
the	O
DFT	B-Algorithm
or	O
other	O
related	O
transforms	O
.	O
</s>
<s>
The	O
Fourier-related	B-Algorithm
transforms	I-Algorithm
that	O
operate	O
on	O
a	O
function	O
over	O
a	O
finite	O
domain	B-Algorithm
,	O
such	O
as	O
the	O
DFT	B-Algorithm
or	O
DCT	B-General_Concept
or	O
a	O
Fourier	O
series	O
,	O
can	O
be	O
thought	O
of	O
as	O
implicitly	O
defining	O
an	O
extension	O
of	O
that	O
function	O
outside	O
the	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
DFT	B-Algorithm
,	O
like	O
the	O
Fourier	O
series	O
,	O
implies	O
a	O
periodic	O
extension	O
of	O
the	O
original	O
function	O
.	O
</s>
<s>
A	O
DCT	B-General_Concept
,	O
like	O
a	O
cosine	O
transform	O
,	O
implies	O
an	O
even	O
extension	O
of	O
the	O
original	O
function	O
.	O
</s>
<s>
However	O
,	O
because	O
DCTs	B-General_Concept
operate	O
on	O
finite	O
,	O
discrete	O
sequences	O
,	O
two	O
issues	O
arise	O
that	O
do	O
not	O
apply	O
for	O
the	O
continuous	O
cosine	O
transform	O
.	O
</s>
<s>
First	O
,	O
one	O
has	O
to	O
specify	O
whether	O
the	O
function	O
is	O
even	O
or	O
odd	O
at	O
both	O
the	O
left	O
and	O
right	O
boundaries	O
of	O
the	O
domain	B-Algorithm
(	O
i.e.	O
</s>
<s>
In	O
particular	O
,	O
consider	O
a	O
sequence	O
abcd	O
of	O
four	O
equally	O
spaced	O
data	B-Application
points	I-Application
,	O
and	O
say	O
that	O
we	O
specify	O
an	O
even	O
left	O
boundary	O
.	O
</s>
<s>
These	O
choices	O
lead	O
to	O
all	O
the	O
standard	O
variations	O
of	O
DCTs	B-General_Concept
and	O
also	O
discrete	O
sine	O
transforms	O
(	O
DSTs	O
)	O
.	O
</s>
<s>
Each	O
boundary	O
can	O
be	O
either	O
even	O
or	O
odd	O
(	O
2	O
choices	O
per	O
boundary	O
)	O
and	O
can	O
be	O
symmetric	O
about	O
a	O
data	B-Application
point	I-Application
or	O
the	O
point	O
halfway	O
between	O
two	O
data	B-Application
points	I-Application
(	O
2	O
choices	O
per	O
boundary	O
)	O
,	O
for	O
a	O
total	O
of	O
2	O
×	O
2	O
×	O
2	O
×	O
2	O
=	O
16	O
possibilities	O
.	O
</s>
<s>
Half	O
of	O
these	O
possibilities	O
,	O
those	O
where	O
the	O
left	O
boundary	O
is	O
even	O
,	O
correspond	O
to	O
the	O
8	O
types	O
of	O
DCT	B-General_Concept
;	O
the	O
other	O
half	O
are	O
the	O
8	O
types	O
of	O
DST	O
.	O
</s>
<s>
These	O
different	O
boundary	O
conditions	O
strongly	O
affect	O
the	O
applications	O
of	O
the	O
transform	O
and	O
lead	O
to	O
uniquely	O
useful	O
properties	O
for	O
the	O
various	O
DCT	B-General_Concept
types	O
.	O
</s>
<s>
Most	O
directly	O
,	O
when	O
using	O
Fourier-related	B-Algorithm
transforms	I-Algorithm
to	O
solve	O
partial	O
differential	O
equations	O
by	O
spectral	B-Algorithm
methods	I-Algorithm
,	O
the	O
boundary	O
conditions	O
are	O
directly	O
specified	O
as	O
a	O
part	O
of	O
the	O
problem	O
being	O
solved	O
.	O
</s>
<s>
Or	O
,	O
for	O
the	O
MDCT	O
(	O
based	O
on	O
the	O
type-IV	O
DCT	B-General_Concept
)	O
,	O
the	O
boundary	O
conditions	O
are	O
intimately	O
involved	O
in	O
the	O
MDCT	O
's	O
critical	O
property	O
of	O
time-domain	O
aliasing	O
cancellation	O
.	O
</s>
<s>
In	O
a	O
more	O
subtle	O
fashion	O
,	O
the	O
boundary	O
conditions	O
are	O
responsible	O
for	O
the	O
"	O
energy	O
compactification	O
"	O
properties	O
that	O
make	O
DCTs	B-General_Concept
useful	O
for	O
image	O
and	O
audio	O
compression	O
,	O
because	O
the	O
boundaries	O
affect	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
any	O
Fourier-like	O
series	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
is	O
well	O
known	O
that	O
any	O
discontinuities	O
in	O
a	O
function	O
reduce	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
the	O
Fourier	O
series	O
,	O
so	O
that	O
more	O
sinusoids	O
are	O
needed	O
to	O
represent	O
the	O
function	O
with	O
a	O
given	O
accuracy	O
.	O
</s>
<s>
The	O
same	O
principle	O
governs	O
the	O
usefulness	O
of	O
the	O
DFT	B-Algorithm
and	O
other	O
transforms	O
for	O
signal	O
compression	O
;	O
the	O
smoother	O
a	O
function	O
is	O
,	O
the	O
fewer	O
terms	O
in	O
its	O
DFT	B-Algorithm
or	O
DCT	B-General_Concept
are	O
required	O
to	O
represent	O
it	O
accurately	O
,	O
and	O
the	O
more	O
it	O
can	O
be	O
compressed	O
.	O
</s>
<s>
(	O
Here	O
,	O
we	O
think	O
of	O
the	O
DFT	B-Algorithm
or	O
DCT	B-General_Concept
as	O
approximations	O
for	O
the	O
Fourier	O
series	O
or	O
cosine	O
series	O
of	O
a	O
function	O
,	O
respectively	O
,	O
in	O
order	O
to	O
talk	O
about	O
its	O
"	O
smoothness	O
"	O
.	O
)	O
</s>
<s>
However	O
,	O
the	O
implicit	O
periodicity	O
of	O
the	O
DFT	B-Algorithm
means	O
that	O
discontinuities	O
usually	O
occur	O
at	O
the	O
boundaries	O
:	O
any	O
random	O
segment	O
of	O
a	O
signal	O
is	O
unlikely	O
to	O
have	O
the	O
same	O
value	O
at	O
both	O
the	O
left	O
and	O
right	O
boundaries	O
.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
DCT	B-General_Concept
where	O
both	O
boundaries	O
are	O
even	O
always	O
yields	O
a	O
continuous	O
extension	O
at	O
the	O
boundaries	O
(	O
although	O
the	O
slope	O
is	O
generally	O
discontinuous	O
)	O
.	O
</s>
<s>
This	O
is	O
why	O
DCTs	B-General_Concept
,	O
and	O
in	O
particular	O
DCTs	B-General_Concept
of	O
types	O
I	O
,	O
II	O
,	O
V	O
,	O
and	O
VI	O
(	O
the	O
types	O
that	O
have	O
two	O
even	O
boundaries	O
)	O
generally	O
perform	O
better	O
for	O
signal	O
compression	O
than	O
DFTs	O
and	O
DSTs	O
.	O
</s>
<s>
In	O
practice	O
,	O
a	O
type-II	O
DCT	B-General_Concept
is	O
usually	O
preferred	O
for	O
such	O
applications	O
,	O
in	O
part	O
for	O
reasons	O
of	O
computational	O
convenience	O
.	O
</s>
<s>
Formally	O
,	O
the	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
is	O
a	O
linear	O
,	O
invertible	O
function	O
(	O
where	O
denotes	O
the	O
set	O
of	O
real	O
numbers	O
)	O
,	O
or	O
equivalently	O
an	O
invertible	O
×	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
There	O
are	O
several	O
variants	O
of	O
the	O
DCT	B-General_Concept
with	O
slightly	O
modified	O
definitions	O
.	O
</s>
<s>
Some	O
authors	O
further	O
multiply	O
the	O
and	O
terms	O
by	O
and	O
correspondingly	O
multiply	O
the	O
and	O
terms	O
by	O
which	O
makes	O
the	O
DCT-I	O
matrix	O
orthogonal	B-Algorithm
,	O
if	O
one	O
further	O
multiplies	O
by	O
an	O
overall	O
scale	O
factor	O
of	O
but	O
breaks	O
the	O
direct	O
correspondence	O
with	O
a	O
real-even	O
DFT	B-Algorithm
.	O
</s>
<s>
The	O
DCT-I	O
is	O
exactly	O
equivalent	O
(	O
up	O
to	O
an	O
overall	O
scale	O
factor	O
of	O
2	O
)	O
,	O
to	O
a	O
DFT	B-Algorithm
of	O
real	O
numbers	O
with	O
even	O
symmetry	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
DCT-I	O
of	O
real	O
numbers	O
is	O
exactly	O
equivalent	O
to	O
a	O
DFT	B-Algorithm
of	O
eight	O
real	O
numbers	O
(	O
even	O
symmetry	O
)	O
,	O
divided	O
by	O
two	O
.	O
</s>
<s>
(	O
In	O
contrast	O
,	O
DCT	B-General_Concept
types	O
II-IV	O
involve	O
a	O
half-sample	O
shift	O
in	O
the	O
equivalent	O
DFT	B-Algorithm
.	O
)	O
</s>
<s>
The	O
DCT-II	O
is	O
probably	O
the	O
most	O
commonly	O
used	O
form	O
,	O
and	O
is	O
often	O
simply	O
referred	O
to	O
as	O
"	O
the	O
DCT	B-General_Concept
"	O
.	O
</s>
<s>
This	O
transform	O
is	O
exactly	O
equivalent	O
(	O
up	O
to	O
an	O
overall	O
scale	O
factor	O
of	O
2	O
)	O
to	O
a	O
DFT	B-Algorithm
of	O
real	O
inputs	O
of	O
even	O
symmetry	O
where	O
the	O
even-indexed	O
elements	O
are	O
zero	O
.	O
</s>
<s>
That	O
is	O
,	O
it	O
is	O
half	O
of	O
the	O
DFT	B-Algorithm
of	O
the	O
inputs	O
where	O
for	O
and	O
for	O
DCT-II	O
transformation	O
is	O
also	O
possible	O
using	O
2	O
signal	O
followed	O
by	O
a	O
multiplication	O
by	O
half	O
shift	O
.	O
</s>
<s>
Some	O
authors	O
further	O
multiply	O
the	O
term	O
by	O
and	O
multiply	O
the	O
rest	O
of	O
the	O
matrix	O
by	O
an	O
overall	O
scale	O
factor	O
of	O
(	O
see	O
below	O
for	O
the	O
corresponding	O
change	O
in	O
DCT-III	O
)	O
.	O
</s>
<s>
This	O
makes	O
the	O
DCT-II	O
matrix	O
orthogonal	B-Algorithm
,	O
but	O
breaks	O
the	O
direct	O
correspondence	O
with	O
a	O
real-even	O
DFT	B-Algorithm
of	O
half-shifted	O
input	O
.	O
</s>
<s>
This	O
is	O
the	O
normalization	O
used	O
by	O
Matlab	B-Language
,	O
for	O
example	O
,	O
see	O
.	O
</s>
<s>
the	O
quantization	B-Algorithm
step	O
in	O
JPEG	O
)	O
,	O
and	O
a	O
scaling	O
can	O
be	O
chosen	O
that	O
allows	O
the	O
DCT	B-General_Concept
to	O
be	O
computed	O
with	O
fewer	O
multiplications	O
.	O
</s>
<s>
Because	O
it	O
is	O
the	O
inverse	O
of	O
DCT-II	O
(	O
up	O
to	O
a	O
scale	O
factor	O
,	O
see	O
below	O
)	O
,	O
this	O
form	O
is	O
sometimes	O
simply	O
referred	O
to	O
as	O
"	O
the	O
inverse	O
DCT	B-General_Concept
"	O
(	O
"	O
IDCT	B-General_Concept
"	O
)	O
.	O
</s>
<s>
Some	O
authors	O
divide	O
the	O
term	O
by	O
instead	O
of	O
by	O
2	O
(	O
resulting	O
in	O
an	O
overall	O
term	O
)	O
and	O
multiply	O
the	O
resulting	O
matrix	O
by	O
an	O
overall	O
scale	O
factor	O
of	O
(	O
see	O
above	O
for	O
the	O
corresponding	O
change	O
in	O
DCT-II	O
)	O
,	O
so	O
that	O
the	O
DCT-II	O
and	O
DCT-III	O
are	O
transposes	O
of	O
one	O
another	O
.	O
</s>
<s>
This	O
makes	O
the	O
DCT-III	O
matrix	O
orthogonal	B-Algorithm
,	O
but	O
breaks	O
the	O
direct	O
correspondence	O
with	O
a	O
real-even	O
DFT	B-Algorithm
of	O
half-shifted	O
output	O
.	O
</s>
<s>
A	O
variant	O
of	O
the	O
DCT-IV	O
,	O
where	O
data	O
from	O
different	O
transforms	O
are	O
overlapped	O
,	O
is	O
called	O
the	O
modified	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
MDCT	O
)	O
.	O
</s>
<s>
DCTs	B-General_Concept
of	O
types	O
I	O
–	O
IV	O
treat	O
both	O
boundaries	O
consistently	O
regarding	O
the	O
point	O
of	O
symmetry	O
:	O
they	O
are	O
even/odd	O
around	O
either	O
a	O
data	B-Application
point	I-Application
for	O
both	O
boundaries	O
or	O
halfway	O
between	O
two	O
data	B-Application
points	I-Application
for	O
both	O
boundaries	O
.	O
</s>
<s>
By	O
contrast	O
,	O
DCTs	B-General_Concept
of	O
types	O
V-VIII	O
imply	O
boundaries	O
that	O
are	O
even/odd	O
around	O
a	O
data	B-Application
point	I-Application
for	O
one	O
boundary	O
and	O
halfway	O
between	O
two	O
data	B-Application
points	I-Application
for	O
the	O
other	O
boundary	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
DCT	B-General_Concept
types	O
I	O
–	O
IV	O
are	O
equivalent	O
to	O
real-even	O
DFTs	O
of	O
even	O
order	O
(	O
regardless	O
of	O
whether	O
is	O
even	O
or	O
odd	O
)	O
,	O
since	O
the	O
corresponding	O
DFT	B-Algorithm
is	O
of	O
length	O
(	O
for	O
DCT-I	O
)	O
or	O
(	O
for	O
DCT-II	O
&	O
III	O
)	O
or	O
(	O
for	O
DCT-IV	O
)	O
.	O
</s>
<s>
The	O
four	O
additional	O
types	O
of	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
correspond	O
essentially	O
to	O
real-even	O
DFTs	O
of	O
logically	O
odd	O
order	O
,	O
which	O
have	O
factors	O
of	O
in	O
the	O
denominators	O
of	O
the	O
cosine	O
arguments	O
.	O
</s>
<s>
the	O
simplest	O
radix-2	O
algorithms	O
are	O
only	O
for	O
even	O
lengths	O
)	O
,	O
and	O
this	O
increased	O
intricacy	O
carries	O
over	O
to	O
the	O
DCTs	B-General_Concept
as	O
described	O
below	O
.	O
</s>
<s>
Using	O
the	O
normalization	O
conventions	O
above	O
,	O
the	O
inverse	O
of	O
DCT-I	O
is	O
DCT-I	O
multiplied	O
by	O
2/	O
( N−1	O
)	O
.	O
</s>
<s>
The	O
inverse	O
of	O
DCT-IV	O
is	O
DCT-IV	O
multiplied	O
by	O
2/N	O
.	O
</s>
<s>
The	O
inverse	O
of	O
DCT-II	O
is	O
DCT-III	O
multiplied	O
by	O
2/N	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Like	O
for	O
the	O
DFT	B-Algorithm
,	O
the	O
normalization	O
factor	O
in	O
front	O
of	O
these	O
transform	O
definitions	O
is	O
merely	O
a	O
convention	O
and	O
differs	O
between	O
treatments	O
.	O
</s>
<s>
Combined	O
with	O
appropriate	O
factors	O
of	O
(	O
see	O
above	O
)	O
,	O
this	O
can	O
be	O
used	O
to	O
make	O
the	O
transform	O
matrix	O
orthogonal	B-Algorithm
.	O
</s>
<s>
Multidimensional	O
variants	O
of	O
the	O
various	O
DCT	B-General_Concept
types	O
follow	O
straightforwardly	O
from	O
the	O
one-dimensional	O
definitions	O
:	O
they	O
are	O
simply	O
a	O
separable	O
product	O
(	O
equivalently	O
,	O
a	O
composition	O
)	O
of	O
DCTs	B-General_Concept
along	O
each	O
dimension	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
two-dimensional	O
DCT-II	O
of	O
an	O
image	O
or	O
a	O
matrix	O
is	O
simply	O
the	O
one-dimensional	O
DCT-II	O
,	O
from	O
above	O
,	O
performed	O
along	O
the	O
rows	O
and	O
then	O
along	O
the	O
columns	O
(	O
or	O
vice	O
versa	O
)	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
2D	O
DCT-II	O
is	O
given	O
by	O
the	O
formula	O
(	O
omitting	O
normalization	O
and	O
other	O
scale	O
factors	O
,	O
as	O
above	O
)	O
:	O
</s>
<s>
The	O
inverse	O
of	O
a	O
multi-dimensional	O
DCT	B-General_Concept
is	O
just	O
a	O
separable	O
product	O
of	O
the	O
inverses	O
of	O
the	O
corresponding	O
one-dimensional	O
DCTs	B-General_Concept
(	O
see	O
above	O
)	O
,	O
e.g.	O
</s>
<s>
Technically	O
,	O
computing	O
a	O
two-	O
,	O
three	O
-	O
(	O
or	O
-multi	O
)	O
dimensional	O
DCT	B-General_Concept
by	O
sequences	O
of	O
one-dimensional	O
DCTs	B-General_Concept
along	O
each	O
dimension	O
is	O
known	O
as	O
a	O
row-column	O
algorithm	O
.	O
</s>
<s>
Owing	O
to	O
the	O
rapid	O
growth	O
in	O
the	O
applications	O
based	O
on	O
the	O
3-D	O
DCT	B-General_Concept
,	O
several	O
fast	O
algorithms	O
are	O
developed	O
for	O
the	O
computation	O
of	O
3-D	O
DCT-II	O
.	O
</s>
<s>
Vector-Radix	O
algorithms	O
are	O
applied	O
for	O
computing	O
M-D	O
DCT	B-General_Concept
to	O
reduce	O
the	O
computational	O
complexity	O
and	O
to	O
increase	O
the	O
computational	O
speed	O
.	O
</s>
<s>
To	O
compute	O
3-D	O
DCT-II	O
efficiently	O
,	O
a	O
fast	O
algorithm	O
,	O
Vector-Radix	O
Decimation	B-Algorithm
in	O
Frequency	O
(	O
VR	O
DIF	O
)	O
algorithm	O
was	O
developed	O
.	O
</s>
<s>
The	O
figure	O
to	O
the	O
adjacent	O
shows	O
the	O
four	O
stages	O
that	O
are	O
involved	O
in	O
calculating	O
3-D	O
DCT-II	O
using	O
VR	O
DIF	O
algorithm	O
.	O
</s>
<s>
The	O
second	O
stage	O
is	O
the	O
butterfly	B-Application
calculation	O
.	O
</s>
<s>
Each	O
butterfly	B-Application
calculates	O
eight	O
points	O
together	O
as	O
shown	O
in	O
the	O
figure	O
just	O
below	O
,	O
where	O
.	O
</s>
<s>
The	O
whole	O
3-D	O
DCT	B-General_Concept
calculation	O
needs	O
stages	O
,	O
and	O
each	O
stage	O
involves	O
butterflies	B-Application
.	O
</s>
<s>
The	O
whole	O
3-D	O
DCT	B-General_Concept
requires	O
butterflies	B-Application
to	O
be	O
computed	O
.	O
</s>
<s>
Each	O
butterfly	B-Application
requires	O
seven	O
real	O
multiplications	O
(	O
including	O
trivial	O
multiplications	O
)	O
and	O
24real	O
additions	O
(	O
including	O
trivial	O
additions	O
)	O
.	O
</s>
<s>
The	O
conventional	O
method	O
to	O
calculate	O
MD-DCT-II	O
is	O
using	O
a	O
Row-Column-Frame	O
(	O
RCF	O
)	O
approach	O
which	O
is	O
computationally	O
complex	O
and	O
less	O
productive	O
on	O
most	O
advanced	O
recent	O
hardware	O
platforms	O
.	O
</s>
<s>
of	O
multiplications	O
associated	O
with	O
the	O
3-D	O
DCT	B-General_Concept
VR	O
algorithm	O
is	O
less	O
than	O
that	O
associated	O
with	O
the	O
RCF	O
approach	O
by	O
more	O
than	O
40%	O
.	O
</s>
<s>
This	O
makes	O
the	O
3-D	O
DCT	B-General_Concept
VR	O
algorithm	O
more	O
efficient	O
and	O
better	O
suited	O
for	O
3-D	O
applications	O
that	O
involve	O
the	O
3-D	O
DCT-II	O
such	O
as	O
video	O
compression	O
and	O
other	O
3-D	O
image	B-Algorithm
processing	I-Algorithm
applications	O
.	O
</s>
<s>
Therefore	O
,	O
although	O
the	O
above	O
proposed	O
3-D	O
VR	O
algorithm	O
does	O
not	O
achieve	O
the	O
theoretical	O
lower	O
bound	O
on	O
the	O
number	O
of	O
multiplications	O
,	O
it	O
has	O
a	O
simpler	O
computational	O
structure	O
as	O
compared	O
to	O
other	O
3-D	O
DCT	B-General_Concept
algorithms	O
.	O
</s>
<s>
It	O
can	O
be	O
implemented	O
in	O
place	O
using	O
a	O
single	O
butterfly	B-Application
and	O
possesses	O
the	O
properties	O
of	O
the	O
Cooley	B-Algorithm
–	I-Algorithm
Tukey	I-Algorithm
FFT	I-Algorithm
algorithm	I-Algorithm
in	O
3-D	O
.	O
</s>
<s>
Hence	O
,	O
the	O
3-D	O
VR	O
presents	O
a	O
good	O
choice	O
for	O
reducing	O
arithmetic	O
operations	O
in	O
the	O
calculation	O
of	O
the	O
3-D	O
DCT-II	O
,	O
while	O
keeping	O
the	O
simple	O
structure	O
that	O
characterize	O
butterfly-style	O
Cooley	B-Algorithm
–	I-Algorithm
Tukey	I-Algorithm
FFT	I-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
The	O
image	O
to	O
the	O
right	O
shows	O
a	O
combination	O
of	O
horizontal	O
and	O
vertical	O
frequencies	O
for	O
an	O
two-dimensional	O
DCT	B-General_Concept
.	O
</s>
<s>
The	O
M-D	O
DCT-IV	O
is	O
just	O
an	O
extension	O
of	O
1-D	O
DCT-IV	O
on	O
to	O
dimensional	O
domain	B-Algorithm
.	O
</s>
<s>
We	O
can	O
compute	O
the	O
MD	O
DCT-IV	O
using	O
the	O
regular	O
row-column	O
method	O
or	O
we	O
can	O
use	O
the	O
polynomial	O
transform	O
method	O
for	O
the	O
fast	O
and	O
efficient	O
computation	O
.	O
</s>
<s>
The	O
main	O
idea	O
of	O
this	O
algorithm	O
is	O
to	O
use	O
the	O
Polynomial	O
Transform	O
to	O
convert	O
the	O
multidimensional	O
DCT	B-General_Concept
into	O
a	O
series	O
of	O
1-D	O
DCTs	B-General_Concept
directly	O
.	O
</s>
<s>
MD	O
DCT-IV	O
also	O
has	O
several	O
applications	O
in	O
various	O
fields	O
.	O
</s>
<s>
One	O
can	O
also	O
compute	O
DCTs	B-General_Concept
via	O
FFTs	O
combined	O
with	O
pre	O
-	O
and	O
post-processing	O
steps	O
.	O
</s>
<s>
In	O
general	O
,	O
methods	O
to	O
compute	O
DCTs	B-General_Concept
are	O
known	O
as	O
fast	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
FCT	O
)	O
algorithms	O
.	O
</s>
<s>
The	O
most	O
efficient	O
algorithms	O
,	O
in	O
principle	O
,	O
are	O
usually	O
those	O
that	O
are	O
specialized	O
directly	O
for	O
the	O
DCT	B-General_Concept
,	O
as	O
opposed	O
to	O
using	O
an	O
ordinary	O
FFT	O
plus	O
extra	O
operations	O
(	O
see	O
below	O
for	O
an	O
exception	O
)	O
.	O
</s>
<s>
However	O
,	O
even	O
"	O
specialized	O
"	O
DCT	B-General_Concept
algorithms	O
(	O
including	O
all	O
of	O
those	O
that	O
achieve	O
the	O
lowest	O
known	O
arithmetic	O
counts	O
,	O
at	O
least	O
for	O
power-of-two	O
sizes	O
)	O
are	O
typically	O
closely	O
related	O
to	O
FFT	O
algorithms	O
–	O
since	O
DCTs	B-General_Concept
are	O
essentially	O
DFTs	O
of	O
real-even	O
data	O
,	O
one	O
can	O
design	O
a	O
fast	O
DCT	B-General_Concept
algorithm	O
by	O
taking	O
an	O
FFT	O
and	O
eliminating	O
the	O
redundant	O
operations	O
due	O
to	O
this	O
symmetry	O
.	O
</s>
<s>
Algorithms	O
based	O
on	O
the	O
Cooley	B-Algorithm
–	I-Algorithm
Tukey	I-Algorithm
FFT	I-Algorithm
algorithm	I-Algorithm
are	O
most	O
common	O
,	O
but	O
any	O
other	O
FFT	O
algorithm	O
is	O
also	O
applicable	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Winograd	O
FFT	O
algorithm	O
leads	O
to	O
minimal-multiplication	O
algorithms	O
for	O
the	O
DFT	B-Algorithm
,	O
albeit	O
generally	O
at	O
the	O
cost	O
of	O
more	O
additions	O
,	O
and	O
a	O
similar	O
algorithm	O
was	O
proposed	O
by	O
for	O
the	O
DCT	B-General_Concept
.	O
</s>
<s>
Because	O
the	O
algorithms	O
for	O
DFTs	O
,	O
DCTs	B-General_Concept
,	O
and	O
similar	O
transforms	O
are	O
all	O
so	O
closely	O
related	O
,	O
any	O
improvement	O
in	O
algorithms	O
for	O
one	O
transform	O
will	O
theoretically	O
lead	O
to	O
immediate	O
gains	O
for	O
the	O
other	O
transforms	O
as	O
well	O
.	O
</s>
<s>
While	O
DCT	B-General_Concept
algorithms	O
that	O
employ	O
an	O
unmodified	O
FFT	O
often	O
have	O
some	O
theoretical	O
overhead	O
compared	O
to	O
the	O
best	O
specialized	O
DCT	B-General_Concept
algorithms	O
,	O
the	O
former	O
also	O
have	O
a	O
distinct	O
advantage	O
:	O
Highly	O
optimized	O
FFT	O
programs	O
are	O
widely	O
available	O
.	O
</s>
<s>
Specialized	O
DCT	B-General_Concept
algorithms	O
,	O
on	O
the	O
other	O
hand	O
,	O
see	O
widespread	O
use	O
for	O
transforms	O
of	O
small	O
,	O
fixed	O
sizes	O
such	O
as	O
the	O
DCT-II	O
used	O
in	O
JPEG	O
compression	O
,	O
or	O
the	O
small	O
DCTs	B-General_Concept
(	O
or	O
MDCTs	O
)	O
typically	O
used	O
in	O
audio	O
compression	O
.	O
</s>
<s>
(	O
Reduced	O
code	O
size	O
may	O
also	O
be	O
a	O
reason	O
to	O
use	O
a	O
specialized	O
DCT	B-General_Concept
for	O
embedded-device	O
applications	O
.	O
)	O
</s>
<s>
In	O
fact	O
,	O
even	O
the	O
DCT	B-General_Concept
algorithms	O
using	O
an	O
ordinary	O
FFT	O
are	O
sometimes	O
equivalent	O
to	O
pruning	O
the	O
redundant	O
operations	O
from	O
a	O
larger	O
FFT	O
of	O
real-symmetric	O
data	O
,	O
and	O
they	O
can	O
even	O
be	O
optimal	O
from	O
the	O
perspective	O
of	O
arithmetic	O
counts	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
type-II	O
DCT	B-General_Concept
is	O
equivalent	O
to	O
a	O
DFT	B-Algorithm
of	O
size	O
with	O
real-even	O
symmetry	O
whose	O
even-indexed	O
elements	O
are	O
zero	O
.	O
</s>
<s>
the	O
method	O
used	O
in	O
FFTPACK	B-Application
and	O
FFTW	B-Application
)	O
was	O
described	O
by	O
and	O
,	O
and	O
this	O
method	O
in	O
hindsight	O
can	O
be	O
seen	O
as	O
one	O
step	O
of	O
a	O
radix-4	O
decimation-in-time	O
Cooley	O
–	O
Tukey	O
algorithm	O
applied	O
to	O
the	O
"	O
logical	O
"	O
real-even	O
DFT	B-Algorithm
corresponding	O
to	O
the	O
DCT-II	O
.	O
</s>
<s>
If	O
the	O
subsequent	O
size	O
real-data	O
FFT	O
is	O
also	O
performed	O
by	O
a	O
real-data	O
split-radix	O
algorithm	O
(	O
as	O
in	O
)	O
,	O
then	O
the	O
resulting	O
algorithm	O
actually	O
matches	O
what	O
was	O
long	O
the	O
lowest	O
published	O
arithmetic	O
count	O
for	O
the	O
power-of-two	O
DCT-II	O
(	O
real-arithmetic	O
operations	O
)	O
.	O
</s>
<s>
So	O
,	O
there	O
is	O
nothing	O
intrinsically	O
bad	O
about	O
computing	O
the	O
DCT	B-General_Concept
via	O
an	O
FFT	O
from	O
an	O
arithmetic	O
perspective	O
–	O
it	O
is	O
sometimes	O
merely	O
a	O
question	O
of	O
whether	O
the	O
corresponding	O
FFT	O
algorithm	O
is	O
optimal	O
.	O
</s>
