<s>
Rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
is	O
a	O
major	O
branch	O
of	O
information	O
theory	O
which	O
provides	O
the	O
theoretical	O
foundations	O
for	O
lossy	B-Algorithm
data	I-Algorithm
compression	I-Algorithm
;	O
it	O
addresses	O
the	O
problem	O
of	O
determining	O
the	O
minimal	O
number	O
of	O
bits	O
per	O
symbol	O
,	O
as	O
measured	O
by	O
the	O
rate	O
R	O
,	O
that	O
should	O
be	O
communicated	O
over	O
a	O
channel	O
,	O
so	O
that	O
the	O
source	O
(	O
input	O
signal	O
)	O
can	O
be	O
approximately	O
reconstructed	O
at	O
the	O
receiver	O
(	O
output	O
signal	O
)	O
without	O
exceeding	O
an	O
expected	O
distortion	O
D	O
.	O
</s>
<s>
Rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
gives	O
an	O
analytical	O
expression	O
for	O
how	O
much	O
compression	O
can	O
be	O
achieved	O
using	O
lossy	B-Algorithm
compression	I-Algorithm
methods	O
.	O
</s>
<s>
Many	O
of	O
the	O
existing	O
audio	O
,	O
speech	O
,	O
image	O
,	O
and	O
video	O
compression	B-General_Concept
techniques	I-General_Concept
have	O
transforms	O
,	O
quantization	B-Algorithm
,	O
and	O
bit-rate	O
allocation	O
procedures	O
that	O
capitalize	O
on	O
the	O
general	O
shape	O
of	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
functions	I-General_Concept
.	O
</s>
<s>
Rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
was	O
created	O
by	O
Claude	O
Shannon	O
in	O
his	O
foundational	O
work	O
on	O
information	O
theory	O
.	O
</s>
<s>
In	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
,	O
the	O
rate	O
is	O
usually	O
understood	O
as	O
the	O
number	O
of	O
bits	O
per	O
data	O
sample	O
to	O
be	O
stored	O
or	O
transmitted	O
.	O
</s>
<s>
In	O
the	O
most	O
simple	O
case	O
(	O
which	O
is	O
actually	O
used	O
in	O
most	O
cases	O
)	O
,	O
the	O
distortion	O
is	O
defined	O
as	O
the	O
expected	O
value	O
of	O
the	O
square	O
of	O
the	O
difference	O
between	O
input	O
and	O
output	O
signal	O
(	O
i.e.	O
,	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
)	O
.	O
</s>
<s>
However	O
,	O
since	O
we	O
know	O
that	O
most	O
lossy	B-Algorithm
compression	I-Algorithm
techniques	O
operate	O
on	O
data	O
that	O
will	O
be	O
perceived	O
by	O
human	O
consumers	O
(	O
listening	O
to	O
music	O
,	O
watching	O
pictures	O
and	O
video	O
)	O
the	O
distortion	O
measure	O
should	O
preferably	O
be	O
modeled	O
on	O
human	O
perception	O
and	O
perhaps	O
aesthetics	O
:	O
much	O
like	O
the	O
use	O
of	O
probability	O
in	O
lossless	B-Algorithm
compression	I-Algorithm
,	O
distortion	O
measures	O
can	O
ultimately	O
be	O
identified	O
with	O
loss	O
functions	O
as	O
used	O
in	O
Bayesian	O
estimation	O
and	O
decision	O
theory	O
.	O
</s>
<s>
In	O
audio	O
compression	O
,	O
perceptual	O
models	O
(	O
and	O
therefore	O
perceptual	O
distortion	O
measures	O
)	O
are	O
relatively	O
well	O
developed	O
and	O
routinely	O
used	O
in	O
compression	B-General_Concept
techniques	I-General_Concept
such	O
as	O
MP3	B-Application
or	O
Vorbis	B-Algorithm
,	O
but	O
are	O
often	O
not	O
easy	O
to	O
include	O
in	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
.	O
</s>
<s>
In	O
image	O
and	O
video	O
compression	O
,	O
the	O
human	O
perception	O
models	O
are	O
less	O
well	O
developed	O
and	O
inclusion	O
is	O
mostly	O
limited	O
to	O
the	O
JPEG	O
and	O
MPEG	O
weighting	O
(	O
quantization	B-Algorithm
,	O
normalization	B-Algorithm
)	O
matrix	O
.	O
</s>
<s>
In	O
the	O
definition	O
of	O
the	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
function	I-General_Concept
,	O
and	O
are	O
the	O
distortion	O
between	O
and	O
for	O
a	O
given	O
and	O
the	O
prescribed	O
maximum	O
distortion	O
,	O
respectively	O
.	O
</s>
<s>
When	O
we	O
use	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
as	O
distortion	O
measure	O
,	O
we	O
have	O
(	O
for	O
amplitude-continuous	O
signals	O
)	O
:	O
</s>
<s>
As	O
the	O
above	O
equations	O
show	O
,	O
calculating	O
a	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
function	I-General_Concept
requires	O
the	O
stochastic	O
description	O
of	O
the	O
input	O
in	O
terms	O
of	O
the	O
PDF	O
,	O
and	O
then	O
aims	O
at	O
finding	O
the	O
conditional	O
PDF	O
that	O
minimize	O
rate	O
for	O
a	O
given	O
distortion	O
.	O
</s>
<s>
The	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
function	I-General_Concept
of	O
any	O
source	O
is	O
known	O
to	O
obey	O
several	O
fundamental	O
properties	O
,	O
the	O
most	O
important	O
ones	O
being	O
that	O
it	O
is	O
a	O
continuous	O
,	O
monotonically	O
decreasing	O
convex	O
(	O
U	O
)	O
function	O
and	O
thus	O
the	O
shape	O
for	O
the	O
function	O
in	O
the	O
examples	O
is	O
typical	O
(	O
even	O
measured	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
functions	I-General_Concept
in	O
real	O
life	O
tend	O
to	O
have	O
very	O
similar	O
forms	O
)	O
.	O
</s>
<s>
One	O
important	O
feature	O
of	O
the	O
SLB	O
is	O
that	O
it	O
is	O
asymptotically	O
tight	O
in	O
the	O
low	O
distortion	O
regime	O
for	O
a	O
wide	O
class	O
of	O
sources	O
and	O
in	O
some	O
occasions	O
,	O
it	O
actually	O
coincides	O
with	O
the	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
function	I-General_Concept
.	O
</s>
<s>
The	O
Blahut	O
–	O
Arimoto	O
algorithm	O
,	O
co-invented	O
by	O
Richard	O
Blahut	O
,	O
is	O
an	O
elegant	O
iterative	O
technique	O
for	O
numerically	O
obtaining	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
functions	I-General_Concept
of	O
arbitrary	O
finite	O
input/output	O
alphabet	O
sources	O
and	O
much	O
work	O
has	O
been	O
done	O
to	O
extend	O
it	O
to	O
more	O
general	O
problem	O
instances	O
.	O
</s>
<s>
When	O
working	O
with	O
stationary	O
sources	O
with	O
memory	O
,	O
it	O
is	O
necessary	O
to	O
modify	O
the	O
definition	O
of	O
the	O
rate	B-General_Concept
distortion	I-General_Concept
function	I-General_Concept
and	O
it	O
must	O
be	O
understood	O
in	O
the	O
sense	O
of	O
a	O
limit	O
taken	O
over	O
sequences	O
of	O
increasing	O
lengths	O
.	O
</s>
<s>
If	O
we	O
assume	O
that	O
is	O
a	O
Gaussian	O
random	O
variable	O
with	O
variance	O
,	O
and	O
if	O
we	O
assume	O
that	O
successive	O
samples	O
of	O
the	O
signal	O
are	O
stochastically	O
independent	O
(	O
or	O
equivalently	O
,	O
the	O
source	O
is	O
memoryless	O
,	O
or	O
the	O
signal	O
is	O
uncorrelated	O
)	O
,	O
we	O
find	O
the	O
following	O
analytical	O
expression	O
for	O
the	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
function	I-General_Concept
:	O
</s>
<s>
Rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
tell	O
us	O
that	O
'	O
no	O
compression	O
system	O
exists	O
that	O
performs	O
outside	O
the	O
gray	O
area	O
 '	O
.	O
</s>
<s>
Nevertheless	O
,	O
even	O
at	O
unit	O
blocklengths	O
one	O
can	O
often	O
find	O
good	O
(	O
scalar	O
)	O
quantizers	B-Algorithm
that	O
operate	O
at	O
distances	O
from	O
the	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
function	I-General_Concept
that	O
are	O
practically	O
relevant	O
.	O
</s>
<s>
This	O
rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
function	I-General_Concept
holds	O
only	O
for	O
Gaussian	O
memoryless	O
sources	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
the	O
Gaussian	O
source	O
is	O
the	O
most	O
"	O
difficult	O
"	O
source	O
to	O
encode	O
:	O
for	O
a	O
given	O
mean	B-Algorithm
square	I-Algorithm
error	I-Algorithm
,	O
it	O
requires	O
the	O
greatest	O
number	O
of	O
bits	O
.	O
</s>
<s>
The	O
rate-distortion	B-General_Concept
function	I-General_Concept
of	O
a	O
Bernoulli	O
random	O
variable	O
with	O
Hamming	O
distortion	O
is	O
given	O
by	O
:	O
</s>
<s>
Plot	O
of	O
the	O
rate-distortion	B-General_Concept
function	I-General_Concept
for	O
:	O
</s>
<s>
Suppose	O
we	O
want	O
to	O
transmit	O
information	O
about	O
a	O
source	O
to	O
the	O
user	O
with	O
a	O
distortion	O
not	O
exceeding	O
D	O
.	O
Rate	B-General_Concept
–	I-General_Concept
distortion	I-General_Concept
theory	I-General_Concept
tells	O
us	O
that	O
at	O
least	O
bits/symbol	O
of	O
information	O
from	O
the	O
source	O
must	O
reach	O
the	O
user	O
.	O
</s>
