<s>
In	O
mathematics	O
,	O
a	O
wavelet	B-Algorithm
series	I-Algorithm
is	O
a	O
representation	O
of	O
a	O
square-integrable	B-Algorithm
(	O
real	O
-	O
or	O
complex-valued	O
)	O
function	O
by	O
a	O
certain	O
orthonormal	B-Algorithm
series	O
generated	O
by	O
a	O
wavelet	O
.	O
</s>
<s>
This	O
article	O
provides	O
a	O
formal	O
,	O
mathematical	O
definition	O
of	O
an	O
orthonormal	B-Algorithm
wavelet	I-Algorithm
and	O
of	O
the	O
integral	B-Algorithm
wavelet	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
A	O
function	O
is	O
called	O
an	O
orthonormal	B-Algorithm
wavelet	I-Algorithm
if	O
it	O
can	O
be	O
used	O
to	O
define	O
a	O
Hilbert	O
basis	O
,	O
that	O
is	O
a	O
complete	O
orthonormal	B-Algorithm
system	O
,	O
for	O
the	O
Hilbert	O
space	O
of	O
square	B-Algorithm
integrable	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
The	O
Hilbert	O
basis	O
is	O
constructed	O
as	O
the	O
family	O
of	O
functions	O
by	O
means	O
of	O
dyadic	B-Algorithm
translations	O
and	O
dilations	B-Algorithm
of	O
,	O
</s>
<s>
this	O
family	O
is	O
orthonormal	B-Algorithm
,	O
it	O
is	O
an	O
orthonormal	B-Algorithm
system	O
:	O
</s>
<s>
Such	O
a	O
representation	O
of	O
f	O
is	O
known	O
as	O
a	O
wavelet	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
This	O
implies	O
that	O
an	O
orthonormal	B-Algorithm
wavelet	I-Algorithm
is	O
self-dual	O
.	O
</s>
<s>
Here	O
,	O
is	O
called	O
the	O
binary	O
dilation	B-Algorithm
or	O
dyadic	B-Algorithm
dilation	B-Algorithm
,	O
and	O
is	O
the	O
binary	O
or	O
dyadic	B-Algorithm
position	O
.	O
</s>
<s>
The	O
fundamental	O
idea	O
of	O
wavelet	B-Algorithm
transforms	I-Algorithm
is	O
that	O
the	O
transformation	O
should	O
allow	O
only	O
changes	O
in	O
time	O
extension	O
,	O
but	O
not	O
shape	O
.	O
</s>
<s>
The	O
larger	O
the	O
extension	O
of	O
the	O
analysis	O
windows	B-Language
is	O
chosen	O
,	O
the	O
larger	O
is	O
the	O
value	O
of	O
.	O
</s>
<s>
Therefore	O
,	O
wavelet-transformation	O
contains	O
information	O
similar	O
to	O
the	O
short-time-Fourier-transformation	B-Algorithm
,	O
but	O
with	O
additional	O
special	O
properties	O
of	O
the	O
wavelets	O
,	O
which	O
show	O
up	O
at	O
the	O
resolution	O
in	O
time	O
at	O
higher	O
analysis	O
frequencies	O
of	O
the	O
basis	O
function	O
.	O
</s>
<s>
The	O
difference	O
in	O
time	O
resolution	O
at	O
ascending	O
frequencies	O
for	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
the	O
wavelet	B-Algorithm
transform	I-Algorithm
is	O
shown	O
below	O
.	O
</s>
<s>
This	O
shows	O
that	O
wavelet	B-Algorithm
transformation	I-Algorithm
is	O
good	O
in	O
time	O
resolution	O
of	O
high	O
frequencies	O
,	O
while	O
for	O
slowly	O
varying	O
functions	O
,	O
the	O
frequency	O
resolution	O
is	O
remarkable	O
.	O
</s>
<s>
Another	O
example	O
:	O
The	O
analysis	O
of	O
three	O
superposed	O
sinusoidal	O
signals	O
with	O
STFT	B-Algorithm
and	O
wavelet-transformation	O
.	O
</s>
<s>
Wavelet	O
compression	O
is	O
a	O
form	O
of	O
data	B-General_Concept
compression	I-General_Concept
well	O
suited	O
for	O
image	B-General_Concept
compression	I-General_Concept
(	O
sometimes	O
also	O
video	O
compression	O
and	O
audio	O
compression	O
)	O
.	O
</s>
<s>
Notable	O
implementations	O
are	O
JPEG	O
2000	O
,	O
DjVu	O
and	O
ECW	O
for	O
still	O
images	O
,	O
JPEG	O
XS	O
,	O
CineForm	B-Algorithm
,	O
and	O
the	O
BBC	O
's	O
Dirac	B-Application
.	O
</s>
<s>
The	O
goal	O
is	O
to	O
store	O
image	O
data	O
in	O
as	O
little	O
space	O
as	O
possible	O
in	O
a	O
file	B-Operating_System
.	O
</s>
<s>
Wavelet	O
compression	O
can	O
be	O
either	O
lossless	B-Algorithm
or	O
lossy	B-Algorithm
.	O
</s>
<s>
Using	O
a	O
wavelet	B-Algorithm
transform	I-Algorithm
,	O
the	O
wavelet	O
compression	O
methods	O
are	O
adequate	O
for	O
representing	O
transients	O
,	O
such	O
as	O
percussion	O
sounds	O
in	O
audio	O
,	O
or	O
high-frequency	O
components	O
in	O
two-dimensional	O
images	O
,	O
for	O
example	O
an	O
image	O
of	O
stars	O
on	O
a	O
night	O
sky	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
transient	O
elements	O
of	O
a	O
data	O
signal	O
can	O
be	O
represented	O
by	O
a	O
smaller	O
amount	O
of	O
information	O
than	O
would	O
be	O
the	O
case	O
if	O
some	O
other	O
transform	O
,	O
such	O
as	O
the	O
more	O
widespread	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
,	O
had	O
been	O
used	O
.	O
</s>
<s>
Discrete	O
wavelet	B-Algorithm
transform	I-Algorithm
has	O
been	O
successfully	O
applied	O
for	O
the	O
compression	O
of	O
electrocardiograph	O
(	O
ECG	O
)	O
signals	O
In	O
this	O
work	O
,	O
the	O
high	O
correlation	O
between	O
the	O
corresponding	O
wavelet	B-Algorithm
coefficients	I-Algorithm
of	O
signals	O
of	O
successive	O
cardiac	O
cycles	O
is	O
utilized	O
employing	O
linear	O
prediction	O
.	O
</s>
<s>
But	O
smooth	O
,	O
periodic	O
signals	O
are	O
better	O
compressed	O
using	O
other	O
methods	O
,	O
particularly	O
traditional	O
harmonic	O
analysis	O
in	O
the	O
frequency	O
domain	O
with	O
Fourier-related	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Vorbis	B-Algorithm
audio	B-Algorithm
codec	I-Algorithm
primarily	O
uses	O
the	O
modified	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
to	O
compress	O
audio	O
(	O
which	O
is	O
generally	O
smooth	O
and	O
periodic	O
)	O
,	O
however	O
allows	O
the	O
addition	O
of	O
a	O
hybrid	O
wavelet	O
filter	O
bank	O
for	O
improved	O
reproduction	O
of	O
transients	O
.	O
</s>
<s>
First	O
a	O
wavelet	B-Algorithm
transform	I-Algorithm
is	O
applied	O
.	O
</s>
<s>
This	O
produces	O
as	O
many	O
coefficients	O
as	O
there	O
are	O
pixels	B-Algorithm
in	O
the	O
image	O
(	O
i.e.	O
,	O
there	O
is	O
no	O
compression	O
yet	O
since	O
it	O
is	O
only	O
a	O
transform	O
)	O
.	O
</s>
<s>
This	O
principle	O
is	O
called	O
transform	B-Algorithm
coding	I-Algorithm
.	O
</s>
<s>
After	O
that	O
,	O
the	O
coefficients	O
are	O
quantized	B-Algorithm
and	O
the	O
quantized	B-Algorithm
values	O
are	O
entropy	B-Algorithm
encoded	I-Algorithm
and/or	O
run	B-Algorithm
length	I-Algorithm
encoded	I-Algorithm
.	O
</s>
<s>
From	O
the	O
perspective	O
of	O
image	B-General_Concept
compression	I-General_Concept
and	O
reconstruction	O
,	O
a	O
wavelet	O
should	O
meet	O
the	O
following	O
criteria	O
while	O
performing	O
image	B-General_Concept
compression	I-General_Concept
:	O
</s>
<s>
Wavelet	O
image	B-General_Concept
compression	I-General_Concept
system	O
involves	O
filters	O
and	O
decimation	O
,	O
so	O
it	O
can	O
be	O
described	O
as	O
a	O
linear	O
shift-variant	O
system	O
.	O
</s>
<s>
A	O
typical	O
wavelet	B-Algorithm
transformation	I-Algorithm
diagram	O
is	O
displayed	O
below	O
:	O
</s>
<s>
The	O
compression	O
and	O
reconstruction	O
system	O
generally	O
involves	O
low	O
frequency	O
components	O
,	O
which	O
is	O
the	O
analysis	O
filters	O
for	O
image	B-General_Concept
compression	I-General_Concept
and	O
the	O
synthesis	O
filters	O
for	O
reconstruction	O
.	O
</s>
<s>
Another	O
important	O
issue	O
for	O
image	B-General_Concept
compression	I-General_Concept
and	O
reconstruction	O
is	O
the	O
system	O
's	O
oscillatory	O
behavior	O
,	O
which	O
might	O
lead	O
to	O
severe	O
undesired	O
artifacts	O
in	O
the	O
reconstructed	O
image	O
.	O
</s>
<s>
So	O
far	O
we	O
have	O
discussed	O
about	O
one-dimension	O
transformation	O
of	O
the	O
image	B-General_Concept
compression	I-General_Concept
system	O
.	O
</s>
<s>
Finally	O
,	O
the	O
peak	O
to	O
first	O
sidelobe	O
ratio	O
and	O
the	O
average	O
second	O
sidelobe	O
of	O
the	O
overall	O
impulse	O
response	O
can	O
be	O
used	O
to	O
evaluate	O
the	O
wavelet	O
image	B-General_Concept
compression	I-General_Concept
performance	O
.	O
</s>
<s>
Wavelets	O
have	O
some	O
slight	O
benefits	O
over	O
Fourier	B-Algorithm
transforms	I-Algorithm
in	O
reducing	O
computations	O
when	O
examining	O
specific	O
frequencies	O
.	O
</s>
<s>
However	O
,	O
they	O
are	O
rarely	O
more	O
sensitive	O
,	O
and	O
indeed	O
,	O
the	O
common	O
Morlet	O
wavelet	O
is	O
mathematically	O
identical	O
to	O
a	O
short-time	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
using	O
a	O
Gaussian	O
window	B-Language
function	I-Language
.	O
</s>
<s>
The	O
exception	O
is	O
when	O
searching	O
for	O
signals	O
of	O
a	O
known	O
,	O
non-sinusoidal	O
shape	O
(	O
e.g.	O
,	O
heartbeats	O
)	O
;	O
in	O
that	O
case	O
,	O
using	O
matched	O
wavelets	O
can	O
outperform	O
standard	O
STFT/Morlet	O
analyses	O
.	O
</s>
<s>
The	O
wavelet	B-Algorithm
transform	I-Algorithm
can	O
provide	O
us	O
with	O
the	O
frequency	O
of	O
the	O
signals	O
and	O
the	O
time	O
associated	O
to	O
those	O
frequencies	O
,	O
making	O
it	O
very	O
convenient	O
for	O
its	O
application	O
in	O
numerous	O
fields	O
.	O
</s>
<s>
Synchro-squeezed	O
transform	O
can	O
significantly	O
enhance	O
temporal	O
and	O
frequency	O
resolution	O
of	O
time-frequency	O
representation	O
obtained	O
using	O
conventional	O
wavelet	B-Algorithm
transform	I-Algorithm
.	O
</s>
