<s>
Multiresolution	B-Algorithm
Fourier	I-Algorithm
Transform	I-Algorithm
is	O
an	O
integral	O
fourier	B-Algorithm
transform	I-Algorithm
that	O
represents	O
a	O
specific	O
wavelet-like	O
transform	O
with	O
a	O
fully	O
scalable	O
modulated	O
window	O
,	O
but	O
not	O
all	O
possible	O
translations	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
one	O
of	O
the	O
most	O
common	O
approaches	O
when	O
it	O
comes	O
to	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
and	O
signal	O
analysis	O
.	O
</s>
<s>
A	O
disadvantage	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
that	O
both	O
sine	O
and	O
cosine	O
function	O
are	O
defined	O
in	O
the	O
whole	O
time	O
plane	O
,	O
meaning	O
that	O
there	O
is	O
no	O
time	O
resolution	O
.	O
</s>
<s>
Certain	O
variants	O
of	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
such	O
as	O
Short	B-Algorithm
Time	I-Algorithm
Fourier	I-Algorithm
Transform	I-Algorithm
(	O
STFT	B-Algorithm
)	O
utilize	O
a	O
window	O
for	O
sampling	O
,	O
but	O
the	O
window	O
length	O
is	O
fixed	O
meaning	O
that	O
the	O
results	O
will	O
be	O
satisfactory	O
only	O
for	O
either	O
low	O
or	O
high	O
frequency	O
components	O
.	O
</s>
<s>
Fast	O
fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
is	O
used	O
often	O
because	O
of	O
its	O
computational	O
speed	O
,	O
but	O
shows	O
better	O
results	O
for	O
stationary	O
signals	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
wavelet	B-Algorithm
transform	I-Algorithm
can	O
improve	O
all	O
the	O
aforementioned	O
downsides	O
.	O
</s>
<s>
It	O
preserves	O
both	O
time	O
and	O
frequency	O
information	O
and	O
it	O
uses	O
a	O
window	O
of	O
variable	O
length	O
,	O
meaning	O
that	O
both	O
low	O
and	O
high	O
frequency	O
components	O
will	O
be	O
derived	O
with	O
higher	O
accuracy	O
than	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
wavelet	B-Algorithm
transform	I-Algorithm
also	O
shows	O
better	O
results	O
in	O
transient	O
states	O
.	O
</s>
<s>
Multiresolution	B-Algorithm
Fourier	I-Algorithm
Transform	I-Algorithm
leverages	O
the	O
advantageous	O
properties	O
of	O
the	O
wavelet	B-Algorithm
transform	I-Algorithm
and	O
uses	O
them	O
for	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
defined	O
representation	O
of	O
with	O
the	O
functions	O
and	O
is	O
called	O
the	O
B-wavelet	O
transform	O
,	O
and	O
is	O
used	O
to	O
define	O
the	O
integral	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
integral	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
then	O
be	O
defined	O
from	O
B-wavelet	O
transform	O
using	O
.	O
</s>
<s>
Now	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
represented	O
via	O
two	O
integral	B-Algorithm
wavelet	I-Algorithm
transforms	I-Algorithm
sampled	O
by	O
only	O
translation	O
parameter	O
:	O
</s>
<s>
Multiresolution	B-Algorithm
Fourier	I-Algorithm
Transform	I-Algorithm
is	O
applied	O
in	O
fields	O
such	O
as	O
image	O
and	O
audio	O
signal	O
analysis	O
,	O
curve	O
and	O
corner	O
extraction	O
,	O
and	O
edge	O
detection	O
.	O
</s>
