<s>
In	O
applied	O
mathematical	O
analysis	O
,	O
shearlets	B-Algorithm
are	O
a	O
multiscale	O
framework	O
which	O
allows	O
efficient	O
encoding	O
of	O
anisotropic	O
features	O
in	O
multivariate	O
problem	O
classes	O
.	O
</s>
<s>
Originally	O
,	O
shearlets	B-Algorithm
were	O
introduced	O
in	O
2006	O
for	O
the	O
analysis	O
and	O
sparse	O
approximation	O
of	O
functions	O
.	O
</s>
<s>
Shearlets	B-Algorithm
are	O
constructed	O
by	O
parabolic	O
scaling	B-Algorithm
,	O
shearing	O
,	O
and	O
translation	B-Algorithm
applied	O
to	O
a	O
few	O
generating	O
functions	O
.	O
</s>
<s>
At	O
fine	O
scales	O
,	O
they	O
are	O
essentially	O
supported	O
within	O
skinny	O
and	O
directional	O
ridges	O
following	O
the	O
parabolic	O
scaling	B-Algorithm
law	O
,	O
which	O
reads	O
length²	O
≈	O
width	O
.	O
</s>
<s>
Similar	O
to	O
wavelets	O
,	O
shearlets	B-Algorithm
arise	O
from	O
the	O
affine	O
group	O
and	O
allow	O
a	O
unified	O
treatment	O
of	O
the	O
continuum	O
and	O
digital	O
situation	O
leading	O
to	O
faithful	O
implementations	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
important	O
properties	O
of	O
shearlets	B-Algorithm
is	O
their	O
ability	O
to	O
provide	O
optimally	O
sparse	O
approximations	O
(	O
in	O
the	O
sense	O
of	O
optimality	O
in	O
)	O
for	O
cartoon-like	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
In	O
imaging	O
sciences	O
,	O
cartoon-like	B-Algorithm
functions	I-Algorithm
serve	O
as	O
a	O
model	O
for	O
anisotropic	O
features	O
and	O
are	O
compactly	O
supported	O
in	O
while	O
being	O
apart	O
from	O
a	O
closed	O
piecewise	B-Algorithm
singularity	O
curve	O
with	O
bounded	O
curvature	O
.	O
</s>
<s>
The	O
decay	O
rate	O
of	O
the	O
-error	O
of	O
the	O
-term	O
shearlet	B-Algorithm
approximation	O
obtained	O
by	O
taking	O
the	O
largest	O
coefficients	O
from	O
the	O
shearlet	B-Algorithm
expansion	O
is	O
in	O
fact	O
optimal	O
up	O
to	O
a	O
log-factor	O
:	O
</s>
<s>
Shearlets	B-Algorithm
are	O
to	O
date	O
the	O
only	O
directional	O
representation	O
system	O
that	O
provides	O
sparse	O
approximation	O
of	O
anisotropic	O
features	O
while	O
providing	O
a	O
unified	O
treatment	O
of	O
the	O
continuum	O
and	O
digital	O
realm	O
that	O
allows	O
faithful	O
implementation	O
.	O
</s>
<s>
Extensions	O
of	O
shearlet	B-Algorithm
systems	O
to	O
are	O
also	O
available	O
.	O
</s>
<s>
A	O
comprehensive	O
presentation	O
of	O
the	O
theory	O
and	O
applications	O
of	O
shearlets	B-Algorithm
can	O
be	O
found	O
in	O
.	O
</s>
<s>
In	O
comparison	O
to	O
curvelets	O
,	O
shearlets	B-Algorithm
use	O
shearings	O
instead	O
of	O
rotations	O
,	O
the	O
advantage	O
being	O
that	O
the	O
shear	O
operator	O
leaves	O
the	O
integer	O
lattice	O
invariant	O
in	O
case	O
,	O
i.e.	O
,	O
This	O
indeed	O
allows	O
a	O
unified	O
treatment	O
of	O
the	O
continuum	O
and	O
digital	O
realm	O
,	O
thereby	O
guaranteeing	O
a	O
faithful	O
digital	O
implementation	O
.	O
</s>
<s>
where	O
denotes	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
For	O
instance	O
,	O
one	O
can	O
choose	O
to	O
be	O
a	O
Meyer	O
wavelet	O
.	O
</s>
<s>
is	O
called	O
a	O
classical	O
shearlet	B-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
the	O
corresponding	O
discrete	O
shearlet	B-Algorithm
system	O
constitutes	O
a	O
Parseval	O
frame	O
for	O
consisting	O
of	O
bandlimited	O
functions	O
.	O
</s>
<s>
Another	O
example	O
are	O
compactly	O
supported	O
shearlet	B-Algorithm
systems	O
,	O
where	O
a	O
compactly	O
supported	O
function	O
can	O
be	O
chosen	O
so	O
that	O
forms	O
a	O
frame	O
for	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
all	O
shearlet	B-Algorithm
elements	O
in	O
are	O
compactly	O
supported	O
providing	O
superior	O
spatial	O
localization	O
compared	O
to	O
the	O
classical	O
shearlets	B-Algorithm
,	O
which	O
are	O
bandlimited	O
.	O
</s>
<s>
Although	O
a	O
compactly	O
supported	O
shearlet	B-Algorithm
system	O
does	O
not	O
generally	O
form	O
a	O
Parseval	O
frame	O
,	O
any	O
function	O
can	O
be	O
represented	O
by	O
the	O
shearlet	B-Algorithm
expansion	O
due	O
to	O
its	O
frame	O
property	O
.	O
</s>
<s>
One	O
drawback	O
of	O
shearlets	B-Algorithm
defined	O
as	O
above	O
is	O
the	O
directional	O
bias	O
of	O
shearlet	B-Algorithm
elements	O
associated	O
with	O
large	O
shearing	O
parameters	O
.	O
</s>
<s>
This	O
effect	O
is	O
already	O
recognizable	O
in	O
the	O
frequency	O
tiling	O
of	O
classical	O
shearlets	B-Algorithm
(	O
see	O
Figure	O
in	O
Section	O
#Examples	O
)	O
,	O
where	O
the	O
frequency	O
support	O
of	O
a	O
shearlet	B-Algorithm
increasingly	O
aligns	O
along	O
the	O
-axis	O
as	O
the	O
shearing	O
parameter	O
goes	O
to	O
infinity	O
.	O
</s>
<s>
This	O
causes	O
serious	O
problems	O
when	O
analyzing	O
a	O
function	O
whose	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
concentrated	O
around	O
the	O
-axis	O
.	O
</s>
<s>
The	O
associated	O
cone-adapted	O
discrete	O
shearlet	B-Algorithm
system	O
consists	O
of	O
three	O
parts	O
,	O
each	O
one	O
corresponding	O
to	O
one	O
of	O
these	O
frequency	O
domains	O
.	O
</s>
<s>
Finally	O
,	O
the	O
scaling	B-Algorithm
function	O
is	O
associated	O
with	O
the	O
low-frequency	O
part	O
.	O
</s>
