<s>
In	O
mathematics	O
,	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
the	O
integral	B-Algorithm
transform	I-Algorithm
which	O
takes	O
a	O
function	O
f	O
defined	O
on	O
the	O
plane	O
to	O
a	O
function	O
Rf	O
defined	O
on	O
the	O
(	O
two-dimensional	O
)	O
space	O
of	O
lines	O
in	O
the	O
plane	O
,	O
whose	O
value	O
at	O
a	O
particular	O
line	O
is	O
equal	O
to	O
the	O
line	O
integral	O
of	O
the	O
function	O
over	O
that	O
line	O
.	O
</s>
<s>
Radon	O
further	O
included	O
formulas	O
for	O
the	O
transform	O
in	O
three	O
dimensions	O
,	O
in	O
which	O
the	O
integral	O
is	O
taken	O
over	O
planes	O
(	O
integrating	O
over	O
lines	O
is	O
known	O
as	O
the	O
X-ray	B-Algorithm
transform	I-Algorithm
)	O
.	O
</s>
<s>
The	O
complex	O
analogue	O
of	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
known	O
as	O
the	O
Penrose	O
transform	O
.	O
</s>
<s>
The	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
widely	O
applicable	O
to	O
tomography	B-Algorithm
,	O
the	O
creation	O
of	O
an	O
image	O
from	O
the	O
projection	O
data	O
associated	O
with	O
cross-sectional	O
scans	O
of	O
an	O
object	O
.	O
</s>
<s>
If	O
a	O
function	O
represents	O
an	O
unknown	O
density	O
,	O
then	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
represents	O
the	O
projection	O
data	O
obtained	O
as	O
the	O
output	O
of	O
a	O
tomographic	B-Algorithm
scan	O
.	O
</s>
<s>
Hence	O
the	O
inverse	O
of	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
can	O
be	O
used	O
to	O
reconstruct	O
the	O
original	O
density	O
from	O
the	O
projection	O
data	O
,	O
and	O
thus	O
it	O
forms	O
the	O
mathematical	O
underpinning	O
for	O
tomographic	B-Algorithm
reconstruction	I-Algorithm
,	O
also	O
known	O
as	O
iterative	B-Algorithm
reconstruction	I-Algorithm
.	O
</s>
<s>
The	O
Radon	B-Algorithm
transform	I-Algorithm
data	O
is	O
often	O
called	O
a	O
sinogram	O
because	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
of	O
an	O
off-center	O
point	O
source	O
is	O
a	O
sinusoid	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
of	O
a	O
number	O
of	O
small	O
objects	O
appears	O
graphically	O
as	O
a	O
number	O
of	O
blurred	O
sine	O
waves	O
with	O
different	O
amplitudes	O
and	O
phases	O
.	O
</s>
<s>
The	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
useful	O
in	O
computed	O
axial	O
tomography	B-Algorithm
(	O
CAT	O
scan	O
)	O
,	O
barcode	O
scanners	O
,	O
electron	O
microscopy	O
of	O
macromolecular	O
assemblies	O
like	O
viruses	O
and	O
protein	O
complexes	O
,	O
reflection	O
seismology	O
and	O
in	O
the	O
solution	O
of	O
hyperbolic	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
The	O
Radon	B-Algorithm
transform	I-Algorithm
,	O
,	O
is	O
a	O
function	O
defined	O
on	O
the	O
space	O
of	O
straight	O
lines	O
by	O
the	O
line	O
integral	O
along	O
each	O
such	O
line	O
as:Concretely	O
,	O
the	O
parametrization	O
of	O
any	O
straight	O
line	O
with	O
respect	O
to	O
arc	O
length	O
can	O
always	O
be	O
written:where	O
is	O
the	O
distance	O
of	O
from	O
the	O
origin	O
and	O
is	O
the	O
angle	O
the	O
normal	O
vector	O
to	O
makes	O
with	O
the	O
-axis	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
quantities	O
can	O
be	O
considered	O
as	O
coordinates	O
on	O
the	O
space	O
of	O
all	O
lines	O
in	O
,	O
and	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
can	O
be	O
expressed	O
in	O
these	O
coordinates	O
by	O
:	O
More	O
generally	O
,	O
in	O
the	O
-dimensional	O
Euclidean	O
space	O
,	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
satisfying	O
the	O
regularity	O
conditions	O
is	O
a	O
function	O
on	O
the	O
space	O
of	O
all	O
hyperplanes	O
in	O
.	O
</s>
<s>
Thus	O
the	O
-dimensional	O
Radon	B-Algorithm
transform	I-Algorithm
may	O
be	O
rewritten	O
as	O
a	O
function	O
on	O
via	O
:	O
It	O
is	O
also	O
possible	O
to	O
generalize	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
still	O
further	O
by	O
integrating	O
instead	O
over	O
-dimensional	O
affine	O
subspaces	O
of	O
.	O
</s>
<s>
The	O
X-ray	B-Algorithm
transform	I-Algorithm
is	O
the	O
most	O
widely	O
used	O
special	O
case	O
of	O
this	O
construction	O
,	O
and	O
is	O
obtained	O
by	O
integrating	O
over	O
straight	O
lines	O
.	O
</s>
<s>
The	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
closely	O
related	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
We	O
define	O
the	O
univariate	O
Fourier	B-Algorithm
transform	I-Algorithm
here	O
as	O
:	O
For	O
a	O
function	O
of	O
a	O
-vector	O
,	O
the	O
univariate	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
:	O
For	O
convenience	O
,	O
denote	O
.	O
</s>
<s>
Thus	O
the	O
two-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
initial	O
function	O
along	O
a	O
line	O
at	O
the	O
inclination	O
angle	O
is	O
the	O
one	O
variable	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
(	O
acquired	O
at	O
angle	O
)	O
of	O
that	O
function	O
.	O
</s>
<s>
This	O
fact	O
can	O
be	O
used	O
to	O
compute	O
both	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
.	O
</s>
<s>
The	O
dual	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
a	O
kind	O
of	O
adjoint	O
to	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Beginning	O
with	O
a	O
function	O
g	O
on	O
the	O
space	O
,	O
the	O
dual	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
the	O
function	O
on	O
Rn	O
defined	O
by	O
:	O
The	O
integral	O
here	O
is	O
taken	O
over	O
the	O
set	O
of	O
all	O
hyperplanes	O
incident	O
with	O
the	O
point	O
,	O
and	O
the	O
measure	O
is	O
the	O
unique	O
probability	O
measure	O
on	O
the	O
set	O
invariant	O
under	O
rotations	O
about	O
the	O
point	O
.	O
</s>
<s>
Concretely	O
,	O
for	O
the	O
two-dimensional	O
Radon	B-Algorithm
transform	I-Algorithm
,	O
the	O
dual	O
transform	O
is	O
given	O
by	O
:	O
In	O
the	O
context	O
of	O
image	O
processing	O
,	O
the	O
dual	O
transform	O
is	O
commonly	O
called	O
back-projection	O
as	O
it	O
takes	O
a	O
function	O
defined	O
on	O
each	O
line	O
in	O
the	O
plane	O
and	O
'	O
smears	O
 '	O
or	O
projects	O
it	O
back	O
over	O
the	O
line	O
to	O
produce	O
an	O
image	O
.	O
</s>
<s>
The	O
Radon	B-Algorithm
transform	I-Algorithm
and	O
its	O
dual	O
are	O
intertwining	O
operators	O
for	O
these	O
two	O
differential	O
operators	O
in	O
the	O
sense	O
that	O
:	O
In	O
analysing	O
the	O
solutions	O
to	O
the	O
wave	O
equation	O
in	O
multiple	O
spatial	O
dimensions	O
,	O
the	O
intertwining	O
property	O
leads	O
to	O
the	O
translational	O
representation	O
of	O
Lax	O
and	O
Philips	O
.	O
</s>
<s>
In	O
the	O
two-dimensional	O
case	O
,	O
the	O
most	O
commonly	O
used	O
analytical	O
formula	O
to	O
recover	O
from	O
its	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
the	O
Filtered	B-Algorithm
Back-projection	I-Algorithm
Formula	O
or	O
Radon	O
Inversion	O
Formula	O
:	O
where	O
is	O
such	O
that	O
.	O
</s>
<s>
The	O
convolution	O
kernel	B-Algorithm
is	O
referred	O
to	O
as	O
Ramp	O
filter	O
in	O
some	O
literature	O
.	O
</s>
<s>
Intuitively	O
,	O
in	O
the	O
filtered	B-Algorithm
back-projection	I-Algorithm
formula	O
,	O
by	O
analogy	O
with	O
differentiation	O
,	O
for	O
which	O
,	O
we	O
see	O
that	O
the	O
filter	O
performs	O
an	O
operation	O
similar	O
to	O
a	O
derivative	O
.	O
</s>
<s>
where	O
is	O
the	O
previously	O
defined	O
adjoint	O
to	O
the	O
Radon	B-Algorithm
Transform	I-Algorithm
.	O
</s>
<s>
Compared	O
with	O
the	O
Filtered	B-Algorithm
Back-projection	I-Algorithm
method	O
,	O
iterative	B-Algorithm
reconstruction	I-Algorithm
costs	O
large	O
computation	O
time	O
,	O
limiting	O
its	O
practical	O
use	O
.	O
</s>
<s>
However	O
,	O
due	O
to	O
the	O
ill-posedness	O
of	O
Radon	O
Inversion	O
,	O
the	O
Filtered	B-Algorithm
Back-projection	I-Algorithm
method	O
may	O
be	O
infeasible	O
in	O
the	O
presence	O
of	O
discontinuity	O
or	O
noise	O
.	O
</s>
<s>
Iterative	B-Algorithm
reconstruction	I-Algorithm
methods	O
(	O
e.g.	O
</s>
<s>
iterative	B-Algorithm
Sparse	I-Algorithm
Asymptotic	I-Algorithm
Minimum	I-Algorithm
Variance	I-Algorithm
)	O
could	O
provide	O
metal	O
artefact	O
reduction	O
,	O
noise	O
and	O
dose	O
reduction	O
for	O
the	O
reconstructed	O
result	O
that	O
attract	O
much	O
research	O
interest	O
around	O
the	O
world	O
.	O
</s>
<s>
Explicit	O
and	O
computationally	O
efficient	O
inversion	O
formulas	O
for	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
and	O
its	O
dual	O
are	O
available	O
.	O
</s>
<s>
The	O
Radon	B-Algorithm
transform	I-Algorithm
in	O
dimensions	O
can	O
be	O
inverted	O
by	O
the	O
formula	O
:	O
where	O
,	O
and	O
the	O
power	O
of	O
the	O
Laplacian	O
is	O
defined	O
as	O
a	O
pseudo-differential	O
operator	O
if	O
necessary	O
by	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
:	O
For	O
computational	O
purposes	O
,	O
the	O
power	O
of	O
the	O
Laplacian	O
is	O
commuted	O
with	O
the	O
dual	O
transform	O
to	O
give	O
:	O
where	O
is	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
with	O
respect	O
to	O
the	O
s	O
variable	O
.	O
</s>
<s>
One	O
can	O
prove	O
directly	O
from	O
the	O
Fourier	B-Algorithm
slice	I-Algorithm
theorem	I-Algorithm
and	O
change	O
of	O
variables	O
for	O
integration	O
that	O
for	O
a	O
compactly	O
supported	O
continuous	O
function	O
of	O
two	O
variables	O
:	O
Thus	O
in	O
an	O
image	O
processing	O
context	O
the	O
original	O
image	O
can	O
be	O
recovered	O
from	O
the	O
'	O
sinogram	O
 '	O
data	O
by	O
applying	O
a	O
ramp	O
filter	O
(	O
in	O
the	O
variable	O
)	O
and	O
then	O
back-projecting	O
.	O
</s>
<s>
As	O
the	O
filtering	O
step	O
can	O
be	O
performed	O
efficiently	O
(	O
for	O
example	O
using	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
techniques	O
)	O
and	O
the	O
back	O
projection	O
step	O
is	O
simply	O
an	O
accumulation	O
of	O
values	O
in	O
the	O
pixels	O
of	O
the	O
image	O
,	O
this	O
results	O
in	O
a	O
highly	O
efficient	O
,	O
and	O
hence	O
widely	O
used	O
,	O
algorithm	O
.	O
</s>
<s>
In	O
algebraic	O
geometry	O
,	O
a	O
Radon	B-Algorithm
transform	I-Algorithm
(	O
also	O
known	O
as	O
the	O
Brylinski	B-Algorithm
–	I-Algorithm
Radon	I-Algorithm
transform	I-Algorithm
)	O
is	O
constructed	O
as	O
follows	O
.	O
</s>
<s>
for	O
the	O
universal	B-Algorithm
hyperplane	I-Algorithm
,	O
i.e.	O
,	O
H	O
consists	O
of	O
pairs	O
(	O
x	O
,	O
h	O
)	O
where	O
x	O
is	O
a	O
point	O
in	O
d-dimensional	O
projective	O
space	O
and	O
h	O
is	O
a	O
point	O
in	O
the	O
dual	O
projective	O
space	O
(	O
in	O
other	O
words	O
,	O
x	O
is	O
a	O
line	O
through	O
the	O
origin	O
in	O
(	O
d+1	O
)	O
-dimensional	O
affine	O
space	O
,	O
and	O
h	O
is	O
a	O
hyperplane	O
in	O
that	O
space	O
)	O
such	O
that	O
x	O
is	O
contained	O
in	O
h	O
.	O
</s>
