<s>
Tomographic	B-Algorithm
reconstruction	I-Algorithm
is	O
a	O
type	O
of	O
multidimensional	O
inverse	O
problem	O
where	O
the	O
challenge	O
is	O
to	O
yield	O
an	O
estimate	O
of	O
a	O
specific	O
system	O
from	O
a	O
finite	O
number	O
of	O
projections	B-Algorithm
.	O
</s>
<s>
The	O
mathematical	O
basis	O
for	O
tomographic	B-Algorithm
imaging	I-Algorithm
was	O
laid	O
down	O
by	O
Johann	O
Radon	O
.	O
</s>
<s>
A	O
notable	O
example	O
of	O
applications	O
is	O
the	O
reconstruction	O
of	O
computed	O
tomography	B-Algorithm
(	O
CT	O
)	O
where	O
cross-sectional	O
images	O
of	O
patients	O
are	O
obtained	O
in	O
non-invasive	O
manner	O
.	O
</s>
<s>
Recent	O
developments	O
have	O
seen	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
and	O
its	O
inverse	O
used	O
for	O
tasks	O
related	O
to	O
realistic	O
object	O
insertion	O
required	O
for	O
testing	O
and	O
evaluating	O
computed	O
tomography	B-Algorithm
use	O
in	O
airport	O
security	O
.	O
</s>
<s>
This	O
article	O
applies	O
in	O
general	O
to	O
reconstruction	O
methods	O
for	O
all	O
kinds	O
of	O
tomography	B-Algorithm
,	O
but	O
some	O
of	O
the	O
terms	O
and	O
physical	O
descriptions	O
refer	O
directly	O
to	O
the	O
reconstruction	O
of	O
X-ray	B-Library
computed	O
tomography	B-Algorithm
.	O
</s>
<s>
The	O
projection	B-Algorithm
of	O
an	O
object	O
,	O
resulting	O
from	O
the	O
tomographic	B-Algorithm
measurement	O
process	O
at	O
a	O
given	O
angle	O
,	O
is	O
made	O
up	O
of	O
a	O
set	O
of	O
line	O
integrals	O
(	O
see	O
Fig	O
.	O
</s>
<s>
A	O
set	O
of	O
many	O
such	O
projections	B-Algorithm
under	O
different	O
angles	O
organized	O
in	O
2D	O
is	O
called	O
sinogram	O
(	O
see	O
Fig	O
.	O
</s>
<s>
In	O
X-ray	B-Library
CT	O
,	O
the	O
line	O
integral	O
represents	O
the	O
total	O
attenuation	O
of	O
the	O
beam	O
of	O
x-rays	B-Library
as	O
it	O
travels	O
in	O
a	O
straight	O
line	O
through	O
the	O
object	O
.	O
</s>
<s>
The	O
simplest	O
and	O
easiest	O
way	O
to	O
visualise	O
the	O
method	O
of	O
scanning	O
is	O
the	O
system	O
of	O
parallel	O
projection	B-Algorithm
,	O
as	O
used	O
in	O
the	O
first	O
scanners	O
.	O
</s>
<s>
For	O
this	O
discussion	O
we	O
consider	O
the	O
data	O
to	O
be	O
collected	O
as	O
a	O
series	O
of	O
parallel	O
rays	O
,	O
at	O
position	O
,	O
across	O
a	O
projection	B-Algorithm
at	O
angle	O
.	O
</s>
<s>
Therefore	O
,	O
generally	O
the	O
total	O
attenuation	O
of	O
a	O
ray	O
at	O
position	O
,	O
on	O
the	O
projection	B-Algorithm
at	O
angle	O
,	O
is	O
given	O
by	O
the	O
line	O
integral	O
:	O
</s>
<s>
This	O
function	O
is	O
known	O
as	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
(	O
or	O
sinogram	O
)	O
of	O
the	O
2D	O
object	O
.	O
</s>
<s>
represents	O
a	O
slice	O
of	O
the	O
2D	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
at	O
angle	O
.	O
</s>
<s>
Using	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
the	O
inverse	B-Algorithm
Radon	I-Algorithm
transform	I-Algorithm
formula	O
can	O
be	O
easily	O
derived	O
.	O
</s>
<s>
In	O
theory	O
,	O
the	O
inverse	O
Radon	B-Algorithm
transformation	I-Algorithm
would	O
yield	O
the	O
original	O
image	O
.	O
</s>
<s>
The	O
projection-slice	B-Algorithm
theorem	I-Algorithm
tells	O
us	O
that	O
if	O
we	O
had	O
an	O
infinite	O
number	O
of	O
one-dimensional	O
projections	B-Algorithm
of	O
an	O
object	O
taken	O
at	O
an	O
infinite	O
number	O
of	O
angles	O
,	O
we	O
could	O
perfectly	O
reconstruct	O
the	O
original	O
object	O
,	O
.	O
</s>
<s>
However	O
,	O
there	O
will	O
only	O
be	O
a	O
finite	O
number	O
of	O
projections	B-Algorithm
available	O
in	O
practice	O
.	O
</s>
<s>
Practical	O
reconstruction	O
algorithms	O
have	O
been	O
developed	O
to	O
implement	O
the	O
process	O
of	O
reconstruction	O
of	O
a	O
3-dimensional	O
object	O
from	O
its	O
projections	B-Algorithm
.	O
</s>
<s>
These	O
algorithms	O
are	O
designed	O
largely	O
based	O
on	O
the	O
mathematics	O
of	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
,	O
statistical	O
knowledge	O
of	O
the	O
data	O
acquisition	O
process	O
and	O
geometry	O
of	O
the	O
data	O
imaging	O
system	O
.	O
</s>
<s>
Assume	O
-projections	O
of	O
are	O
generated	O
at	O
equally	O
spaced	O
angles	O
,	O
each	O
sampled	O
at	O
the	O
same	O
rate	O
.	O
</s>
<s>
The	O
Discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
on	O
each	O
projection	B-Algorithm
will	O
yield	O
sampling	O
in	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
Combining	O
all	O
the	O
frequency-sampled	O
projections	B-Algorithm
would	O
generate	O
a	O
polar	O
raster	O
in	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
The	O
polar	O
raster	O
will	O
be	O
sparse	O
so	O
interpolation	O
is	O
used	O
to	O
fill	O
the	O
unknown	O
DFT	O
points	O
and	O
reconstruction	O
can	O
be	O
done	O
through	O
inverse	B-Algorithm
Discrete	I-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
concentric	O
square	O
raster	O
in	O
the	O
frequency	O
domain	O
can	O
be	O
obtained	O
by	O
changing	O
the	O
angle	O
between	O
each	O
projection	B-Algorithm
as	O
follow	O
:	O
</s>
<s>
In	O
practice	O
of	O
tomographic	B-Algorithm
image	O
reconstruction	O
,	O
often	O
a	O
stabilized	O
and	O
discretized	B-Algorithm
version	O
of	O
the	O
inverse	B-Algorithm
Radon	I-Algorithm
transform	I-Algorithm
is	O
used	O
,	O
known	O
as	O
the	O
filtered	O
back	O
projection	B-Algorithm
algorithm	O
.	O
</s>
<s>
where	O
is	O
the	O
angular	O
spacing	O
between	O
the	O
projections	B-Algorithm
and	O
is	O
radon	O
kernel	O
with	O
frequency	O
response	O
.	O
</s>
<s>
The	O
name	O
back-projection	O
comes	O
from	O
the	O
fact	O
that	O
1D	O
projection	B-Algorithm
needs	O
to	O
be	O
filtered	O
by	O
1D	O
Radon	O
kernel	O
(	O
back-projected	O
)	O
in	O
order	O
to	O
obtain	O
a	O
2D	O
signal	O
.	O
</s>
<s>
Reconstruction	O
using	O
back-projection	O
allows	O
better	O
resolution	O
than	O
interpolation	O
method	O
described	O
above	O
.	O
</s>
<s>
Let	O
be	O
the	O
number	O
of	O
projections	B-Algorithm
,	O
be	O
the	O
distortion	O
operator	O
for	O
th	O
projection	B-Algorithm
taken	O
at	O
an	O
angle	O
.	O
</s>
<s>
An	O
alternative	O
family	O
of	O
recursive	O
tomographic	B-Algorithm
reconstruction	I-Algorithm
algorithms	O
are	O
the	O
Algebraic	B-Algorithm
Reconstruction	I-Algorithm
Techniques	I-Algorithm
and	O
iterative	B-Algorithm
Sparse	I-Algorithm
Asymptotic	I-Algorithm
Minimum	I-Algorithm
Variance	I-Algorithm
.	O
</s>
<s>
Fan	O
beams	O
will	O
generate	O
series	O
of	O
line	O
integrals	O
,	O
not	O
parallel	O
to	O
each	O
other	O
,	O
as	O
projections	B-Algorithm
.	O
</s>
<s>
Back	O
projection	B-Algorithm
follows	O
a	O
similar	O
2	O
step	O
procedure	O
that	O
yields	O
reconstruction	O
by	O
computing	O
weighted	O
sum	O
back-projections	O
obtained	O
from	O
filtered	O
projections	B-Algorithm
.	O
</s>
<s>
Deep	O
learning	O
methods	O
are	O
widely	O
applied	O
to	O
image	O
reconstruction	O
nowadays	O
and	O
have	O
achieved	O
impressive	O
results	O
in	O
various	O
image	O
reconstruction	O
tasks	O
,	O
including	O
low-dose	O
denoising	O
,	O
sparse-view	O
reconstruction	O
,	O
limited	O
angle	O
tomography	B-Algorithm
and	O
metal	O
artifact	B-Algorithm
reduction	O
.	O
</s>
<s>
Artifact	B-Algorithm
reduction	O
using	O
the	O
U-Net	O
in	O
limited	O
angle	O
tomography	B-Algorithm
is	O
such	O
an	O
example	O
application	O
.	O
</s>
<s>
For	O
example	O
,	O
direct	O
image	O
reconstruction	O
from	O
projection	B-Algorithm
data	O
can	O
be	O
learnt	O
from	O
the	O
framework	O
of	O
filtered	B-Algorithm
back-projection	I-Algorithm
.	O
</s>
<s>
For	O
flexible	O
tomographic	B-Algorithm
reconstruction	I-Algorithm
,	O
open	O
source	O
toolboxes	O
are	O
available	O
,	O
such	O
as	O
PYRO-NN	O
,	O
TomoPy	O
,	O
CONRAD	O
,	O
ODL	O
,	O
the	O
ASTRA	O
toolbox	O
,	O
and	O
TIGRE	O
.	O
</s>
<s>
TomoPy	O
is	O
an	O
open-source	O
Python	O
toolbox	O
to	O
perform	O
tomographic	B-Algorithm
data	O
processing	O
and	O
image	O
reconstruction	O
tasks	O
at	O
the	O
Advanced	O
Photon	O
Source	O
at	O
Argonne	O
National	O
Laboratory	O
.	O
</s>
<s>
The	O
ASTRA	O
Toolbox	O
is	O
a	O
MATLAB	O
and	O
Python	O
toolbox	O
of	O
high-performance	O
GPU	O
primitives	O
for	O
2D	O
and	O
3D	O
tomography	B-Algorithm
,	O
from	O
2009	O
to	O
2014	O
developed	O
by	O
iMinds-Vision	O
Lab	O
,	O
University	O
of	O
Antwerp	O
and	O
since	O
2014	O
jointly	O
developed	O
by	O
iMinds-VisionLab	O
(	O
now	O
imec-VisionLab	O
)	O
,	O
UAntwerpen	O
and	O
CWI	O
,	O
Amsterdam	O
.	O
</s>
<s>
A	O
large	O
number	O
of	O
reconstruction	O
algorithms	O
are	O
available	O
through	O
TomoPy	O
and	O
the	O
ASTRA	O
toolkit	O
,	O
including	O
FBP	O
,	O
Gridrec	O
,	O
ART	B-Algorithm
,	O
SIRT	O
,	O
SART	O
,	O
BART	O
,	O
CGLS	O
,	O
PML	O
,	O
MLEM	O
and	O
OSEM	O
.	O
</s>
<s>
By	O
integrating	O
the	O
ASTRA	O
toolbox	O
in	O
the	O
TomoPy	O
framework	O
,	O
the	O
optimized	O
GPU-based	O
reconstruction	O
methods	O
become	O
easily	O
available	O
for	O
synchrotron	O
beamline	O
users	O
,	O
and	O
users	O
of	O
the	O
ASTRA	O
toolbox	O
can	O
more	O
easily	O
read	O
data	O
and	O
use	O
TomoPy	O
's	O
other	O
functionality	O
for	O
data	O
filtering	O
and	O
artifact	B-Algorithm
correction	O
.	O
</s>
<s>
Shown	O
in	O
the	O
gallery	O
is	O
the	O
complete	O
process	O
for	O
a	O
simple	O
object	O
tomography	B-Algorithm
and	O
the	O
following	O
tomographic	B-Algorithm
reconstruction	I-Algorithm
based	O
on	O
ART	B-Algorithm
.	O
</s>
