<s>
In	O
mathematics	O
,	O
the	O
projection-slice	B-Algorithm
theorem	I-Algorithm
,	O
central	B-Algorithm
slice	I-Algorithm
theorem	I-Algorithm
or	O
Fourier	B-Algorithm
slice	I-Algorithm
theorem	I-Algorithm
in	O
two	O
dimensions	O
states	O
that	O
the	O
results	O
of	O
the	O
following	O
two	O
calculations	O
are	O
equal	O
:	O
</s>
<s>
using	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
)	O
it	O
onto	O
a	O
(	O
one-dimensional	O
)	O
line	O
,	O
and	O
do	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
that	O
projection	O
.	O
</s>
<s>
Take	O
that	O
same	O
function	O
,	O
but	O
do	O
a	O
two-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
first	O
,	O
and	O
then	O
slice	O
it	O
through	O
its	O
origin	O
,	O
which	O
is	O
parallel	O
to	O
the	O
projection	O
line	O
.	O
</s>
<s>
F1	O
and	O
F2	O
are	O
the	O
1	O
-	O
and	O
2-dimensional	O
Fourier	B-Algorithm
transform	I-Algorithm
operators	O
mentioned	O
above	O
,	O
</s>
<s>
up	O
a	O
complete	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
that	O
density	O
.	O
</s>
<s>
In	O
addition	O
to	O
generalizing	O
to	O
N	O
dimensions	O
,	O
the	O
projection-slice	B-Algorithm
theorem	I-Algorithm
can	O
be	O
further	O
generalized	O
with	O
an	O
arbitrary	O
change	O
of	O
basis	O
.	O
</s>
<s>
The	O
projection-slice	B-Algorithm
theorem	I-Algorithm
is	O
easily	O
proven	O
for	O
the	O
case	O
of	O
two	O
dimensions	O
.	O
</s>
<s>
Using	O
a	O
shifted	O
line	O
(	O
in	O
y	O
)	O
gives	O
the	O
same	O
projection	O
and	O
therefore	O
the	O
same	O
1D	O
Fourier	B-Algorithm
transform	I-Algorithm
results	O
.	O
</s>
<s>
The	O
rotated	O
function	O
is	O
the	O
Fourier	O
pair	O
of	O
the	O
rotated	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
for	O
which	O
the	O
theorem	O
again	O
holds	O
.	O
</s>
<s>
which	O
is	O
just	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
p(x )	O
.	O
</s>
<s>
will	O
be	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
of	O
f(r )	O
.	O
</s>
<s>
of	O
f(r )	O
will	O
be	O
a	O
circularly	O
symmetric	O
function	O
given	O
by	O
the	O
zeroth-order	O
Hankel	B-Algorithm
transform	I-Algorithm
of	O
f(r )	O
,	O
which	O
will	O
therefore	O
also	O
represent	O
any	O
slice	O
through	O
the	O
origin	O
.	O
</s>
<s>
The	O
projection-slice	B-Algorithm
theorem	I-Algorithm
is	O
suitable	O
for	O
CT	O
image	O
reconstruction	O
with	O
parallel	O
beam	O
projections	O
.	O
</s>
