<s>
In	O
mathematics	O
,	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
,	O
named	O
for	O
Niels	O
Henrik	O
Abel	O
,	O
is	O
an	O
integral	B-Algorithm
transform	I-Algorithm
often	O
used	O
in	O
the	O
analysis	O
of	O
spherically	O
symmetric	O
or	O
axially	O
symmetric	O
functions	O
.	O
</s>
<s>
In	O
image	B-General_Concept
analysis	I-General_Concept
,	O
the	O
forward	O
Abel	B-Algorithm
transform	I-Algorithm
is	O
used	O
to	O
project	O
an	O
optically	O
thin	O
,	O
axially	O
symmetric	O
emission	O
function	O
onto	O
a	O
plane	O
,	O
and	O
the	O
inverse	O
Abel	B-Algorithm
transform	I-Algorithm
is	O
used	O
to	O
calculate	O
the	O
emission	O
function	O
given	O
a	O
projection	O
(	O
i.e.	O
</s>
<s>
In	O
absorption	O
spectroscopy	O
of	O
cylindrical	O
flames	O
or	O
plumes	O
,	O
the	O
forward	O
Abel	B-Algorithm
transform	I-Algorithm
is	O
the	O
integrated	O
absorbance	O
along	O
a	O
ray	O
with	O
closest	O
distance	O
y	O
from	O
the	O
center	O
of	O
the	O
flame	O
,	O
while	O
the	O
inverse	O
Abel	B-Algorithm
transform	I-Algorithm
gives	O
the	O
local	O
absorption	O
coefficient	O
at	O
a	O
distance	O
r	O
from	O
the	O
center	O
.	O
</s>
<s>
Abel	B-Algorithm
transform	I-Algorithm
is	O
limited	O
to	O
applications	O
with	O
axially	O
symmetric	O
geometries	O
.	O
</s>
<s>
For	O
more	O
general	O
asymmetrical	O
cases	O
,	O
more	O
general-oriented	O
reconstruction	O
algorithms	O
such	O
as	O
algebraic	B-Algorithm
reconstruction	I-Algorithm
technique	I-Algorithm
(	O
ART	O
)	O
,	O
maximum	O
likelihood	O
expectation	O
maximization	O
(	O
MLEM	O
)	O
,	O
filtered	B-Algorithm
back-projection	I-Algorithm
(	O
FBP	O
)	O
algorithms	O
should	O
be	O
employed	O
.	O
</s>
<s>
In	O
recent	O
years	O
,	O
the	O
inverse	O
Abel	B-Algorithm
transform	I-Algorithm
(	O
and	O
its	O
variants	O
)	O
has	O
become	O
the	O
cornerstone	O
of	O
data	O
analysis	O
in	O
photofragment-ion	O
imaging	O
and	O
photoelectron	O
imaging	O
.	O
</s>
<s>
Among	O
recent	O
most	O
notable	O
extensions	O
of	O
inverse	O
Abel	B-Algorithm
transform	I-Algorithm
are	O
the	O
"	O
onion	O
peeling	O
"	O
and	O
"	O
basis	O
set	O
expansion	O
"	O
(	O
BASEX	O
)	O
methods	O
of	O
photoelectron	O
and	O
photoion	O
image	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
F(y )	O
can	O
be	O
interpreted	O
as	O
the	O
projection	O
of	O
a	O
circularly	O
symmetric	O
function	O
f(r )	O
along	O
a	O
set	O
of	O
parallel	O
lines	O
of	O
sight	O
at	O
a	O
distance	O
y	O
from	O
the	O
origin	O
.	O
</s>
<s>
which	O
yields	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
of	O
f(r )	O
.	O
</s>
<s>
The	O
Abel	B-Algorithm
transform	I-Algorithm
may	O
be	O
extended	O
to	O
higher	O
dimensions	O
.	O
</s>
<s>
If	O
we	O
have	O
an	O
axially	O
symmetric	O
function	O
f(ρ,z )	O
,	O
where	O
ρ2	O
=	O
x2+y2	O
is	O
the	O
cylindrical	O
radius	B-Protocol
,	O
then	O
we	O
may	O
want	O
to	O
know	O
the	O
projection	O
of	O
that	O
function	O
onto	O
a	O
plane	O
parallel	O
to	O
the	O
z	O
axis	O
.	O
</s>
<s>
which	O
is	O
just	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
of	O
f(ρ,z )	O
in	O
ρ	O
and	O
y	O
.	O
</s>
<s>
which	O
is	O
again	O
,	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
of	O
f(r )	O
in	O
r	O
and	O
s	O
.	O
</s>
<s>
Now	O
substitute	O
this	O
into	O
the	O
inverse	O
Abel	B-Algorithm
transform	I-Algorithm
formula	O
:	O
</s>
<s>
The	O
Abel	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
f(r )	O
is	O
under	O
these	O
circumstances	O
again	O
given	O
by	O
:	O
</s>
<s>
The	O
extended	O
version	O
of	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
for	O
discontinuous	O
F	O
is	O
proven	O
upon	O
applying	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
to	O
shifted	O
,	O
continuous	O
,	O
and	O
it	O
reduces	O
to	O
the	O
classical	O
Abel	B-Algorithm
transform	I-Algorithm
when	O
.	O
</s>
<s>
If	O
has	O
more	O
than	O
a	O
single	O
discontinuity	O
,	O
one	O
has	O
to	O
introduce	O
shifts	O
for	O
any	O
of	O
them	O
to	O
come	O
up	O
with	O
a	O
generalized	O
version	O
of	O
the	O
inverse	O
Abel	B-Algorithm
transform	I-Algorithm
which	O
contains	O
n	O
additional	O
terms	O
,	O
each	O
of	O
them	O
corresponding	O
to	O
one	O
of	O
the	O
n	O
discontinuities	O
.	O
</s>
<s>
The	O
Abel	B-Algorithm
transform	I-Algorithm
is	O
one	O
member	O
of	O
the	O
FHA	B-Algorithm
cycle	I-Algorithm
of	O
integral	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
the	O
Hankel	B-Algorithm
transform	I-Algorithm
to	O
that	O
function	O
.	O
</s>
<s>
Abel	B-Algorithm
transform	I-Algorithm
can	O
be	O
viewed	O
as	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
of	O
an	O
isotropic	O
2D	O
function	O
f(r )	O
.	O
</s>
<s>
As	O
f(r )	O
is	O
isotropic	O
,	O
its	O
Radon	B-Algorithm
transform	I-Algorithm
is	O
the	O
same	O
at	O
different	O
angles	O
of	O
the	O
viewing	O
axis	O
.	O
</s>
<s>
Thus	O
,	O
the	O
Abel	B-Algorithm
transform	I-Algorithm
is	O
a	O
function	O
of	O
the	O
distance	O
along	O
the	O
viewing	O
axis	O
only	O
.	O
</s>
