<s>
In	O
the	O
mathematical	O
field	O
of	O
integral	O
geometry	O
,	O
the	O
Funk	B-Algorithm
transform	I-Algorithm
(	O
also	O
known	O
as	O
Minkowski	B-Algorithm
–	I-Algorithm
Funk	I-Algorithm
transform	I-Algorithm
,	O
Funk	O
–	O
Radon	B-Algorithm
transform	I-Algorithm
or	O
spherical	O
Radon	B-Algorithm
transform	I-Algorithm
)	O
is	O
an	O
integral	B-Algorithm
transform	I-Algorithm
defined	O
by	O
integrating	O
a	O
function	O
on	O
great	O
circles	O
of	O
the	O
sphere	O
.	O
</s>
<s>
It	O
is	O
closely	O
related	O
to	O
the	O
Radon	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
original	O
motivation	O
for	O
studying	O
the	O
Funk	B-Algorithm
transform	I-Algorithm
was	O
to	O
describe	O
Zoll	O
metrics	O
on	O
the	O
sphere	O
.	O
</s>
<s>
The	O
Funk	B-Algorithm
transform	I-Algorithm
is	O
defined	O
as	O
follows	O
.	O
</s>
<s>
The	O
Funk	B-Algorithm
transform	I-Algorithm
annihilates	O
all	O
odd	O
functions	O
,	O
and	O
so	O
it	O
is	O
natural	O
to	O
confine	O
attention	O
to	O
the	O
case	O
when	O
ƒ	O
is	O
even	O
.	O
</s>
<s>
In	O
that	O
case	O
,	O
the	O
Funk	B-Algorithm
transform	I-Algorithm
takes	O
even	O
(	O
continuous	O
)	O
functions	O
to	O
even	O
continuous	O
functions	O
,	O
and	O
is	O
furthermore	O
invertible	O
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
to	O
formulate	O
the	O
Funk	B-Algorithm
transform	I-Algorithm
in	O
a	O
manner	O
that	O
makes	O
it	O
invariant	O
under	O
the	O
special	O
linear	O
group	O
SL(3,R )	O
.	O
</s>
<s>
The	O
function	O
Fƒ:Λ2R3	O
→	O
R	O
agrees	O
with	O
the	O
Funk	B-Algorithm
transform	I-Algorithm
when	O
ƒ	O
is	O
the	O
degree	O
2	O
homogeneous	O
extension	O
of	O
a	O
function	O
on	O
the	O
sphere	O
and	O
the	O
projective	O
space	O
associated	O
to	O
Λ2R3	O
is	O
identified	O
with	O
the	O
space	O
of	O
all	O
circles	O
on	O
the	O
sphere	O
.	O
</s>
<s>
Alternatively	O
,	O
Λ2R3	O
can	O
be	O
identified	O
with	O
R3	O
in	O
an	O
SL(3,R )	O
-invariant	O
manner	O
,	O
and	O
so	O
the	O
Funk	B-Algorithm
transform	I-Algorithm
F	O
maps	O
smooth	O
even	O
homogeneous	O
functions	O
of	O
degree	O
2	O
on	O
R3\{0}	O
to	O
smooth	O
even	O
homogeneous	O
functions	O
of	O
degree	O
1	O
on	O
R3\{0}	O
.	O
</s>
<s>
The	O
Funk-Radon	B-Algorithm
transform	I-Algorithm
is	O
used	O
in	O
the	O
Q-Ball	O
method	O
for	O
Diffusion	B-Algorithm
MRI	I-Algorithm
introduced	O
by	O
.	O
</s>
