2.2
Filtered Back projection
Recall
f
(
x
)=
1
2
π
°
π
0
A
(
Rf
)(
x
·
ω,ω
)
dω,
where
A
(
Rf
)(
t,ω
)=
1
2
π
°
∞
−∞
±
Rf
(
r, ω
)
e
irt

r

dr
=
−
i
H
∂
t
(
Rf
)(
t,ω
)
.
Here
H
is the Hilbert transform. We note that
A
(
Rf
)(
t,ω
) is a onedimensional shift invari
ant Flter which does not depend on
ω
. This Flter can be built into the scanner once the data
{
Rf
(
t,ω
k
)
}
are collected for each
ω
k
, giving
A
(
Rf
)(
t,ω
k
), and then
f
can be approximated
with the formula
f
(
x
n
)
≈
∆
θ
2
π
M
²
k
=0
A
(
Rf
)((
x
n
·
ω
k
)
,ω
k
)
,
(5)
which requires
O
(
M
) operations for each
n
=1
,
···
K
. In the end the total complexity is
O
(
MK
). Discretely, suppose
A
(
Rf
)(
t
j
,ω
k
) is known, then in computing (5), interpolation
is need to compute
A
(
Rf
)((
x
n
·
ω
k
)
,ω
k
).
There is one other draw back to computing
A
(
Rf
)(
t,ω
): Suppose that there is noise
measuring
Rf
, that is we measure
±
Rf
+ˆ
v
,where
v
is additive noise. In computing
A
(
Rf
),
the noise is ampliFed by ˆ
v
(
r
)

r

. To overcoming this problem, we need to apply a smoothing
Flter to
A
(
Rf
).
3
RamLak Filters and other approaches
In the Flter
A
, we would like to replace

r

with a
A
(
r
)

r

where
A
(
r
)
→
0as

r
→∞
.L
e
t
φ
be the function such that
ˆ
φ
(
r
)=
A
(
r
)

r

. DeFne
A
φ
(
Rf
)(
t,ω
)=
1
2
π
°
∞
−∞
±
Rf
(
r, ω
)
ˆ
φ
(
r
)
e
irt
dr.
Then we can rewrite
A
φ
((
Rf
)(
t,ω
)=
°
∞
−∞
Rf
(
r, ω
)
φ
(
t
−
r
)
dr.
Then
f
is approximated by
A
φ
(
Rf
)(
t,ω
)as
f
(
x
)
≈
f
φ
(
x
)=
1
2
π
°
π
0
A
φ
(
Rf
)((
x
·
ω
)
,ω
)
dθ.
Suppose
ω
are sampled at
Ω
M
=
³
ω
k
=
ω
(
k
∆
θ
):
k
=1
,
···
,M,
∆
θ
=
π
M
+1
´