Biomedical Engineering Reference
In-Depth Information
dened in Fourier space by jjjj. The algorithm goes by the name of {ltered
layergram, where of course stands for the norm of .
While this is perfectly implementable, a variant of this algorithm is usually
employed. Instead of performing the filter step on the image delivered by
the backprojection, it can be pulled through the operator R
and performed
directly on the data. Thus, the inversion formula now reads
1
4
(R
h)(x)
f(x) =
where
h(;) = jjRf(;)
and the Fourier transform again refers to the second variable only.
Since this is the main theorem of the chapter, we take the liberty to give
the very short proof:
Z
1
4
(R
h)(x)
1
4
=
h(;x)d
S
1
Z
Z
1
h(;)e
ix
dd
=
p
2
2
S
1
IR
Z
Z
1
jjRf(;)e
ix
dd
p
=
2
2
S
1
IR
Z
Z
1
4
jjf()e
ix
dd
=
S
1
IR
Z
1
2
f()e
ix
d
=
IR
2
= f(x)
where we have used the Fourier slice theorem and changed variables from
polar to rectangular coordinates. The algorithm for filtered backprojection
thus reads:
1. For a xed measurement direction , compute the data's Fourier trans-
form f.
2. Multiply f() by jj=(4).
3. Compute the backprojection of the result.
An equivalent formulation can be derived for the X-ray transform in 3D.
Z
1
(4)
2
1
(4)
2
(P
h)(x)
S
2
h(;x
0
)dx =
f(x)
=
h(;)
= jjjj
[
(Rf)(;)
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