PET Image Reconstruction - Correction Techniques in Emission Tomography - page 37

where again x 0 is the orthogonal projection of x on ? . Note that the integra-
tion runs over all lines through x.
The algorithm for a point source is in Figure 3.2.

0.8 0.6

0.35

0.7 0.3

0.5

0.25

0.6

0.4

0.2

0.5

0.15 0.3

0.4

0.1

0.2

0.05

0.3

0 0.1

0.2

−0.05

0

0.1

−0.1

0 −0.15 −0.1

0 50 100 150 200 0 50 100 150 200 0 20 40 60 80 100 120 140

(a) (b) (c)

FIGURE 3.2: Filtered backprojection of a point source. (a) Measured data
for a point source in one direction. (b) Same, with the Ram{Lak lter applied.
(c) Point source backprojection cross section.

3.2.3 Implementation: Resolution and complexity
Note that the inversion of the Radon transform is an ill{posed problem,
implying that small measurement errors can lead to arbitrarily large errors in
the reconstruction. However, the degree of ill{posedness for n = 2 is 1=2 on a
Sobolev scale, with 1 being the degree of ill{posedness of the rst derivative.
In other words, applying the Radon transform twice in a row would be only
as dicult, with respect to measurement errors, as taking the first derivative.
So, we can expect reasonable reconstructions even from mildly polluted data.
However, we still have to take ill{posedness into account. In the backpro-
jection algorithm, the source of ill{posedness is easily seen: high frequency-
oscillations in the data are multiplied by the absolute value of the frequency,
resulting in arbitrarily large errors. Consequently, the filter function is set to
zero away from a limit frequency . This can be done using a sharp cuto
(ramp) filter (cf. Ram-Lak filter) or in a smooth way, resulting in a selection
of filters with more or less comparable output. We thus arrive at the final
formulation for the filtered backprojection formula

f (x) = (R h)(x)
h(;) = ()jjRf(;)
() = 0 forjj >

where we choose () = 1, jj for the Ram{Lak lter.
The implementation is obvious; simply use the same algorithm as before,
but set the value of the Fourier transform to zero beyond .
However, this means that the output of our reconstruction formula is no