PET Image Reconstruction - Correction Techniques in Emission Tomography - page 33

For a suciently smooth, fast{decaying function f in IR n we dene the
Radon transform as the integral over L(;s)

Z

Rf(;s) = f(x) dx

L(;s)

and its companion, the X{ray transform, as the line integral over the line in
direction passing through x

Z

f(x + t) dt; x 2 IR n ; x 2 ? :

Pf(;x) =

IR

Note that for n = 2, we have Rf(;s) = Pf( ? ;s), so both are equivalent.
For n = 2, both Radon and X{ray transforms are the mathematical realization
of the physical PET measurement process, for n = 3, X{ray transform gives
the correct model.
We define the backprojection operator R as

Z

(R g)(x) = g(;x) d

S n 1

or (R g)(x) the integral of g over all hyperplanes passing through x. Back-
projection is the simplest approach for doing inversion on measured data: in
order to recover the activity in a point x, simply average all measurements
over lines through x. We will analyze this approach later; a simple example is
in Figure 3.1.
Backprojection for P is defined accordingly for line integrals

Z

(P h)(x) = h(;x 0 )d

S n 1

where x 0 is the projection of x onto ? .

1.4

1.2

1

x

0.8

0.6

0.4

0.2

0

0 20 40 60 80 100 120 140

(a) (b) (c)

FIGURE 3.1: Unfiltered backprojection. (a) Integrate over all line integrals
through x. (b) Simple unfiltered backprojection of a point source. (c) Point
source backprojection cross section.