PET Image Reconstruction - Correction Techniques in Emission Tomography - page 34

For x; 2 IR n we dene the Fourier transform f by
f() = (2) n=2 Z

f(x)e ix dx

IR n

and its inverse f = e f by
f(x) = (2) n=2 Z
IR n

f()e ix d:

The convolution of two functions f and g is defined by

Z
IR n f(y)g(xy) dy;
and we have the convolution theorem

(f g)(x) =

\ (f g)() = (2) n=2 f()g():

Its main use is visible in its corollary of deconvolution: Assume that instead of
a data function g(x), we can observe only a function h(x) as a convolution of
g with a known function (x). Then, at least mathematically, we can recover
g(x) via

g(x) = ^

h()=b ()

(x):

Essentially, the theorem says that if rather than a true signal we measure
a smoothed version of it, we can recover the original, provided the transfer
function is known.
Note, however, that this will work in practice only when jb j > for a
reasonable , which it usually is not, so this will usually not be implementable
out of the box.
Now we have all the tools for the projection slice theorem.

Theorem 3.2.1 (Fourier Slice) Assume that f is a fast decaying, smooth
function on IR n . Then, for 2S n1 , 2 IR,

Rf(;) = (2) (n1)=2 f()

where Rf is a Fourier transform with respect to the second variable. For the
X-ray transform, we have

Pf(;) = (2) 1=2 f()

where ? , and Pf is a (n1){dimensional Fourier transform in the second
variable with respect to ? .