
8.3
The Filtered-Backprojection Algorithm
379
By the convolution theorem (39) can be written as
Qe(t)= Γ Pe(t')Kt-t')dt' (44)
J
—
OO
Substituting (43) in (44) we get the following result for the values of the
filtered projections at the sampling points:
OO
Q
e
(m) = T Σ Km - kx)P
9
(kz) (45)
k = - oo
In practice each projection is of only finite extent. Suppose that each P
e
(kx)
is zero outside the index range k = 0, 1,..., Ν — 1. We may now write
the following two equivalent forms of (45):
Ν-
1
Q
e
(nz) = τ X h(m - kx)P
e
{kx\ η = 0, 1, ..., Ν - 1 (46a)
k =
0
or
Ν
— 1
Q
e
(nx) = τ £ h{kx)P
e
{m - kx\ η = 0, 1,..., Ν - 1 (46b)
Fig. 14 The transfer functio ...