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8 Reconstruction
8.5 NOISE IN RECONSTRUCTED IMAGES
8.5.1 The Continuous Case
Consider the case where each projection, P
e
(t\ is corrupted by additive
noise ν
θ
(ί)· The measured projections,
P
0m
(O>
are now given by
PAO = Peit) + ν
θ
(ί) (48)
We will assume that the noise is a stationary zero-mean random process and
that its values are uncorrelated for any two rays in the system. Therefore,
£{ν
θι
(ί!)ν
θ2
(ί
2
)}
= S
0
<5(0i " θ
2
) S(t
x
- t
2
) (49)
The reconstruction from the measured projection data is obtained by first
filtering each projection:
ÖA0 = |°
SAw)
I
w
I
G(w)e
j2nwt
dw (50)
where S
e
m
(w) is the Fourier transform of Ρ AO
an
d G(w)
IS
th
e
smoothin ...