Page 321 - Biomedical Engineering and Design Handbook Volume 2, Applications

THE PRINCIPLES OF X-RAY COMPUTED TOMOGRAPHY 299

where the object contrast [Eq. (10.101)], or modulation, is
σ
σ
L () − L ()
σ
Γ () = max min (10.104)
0 r
σ
σ
L () + L ()
max min
and
σ
σ
L() max + L() min
L =
0
2
For the object contrast we have the condition Γ (s) = 1 for L(s) = 0.
0 r min
In seeking a relationship for the value of the MTF for a system, we consider the contribution from
a single component and offer the output as the input for the next component in the system chain.
Hence, the incoherent image I(x) of the rectangular wave grating, formed by convolution of the
object L(x) with the line-spread function h(x) of the first component, is given by
∞
x dx′
( ′
Ix() = ∫ −∞ h x L x − ′ ) (10.105)
(
)
If h(x) possesses symmetry, the modulation transfer function value M [(2k − 1)s], for the spatial
n
frequencies (2k − 1)s, according to Eq. (10.100) may be written as
+∞
π
−
∫ hx()cos[22 k 1 )σ x dx′
′
]
(
σ
−
M [(2 k 1 ) ] = −∞ +∞
n
∫ −∞ h( ′ xxdx) ′
Substituting the expression for the object function Eq. (10.103) into Eq. (10.105), we obtain
⎧ ⎪ 4 ∞ () k+1 M [(2 k −1 σ ⎫
) ]
⎪
−1
σ
π
Ix() = L H × 1 ⎨ + Γ 0 ( ) r∑ n coos[22k − ) 1 σ x ⎬ (10.106)
]
(
0
0
⎩ ⎪ π k= 1 2 k −1 ⎭ ⎪
where
+∞
0 ∫
H = h x dx′ = O 0]
′
()
[
−∞
The image contrast Γ (v) can be written in a similar manner to the object contrast [Eq. (10.104)], so
i r
that an overall rectangular-wave response, at the spatial frequency s = 1/d may be written as
Γ ()
σ
M() = i r
σ
r (10.107)
Γ ()
σ
0 r
where
σ
σ
I() − I()
σ
Γ () = max min
σ
i r
σ
I() + I()
max min
Substituting for L(x) , L(x) from Eq. (10.103) and I(x) , I(x) from Eq. (10.106), we find the
min max min max
overall response [Eq. (10.107)] becomes
∞ k+1
−1
4
M() = ∑ () M [(2k −1 ) ]
σ
σ
r π 2k −1 n
k=1
(10.108)
4 ⎡ 1 1 ⎤
σ
−
= ⎢ M []− M [ 3 ]σ + M [ 5 ]σ − . .. ⎥
π ⎣ n 3 n 5 n ⎦