Page 320 - Biomedical Engineering and Design Handbook Volume 2, Applications

298 DIAGNOSTIC EQUIPMENT DESIGN

transfer function O[u, v] (OTF). 22 This is a spatial frequency-dependent complex function with a
modulus component called the modulation transfer function M[u, v] (MTF) and a phase component
called the phase transfer function Φ[u, v] (PTF). The MTF is the ratio of the image-to-object mod-
ulation, while the PTF is a measure of the relative positional shift from object to image.
Taking a one-dimensional distribution for simplicity, we have
v
≡
[]e
ℑ{ ( )} O [] = Mv iΦ () (10.99)
hx
v
where M[v] and Φ(v) are MTF and PTF, respectively. It is customary to define a set of normalized
transfer functions by dividing by its zero spatial frequency value
+∞
O[] = ∫ −∞ h x dx
(
)
0
according to
ℑ {(
hx)}
v
hx)} =
Ov[] = =ℑ {( Mv e i (Φ v)
[
]
n +∞ n n (10.100)
∫ −∞ hx dx
()
where
hx()
hx() = +∞
n
∫ −∞ hx dx
()
is the normalized line-spread function and
Mv[]
Mv[] =
n
O[]
0
is the normalized modulation transfer function. If the line-spread function is symmetrical, there is no
phase shift so that Im O [v] = 0, M [v] = Re O [v], and Φ(v) = 0. A useful parameter in evaluating
n
n
n
the performance of a system is the contrast or modulation, defined by
Iv − Iv
()
()
Γ [] = max min
v
()
()
Iv + Iv (10.101)
max min
where I(v) max is the peak of the profile and I(v) min is the trough of the profile. Since the MTF is the
ratio of the image-to-object modulation, we may write, for sinusoids of period l and corresponding
frequency v = 1/l the expression
Γ v ()
Mv[] = image (10.102)
n
Γ v () object
It is a common practice to use the MTF to quantify the resolution capability of linear systems asso-
ciated with image processing. This is because if the individual MTFs are known for the components
of a system, the overall MTF is often simply their product. A natural resolution standard would com-
prise a continuous series of parallel lines with sine wave intensity profiles of increasing frequency.
However, this is difficult to produce in practice and it is usual to adopt a simpler pattern known as
the Sayce chart, which has a parallel line, or rectangular, wave grating with a decreasing spatial
23
period d and hence increasing spatial frequency s = 1/d. The rectangular pattern profile L(x) may
be expressed in terms of its Fourier series components [Eq. (10.34)], according to
⎧ ⎪ 4 ∞ ∞ () k+1 ⎫ ⎪
−
1
σ
π
Lx() = L ⎨ 1 + Γ 0 ( ) r∑ cos22 k −1 )σ ⎬ (10.103)
(
0
⎩ ⎪ π k=1 2 k −1 ⎭ ⎪