Digital Signal Processing Reference
In-Depth Information
y
f(x,y)
x
l
l
L(l,θ)
0
θ
Figure 1.7
Geometric representations of lines and projections.
x
(
s
)=
l
cos
θ
−
s
sin
θ
(1.5)
y
(
s
)=
l
sin
θ
+
s
cos
θ
(1.6)
Thus, the line integral of a function
f
(
x, y
)isgivenas
g
(
l, θ
)=
∞
−∞
f
(
x
(
s
)
,y
(
s
))
ds
(1.7)
For a fixed angle
θ
,
g
(
l, θ
) represents a projection, while for all
l
and
θ
it is called the
2-D radon transformation
of
f
(
x, y
).
The imaging equation for SPECT, ignoring the effect of the attenu-
ation term, is:
ϕ
(
l, θ
)=
∞
−∞
A
(
x
(
s
)
,y
(
s
))
ds
(1.8)
where
A
(
x
(
s
)
,y
(
s
)) describes the radioactivity within the 3-D body
and is the inverse 2-D Radon transform of
ϕ
(
l, θ
). Therefore, there is
no closed-form solution for attenuation correction in SPECT. SPECT
represents an important imaging technique by providing an accurate
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