Biomedical Engineering Reference
In-Depth Information
density
β
0
and collection of solid objects with density
β
1
. We denote the (open)
set of points in those regions as
, the closure of that set, the surface, as
S
.
The projection of a 2D signal
f
(
x
,
y
) produces a sinogram given by the radon
transform as
+∞
+∞
p
(
s
,θ
)
=
f
(
x
,
y
)
δ
(
R
θ
x
−
s
)
d
x
,
(8.33)
−∞
−∞
where
R
θ
x
=
x
cos(
θ
)
+
y
sin(
θ
) is a rotation and projection of a point
x
=
(
x
,
y
)
onto the imaging plane associated with
θ
. The 3D formulation is the same, except
that the signal
f
(
x
,
y
,
z
) produces a collection of images. We denote the projec-
tion of the model, which includes estimates of the objects and the background, as
p
(
s
,θ
). For this work we denote the angles associated with a discrete set of pro-
jections as
θ
1
,...,θ
N
and denote the domain of each projection as
S
=
s
1
,...
s
M
.
Our strategy is to find
,
β
0
, and
β
1
by maximizing the likelihood.
If we assume the projection measurements are corrupted by independent
noise, the log likelihood of a collection of measurements for a specific shape
and density estimate is the probability of those measurements conditional on
the model,
ln
P
(
p
(
s
1
,θ
1
)
,
p
(
s
2
,θ
1
)
,...,
p
(
s
M
,θ
N
)
|
S,β
0
,β
1
)
=
ln
P
(
p
(
s
j
,θ
i
)
|
S,β
0
,β
1
)
.
(8.34)
i
j
We call the negative log likelihood the
error
and denote it
E
data
. Normally, the
probability density of a measurement is parameterized by the ideal value, which
gives
N
M
E
p
ij
,
p
ij
,
E
data
=
(8.35)
i
=
1
j
=
1
where
E
(
p
i
,
j
,
p
i
,
j
)
=−
ln
P
(
p
i
,
j
,
p
i
,
j
) is the error associated with a particular
point in the radon space, and
p
i
,
j
=
p
(
s
j
,θ
i
). In the case of independent Gaussian
noise,
E
is a quadratic, and the log likelihood results in a weighted least-squares
in the radon space. For all of our results, we use a Gaussian noise model. Next
we apply the object model, shown in Fig. 8.17, to the reconstruction of
f
.Ifwe
let
g
(
x
,
y
) be a binary inside-outside function on
, then we have the following
approximation to
f
(
x
,
y
):
f
(
x
,
y
)
≈
β
0
+
[
β
1
−
β
0
]
g
(
x
,
y
)
.
(8.36)
