Biomedical Engineering Reference
In-Depth Information
Ω
Expansion
Contraction
Model projection
Measured data
Figure 8.19: The model expands or contracts based on the difference in the
sinograms between the projected model and the measured data.
motion is
N
dE
data
d
x
=
(
β
0
−
β
1
)
x
t
=−
e
i
(
x
)
n
(
x
)
.
(8.42)
i
=
1
Thus, at a point
x
∈
S
, the
i
th projection has the effect of causing the surface
to expand or contract according to the difference between the projected model
values and the measured data at the point
s
i
(
x
), the projection of
x
(Fig. 8.19). The
surface motion prescribed by a collection of projections is the sum of motions
from the individual projections. In the case of continuous set of angles, the
surface motion at a point is proportional to the sinusoidal line integral on the
error sinogram
, which is
e
(
s
,θ
).
8.6.2.1 Density Parameter Estimation
The density parameters also affect the error term in Eq. (8.37). We propose
to update the estimate of the surface model iteratively, and at each iteration
we re-estimate the quantities
β
0
and
β
1
in such a way that the energy
E
data
is
minimized. Treating
as fixed, Eq. (8.37) has two unknowns,
β
0
and
β
1
, which
are computed from the following system:
∂
E
data
∂β
0
=
0
,
∂
E
data
∂β
1
=
0
.
(8.43)
