Biomedical Engineering Reference
In-Depth Information
In the case of a Gaussian noise model, (8.43) is a linear system. Because of
variations in instrumentation, the contrast levels of images taken at different
angles can vary. In such cases we estimate sets of such parameters, i.e.,
β
0
(
θ
i
)
and
β
1
(
θ
i
) for
i
=
1
,...,
N
.
To extend the domain to higher dimensions, we have
x
∈
IR
n
, and
S
⊂
IR
n
−
1
and the mapping
s
i
:IR
n
→
S
models the projective geometry of the imaging
system (e.g. orthographic, cone beam, or fan beam). Otherwise, the formulation
is the same as in 2D.
One important consideration is to model more complex models of density.
If
β
0
and
β
1
are smooth, scalar functions defined over the space in which the
surface model deforms and
g
is a binary function, the density model is
f
(
x
)
=
β
0
(
x
)
+
(
β
1
(
x
)
−
β
0
(
x
))
g
(
x
,
y
)
.
(8.44)
The first variation of the boundary is simply
N
d
x
dt
=
[
β
1
(
x
)
−
β
0
(
x
)]
e
i
(
x
)
n
(
x
)
.
(8.45)
i
=
1
Note that this formulation is different from that of Yu
et al.
[95], who address the
problem of reconstruction from noisy tomographic data using a single density
function
f
with a smoothing term that interacts with a set of deformable edge
models
. The edges models are surfaces, represented using level sets. In that
case the variational framework for deforming
requires differentiation of
f
across the edge, precisely where the proposed model exhibits (intentionally) a
discontinuity.
8.6.2.2 Prior
The analysis above maximizes the likelihood. For a full MAP estimation, we in-
clude a prior term. Because we are working with the logarithm of the likelihood,
the effect of the prior is additive:
dE
data
d
x
−
dE
prior
d
x
.
x
t
=−
(8.46)
Thus in addition to the noise model, we can incorporate some knowledge about
the kinds of shapes that give rise to the measurements. With appropriately fash-
ioned priors, we can push the solution toward desirable shapes or density val-
ues, or penalize certain shape properties, such as roughness or complexity. The
