Biomedical Engineering Reference
In-Depth Information
0
1 AU pinhole
2 AU pinhole
5 AU pinhole
−2
−4
t
−8
−10
−12
−14
−16
−18
−20
−60
−40
−20
0
20
40
60
Gradient
∇
x
i
(
x
)
Fig. 4.9
Histograms of the gradient along the
x
-direction of the specimen under three different
confocal pinhole settings. There is large difference between the histograms for 1 and 2 AU but
very less difference between 2 and 5 AU
is not differentiable in zero. An approach
to circumvent this problem is to instead consider the smoothed isotropic, discrete
definition (by approximating the signal derivative in terms of finite differences over
the sampling distance) as:
From a numerical perspective,
|∇o
(
x
)
|
|
ε
=
|∇o
(
x, y, z
)
|∇o
(
x
)
|
2
+
ε,
(4.25)
where
ε
is an arbitrarily small value (
<
10
−
3
). When this is used as an a priori model
for the object, we have the following smoothed TV regularization functional [
63
]:
)) =
λ
x
J
reg
(
o
(
x
|∇o
(
x
|
ε
.
)
(4.26)
For the
partition function
,
Z
λ
=
o∈O
exp
(
−λ
x
|∇o
(
x
)
|
ε
),tobefinite,we
restrict the possible values of
o
(
x
) so that the numerical gradient
|∇o
(
x
)
|
is also
bounded. The resulting functional
J
(
o
(
x
)) (Eq.
4.21
) is the sum of Eqs. (
4.16
)and
(
4.26
)
)) =
x
∈Ω
s
J
(
o
(
x
((
h ∗ o
)(
x
)+
b
(
x
−
i
(
x
)log((
h ∗ o
)(
x
)+
b
(
x
))
))
∈Ω
s
X
+
λ
x
∈Ω
s
|∇o
(
x
)
|
ε
.
(4.27)
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