Biomedical Engineering Reference
In-Depth Information
|
BV
=
J
(
u
)=
|
u
Ω
|
Du
|
(3)
being
Ω
|
the Total variation of
u
with
Du
its generalized gradient (a vector
bounded Radon measure). When
u
Du
|
W
1
,
1
(
Ω
)
we have
Ω
|
=
Ω
|∇
∈
Du
|
u
|
dx
.The
λ
parameter in (2) is a scale parameter tuning the model.
The data term
H
(
u,f
)
is a fitting functional which is nonnegative with respect to
u
for fixed
f
. To model Rician noise the form of
H
(
u,f
)
has been deduced in [6] in the
context of weighed diffusion tensor MR images. The Rician likelihood term is of the
form:
u
2
+
f
2
2
σ
2
uf
σ
2
log
f
σ
2
dx
H
(
u,f
)=
−
log
I
0
−
(4)
Ω
where
σ
is the standard deviation of the Rician noise of the data and
I
0
is the modified
zeroth-order
Bessel
function
of
the
first
kind.
Notice
that
the
constant
terms
and
Ω
log
f/σ
2
appearing in (4) do not affect the minimization prob-
lem. Dropping these terms (which do not allows to define the energy
H
(
u,
0)
corre-
sponding to a black image
f
(1
/
2
σ
2
)
2
2
f
≡
0
)wehave:
uf
σ
2
dx
1
2
σ
2
u
2
dx
H
(
u,f
)=
−
log
I
0
(5)
Ω
Ω
with
H
(
u,
0) = (1
/
2
σ
2
)
u
2
and
H
(0
,f
)=0
for any given
f ≥
0
. Using (2), (3) and
(5) the minimization problem is formulated as:
Fixed
λ
and
σ
and given a noisy image
f
L
∞
(
Ω
)
∈
∩
[0
,
1]
recover
u
∈
BV
(
Ω
)
∩
L
∞
(
Ω
)
∩
[0
,
1]
minimizing the energy:
uf
σ
2
dx
λ
Ω
λ
2
σ
2
u
2
dx
J
(
u
)+
λH
(
u,f
)=
|
Du
|
(
Ω
)+
−
log
I
0
(6)
Ω
Due to the fact that the functional in (3) (hence in (6)) is not differentiable at the origin
we introduce the subdifferential of
J
(
u
)
at a point
u
by
BV
(
Ω
)
∗
|
∂J
(
u
)=
{
p
∈
J
(
v
)
≥
J
(
u
)+
<p,v
−
u>
}
for all
v
BV
(
Ω
)
, to give a (weak and multivalued) meaning to the Euler-Lagrange
equation associated to the minimization problem (6). In fact the first order optimality
condition reads
∈
∂J
(
u
)+
λ∂
u
H
(
u,f
)
0
(7)
with (Gateaux) differential
∂
u
H
(
u,f
)=
u
σ
2
I
1
uf
σ
2
/I
0
uf
σ
2
f
σ
2
−
where
I
1
is the modified first-order Bessel function of the first kind and verifies ([10])
0
ξ>
0
. This model, first proposed in [4], differs from [6]
because of the geometric prior (the TV-based regularization term) which substitutes
≤
I
1
(
ξ
)
/I
0
(
ξ
)
<
1
,
∀
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