Biomedical Engineering Reference
In-Depth Information
4
A Semi-implicit Formulation
In the previous section we considered the approximated Euler-Lagrange equation (8)
associated to the minimization of the energy (6). This is a modelling approximation and
we can get rid of it. In fact, considering the original Euler-Lagrange equation associated
to the energy (6) we have (with abuse of notation for the diffusive term)
div
∇
+
λ
σ
2
u
−
[
u
−
r
(
u,f
)
f
]=0
(12)
|∇
u
|
with
r
(
u,f
)=
I
1
(
uf/σ
2
)
/I
0
(
uf/σ
2
)
. A rigorous treatment of equation (12) should
follow the multivalued formalism of (7).
Using again a gradient descent scheme we have to solve the parabolic problem:
∂t
=
div
∇
∂u
u
λ
σ
2
[
u
−
−
r
(
u,f
)
f
]
(13)
|∇
u
|
together with Neumann homogeneous boundary conditions
∂u/∂n
=0
and initial con-
dition
u
(0
,x
)=
u
0
(
x
)
. For comparison purposes we used
u
0
(
x
)=
u
(
x
)
in all numer-
0
ical tests.
Using forward finite difference for the temporal derivative in (13) and a semi-implicit
scheme where only the term depending on the ratio of the Bessel's functions is delayed,
results in the numerical scheme:
1+
τ
λ
σ
2
u
n
+1
=
u
n
+
τ
div
∇u
n
+1
|∇
+
λ
σ
2
r
(
u
n
,f
)
f
(14)
u
n
+1
|
where the diffusive term is (formally) exact and implicitly considered (compare with
(11)). Defining
β
=(
τλ
)
/σ
2
,
γ
=(1+
β
)
/τ
and
g
n
=
1
1+
β
u
n
+
β
1+
β
r
(
u
n
,f
)
f
(15)
we can write:
div
∇
+
1
γ
u
n
+1
−
g
n
=0
u
n
+1
−
(16)
|∇
u
n
+1
|
which is the Euler-Lagrange equation of the ROF energy functional ([7]):
E
n
(
u
)=
+
1
2
γ
g
n
)
2
dx
Ω
|
Du
|
(
u
−
(17)
Ω
for any positive integer
n>
0
, with (artificial) time
t
n
=
nτ
. Hence, at each gradient
descent step
τ
, we can solve a ROF problem associated to the minimization of the
energy (17) in the space
BV
(
Ω
)
[0
,
1]
. This problem is mathematically well-posed
and it can be numerically solved by very efficient methods, when formulated using well
known duality arguments as can be seen in [8], in order to apply this algorithm we must
introduce the discrete setting for the Total Variation Rician Denoising problem.
∩
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