Biomedical Engineering Reference
In-Depth Information
Numerical Resolution for Rician Denoising
We consider a regular Cartesian grid of size
N
M
:
(
ih,jh
)
1
≤i≤N,
1
≤j≤M
where
h
denotes the size of the spacing. The matrix
(
u
i,j
)
represents a discrete image where
each pixel
u
i,j
is located in the correspondent node
(
ih,jh
)
. In what follows, we shall
choose
h
=1
because it only causes a rescaling of the energy. Henceforth we shall
drop the dependence of the mesh size and
u
h
=
u
.Let
X
=
×
N×M
be the vectorial
R
h
=
solutions space. We introduce the discrete gradient
∇
∇
defined as:
u
)
i,j
=
(
∂
x
u
)
i,j
=
u
i
+1
,j
−
u
i,j
(
∇
(18)
(
∂
y
u
)
i,j
u
i,j
+1
−
u
i,j
except at the boundaries
i
=
N
where
(
∂
x
u
)
N,j
=0
,and
j
=
M
with
(
∂
y
u
)
i,j
=0
.
Hence the (discrete)
operator is a linear map from
X
to
Y
=
X × X
. The discrete
version of the isotropic Total Variation semi-norm is:
∇
||
1
=
||∇
u
i.j
|
(
∇
u
)
i.j
|
,
with
=
((
u
)
i.j
)
2
+((
u
)
i.j
)
2
|
(
∇
u
)
i.j
|
∇
∇
The discrete energy for Rician denoising (6) reads as:
u
i,j
2
σ
2
−
u
i,j
f
i,j
σ
2
i.j
|
+
λ
i.j
(
∇
u
)
i.j
|
log
I
0
(19)
where the matrix
(
f
i,j
)
represents the discrete noisy image, with each pixel
f
i,j
lo-
cated at the node
(
i,j
)
. In the same way we can define the discrete version of the ROF
problem deduced in (17) which is (at a generic time step
t
n
)
i.j
|
+
1
2
β
u
i,j
−
g
i,j
2
(
∇
u
)
i,j
|
(20)
i.j
The algorithm presented in [8] is based in the dual formulation of the ROF problem
then if we endow the spaces
X
and
Y
with the standard Euclidean scalar product, the
adjoint operator of the discrete gradient
h
(see 18) denoted by
div
h
=
div is defined
∇
−
as
<
∇
u,p >
Y
=
−
<u,
div
p>
X
(21)
X
and (with
p
h
=
p
)say
p
=(
p
i,j
,p
i,j
)
for any
u
∈
∈
Y
, and it is given by the
following formulas:
(
div
p
)
i,j
=(
p
i,j
−
p
i−
1
,j
)+(
p
i,j
−
p
i,j−
1
)
1
.Theterm(
p
i,j
−
p
i−
1
,j
) is replaced with
p
i,j
if
i
=1
and with
for
2
≤
i,j
≤
N
−
p
i−
1
,j
if
i
=
N
, while the term (
p
i,j
−
p
i,j−
1
) is replaced with
p
i,j
if
j
=1
and with
−
p
i,j−
1
if
j
=
N
. The final algorithm for Rician Denoising is as follows:
−
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