Biomedical Engineering Reference
In-Depth Information
with proper diffusion flows. By doing this, the following multiscale represen-
tation is proposed for 2D images, which can be straightforwardly extended to
3D. Let
B
:
D
⊂
×
+
→
S
m
−
1
, the image bright-
ness and chromaticity, respectively ((
S
m
−
1
) being the (
m
−
1)-dimensional unit
sphere), such that:
2
×
+
→
+
and
C
:
D
⊂
2
i
=
1
i
(
u
1
,
u
2
)
,
m
B
(
u
1
,
u
2
,
0)
=
(7.39)
1
B
(
u
1
,
u
2
,
0)
(
u
1
,
u
2
)
,
C
(
u
1
,
u
2
,
0)
=
(7.40)
and, at time
t
, the former will be given by the following anisotropic diffusion
flow:
B
u
1
u
1
B
u
2
−
2
B
u
1
B
u
2
B
u
1
u
2
+
B
u
2
u
2
B
u
1
1
/
3
1
+
∇
B
∂
B
∂
t
=
(7.41)
,
which is motivated by the affine-invariant denoising method proposed in [51,
58]. The above flow can be interpreted by observing that the level sets of the
brightness function have curvature
K
that can be written as (see expression
(7.21) also):
B
u
1
u
1
B
u
2
−
2
B
u
1
B
u
2
B
u
1
u
2
+
B
u
2
u
2
B
u
1
∇
B
K
=
(7.42)
.
3
Thus, the desired effect is to get an affine-invariant diffusion without smoothing
the brightness field across edges (see [51, 57] for more details).
The chromaticity is the solution of the variational problem given by:
D
∇
C
p
du
1
du
2
,
min
(7.43)
2
→
S
m
−
1
C
:
where
p
≥
1 and
∇
C
is:
m
∂
C
i
∂
u
1
2
1
/
2
2
∂
C
i
∂
u
2
∇
C
=
(7.44)
+
.
i
=
1
The scheme for the chromaticity comes from the theory of harmonic maps in
liquid crystals [66]. The optimization problem can be solved by Euler-Lagrange
equations or even in the content of weak solution. In [67] some results are
reported for 2D images and open questions related to the mathematical formu-
lation are presented.
