Biomedical Engineering Reference
In-Depth Information
2.2
3D-Nonlinear Complex Diffusion Filter
The application of diffusion filters to image processing is based on the analogy to
physical diffusion processes, the rationale being to balance different concentrations
without creation or destruction of mass/energy [
36
].
Here, the concentration becomes the image intensity and the diffusion equa-
tion becomes:
@I
@t
Dr
.D
r
I/;
(1)
where the initial condition is given by the original image (I
t D0
D
I
0
), D is the
diffusion coefficient,
is the divergence operator.
Commonly, the diffusion coefficient is chosen to be dependent on the image
gradient [
24
], hence
r
is the gradient operator and
r
D
D
d.
jr
I
j
/;
(2)
where
denotes the magnitude.
Fernandez et al. [
11
] and Salinas et al. [
28
] shown that a nonlinear complex
diffusion filter can be successfully applied to remove the OCT speckle noise while
preserving image features. Both defined the diffusion coefficient as
jj
e
i
d.Im.I //
D
2
;
(3)
Im.I /
k
1
C
p
where Im.
/ is the imaginary value, i
D
1, k a threshold parameter and a
phase angle [
28
].
This choice relies on the fact that for small the imaginary part can be considered
as a smoothed second derivative of the initial signal factored by and time (t )[
13
]
Im.I/
lim
!0
D
tg
I
0
;
(4)
where is the laplacian, g is a gaussian and
the convolution operator.
This formulation does not require to compute expressly derivatives of the image,
thus avoiding the numerical instabilities due to noise at early stages and is a good
choice regarding edge preservation.
In the work herewith presented we extended this application from 2- to 3-
dimensions taking advantage of the volumetric information provided by the Cirrus
HD-OCT.
The extension to 3D is proposed by implementing a forward in time and centered
in space (FTCS) finite difference scheme, being the iterative update given by
t
l;j;m
,
I
.nC1/
l;j;m
I
.n/
D
.n/
l;j;m
h
I
.n/
l;j;m
Cr
h
D
.n/
l;j;m
r
h
I
.n/
D
l;j;m
C
(5)
