ANALYTICAL APPROXIMATIONS FOR NONLINEAR DIFFUSION
TIME IN MULTISCALE EDGE ENHANCEMENT
C. Platero, J. Sanguino, M. C. Tobar, J. M. Poncela and G. Asensio
Applied Bioengineering Group, Polytechnic University of Madrid, Spain
Keywords:
Nonlinear diffusion filter, Multiscale edge enhancement, Segmentation.
Abstract:
The image simplification, noise elimination and edge enhancement steps are all fundamental to segmentation
tasks. These processing techniques usually require the tuning of their control parameters; a procedure known
to be incompatible with automatic segmentation. The aim of this paper is to adopt a procedure, based on
nonlinear diffusion, that is capable of auto tuning by means of analytical expressions that relate diffusion
times to the gradient module. The numerical method and experimental results are shown in 1D, 2D and 3D.
1 INTRODUCTION
One of the objectives of image analysis is its segmen-
tation. Firstly, edge enhancement and smoothing of
the different areas of the image are essential, and rep-
resent the means by which to eliminate noise and in-
crease contrast. A review of these techniques can be
found in Buades et al(Buades et al., 2005). In this
paper, we will adopt procedures that are connected
to the nonlinear diffusion equation(Perona and Ma-
lik, 1990). Starting with an initial image u
0
: R
defined over a domain R
n
, another image u(x) is
obtained as the solution of a nonlinear diffusion equa-
tion with initial and Neumann boundary conditions:
u
t
= div (g(kuk)u) x t > 0 (1)
with u(x,0) = u
0
(x) when x as initial conditions
and u
n
= 0 when x ∂Ω as boundary conditions,
with g(kuk) further representing diffusivity, which
is a non-negative function usually decreasing. The
properties of the nonlinear diffusion filter are clearer
when set out in a new orthonormal basis in which
one of the axes is determined by the gradient vector
η = u/kuk where kuk 6= 0 , which together with
ξ and ζ form the curve/surface at a level perpendicu-
lar to η(Keeling and Stollberger, 2002; Teboul et al.,
1998; Weickert and Benhamouda, 1997):
u
t
= g(kuk)(u
ξξ
+ u
ζζ
)+
+
g(kuk) + g
(kuk) · kuk
u
ηη
(2)
where u
ηη
represents the second derivative of u in the
direction of η.
In the one-dimensional case, enhancement is
achieved by the absolute and dynamic increase of
edge slopes which imposes the condition that the
coefficient u
ηη
in (2) must be negative(Keeling and
Stollberger,2002). This conclusion can also be gener-
alised to 2D and 3D. The drawback to this approach,
however, is attributed to the fact that the continuous
enhancement model gives rise to an ill-posed prob-
lem(Kichenassamy, 1997; Catte et al., 1992). This
scenario, in the discrete instance, may change under
certain data conditions, givingrise to convergentsolu-
tions as referred to in (Catte et al., 1992) and analysed
in more detail in (Weickert and Benhamouda, 1997).
This paper proposes, primarily, a diffusivity with-
out tuning parameters which is capable of guarantee-
ing the balance between what has been smoothed and
what has been enhanced. The paper will look sub-
sequently at the theoretical research, offering an ana-
lytical solution to the nonlinear diffusion process be-
tween 3 one-dimensional pixels. We will then draw
conclusions that link diffusion time to enhancement
tasks. The extension to n-pixels will be experimen-
tally validated. Its application to the highest dimen-
sion will be thoroughly tested by applying operators
based on the orthogonal decomposition of the diver-
gence (Weickert et al., 1998). Finally, our filter shall
be used in different examples, allowing us to compare
the conclusions drawn and their subsequent applica-
78
Platero C., Sanguino J., C. Tobar M., M. Poncela J. and Asensio G. (2009).
ANALYTICAL APPROXIMATIONS FOR NONLINEAR DIFFUSION TIME IN MULTISCALE EDGE ENHANCEMENT.
In Proceedings of the Fourth International Conference on Computer Vision Theor y and Applications, pages 78-81
DOI: 10.5220/0001789200780081
Copyright
c
SciTePress
tion to medical images.
2 NONLINEAR DIFFUSION
FILTER WITHOUT CONTROL
PARAMETERS
One of the major features of TV diffusivity(Andreu
et al., 2002) is the absence of control parameters.
However, this advantage is negligible since it does
not allow us to increase enhancement of the image,
due to the fact that the coefficient
ηη
u in (2) is not
negative. More recently there has been a growing in-
terest in unbounded diffusion family which does not
require ad hoc adjustments (Tsurkov, 2000; Keeling
and Stollberger, 2002; Welk et al., 2008): g(kuk) =
1
kuk
p
, p 0. Where p = 0 represents linear diffusion,
p = 1 corresponding to TV and p = 2 to BFB (Bal-
ance Forward Backward)(Keeling and Stollberger,
2002). The condition p > 1 in these single-parameter
families of diffusion achieves the objective of en-
hancement. To find a continuous solution is diffi-
cult. One approximation to this type of equation can
be found in Tsurkov(Tsurkov, 2000) using techniques
that have already been applied to porous media mod-
els.
2.1 Semi-discrete Formulation
It is known that ill-posed problems in the continu-
ous case can be studied to a certain degree of success
in the semi-discrete and discrete cases (Kichenas-
samy, 1997; Weickert and Benhamouda, 1997).
Hence, spatial discretisation is performed on the one-
dimensional equation (1) using the proposed single-
parameter diffusion function. For this purpose, an ap-
proximation on finite differences based on the average
distance between pixels is used, which subsequently
gives rise to an autonomoussystem of ordinary differ-
ential equations:
˙
U
i
(t) = h
p2
U
i+1
U
i
|U
i+1
U
i
|
p
U
i
U
i1
|U
i
U
i1
|
p
(3)
with h = x,i = 2,...n 1. On carrying out the
operation we get an autonomous matrix ordinary dif-
ferential expression of the type
dU
dt
(t) = f(U(t)) =
A(U(t))U(t). The generalisation to highest dimen-
sions is direct (Weickert et al., 1998).
3 STUDY OF SEMI-ANALYTICAL
SOLUTIONS FOR NONLINEAR
DIFFUSION
3.1 Background
System resolution (3) has already been covered in
Steidl (Steidl et al., 2004) and subsequently in
Welk (Welk et al., 2008). It would appear that semi-
discrete formulation (3) of the equation (1), gives rise
to a singularity on the system for gradientvalues close
to zero. To avoid this we introduce a positive con-
stant ε almost zero. This regularisation leads us to
consider a new diffusion function g
ε
(s) =
1
(s+ε)
p
1
ε
p
with s 0,conforming to g
ε
g where ε 0.
It follows that an explicit Euler method places a
major restriction on the increase of time. One alterna-
tive is the two-pixel method (Welk et al., 2008). It has
been observed, however, that the semi-discrete regu-
larisation method is more effective for longer time in-
crement, whilst the two pixel method is more accurate
for shorter time increments.
One limitation of the semi-implicit method is the
need to manually tune the diffusion times for en-
hancement tasks.
3.2 Analytical Approach to Nonlinear
Diffusion Time
Let’s suppose, initially, a 1D signal comprised of just
three pixels in which the gradients are not zero. Ap-
plying the semi-implicit Euler method whose coeffi-
cient matrix can be reversed creates the expression:
U
n+1
1
U
n+1
2
U
n+1
3
=
1
d
B
U
n
1
U
n
2
U
n
3
(4)
where
B=
α
p
β
p
+ 2rα
p
+ rβ
p
+ r
2
r(β
p
+ r) r
2
r(β
p
+ r) α
p
β
p
+ rα
p
+ rβ
p
+ r
2
r(α
p
+ r)
r
2
r(α
p
+ r) α
p
β
p
+ rα
p
+ 2rβ
p
+ r
2
(5)
where α = |U
2
U
1
| 6= 0, β = |U
3
U
2
| 6= 0 and r =
kh
p2
with k = t and d = α
p
β
p
+2rα
p
+2rβ
p
+3r
2
.
It is interesting to observe the interaction between
the three pixels within the established dynamic. Ir-
respective of the initial pixel values, and a finite time,
the matrix coefficients are all equal to 1/3, directing
it towards linear diffusion with a total variation di-
minishing dynamic. However, the issue lies in how
to determine the nonlinear diffusion time so that it
produces diffusion between the low gradient mod-
ule pixels without transferring diffusion to the pix-
els that have a high value on the gradient module.
ANALYTICAL APPROXIMATIONS FOR NONLINEAR DIFFUSION TIME IN MULTISCALE EDGE
ENHANCEMENT
79
Without loss of overall applicability, in (4) it is im-
posed α β, so as to favour diffusion between pixel
2 and 3 and as a way to maintain the value of the first.
This evolution means that the matrix (5) tends to be:
1 0 0
0 1/2 1/2
0 1/2 1/2
which forces r < 2α
p
. This result sup-
ports the notion that it is possible to dimension the
nonlinear diffusiontime in such a way so as to smooth
over areas of low contrast and enhance areas of multi-
scale edges. The next section experimentally analyses
the validity of the conclusions drawn on n-pixels.
4 NUMERICAL METHODS,
EXPERIMENTS AND
APPLICATIONS
We firstly propose a comparative study between the
analytical method of (4) and the numerical method
using a semi-implicit scheme. On the basis of (3)
and by carrying out an explicit time discretisation, the
discrete dynamic is reflected in a matrix
I rA(U
k
)
which is defined as positive and tridiagonal.The in-
version of this matrix can be effectively solved using
the Thomas algorithm. For its implementation within
the scope of nonlinear diffusion see (Weickert et al.,
1998). The diffusivities have been regularised with
ε = 10
3
and [0,1] has been established for the signal
dynamic range.
With the aim of reducing the degrees of freedom
during the diffusion process, the spatial increase has
been fixed as a single-unit. In order to verify the en-
hancement conclusions duringthe experimentalwork,
we employed an order of magnitude lower than the
odds ratio over the diffusion time. Once the diffusion
time had been set, we observed that by using the nu-
merical methods implemented, the dynamic tends to
be consistent, using at least four iterations, thereby
resulting in a computational cost saving:
k
2α
p
th
10n
iter
(6)
Where h = 1, n
iter
denotes the number of itera-
tions and α
th
is the absolute value of difference be-
tween pixels, this being the slope threshold on which
enhancement is achieved. In the experiments carried
out, we have employed 10 iterations for 1D and 2D
and 5 iterations for 3D. The diffusion results with
three pixels using analytical expression of (4) and the
Thomas numerical scheme are practically identical
for any value of p.
In order to validate the conclusions drawn on the
analytical diffusion model with 3-pixels to n-pixels,
Figure 1: Comparative study of the dynamic in terms of p
with k = 20ms a) p = 2, b) p = 3
the analytical evolution of the diffusion of the central
pixel of (4) has been extended to all others. Both nu-
merical methods are tested with a synthetic border of
0 to 1, hence the value of α
th
shall be the unit, then
the iteration time for any p, shall be 20ms. White
noise has been added to the signal and the resolution
has been adjusted in order to increase the number of
pixels.
We observed that the analytical model and the
numerical model differ to the extent that the value
of p increases. For p lower than 3, the analytical
model sets the trend for the diffusion process, show-
ing that the conclusions drawn can be approximated
to n-pixels. The explanation is based on the equa-
tion (2) high diffusion coefficients to the extent that
p increases. Moreover, it confirms that the tridiago-
nal matrix inversion - despite increasing the computa-
tional cost - shows an interaction that is much wider
than a neighbourhood of only the closest neighbours.
The second objective is related to the election of
p. Although p must be greater than the unit in order to
achieve enhancement, what would be the best value?.
We have seen that the increase of p inhibits the stair-
case but also gives rise to a reduction in the signal
dynamic range. Furthermore, the validity of the equa-
tion (6) is based on the approximation between the
analytical and numerical model and on which basis
we conclude that a compromise value could be p = 3.
Extension to a highest dimension is carried out by
applying AOS (Additive Operator Splitting) (Weick-
ert et al., 1998). Using cameraman image contami-
nated with white noise and the use of α
th
= 0.3 hav-
ing been incorporated, we have observed that if p = 3
the noise disappears and all objects with a difference
greater than 0.3 in greyscales are enhanced. In this
image, the tower is feathered against the sky since it
VISAPP 2009 - International Conference on Computer Vision Theory and Applications
80
Figure 2: 2D filter with AOS a) Original, b) p = 2, k =
1.8ms, c) p = 3, k = 0.54ms
Figure 3: 3D Diffusion where p = 3, k = 0.04ms (original
on the left and on the right, processed)
does not exceed the threshold. If p = 2 the noise in-
cidence is higher and the contrast is accentuated to a
lesser degree.
This procedure has been applied to the multi-
phase segmentation of the liver based on magnetic
resonance (Platero et al., 2008). Due to the high vol-
ume of information, the number of iterations is re-
duced to 5. The figure below illustrates just six con-
secutive slices showing a hepatic lesion. We have se-
lected α
th
= 0.1. The slices show the increase in con-
trast of both the organ and the tumour.
5 CONCLUSIONS
The proposed objective is to determine a numerical
method that allows for images to be automatically
enhanced at a low computational cost. In this in-
stance, the method is based on the nonlinear diffu-
sion filter. We have selected a family of diffusivities
without control parameters. Based on the analytical
expression, obtained on the discrete evolution of 3
pixels through the resolution of a semi-implicit Eu-
ler method, we have experimentally verified stability,
consistency and enhancement properties. Using the
analytical model, we have determined the relationship
between the diffusion time and the gradient module.
Experimentally, the value of p = 3 has been consid-
ered the most suitable based on the convergence to the
analytical model presented, to the conclusions drawn
and the lower incidence of the staircase.
ACKNOWLEDGEMENTS
This work is supported DPI-2007-63654 project of
Spanish Ministry of Science.
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