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

: Ω → R

deﬁned over a domain Ω ⊂ R

, another image u(x) is

obtained as the solution of a nonlinear diffusion equa-

tion with initial and Neumann boundary conditions:

= div (g(k∇uk)∇u) x ∈ Ω t > 0 (1)

with u(x,0) = u

(x) when x ∈ Ω as initial conditions

and u

= 0 when x ∈ ∂Ω as boundary conditions,

with g(k∇uk) 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/k∇uk where k∇uk 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):

= g(k∇uk)(u

ξξ

+ u

ζζ



g(k∇uk) + g

′

(k∇uk) · k∇uk



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-

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

 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(k∇uk) =

k∇uk

, 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:

(t) = h

p−2



i+1

−U

i+1

−U

−

−U

i−1

−U

i−1



(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

(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

(s) =

(s+ε)

≤

with s ≥ 0,conforming to g

→ 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:







n+1



















(4)

where







+ 2rα

+ rβ

+ r

r(β

+ r) r

r(β

+ r) α

+ rα

+ rβ

+ r

r(α

+ r)

r(α

+ r) α

+ rα

+ 2rβ

+ r







(5)

where α = |U

−U

| 6= 0, β = |U

−U

| 6= 0 and r =

p−2

with k = ∆t and d = α

+2rα

+2rβ

+3r

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

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







which forces r < 2α

. 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

)



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

10n

iter

(6)

Where h = 1, n

iter

denotes the number of itera-

tions and α

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

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

= 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

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

= 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.

REFERENCES

Andreu, F., Caselles, V., Daz, J., and Mazn, J. (2002). Some

qualitative properties for the total variation ﬂow. Jour-

nal of Functional Analysis, 188:516–547.

Buades, A., Coll, B., and Morel, J. M. (2005). A review

of image denoising algorithms with a new one. Multi-

scale Modeling & Simulation, 4:490–530.

Catte, F., Lions, P. L., Morel, J. M., and Coll, T. (1992). Im-

age selective smoothing and edge detection by nonlin-

ear diffusion. SIAM Journal on Applied Mathematics,

29(1):182–193.

Keeling, S. L. and Stollberger, R. (2002). Nonlinear

anisotropic diffusion ﬁlters for wide range edge sharp-

ening. Inverse Problems, 18:175–190.

Kichenassamy, S. (1997). The perona-malik paradox. SIAM

Journal on Applied Mathematics, 57(5):1328–1342.

Perona, P. and Malik, J. (1990). Scale-space and edge detec-

tion using anisotropic diffusion. IEEE Transaction on

pattern analysis and machine intelligence, 12(7):629–

639.

Platero, C., Poncela, J. M., Gonzalez, P., Tobar, M. C., San-

guino, J., Asensio, G., and Santos, E. (2008). Liver

segmentation for hepatic lesions detection and char-

acterisation. Biomedical Imaging: From Nano to

Macro, 2008.ISBI 2008.5th IEEE International Sym-

posium on, pages 13–16.

Steidl, G., Weickert, J., Brox, T., Mrzek, P., and Welk, M.

(2004). On the equivalence of soft wavelet shrinkage,

total variation diffusion, total variation regularitation,

and sides. SIAM Journal on Applied Mathematics,

42(2):686–713.

Teboul, S., Blanc-Feraud, L., Aubert, G., and Barlaud, M.

(1998). Variational approach for edge-preserving reg-

ularization using couple pdes. IEEE Trasanctions on

Image Processing, 7:387–397.

Tsurkov, V. I. (2000). An analytical model of edge protec-

tion under noise suppression by anisotropic diffusion.

Journal of Computer and Systems Sciences Interna-

tional, 39(3):437–440.

Weickert, J. and Benhamouda, B. (1997). A semidiscrete

nonlinear scale-space theory and its relation to the

perona-malik paradox. Advances in Computer Vision,

pages 1–10.

Weickert, J., ter Haar Romeny, B., and Viergever, M. A.

(1998). Efﬁcient and reliable schemes for nonlinear

diffusion ﬁltering. IEEE Trasanctions on Image Pro-

cessing, 7(3):398–410.

Welk, M., Steidl, G., and Weickert, J. (2008). Locally an-

alytic schemes: A link between diffusion ﬁltering and

wavelet shrinkage. Applied and Computational Har-

monic Analysis, 24(2):195–224.

ANALYTICAL APPROXIMATIONS FOR NONLINEAR DIFFUSION TIME IN MULTISCALE EDGE

ENHANCEMENT