MEMORY-BASED SPECKLE REDUCING ANISOTROPIC

DIFFUSION

Walid Ibrahim and Mahmoud R. El-Sakka

Computer Science Department, The university of Western Ontario, London, Ontario, N6A-5B7, Canada

Keywords: Diffusion, Anisotropic diffusion, Diffusion reaction, Speckle reduction, Edge detection, Instantaneous

coefficient of variation.

Abstract: Diffusion filters are usually modelled as partial differential equations (PDEs) and used to reduce image

noise without affecting the image main features. However, they have a drawback of broadening object

boundaries and dislocating edges. Such drawbacks limit the ability of diffusion techniques applied to image

processing. Yu and Acton. introduced the speckle reducing anisotropic diffusion (SRAD) to reduce speckle

noise from ultrasound (US) and synthetic aperture radar (SAR) images. Incorporating the instantaneous

coefficient of variation (ICOV) as the diffusion coefficient and edge detector, SRAD gives significantly

enhanced images where most of the speckle noise is reduced. Yet, SRAD still faces the same problem of

ordinary diffusion filters where the boundary broadening and edge dislocation affect its overall

performance. In this paper, we introduce a novel approach to the diffusion filtering process, where a

memory term is introduced as a reaction-diffusion term. By applying our new memory-based diffusion to

SRAD, we significantly reduced the boundary broadening and edge dislocation effect and enhanced the

diffusion process itself. Experimental results showed that the performance of our proposed memory-based

scheme surpass other diffusion filters like normal SRAD and Perona-Malik filter as well as various adaptive

linear de-noising filters.

1 INTRODUCTION

Diffusion has been widely used in image processing

for smoothing and reducing noise. Sharing the

physical properties of the diffusion process and

being modelled as partial differential equation

(PDE), diffusion arises as a powerful tool in various

fields of image enhancement. However, the usual

drawbacks of the diffusion process (e.g., the

broadening of objects boundaries and edges

dislocation) are hindering its applications. Weickert

gave an in-depth analysis of the diffusion process

and its application in image processing(Weickert,

1997).

Perona-Malik introduced one of the earliest

edge-sensitive diffusion filter for additive noise

reduction (Perona and Malik, 1990). Using nonlinear

anisotropic diffusion, the filter greatly reduced the

additive noise, where weighted image gradient is

used as the diffusion coefficient. . For correcting

Perona-Malik feature distortion effect and

preserving edges, a nonlinear edge enhanced

anisotropic diffusion is introduced (Fu et al., 2005).

Yu et. al. introduced the speckle reducing

anisotropic diffusion (SRAD). SRAD (Yu and

Acton, 2002) combined both, the ordinary nonlinear

anisotropic diffusion process proposed by Perona-

Malik, as well as the adaptive speckle multiplicative

noise filters of Lee (Lee, 1980) and Frost (Frost et.

al., 1982). SRAD alleviates the reliance of adaptive

filters of Lee (Lee, 1980) and Frost (Frost et. al.,

1982) on the window size (i.e. mask size) of the

filter.

On contrary to Perona-Malik filter, SRAD uses

instantaneous coefficient of variation (ICOV) (Yu

and Acton, 2004) of the image as the diffusion

coefficient instead of the image gradient. ICOV has

superior edge maps compared to ordinary edge

detectors due to its incorporation as the diffusion

coefficient into SRAD. SRAD enhances the

reduction of speckle noise while ICOV extracts

edges.

However, SRAD suffers from the drawbacks of

ordinray diffusion (boundary broadening and edges

migration). It produces a set of a coarse to fine

images. The features identified at the finer scale are

Ibrahim W. and El-Sakka M. (2009).

MEMORY-BASED SPECKLE REDUCING ANISOTROPIC DIFFUSION.

In Proceedings of the First International Conference on Computer Imaging Theory and Applications, pages 64-69

DOI: 10.5220/0001803500640069

 SciTePress

distorted and having dislocated edges. Meanwhile,

features identified at coarse scale are noisy.

Trying to limit SRAD boundary broadening

effect, a more robust diffusion coefficients tensor is

introduced to further stop diffusion across main

edges (Tauber et al., 2004). Acton introduced

deconvolutional SRAD (DeSpeRADo) filter (Acton,

2005), where a deblurring is performed at at the

same time with diffusion. DeSperado showed

significant improved results when applied to

synthesized images. However, the poor estimation of

the point spread function (PSF) of the imaging

device (assumed to cause the boundary broadening

effect) limited its application on real data. Yu et. al.

developed a regularized SRAD (Reg-SRAD) for

enhancing point, linear and regional features (Yu

and Yadegar, 2006). Reg-SRAD required the correct

estimation of a threshold value for bright image

features.

In this paper, we propose memory-based SRAD

(MSRAD) where memory is integrated into the

diffusion process through the reaction term. The

incorporated memory provides feedback between

diffusion stages, reminding the newly diffused

image with the correct edge location found in

previously diffused images. MSRAD will enhance

the diffusion process providing a balance between

diffusion and correct edge localization by maintaing

features’ sizes.

The organization of this paper is as follows; in

Section 2, we first give a brief introduction to the

diffusion process, its physical background and the

Perona-Malik diffusion model. Then, we outline the

original SRAD and ICOV models, and previous

refinements made to them. In Section 3, we

introduce our MSRAD technique. In Section 4, we

outline the results obtained by MSRAD. Finally, In

Section 5, we conclude our work.

2 DIFFUSION FILTERING

Diffusion is a physical process that equilibrates

concentration differences without creating or

destroying mass. One of the well known physical

diffusion equation is Fick’s law (Weickert, 1997)

stating that a concentration gradient causes a flux in

order to compensate for this gradient. A diffusion

tensor (D) governs the relation between

concentration gradient and the produced flux.

In image processing, the concentration gradient

can be expressed as image gradient. A constant

diffusion tensor (D) applied over the whole image

domain causes homogenous diffusion or isotropic

diffusion. In addition, a space-dependant D on the

image domain causes inhomogeneous (anisotropic)

diffusion. Linear diffusion happens when D is a

function of the differential structure (image gradient)

of the original image, while non-linear diffusion has

the diffusivity matrix D dependant on the

successively diffused image differential structure

(Weickert, 1997).

Throughout this proposal, the notation used for

diffusion time is t, where a time dependent variable

will have t as its superscript. I indicates the original

image, u refers to the diffused image, u

indicates the

diffused image at time t, where u

t=0

is the original

image I. The subscript x is used to represent the

pixel coordinates (i,j) of the image in the Cartesian

domain, and it is assumed to exist wherever I or u

terms are used.

The general diffusion equation is given by

(1),

where div is the divergence operator, D is the

diffusivity tensor, u is the diffused image,

u is the

image gradient. Changing the diffusivity tensor

defines the kind of diffusion applied to the image

whether linear, nonlinear, isotropic, or anisotropic.

The Perona-Malik model uses a rapidly

decreasing diffusivities D as shown in (2),

where λ is the edge magnitude parameter, D is a

function that gives low values (near zero) for

gradient values >> λ inhibiting diffusion near edges

(Perona and Malik, 1990). Using (2) as the

diffusivity coefficient of (1), the model sharpens

edges if their gradient is larger than the edge

magnitude parameter λ by inhibiting diffusion. For

gradient values << λ, D approaches one and isotropic

diffusion smoothes homogenous regions of the

image converging equation (2) to a linear

homogenous diffusion similar. The correct choice of

λ greatly affects the filter operation. As for large

values of λ, D will be always close to one

independent on the gradient value. While for smaller

values of λ, D will be nearly equal to zero inhibiting

diffusion.

)(

uDdiv

∇×=

∂

(1)

),0(

)(

≠

⎟

⎠

⎞

⎜

⎝

⎛

∇

=∇=

ugD

(2)

MEMORY-BASED SPECKLE REDUCING ANISOTROPIC DIFFUSION

2.1 Instantaneous Coefficient of

Variation (ICOV)

Yu and Acton (2002) (2004) introduced ICOV as the

edge detector operator. ICOV operator is given by

(3),

where |.| is the absolute operator, ||.|| is the

magnitude operator,V is the gradient operator,V

the Laplacian operator, δ, ω, and, χ are weighting

parameters responsible for sharpening edge response

and reduce edge position bias. They are usually

taken to be equal 1/2, 1/16 and, 1/4, respectively.

ICOV is an edge detector utilizing the

normalized gradient and Laplacian operators. It

optimizes edge detection in speckle imagery through

decreasing the probability of false edge detection

and improving the edge localization accuracy.

2.2 Speckle Reducing Anisotropic

Diffusion (SRAD)

Yu et. al. incorporated Lee (Lee, 1980) and Frost

(Frost et. al., 1982) filters along with the anisotropic

diffusion filter of Perona-Malik to come up with a

novel speckle de-noising partial differential equation

called speckle reducing anisotropic diffusion

(SRAD) filter (Yu and Acton, 2002). SRAD is given

by (4),

where, t is diffusion time index where u

t=0

is the

original image I. ∆t is the time step (usually taken in

the range from 0.05 to 0.25) and it is responsible for

the convergence rate of the diffusion process, g(.) is

the diffusion tensor function and is given by (5),

P is a function in the ICOV of the diffused image as

shown in (6),

where q

is the measure of speckle coefficient of

variation in a homogenous region of the image.

ICOV serves as the edge detector for the

diffusion process. It gives high response at edges

and low response in homogenous regions. q

weights

the amount of diffusion applied by SRAD to the

image similar to λ in (2). For simplicity, the form in

(7) is used for D,

The behaviour of SRAD allows diffusion in the

direction parallel to the edge. Negative diffusion is

allowed in the direction normal to the edge. SRAD

outperforms normal anisotropic diffusion filters by

enhancing edge strength and reducing speckle noise

along image contours. However, SRAD still suffers

from ordinary diffusion drawbacks distorting the

size of image features with the increase of diffusion.

In the following section, we introduced our

modification to SRAD to lower its smoothing effect.

3 MEMORY-BASED SPECKLE

REDUCING ANISOTROPIC

DIFFUSION (MSRAD)

SRAD efficiently reduces speckle noise from

images, where the incorporation of ICOV as the

diffusion coefficient provides clear edge maps.

Memory-based SRAD provides features tracking

feedback between the generated set of images

through diffusion varying from coarse to fine scale.

At the beginning of the diffusion process, the coarse

images produce noisy edge maps and provide

correctly located edges, as the effect of feature

broadening is not yet severe. As the diffusion

proceeds with time, the finer images are smoother

and generate more enhanced, highly connected edge

maps but they suffer from dislocated edges due to

feature broadening.

MSRAD introduced memory reminds each

diffused image with the correct edge location and

feature size from previously diffused images.

MSRAD enhance the diffusion process providing

memory feedback balancing diffusion (smoothing),

edge localization, and, feature allocation throughout

different diffusion stages.

MSRAD equation is given in 0,

)(

uICOV

∇×+

∇×−∇×

ωδ

(3)

],))(([

)(

ttt

uuICOVgdiv

uuSRAD

∇×=

Δ+

(4)

)(

))((

euICOVgD

−

(5)

()

)(

uICOV

−

⎟

⎠

⎞

⎜

⎝

⎛

(6)

uICOVgD

))((

(7)

0),()1(

)(

),(

>×−+×=

tuSRADu

uuMSRAD

uSRADu

αα

(

IMAGAPP 2009 - International Conference on Imaging Theory and Applications

where α is a weighting parameter. Comparing 0 to

memory-less SRAD in (4), MSRAD incorporates the

weighted average of the currently diffused image

with the set of the previously generated diffused

images. It requires the determination of a single

weighting parameter α.

The proper choice of α favours either more

diffusion or more adhering to image features. The

original and successively the coarse images exhibit

correct edge locations and feature sizes. As diffusion

proceeds with time towards the finer set of images α

provides coupling between the fine and coarse

images. We empirically choose α to be in the range

from 0.15 to 0.85 depending on the amount of

diffusion needed

3.1 MSRAD as Diffusion-Reaction

Term

Reformulating MSRAD as a diffusion-reaction term

0 can be rewritten as 0,

where MSRAD resembles the diffusion-reaction

model (Weickert, 1997). Memory-less SRAD and

consequently ICOV extracted edge maps are highly

sensitive to the time step ∆t determining SRAD rate

of convergence (stopping criteria). MSRAD

alleviate this reliance by incorporating memory to

the diffusion process through the reaction term as

shown in 0.

3.2 MSRAD versus DeSpeRADo and

Reg-SRAD

MSRAD along with DeSpeRADo (Acton, 2005) and

Reg-SRAD (Yu and Yadegar, 2006) tackled the

problem of feature broadening and edge dislocation

exhibited by normal SRAD.

DeSpeRADo required the exact estimation of the

PSF of the imaging device assumed to cause speckle

noise. This estimation makes the real utilization of

DeSpeRADo impractical and dependant on the

imaging device.

Reg-SRAD depends on the determination of a

threshold value along with other two weighting

parameters. The threshold value depends on the

bright regions intensity of the image. Thus, the

correct choice of the threshold value is highly

dependant on the processed image.

MSRAD requires only the determination of a

single weighting parameter. This parameter is

independent neither of the imaging device used nor

of the image features’ intensities. Thus, MSRAD

provides more convenient and easy to determine

weighting parameter providing balance between

diffusion and features perseverance. The lack of

code and/or test data for both DeSpeRADo and Reg-

SRAD limited our ability to compare our results

with theirs. However, in Section 4 we give a

thoroughly measure of MSRAD performance.

4 RESULTS

In this section, the performance of MSRAD is

compared to adaptive linear noise reduction filters of

Lee (Lee, 1980), Frost (Frost et. al., 1982), and,

Weiner (Wiener, 1976). Also, MSRAD is compared

to the diffusion filters of Perona-Malik and normal

SRAD. The evaluation will be made in terms of

feature perseverance and noise reduction.

For evaluating the MSRAD performance, we

generated a synthesized image shown in Figure 1(a).

The synthesized image is of 150 column width and

150 column height. It consists of a unit step function

in the range from column 15 to column 65 and a

ramp function from column 85 to column 135. A

speckled version of the synthesized image is shown

in Figure 1(b), where a Gaussian distributed speckle

noise of zero mean and variance of 0.1 is added.

In terms of noise reduction and feature

perseverance, Figure 1(c), (d), (e), and (f) shows the

results of de-noising the synthesized speckled image

shown in Figure 1(b) by Lee, Frost, Wiener, and

Perona-Malik filters, respectively. The results where

obtained using 3×3 window for Lee and Frost filters

and 5×5 for Weiner filter. For Perona-Malik filter

the edge magnitude parameter λ, was taken equal to

0.03, with a time step ∆t = 0.1. MSRAD, SRAD,

and, Perona-Malik results were obtained after 200

iterations, where SRAD result is shown in Figure

1(g), and MSRAD result shown in Figure 1(h). Both

MSRAD and SRAD results were obtained using a

time step ∆t = 0.25.

Compared to adaptive linear filters (i.e. Frost,

Lee, Wiener) and Perona-Malik filter, MRSAD

showed superior noise reduction effect. Original

SRAD suffer from boundary broadening and

distortion of features. MSRAD result showed

significant perseverance of the features’ sizes.

Figure 2 inspects the results of applying Lee,

Frost, Wiener, Perona-Malik, SRAD, and MSRAD

over a horizontal scan line extracted from the images

at row 71 in Figure 1. The results show that MSRAD

virtually approximated the original signal shown in

0)),(()(

)(

>−×+=

tuSRADuuSRAD

uuMSRAD

ttt

(9)

MEMORY-BASED SPECKLE REDUCING ANISOTROPIC DIFFUSION

Figure 2(a). Lee, Frost, Wiener and, Perona-Malik

filters have limited noise reducing responses. Yet,

they do not suffer from feature broadening effects.

While SRAD suffer from severe boundary

broadening and feature merging effect. MSRAD

shows more consistent features along with good

approximation of original signal.

The adaptive linear filters of Lee, Frost, Weiner

depend totally on the window (mask) size. Perona-

Malik filter depends on the edge magnitude

parameter λ, while SRAD depends on the diffusion

step ∆t. MSRAD depends only on a single weighting

parameter, α, maintaining a good balance between

image smoothing and boundary allocation.

5 CONCLUSIONS

In this paper, memory-based SRAD was introduced

as feature perseverance SRAD. The introduced

memory through the reaction term balanced the

effect of diffusion and correct boundaries allocation.

MSRAD showed significant noise reduction effect

over linear filters of Lee, Frost, and, Wiener, as well

as over the diffusion filter of Perona-Malik.

Compared to the original SRAD, MSRAD

maintained the correct sizes of features and reduced

speckle noise. MSRAD requires the determination of

a single weighting parameter compared to estimating

PSF of DEspeRADo or the image dependant

threshold parameter controlling Reg-SRAD.

ACKNOWLEDGEMENTS

This research is partially funded by the National

Sciences and Engineering Research Council of

Canada (NSERC). This support is greatly

appreciated.

(a) (b)

(f) (g) (h)

Figure 1: Synthesized image along with the results of applying various de-noising filters and MSRAD. (a) Original

synthesized image. (b) Speckled synthesized image. (c) Lee filter result (d) Frost filter result. (e) Wiener filter result (f)

Perona-Malik filter result. (g) SRAD result (h) MSRAD result.

IMAGAPP 2009 - International Conference on Imaging Theory and Applications

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Figure 2: MSRAD versus various de-noising filters in terms of smoothing over a horizontal scan line of the images in

Figure 1. (a) Original signal. (b) Speckled signal. (c) Lee filter signal result. (d) Frost filter signal result. (e) Wiener filter

signal result. (f) Perona-Malik signal result. (g) SRAD result signal. (h) MSRAD signal result.

MEMORY-BASED SPECKLE REDUCING ANISOTROPIC DIFFUSION