A FUZZY SCHEME FOR IMAGE NOISE REDUCTION

Philippe Vautrot

, Michel Herbin

and Laurent Hussenet

CReSTIC EA 3804, University of Reims Champagne Ardenne, Department of Informatique, IUT Info

Rue des Cray`eres, BP 1035, 51687 Reims Cedex 2, France

IUT RCC, Chauss´ee du port, BP 541, 51012 Chˆalons-en-Champagne, France

Keywords:

Image noise reduction, Bilateral ﬁltering, Fuzzy ﬁlter.

Abstract:

The improvement of acquisition devices increases the need for processing of multicomponent images. In

this context, the noise reduction is a preliminary preprocessing step affecting the results of the other image

operations. This paper proposes a framework explaining usual noise reduction methods by the means of two

fuzzy logic techniques: ﬁrst a pixel fuzziﬁcation and second a defuzziﬁcation for estimating the ﬁltered values.

A new density-based ﬁlter is built for removing both impulse noise and Gaussian noise. The ﬁlter we propose

is robust against outliers and it improves the classical bilateral approach for noise reduction of multicomponent

images.

1 INTRODUCTION

In the framework of image processing, one of the ﬁrst

tasks consists in removing or reducing noise from the

images (Gonzales and Woods, 1992). The improve-

ment of acquisition devices increases the need for pro-

cessing multicomponent images obtained from differ-

ent channels (Kotropoulos and Pitas, 2001; Bovik,

2000). The independent processing of image compo-

nents turns out to be inappropriate and leads to strong

artifacts (Lukac et al., 2006). Thus the noise reduc-

tion of multicomponent images is an active ﬁeld of

research in satellite remote sensing, robot guidance,

electron microscopy, medical imaging, color process-

ing and real-time applications (Lin and Hsueh, 2000;

Wong et al., 2004; Gallegos-Funes and Ponomaryov,

2004). This paper focuses on this preprocessing step

for reducing both additive Gaussian noise and im-

pulse noise. Additive Gaussian noise corrupts images

because of the imprecision of acquisition devices. Im-

pulse noise is generally produced by the transmission

devices (Bovik, 2000).

The noise reduction consists in ﬁltering the im-

age, classically by computing a barycenter within a

window. The selection of barycentric coordinates is

the main key of noise reduction methods. The fuzzy

techniques also addresses this issue of noise reduc-

tion (Ville et al., 2003; Morillas et al., 2009; Ca-

marena et al., 2010). In this paper, we consider that

the ﬁltering window is a fuzzy set. First we determine

these fuzzy sets associated to each pixel. This step

corresponds to a fuzziﬁcation of the pixels. Second

the estimation of the ﬁltered value corresponds to a

defuzziﬁcation (Leekwijck and Kerre, 1999). More-

over the pixels of a multi-component image have both

2-dimensional spatial coordinates and n-dimenional

photometric coordinates associated with the n compo-

nents of the image. The bilateral ﬁltering is a classical

way taking into account both the spatial aspect and

the photometric aspect of images in image process-

ing. Bilateral ﬁlter of Tomasi and Manduchi (Tomasi

and Manduchi, 1998) is the archetype of such bilateral

approach. Thanks to the agregation operators (De-

tyniecki, 2001), the fuzzy logic enables us to general-

ize the bilateral approach of ﬁltering. Unfortunately

Bilateral ﬁlter is not robust against outliers. Thus this

paper proposes a new bilateral ﬁlter based on density

estimation that provides robustness against outliers.

The paper is organized as follows: Section 2

presents the general frameworkselecting fuzzy neigh-

borhood of each pixel for image ﬁltering. Section 3

is devoted to the defuzziﬁcation step for estimating

the ﬁltered value of a pixel. In Section 4 we study

the combination of fuzzy neighborhood improving

the classical bilateral ﬁltering (Tomasi and Manduchi,

1998). This approach is applied to reduce Gaussian

noise and impulse noise in color images. The last Sec-

tion proposes a discussion and concludes this paper.

441

Vautrot P., Herbin M. and Hussenet L..

A FUZZY SCHEME FOR IMAGE NOISE REDUCTION.

DOI: 10.5220/0003671604410445

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 441-445

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

2 FUZZY NEIGHBORHOOD OF A

PIXEL

Let p be a pixel of a multicomponent image I with d

components. Let I(p) be its photometric vector. Re-

ducing the noise, I(p) is replaced by the ﬁltered value

∗

(p) which is estimated within a window W

cen-

tered on p. Let p

, p

, ...p

be the N pixels of W

(N = n × n). I

∗

(p) is usually a barycenter of I(p

I(p

),... I(p

) deﬁned by:

∗

(p) =

∑

1≤i≤N

µ(i)

∑

1≤i≤N

µ(i)I(p

) (1)

where µ(i) are the barycentric coordinates of I

∗

(p).

In the fuzzy logic frame, µ(i) becomes the mem-

bership value of the pixel p

to a fuzzy set ep. This

fuzzy set has its support in W

. Then the ﬁrst step of

the ﬁltering procedure consists in selecting this fuzzy

neighborhood of p. This fuzziﬁcation step is detailed

in the following subsections.

2.1 Fuzzy Spatial Neighborhood

When the membership values µ(i) depend only on the

spatial locations of the pixels p

, then a fuzzy spa-

tial neighborhood ep

spat

is deﬁned for ﬁltering. Gaus-

sian ﬁlter is the archetype of these spatial ﬁlters. The

membership values µ

spat

(i) are deﬁned by:

spat

(i) = exp



−

dist

spat

(p, p

)

2σ

spat



(2)

where dist

spat

is the Euclidean distance and σ

spat

the standard deviation of the Gaussian ﬁlter. Note that

these fuzzy neighborhoods are normalized fuzzy sets

(Bouchon-Meunier, 1995) and their largest member-

ship values are equal to 1.

2.2 Fuzzy Photometric Neighborhood

When the membership values depend only on the

closeness between the photometric values I(p

) and

I(p), then the fuzzy neighborhood of p is designed

in the photometric space. Rank ﬁlter or vector me-

dian ﬁlters (Astola et al., 1990) give examples of such

photometric ﬁlters. They are obtained by ordering the

vectors I(p

), I(p

),... I(p

). The estimation of I

∗

(p)

is based on the ranks of I(p

) vectors. In such cases,

the membership values µ

phot

(i) of the fuzzy photo-

metric neighborhood ep

phot

ignore the spatial location

of the pixels p

By analogy to the fuzzy spatial neighborhood, the

Gaussian distribution also permits to give another def-

inition of ep

phot

. The support of the fuzzy set remains

. But the distance dist

phot

is computed in the pho-

tometric space (e.g. Euclidean distance). Then the

membership function is deﬁned by:

phot

(i) = exp



−

dist

phot

(I(p), I(p

))

2σ

phot



(3)

where σ

phot

is the standard deviation of the Gaussian

distribution in the photometric domain.

Because of the noise, I(p) could be inappropriate

as the center of a photometric neighborhood. There-

fore we propose another approach for deﬁning a fuzzy

photometric neighborhood of p.

2.3 Fuzzy Neighborhood based on

Density

For each pixel q in W

phot

is a fuzzy neighborhood

of p centered on I(q). The membership functions of

phot

are deﬁned by:

phot

(i) = exp



−

dist

phot

(I(q), I(p

))

2σ

phot



(4)

where q ∈ W

. These N fuzzy sets are aggregated

using the arithmetic mean of their membership func-

tions. Then the function µ

dens

we obtain corresponds

to a local estimation of a probability density func-

tion (PDF). Improving PDF estimation we preserve

against outliers and noise by ruling out ep

phot

(i.e.

phot

) when estimating the density (Herbin and Bon-

net, 2002). The membership function of this new

fuzzy set based on density is deﬁned by:

dens

(i) =

∑

q∈W

,q6=p

phot

(i) (5)

where C is a normalization coefﬁcient. This paper

proposes this approach through robust density estima-

tion to deﬁne a new fuzzy photometric neighborhood.

2.4 Bilateral Approach of Fuzzy

Neighborhood

To keep the advantage of both spatial and photomet-

ric approaches, the t-norms (Bouchon-Meunier,1995)

(i.e. a conjunction operator) permit to combine the

fuzzy spatial neighborhood and the fuzzy photomet-

ric neighborhood. Tomasi and Manduchi (Tomasi and

Manduchi, 1998) use the algebraic t-norm for com-

puting their bilateral ﬁlter. Then the membership val-

ues of ep

bilat

is deﬁned by:

bilat

(i) = µ

spat

(i) × µ

phot

(i) (6)

FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications

442

In this paper, we use the classical minimum oper-

ator as t-norm combining both spatial and photomet-

ric density-based neighborhoods. The fuzzy bilateral

neighborhood ep

bidens

we propose is the conjunction

of these two fuzzy sets. Therefore the membership

function µ

bidens

is deﬁned by:

bidens

(i) = min



spat

(i), µ

dens

(i)



. (7)

3 DEFUZZIFICATION

The goal of this section is to estimate the ﬁltered

value I

∗

(p) from the fuzzy neighborhoods of p. This

step corresponds to a defuzziﬁcation process (see a

review of the defuzziﬁcation methods in (Leekwijck

and Kerre, 1999)). The defuzziﬁcation is obtained us-

ing two stages: the ﬁrst one operates in the spatial do-

main and the second one operates in the photometric

domain.

The most classical defuzziﬁcation method is based

on the maximum of membership values. In the con-

text of multicomponent images, the maxima method

in the spatial domain consists in selecting the pixel p

for which the membership value µ(i) is maximal. Let

p be this pixel deﬁned by:

p = arg max

∈W



µ(i)



(8)

In this paper, the membership function µ

bidens

is used

to determine p. Therefore p corresponds to the mode

of our density estimation.

Another usual defuzziﬁcation method consists in

computing the center of gravity of a fuzzy set where

the weights are the membership values. This method

is used in the photometric domain. I(p) is considered

as the center of the fuzzy photometric neighborhood

of p. Then the ﬁltered value I

∗

(p) is deﬁned by:

∗

(p) =

∑

1≤i≤N

phot

(i)

∑

1≤i≤N

phot

(i)I(p

) (9)

Indeed this barycenter inside the windowW

is the ﬁl-

tered value we propose to reduce noise in multicom-

ponent images.

4 APPLICATION TO COLOR

IMAGES

To assess our method, we use color images with

three components: Red, Green and Blue. Images are

(a) Reference Image (Parrots)

(b) Noised Image

(d) Bilateral Filter

(e) Density-based Filter

Figure 1: Comparison of noise reduction ﬁlters: (a) Refer-

ence image (Parrots), (b) Part of Noised image (Noised),

tered image (BILAT), (e) Fuzzy Density-based ﬁltered im-

age (DENS).

corrupted with two kinds of independent and identi-

cally distributed noise. A low level noise is designed

through additive Gaussian noise, and high level noise

is modeled by impulse noise. The goal is to reduce

both low level noise and high level noise by ﬁltering

corrupted images.

The classical mean squared error (MSE) evaluates

the results by averaging the squared differences of ﬁl-

tered and reference images. In this context MSE is

deﬁned by:

MSE(I

∗

) =

∑

p∈I

dist

phot

∗

(p), I(p))

(10)

where #I is the number of pixels of the images. We

A FUZZY SCHEME FOR IMAGE NOISE REDUCTION

443

separate MSE into two parts MSE

−

and MSE

MSE

−

is deﬁned by:

MSE

−

∗

) =

−

∑

δ(p)≤T

(p) (11)

where I

−

= {p : δ(p) ≤ T}, and MSE

is deﬁned by:

MSE

∗

) =

∑

δ(p)>T

(p) (12)

where I

= {p : δ(p) > T} . In this paper, the thresh-

old T = 10 is used to separate low levelnoise and high

level noise.

Table 1: Assessments of noised image (Noised), vector me-

dian ﬁltered image (VM), bilateral ﬁltered image (BILAT)

and fuzzy density-based ﬁltered image (DENS) using MSE

with 388, 112 pixels, MSE

−

with N

−

pixels, and MSE

with N

pixels (N

+ N

= 388, 112).

Image MSE MSE

−

Noised 2873.6 256,921 46.1

VM 98.2 338,773 32.9

BILAT 2803.0 336,865 28.6

DENS 69.0 370,588 21.3

Image MSE MSE

Noised 2873.6 131,191 8410.9

VM 98.2 49,339 545.6

BILAT 2803.0 51,247 21040.3

DENS 69.0 17,524 1077.5

In this paper, the ﬁltering windows has 5× 5 pix-

els. Estimating the density in the photometric space,

a large value of σ

phot

is prefered for smoothing PDF

estimation. Then we use σ

phot

= 30.0. In the spatial

domain, σ

spat

is empirically determined (σ

spat

= 0.5).

In the defuzziﬁcation process, σ

phot

value controls the

smoothing when ﬁltering. The value which gives the

best results is σ

phot

= 10.

Evaluating the results, we compare a corrupted

image (Noised), a classical bilateral ﬁltered image

(BILAT), a vector median ﬁltered image (VM) and

our fuzzy density-based ﬁltered image (DENS). Ta-

ble 1 gives the mean square errors obtained when as-

sessing the noise reduction. These results show that

the bilateral ﬁlter is inappropriate in the case of high

level noise (i.e. outliers) and vector median ﬁlter can-

not smooth enough the image for reducing low level

noise when preserving the edges. Figure 1 conﬁrm

these results.

The defuzziﬁcation process uses a weighted mean

of the photometric vectors which permits to smooth

the image. The higher σ

phot

value, the smoother

the image. If σ

phot

is too small, then the ﬁlter does

not smooth the ﬁltered image. Therefore it does not

(a) Reference

(b) σ

phot

= 5

phot

= 10 (d) σ

phot

= 20

Figure 2: Reducing noise and level of texture: (a) part of

a reference image and fuzzy density-baser ﬁltered images

with (b) σ

phot

= 5, (c) σ

phot

= 10, (d) σ

phot

= 20.

enough reduce low level noise. If σ

phot

is too large,

the ﬁne details could disappear when ﬁltering because

of a too large smoothing. Figure 2 displays the results

obtained with σ

phot

= 5, σ

phot

= 10, and σ

phot

= 20.

The value σ

phot

= 10 gives convenient results be-

tween smoothing for reducing low levelnoise and pre-

serving details.

5 CONCLUSIONS

This paper adapts the classical fuzzy scheme for data

analysis in the framework of noise reduction for mul-

ticomponent images. This scheme consists in a data

fuzziﬁcation following by a defuzziﬁcation allowing

the decision. The approach we propose is based on

ﬁrst the selection of adaptive fuzzy neighbourhoods

of the pixels (i.e. the fuzziﬁcation) and second a de-

fuzziﬁcation taking into account both spatial and pho-

tometric aspects of images. This fuzzy logic approach

allows us to model the most classical ﬁlters used in the

framework of image processing. Therefore this fuzzy

scheme offers new angles for noise reduction of mul-

ticomponent images. The new density-based ﬁlter we

propose reduces both high level noise (impulse noise)

and low level noise (Gaussian noise). Like Bilateral

ﬁlter, our ﬁlter reduces low level noise preserving de-

tails because its anisotropic nature. But it is also as

robust against outliers (i.e. high level noise) as the

FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications

444

vector median based ﬁlters are. Therefore the fuzzy

scheme permits us to design a new ﬁlter taking into

account the advantages of two classic ﬁlters for re-

ducing both high and low level noise.

REFERENCES

Astola, J., Haavisto, P., and Neuovo, Y. (1990). Vector me-

dian ﬁlters. IEEE Proceedings, 78:678–689.

Bouchon-Meunier, B. (1995). La logique ﬂoue et ses appli-

cations. Addison-Wesley, Paris.

Bovik, A. (2000). Handbook of image and video processing.

San Diego, CA, Academic Press.

Camarena, J.-G., Gregori, V., Morillas, S., and Sapena, A.

(2010). Two-step fuzzy logic-based method for im-

pulse noise detection in colour images. Pattern Recog-

nition Letters, 31:1842–1849.

Detyniecki, M. (2001). Mathematical aggregation operators

and their application to video querying. In Research

Report, LIP6, PARIS.

Gallegos-Funes, F. and Ponomaryov, V. (2004). Real-time

image ﬁltering scheme based on robust estimators

in presence of impulsive noise. Real-Time Imaging,

10:69–80.

Gonzales, R. and Woods, R. (1992). Digital Image Process-

ing. Addison-Wesley, USA.

Herbin, M. and Bonnet, N. (2002). A new adaptive ker-

nel density estimation. In Information Processing and

Management of Uncertainty (IPMU), Annecy.

Kotropoulos, C. and Pitas, I. (2001). Nonlinear model-

based image/video processing and analysis. New

York, NY, Wiley.

Leekwijck, W. V. and Kerre, E. (1999). Defuzziﬁca-

tion: criteria and classiﬁcation. Fuzzy Sets System,

108(2):159–178.

Lin, R. and Hsueh, Y. (2000). Multichannel ﬁltering by

gradient information. Signal Processing, 80:279–293.

Lukac, R., Smolka, B., Plataniotis, K. N., and Venet-

sanopoulos, A. N. (2006). Vector sigma ﬁlters for

noise detection and removal in color images. Journal

of Visual Communication and Image Representation,

17:1–26.

Morillas, S., Gregori, V., and Herv´as, A. (2009). Fuzzy peer

groups for reducing mixed gaussian-impulse noise

from color images. IEEE Transactions on Image Pro-

cessing, 18(7):1452–1466.

Tomasi, C. and Manduchi, R. (1998). Bilateral ﬁltering for

gray and color images. In Proceedings of IEEE Con-

ference on Computer Vision, Bombay, India.

Ville, D. V. D., Nachtegael, M., der Weken, D. V., Kerre, E.,

Philips, W., and Lemahieu, I. (2003). Noise reduction

by fuzzy image ﬁltering. IEEE Transaction on Fuzzy

Systems, 11(4):429–436.

Wong, W., Chung, A., and Yu, S. (2004). Trilateral ﬁltering

for biomedical images. In IEEE proceedings.

A FUZZY SCHEME FOR IMAGE NOISE REDUCTION

445