NON-LINEAR LOW-LEVEL IMAGE PROCESSING IMPROVEMENT

BY A PURPOSELY INJECTION OF NOISE

A. Histace

ETIS UMR CNRS 8051, ENSEA-UCP, 6 avenue du Ponceau, 95014 Cergy, France

Keywords:

Stochastic resonance, Low-level image processing, Binarization.

Abstract:

It is progressively realized that noise can play a constructive role in nonlinear formation processes. The

starting point of the investigation of such useful noise effect has been the study of the Stochastic Resonance

(SR) effect. The goal of this article is to propose a direct application of SR phenomenon in image processing,

for the interest of SR in that domain is growing-up. As a prolongation of previous work already presented in

the literature by author, we propose to quantitatively show that a purposely injection of a gaussian noise in a

classical nonlinear image process, as image binarization, can play a constructive action. This work can also

be interpreted as a ﬁrst step for a better understanding of SR in image processing relating it to classical results

obtained in a nonlinear signal processing framework for classical low-level image processing tool.

1 INTRODUCTION

It is progressively realized that noise can play a con-

structive role in nonlinear formation processes. The

starting point of the investigation of such useful noise

effect has been the study of the Stochastic Resonance

(SR) effect. This paradoxical effect was ﬁrst intro-

duced some twenty years ago in the domain of cli-

mate dynamics, as an explanation for the regular re-

currences of ice ages (Benzi et al., 1982). Following

this, SR effect has been introduced in nonlinear signal

processing to describe the mechanism of a construc-

tive action of a white Gaussian noise in the transmis-

sion of a sinusoid by a nonlinear dynamic system gov-

erned by a double-well potential. From this time, the

phenomenon of stochastic resonance has experienced

large varieties of extensions with variations concern-

ing the type of noise, the type of information carry-

ing signal or the type of nonlinear system interacting

with the signal-noise mixture (see for example (Gam-

maitoni et al., 1998) for a review in physics, (Harmer

et al., 2002) for an overview in electrical engineering

and (Chapeau-Blondeau and Rousseau, 2002) for the

domain of signal processing). All these extensions of

the original setup preserve the possibility of improv-

ing the processing of a signal by means of an increase

in the level of the noise coupled to this signal. At the

moment, new forms of useful noise effect, related to

stochastic resonance, continue to be demonstrated. A

recent speciﬁc domain of interest for the study of this

useful noise effect is nonlinear image processing (see

(Morfu et al., 2008) for instance).

The goal of this article is to propose a direct appli-

cation of SR phenomenonin image processing, for the

interest of SR in that domain is growing-up. As au-

thors have already shown it in (Histace and Rousseau,

2006; Histace and Rousseau, 2010), a possible appli-

cation ﬁeld is nonlinear image restoration. Neverthe-

less, in order to have a better understanding of the

process and the constructive action of noise, we pro-

pose to study a more simple image processing tool:

Image binarisation.

The main layout of this article is the following:

Second section proposes a presentation of the global

framework of stochastic resonance non limited to im-

age processing. Whereas such presentation has al-

ready been made in various articles, it bears important

historical and conceptual signiﬁcance. For this rea-

son, we choose to remain it. Third section deals with

application of SR phenomenon to image binarization.

Finally, results will be concluded and discussed.

2 A COMMON FRAMEWORK

FOR STUDY OF SR EFFECT IN

NONLINEAR PROCESSING

Today, it is now widely assumed that a global frame-

work for SR can be deﬁned for all type of demon-

strated effects (Chapeau-Blondeau, 2000).

Stochastic resonance involves four essential ele-

226

Histace A..

NON-LINEAR LOW-LEVEL IMAGE PROCESSING IMPROVEMENT BY A PURPOSELY INJECTION OF NOISE.

DOI: 10.5220/0003399202260229

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2011), pages 226-229

ISBN: 978-989-8425-47-8

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

ments: (i) an information-carrying or coherent signal

s: it can be deterministic, periodic or non, or random;

(ii) a noise η, whose statistical properties can be of

various kinds (white or colored, Gaussian or non,... );

(iii) a transmission system, which generally is nonlin-

ear, receiving s and η as inputs under the inﬂuence

of which it produces the ouput signal y; (iv) a per-

formance or efﬁcacy measure, which quantiﬁes some

“similarity” between the output y and the coherent in-

put s (it may be a signal-to-noise ratio, a correlation

coefﬁcient, a Shannon mutual information, ...). SR

takes place each time it is possible to increase the

performance measure by means of an increase in the

level of the noise η. Historically, the developments

of SR have proceeded through variations and exten-

sions over these four basic elements. From the origin

and as it has already been mentioned in previous sec-

tion, SR studies have concentrated on a periodic co-

herent signal s, transmitted by nonlinear systems of a

dynamic and bistable type (McNamara and Wiesen-

feld, 1989). This form of SR now appears simply as a

special form of SR. This primary form of SR will not

be entirely described in this article but a complete de-

scription can be found in (Chapeau-Blondeau, 2000)

for instance. For illustration, we propose to illustrate

phenomenon of SR in the framework of image trans-

mission as it was formerly proposed in (Chapeau-

Blondeau, 2000). This example has the advantage of

its simplicity which makes both theoretical and ex-

perimental analysis possible. Leaning again on the

general scheme of SR phenomenon, author considers

this time that the coherent information-carrying sig-

nal s is a bidimensional image where the pixels are

indexed by integer coordinates (i, j) and have inten-

sity s(i, j). For a simple illustration, a binary image

with s(i, j) ∈ {0, 1} is considered for experiment. A

noise η(i, j), statistically independent of s(i, j), lin-

early corrupts each pixel of image s(i, j). The noise

values are independent from pixel to pixel, and are

identically distributed with the cumulative distribu-

tion function F

(u) = Pr{η(i, j) ≤ u}. A nonlinear

detector, that it is taken as a simple hard limiter with

threshold θ, receives the sum s(i, j) + η(i, j) and pro-

duces the output image y(i, j) according to:

If s(i, j) + η(i, j) > θ then y(i, j) = 1,

else y(i, j) = 0.

(1)

When the intensity of the input image s(i, j) is low

relative to the threshold θ of the detector, i.e. when

θ > 1, then s(i, j) (in the absence of noise) remains

undetected as the output image y(i, j) remains a dark

image. Addition of the noise η(i, j) will then allow

a cooperation between the intensities of images s(i, j)

and η(i, j) to overcome the detection threshold. The

result of this cooperative effect can be visually appre-

ciated on Fig. 1, where an optimal nonzero noise level

maximizes the visual perception.

Figure 1: The image y(i, j) at the output of the detector of

Eq. (1) with threshold θ = 1.2, when η(i, j) is a zero-mean

Gaussian noise with rms amplitude 0.1 (left), 0.5 (center)

and 2 (right).

To quantitatively characterize the effect visually

perceived in Fig. 1, an appropriate quantitative mea-

sure of the similarity between input image s(i, j) and

output image y(i, j), is provided by the normalized

cross-covariancedeﬁned in (Vaudelle et al., 1998) and

given by:

h(s− hsi)(y− hyi)i

h(s− hsi)

ih(y− hyi)

, (2)

where h.i denotes an average over the images.

can be experimentally evaluated through pix-

els counting on images similar to those of Fig. 1.

Also, for the simple transmission system of Eq. (1),

can receive explicit theoretical expressions, as a

function of p

= Prs(i, j) = 1 the probabilty of a pixel

at 1 in the binary input image s(i, j), and as a function

of the properties of the noise conveyed by F

(u) as

mentioned in (Vaudelle et al., 1998).

Considering the above scenario, Fig. 2 showsvari-

ations of C

function of rms amplitude of the input

noise η.

As one can see on Fig. 2, measure of cross-

covariance as deﬁned Eq. (2) identify a maximum

efﬁcacy in image transmission for an optimal nonzero

noise level. This simple example is interpreted here as

the ﬁrst formalized instance of SR for aperiodic bidi-

mensionnal input signal s (even if it is not clearly an

image processing application).

We are now going to show that this kind of ap-

proach can be successfully transposed in a classical

low-level image processing tool.

3 NOISE-AIDED IMAGE

BINARIZATION

Let’s consider image of Fig. 3. Let’s now consider

that our main goal is to binarize image of Fig. 3 in or-

der to automatically extract barycenter of each coin.

NON-LINEAR LOW-LEVEL IMAGE PROCESSING IMPROVEMENT BY A PURPOSELY INJECTION OF NOISE

227

Figure 2: Input-output cross-covariance of Eq. (2) between

input image s(i, j) and output image y(i, j), as a function

of the rms amplitude of the noise η(i, j) chosen zero-mean

Gaussian. The crosses are experimental evaluations through

pixels counting on images, the solid lines are the theoreti-

cal predictions (p

= 0.6) calculated by authors (Chapeau-

Blondeau, 2000).

Figure 3: “coins” Image.

Classically, this task can be tackled by an automatic

estimation of the optimal threshold corresponding to

data to binarize. For instance, existing functions usu-

ally use Otsu’s method (Otsu, 1979), which chooses

the threshold to minimize the intraclass variance of

the black and white pixels. Applying that kind of

function to “coins” image leads to binarization results

of Fig.4.

Figure 4: Binarized “coins” image using classical Otsu’s

method for automatic computation of optimal threshold

level.

As one can notice, this approach is not satisfying

since one coin (characterized by an average grey-level

less important than other coins) will not be clearly de-

tected during following processing steps. Empirical

manual setting of the threshold can lead to more in-

teresting results as shown Fig. 5.

Let’s now consider, that existing method of Fig. 4

is a black box with no possibility to manually adjust

threshold value to reach optimal result of Fig. 5.

Considering the classical framework of SR phe-

nomenon, we now purposely corrupt original “coins”

image with a white gaussian noise η of tunable stan-

dard deviation σ

. Proposed process is then described

by same equations as Eq. (1) with threshold θ corre-

sponding to the value automatically computed thanks

to Otsu’s approach. In order to quantify the possible

beneﬁt of such addition, we choose to perform a mea-

sure of the normalized cross covariance of Eq. (2)

between optimal result of Fig. 5 and obtained results

for each value of σ

. Result of this quantitative study

is presented Fig. 6.

Figure 5: Binarized “coins” image using manual setting for

threshold value.

0 0.05 0.1 0.15 0.2 0.25

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

similarity measure

Figure 6: Normalized cross-covariance of Eq. (2) function

of uniform purposely injected noise level σ

As one can notice on Fig. 6, it is possible to reach

an optimal value of the normalized cross-covariance

for a non zero amount of noise. This is the classi-

cal signature of a SR phenomenon as demonstrated in

previous section for nonlinear binary image transmis-

sion. One can also notice that the normalized cross-

covariance fastly decrease once the optimal value is

VISAPP 2011 - International Conference on Computer Vision Theory and Applications

228

Figure 7: Optimal binarization result obtained for σ

0.19.

reached: The large amount of corresponding noise,

ﬁnally, completely degrade image information. Visu-

ally speaking, the optimal corresponding result pre-

sented Fig. 7 is very close from the optimal one of

Fig. 5. Detection of the whole set of coins is now

possible thanks to the purposely injection of η noise.

Moreover, this experiment shows that it is now pos-

sible to adjust the inner threshold value of the bina-

rization process thanks to an external tuning of noise

level σ

4 CONCLUSIONS

In this article, we show that the now well known SR

phenomenon can ﬁnd application in low-level image

processing. More precisely, we show, thanks to a sim-

ple experiment, that a purposely injection of noise in

a classical non tunable binarization process can lead

to interesting results in term of optimal parameters

setting. This is shown here as a proof of feasibility

and more experiment will be made in order to clearly

identify both theoretically and quantitatively the ben-

eﬁt of such an approach for more complex image pro-

cessing tools as nonlinear image restoration for in-

stance. This work can also be interpreted as a ﬁrst step

for a better understanding of SR in image processing

relating it to classical results obtained in a nonlinear

signal processing framework.

ACKNOWLEDGEMENTS

Author wants to thank Dr. David Rousseau from Uni-

versity of Angers (LISA) for his precious help and

advises about this work.

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