A NOISE REMOVAL MODEL WITH

ANISOTROPIC DIFFUSION BASED ON VISUAL GRADIENT

Li Shi-Fei, Wang Ping and Shen Zhen-Kang

ATR Lab, National University of Defense Technology, Deya Road, Changsha, P. R. China

Keywords: Diffusion filter, Visual gradient, Noise removal.

Abstract: In recent years considerable amount of researchers have been devoted to anisotropic diffusion method and

achieved a series of important development. However, human visual system which perceived and

interpreted images has been paid little attention to in all these models. In this paper, we define a visual

gradient, which is looked as a generalization of the image gradient. After that we substitute the visual

gradient for the image gradient in the anisotropic diffusion model to keep to some extent consistent with

human visual system for the first time. Finally numerical results show the proposed method’s performance.

1 INTRODUCTION

Since Perona (Perona and Malik, 1990) introduced

the anisotropic diffusion to image processing and

proposed a multiscale smoothing edge detection

model first, a considerable amount of researchers

have been devoted to theoretical and practical

understanding of this and related methods. The idea

of anisotropic diffusion is that if the gradient of a

point is large the smoothing process will be low and

therefore the exact localization of the edges will be

preserved. Accordingly, anisotropic diffusion has a

good property of eliminating noise while preserving

high frequency components.

However, it has some disadvantages. For

example, Perona’s model is ill-posed and unstable

and it is difficult to confirm the model’s parameters.

In the recent years there are many modified versions

of Perona’s model have been presented. Nordström

(Nordström,

1990) proposed a biased diffusion to

regulate the ill-posed nature of the function. Catté et

al. (Catté et al.,

1992) have given a thorough

derivation of the process and proposed a stable

scheme for implementation, eliminating the problem

of choosing the number of required iterations. Other

researches have been achieved by such as Alvarez et

al. (Alvarez et al., 1

992), You (Yu-Li, Y and

Wenyuan 1996) and Barcelos et al. (Barcelos et al. ,

2003).

As we all know, all images are eventually

perceived and interpreted by the HVS (Human

Visual System). But these modified models consider

little the influence of human visual system. In fact,

Perona’s anisotropic diffusion model bases on the

gradient of the image which is not integrated the

information of human visual system. So we define

the visual gradient by using the properties of HVS

and develop an anisotropic diffusion model based on

visual gradient to remove image noise without losing

the boundaries or edges.

2 ANISOTROPIC DIFFUSION

Let u be the representation of the reconstructed

image. This representation can be defined as a

function of

2

Ω⊂ →RR

that associate the

pixel

2

(, )xy∈R to its grey level image

(, )uxy

;

Ω

is the image support (generally, a

rectangle). Perona (

1990) substituted the standard

heat equation by the following anisotropic diffusion

equation (for the sake of briefness, in all models we

omit boundary and initial conditions):

(( ) )

t

udivfuu

=

∇∇

(1)

where

f

is a non-increasing smooth function such

that

() 0fs≥ , (0) 1f

=

and lim ( ) 0

s

fs

→∞

= . The

idea is that if the gradient is large, then the diffusion

will be low, therefore the exact localization of the

edges will be preserved.

However, this model still has several theoretical

and practical difficulties. For instance, if the image

61

Shi-Fei L., Ping W. and Zhen-Kang S. (2009).

A NOISE REMOVAL MODEL WITH ANISOTROPIC DIFFUSION BASED ON VISUAL GRADIENT.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 61-64

DOI: 10.5220/0001653700610064

Copyright

c

SciTePress

is very noisy, the gradient will be very large; as a

result the function will be close to zero at almost

every point. Consequently almost all noise will

remain when we use this model smooth the gravely

noisy image.

Aiming to this operator’s theoretical and

practical difficulties, considerable amount of

researchers proposed many modified models. We list

some typical modified models and study their

properties.

Catté (

1992) proposed and studied the following

model:

(( ) )

t

udivgGuu

σ

=∇∗∇ (2)

Where

Gu

σ

∇∗ denotes a convolution of the image

at time

t with a Gaussian kernel of scale

σ

which is

to be given a priori. This model alleviates

Perona’s

model’s ill-posedness, but it inducts a new

parameter

σ

.Moreover proper selection of this

parameter is critical to the success of the proposed

anisotropic diffusion.

From a geometric point of view, Alvarez et al.

(1

992) modify the diffusion operator in a way that

the diffusion process becomes more intense along

the edges and less intense along the perpendicular

direction of the edges:

()()

t

u

ugGu udiv

u

σ

∇

=∇∗∇

∇

(3)

After that ,there are other modified model In (

Yu-

Li,1996; Barcelos et al.,2003. etc.) Following the

idea in (

Perona, 1990; Nordström, 1990; Alvarez, et al.,

1992), Barcelos et al. (2003) proposed the well-

balanced equation:

()(1)()

t

u

ugudiv guI

u

λ

∇

=∇ − − −

∇

(4)

All these modified models have the same difficulty

of selection of Gauss kernel

G

σ

.As a matter of the

fact, using

Gu

σ

∇∗is a regularization process

which complicates the model and increases

calculation. Furthermore their studies may be more

reasonable if they had paid attention to the

properties of human visual system.

3 ANISOTROPIC DIFFUSION

BASED ON HVS

3.1 Properties of the HVS

As information carriers, all images are eventually

perceived and interpreted by human visual system.

As a result, human vision psychology and

psychophysics play an important role in the

successful communication of image information.

This is an important new area that needs to be

further explored, this paper try to have a first attempt

of integrating human visual information into

anisotropic diffusion model.

(i,j)

(i+1,j+1)

(i,j+1)

(i-1,j)

(i+1,j)

(i+1,j-1)

(i,j-1)

(i-1,j-1) (i-1,j+1)

Figure 1: point (, )ij and its eight neighbors.

Weber’s law (Pratt,1991) is the famous portrait

of the function of human visual system in spatial

domain. This law reveals the universal influence of

the background stimulus

u on human’s sensitivity to

the intensity increment

g

δ

, or the so-called WPD

(

Weber Perceptive Different) which is denoted as

w

. It

claims that the fraction

/gg

δ

is a constant at a

great range of luminance:

/g g const

δ

=

(5)

But when the background luminance is very strong

or week the fraction is not a constant. In other

words, the WPD

w is a function of the background

luminance

g .The expression is as following (Lihua,

2005):

2

20 12

088

88

( ) 0.002( 88) 8 88 138

7( 138)

8 138 255

255 138

g

g

wg g g

g

g

−

⎧

≤<

⎪

⎪

⎪

=−+≤≤

⎨

⎪

−

⎪

+<≤

−

⎪

⎩

(6)

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

62

In(6) a small neighborhood around each pixel,

i.e. a pixel

33× (or55× et al.) region is considered

(see Figure 1). The region consists of the central

pixel

(, )ij which gray level is denoted using

(, )gij and a surround consisting of the eight

neighboring with a mean gray level, we use the

mean gray value, which is denoted using

(, )gij, of

the surrounding eight neighboring points standing

for the background luminance:

1, 0 1, 0

1

(, ) ( , )

8

llkk

gij gi l j k

=− ≠ =− ≠

=++

∑∑

(7)

0 50 100 150 200 250 300

8

10

12

14

16

18

20

Figure 2: the Weber perceptive different function.

3.2 Visual Gradient

As we all know, anisotropic diffusion model bases

on the Gradient of the image. In order to using the

information of human visual system, we integrate

expression (13) into the gradient and define visual

gradient:

() / ()

v

Gg gwg

α

=⋅∇ (8)

where

g∇

denotes the gradient of the image and

()wg

is the WPD in the background g ,

α

is an

adjustable parameters.

From Figure 2 we know that at the background

luminance 88, the WPD is the minimum, that is,

human visual system is most sensitive to the

gradient, so the corresponding visual gradient must

be greater than the gradient. However, at the

background luminance 255, human visual system’s

sensitivity is weakest, so the corresponding visual

gradient must be greater than the gradient.

In fact, in the definition of visual gradient if

() 1wg

α

⋅≡

, the visual gradient degenerates to

general gradient. Therefore the visual gradient is

essentially a generalization of image gradient.

3.3 New Model and its Properties

Base on the definition of the visual gradient which is

closer to human visual system than gradient, we

substitute the visual gradient for gradient in

anisotropic diffusion model’s diffusion function and

get the new model based on the visual gradient:

(( ()) )

() ()

tv

v

udivfGgg

Gg wg g

α

=

∇

⎧

⎨

=

⋅⋅∇

⎩

(9)

where ()

v

Gg, g denote correspondingly the visual

gradient, gray level of an arbitrary point and

g

denotes its eight neighbors’ mean gray level. In the

discrete numerical implementation, they are

substituted by

((,))

v

Ggij , (, )gij and (, )gij .

Diffusion function is a non-increasing smooth

function such that

() 0fs≥ , (0) 1f = and

lim ( ) 0

s

fs

→∞

=

.Two choices suggested by Perona

are

()

2

() exp /

f

ssk

⎡

⎤

=−

⎣

⎦

(10)

and

2

() 1/(1 (/ ))

f

ssk=+ (11)

where

k is a constant to be tuned for a particular

application.

In the proposed method, we use visual gradient

rather than image gradient controlling the

anisotropic diffusion model’s diffusion coefficient.

Ours idea is that in the different background the

gradient is also different and the WPD must obey

some rule. Triggered by this idea, we use the

expression (6) generalizing the gradient to keep to

some extent consistent with human visual system.

4 EXPERIMENTAL RESULTS

We use Perona’s discrete scheme(Perona, 1990) and

obtain the formulation:

[

1

,,

((,))

nn n

ij ij v N S

uu fGij uu

λ

+

=

+∇+∇

]

,

n

EW

ij

uu+∇ + ∇ (12)

where

01/4

λ

<

< for the numerical scheme to be

stable and

, 1,, , 1,,

,

Nij i j ij Sij i j ij

uu u uu u

−+

∇

=−∇=−

,,1, ,,1,

,

E ij ij ij W ij ij ij

uu u uu u

+−

∇

=−∇=−

The conduction function is updated at every

iteration as the function of the visual gradient(10)

(11):

(

)( )

2

(, ) exp (, )/

nn

vv

f

Gij Gijk

⎡⎤

=−

⎢⎥

⎣⎦

A NOISE REMOVAL MODEL WITH ANISOTROPIC DIFFUSION BASED ON VISUAL GRADIENT

63

or

2

((,))1/(1((,)/))

nn

vv

f

Gij Gijk=+

(a) (b)

(c) (d)

Figure 3: different noise removal models’ experimental

results. (a) blurred image by Gauss noised with zero mean

and 0.01 variance, (b) filtered by Perona’s model with

iteration 14,(c) smoothed by ALM model with iteration

25,(d)smoothed by the proposed model with iteration 8.

(a) (b)

(c) (d)

Figure 4: different noise removal models’ experimental

results.(a) noised image by salt & pepper noise which

density is 0.05, (b) smoothed by Perona’s model with

iteration 17, (c) filtered by ALM model with iteration 45,

(d) smoothed by the proposed model with iteration 10.

In the experiments, we applied Perona’s model,

ALM model (Alvarez, et al., 1

992) and the proposed

model to smooth the cameraman image with zero

mean and 0.02 variance noise (Figure 3). In Figure 4

we use salt & pepper noise which density is 0.05

blurred the origin image. Among the different

filtered images the reconstructed image using the

proposed model keep to more extent consistent with

human visual system.

5 CONCLUSIONS

In this work, an anisotropic diffusion model for

image smoothing based on the visual gradient is

presented. Our model uses a visual gradient which is

a generalization of the image gradient. Numerical

results show the proposed method’s performance.

As a tentative study of integrating HVS

information into anisotropic diffusion model for the

first time, the proposed model’s performance is

expected to be improved in the further researches.

6 REFERENCES

Perona, P., Malik, J., 1990. Scale-space and edge detection

using anisotropic diffusion, Pattern Analysis and

Machine Intelligence, IEEE Transactions on, vol. 12,

pp. 629-639.

Nordström, K. N, 1990. Biased anisotropic diffusion: a

unif-ied regularization and diffusion approach to edge

detection, Image and Vision Computing, pp. 318 –

327.

Catté, F., Lions, P. L., Morel, J. M. and Coll, T., 1992.

Image Selective Smoothing and Edge-Detection by

Nonlinear Diffusion, Siam Journal on Numerical

Analysis, vol. 29, pp. 182-193.

Alvarez, L., Lions, P. L. and Morel, J. M, 1992. Image

Selective Smoothing and Edge-Detection by Nonlinear

Diffusion .2, Siam Journal on Numerical Analysis,

vol. 29, pp. 845-866.

Yu-Li, Y. Wenyuan, X., 1996. A. Tannenbaum, and M.

Kaveh, Behavioral analysis of anisotropic diffusion in

image processing, Image Processing, IEEE

Transactions on, vol. 5, pp. 1539-1553.

Barcelos, C. A. Z., Boaventura, M. and Silva, Jr., E. C.,

2003. A well-balanced flow equation for noise

removal and edge detection, Image Processing, IEEE

Transactions on, vol. 12, pp. 751-763,

Pratt.W.K., 1991. Digital image processing, John Wiley &

Sons. New York, 2

nd

edition.

Lihua, G. Jianhua, L. and Shutang,Y,2005. Adaptive

Image Segmentation Combining Region and Boundary

Information. Journal Of Shanghai Jiao Tong

University, Vol.39,pp, 522-526 (in Chinese).

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