OPTIMAL NONLINEAR IMAGE DENOISING METHODS IN

HEAVY-TAILED NOISE ENVIRONMENTS

Hee-il Hahn

Dept. Information and Communications Eng. Hankuk University of Foreign Studies, Yongin, Korea

Keywords: Nonlinear denoising, robust statistics, robust

estimation, maximum likelihood estimation, myriad filter,

Cauchy distribution, amplitude-limited sample average filter, amplitude-limited myriad filter.

Abstract: The statistics for the neighbor differences between the particular pixels and their neighbors are introduced.

They are incorporated into the filter to enhance images contaminated by additive Gaussian and impulsive

noise. The derived denoising method corresponds to the maximum likelihood estimator for the heavy-tailed

Gaussian distribution. The error norm corresponding to our estimator from the robust statistics is equivalent

to Huber’s minimax norm. This estimator is also optimal in the respect of maximizing the efficacy under the

above noise environment. It is mixed with the myriad filter to propose an amplitude-limited myriad filter. In

order to reduce visually grainy output due to impulsive noise, Impulse-like signal detection is introduced so

that it can be processed in different manner from the remaining pixels. Our approaches effectively remove

both Gaussian and impulsive noise, not blurring edges severely.

1 INTRODUCTION

Noise introduced into images via image acquisition

devices such as digital cameras can be adequately

assumed to be additive zero-mean Gaussian

distributed. Such impulsive noise as caused by

transmission of images can be more approximated as

stable distribution. In general, the noise with

zero-mean and independent properties can be easily

removed by locally averaging pixel values. A mean

filter is known to be a maximum likelihood

estimator for additive Gaussian noise and is optimal

in the sense of minimizing mean square error. This

filter, however, tends to degrade the sharpness of the

boundaries between regions of an image although it

effectively removes noise inside the smooth regions.

Basically linear filters can not overcome this

problem. That is why nonlinear methods should be

employed for this purpose. One of the simplest

nonlinear filtering algorithms is the median-based

filter. It is a maximum likelihood estimator for

Laplacian distribution. It has a relatively good

property of preserving fine details except for thin

lines and corners. It is known to be robust to

impulsive noise. Stack filter, weighted median and

relaxed median are among its variations to improve

the performance. Median-based methods basically

select one of the samples in the input window. Thus,

it is known that they can not reduce noise effectively.

Motivated by the above limitations, several kinds of

myriad filters have been proposed, which are known

to be maximum likelihood estimator under Cauchy

distribution (Gonzalez, Arce, 2001), (Zurbach, et al.,

1996). Optimality of myriad filters are presented

under

stable distributions (Gonzalez, Arce, 2001).

(Hamza and Krim, 2001)

proposed mean-relaxed

median and mean-LogCauchy filters by combining a

mean filter with a relaxed median or LogCauchy

filter. They are maximum likelihood estimators

under the assumption that the noise probability

distribution is a linear combination of normal

distribution and heavy-tailed distribution such as

Laplacian or Cauchy distribution. Another popular

methods are the anisotropic diffusion techniques into

which a variety of research has been devoted since

the work of (Perona and Malik, 1990). Recent

researches have shown that nonlinear methods such

as median filters and anisotropic diffusions can be

reinterpreted using the theory of robust statistics

(Huber, 1981). Robust-statistics-based denoising

algorithms are developed, which deal with intensity

discontinuities to adapt the analysis window size

(Rabie, 2005). He chose a Lorenzian redescending

estimator in which the influence function tends to

zero with increasing distance.

424

Hahn H. (2007).

OPTIMAL NONLINEAR IMAGE DENOISING METHODS IN HEAVY-TAILED NOISE ENVIRONMENTS.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 424-429

DOI: 10.5220/0001650604240429

 SciTePress

A large number of image denoising algorithms

proposed so far are limited to the case of Gaussian

noise or impulsive noise, not to both of them. The

algorithms tuned for Gaussian noise or impulsive

noise alone present serious performance degradation

in case images are corrupted with both kinds of

noise. To tackle the problem, an amplitude-limited

sample average filter is proposed. It is also a

maximum likelihood estimator in the density

function which is Gaussian on

()

−

, but

Laplacian outside the region. Its idea is incorporated

into the myriad filter to propose an amplitude-

limited myriad filter. In order to reduce visually

grainy output due to impulsive noise, Impulse-like

signal detection is introduced so that it can be

processed in different manner from the remaining

pixels. Our approaches effectively remove both

Gaussian and impulsive noise, not blurring edges

severely.

After reviewing the problems of finding the best

estimate of a model in terms of maximum likelihood

estimate (MLE), given a set of data measurements,

our estimators are interpreted based on the theory of

robust estimation in both Gaussian and impulsive

noise environment.

2 NOISE STATISTICS

In deriving our robust denoising filter, we employ an

observed image model corrupted with additive

Gaussian and impulsive noise

(1)

i ∈ Z

where

is a zero-mean additive white Gaussian

noise plus impulsive noise.

is uncorrelated to the

image sequence

and

is the observed noise-

contaminated image sequence. In this case,

can

be assumed to have a density function whose tails

are heavier than Gaussian. To ensure the

unbiasedness of the maximum likelihood estimator,

its density function is assumed to be symmetric. The

density function of

is assumed to be Gaussian on

(

)

−

, but Laplacian outside the region It has a

shape of Gaussian distribution with heavier

exponential tails given by

(2)

where, of course,

should be chosen so that the

density

(

)

has unit area by proper adjustment of

and

. Its statistics can be modelled as symmetric

stable

()SS

distribution.

3 OUR PROPOSED FILTERS

3.1 Amplitude-Limited Sample

Average Filter

Let us found out the MLE of the mean of a normal

random variable with known variance from

independent observations. The density function for

independent observations is

. (3)

()

px e

πσ

−−

∑

The MLE of

that maximizes the above density

function is given by

. . (4)

()

arg min

== −

∑∑

The MLE is just the sample mean and

is known to

be a minimum variance unbiased and consistent

estimate. This means that the MLE for estimating

the signal under the additive Gaussian model is a

mean filter. It can be interpreted as optimum filter in

the sense of mean-square errors.

Likewise, when the observations have a density of

Laplacian instead of Gaussian, the density function

for

independent observations is

(5)

and the MLE of

that maximizes the above

equation is given by

(6)

Its MLE corresponds to the median filter which

selects the sample located at the center after

arranging the observations in the ascending order.

Thus, combining the results given in Eq. (4) and (6)

we obtain the MLE of

for the density function

given in Eq. (2).

()

aax

Ce e x

fx Ce x

Ce e x

δδδ

−−−

−

−+

=−≤≤

<−

⎧

⎪

⎨

⎪

⎩

()

px e

−−

∑

arg min

−

∑

n=+,

iii

OPTIMAL NONLINEAR IMAGE DENOISING METHODS IN HEAVY-TAILED NOISE ENVIRONMENTS

425

(7)

The corresponding filter can be easily implemented

(8)

where (9)

We call this filter as an amplitude-limited sample

average filter (ALSAF). The efficacy of the estimate

can be found out as follows,

(10)

In the above equation,

(

)

, given in Eq.(2),

represents the density function for each observation.

Since , Efficacy

has the

maximum value. Thus, the ALSAF given above is

the optimal estimate in terms of maximizing the

efficacy under the above noise environment. The

error norm corresponding to our estimator from the

robust statistics is given by

(11)

This is equivalent to Huber’s minimax norm (Huber,

1981), (Black, et al., 1998). To apply our denoising

filter, we need to choose the variables

and

given in Eq. (2) and Eq. (9), which depends on the

statistics of the noisy images. The value of

inversely proportional to the amount of outliers such

as impulsive noise. If the value of

is equal to the

standard deviation

of the density function given in

Eq. (2), the distribution will be similar to Gaussian,

which means that the outliers rarely exist. Thus,

should be less than

(typically

0.8

). The

probability

that the noise is greater than

computed as

()

arg min

δδ

θθ

≤>

=−+

⎧⎫

⎨⎬

⎩⎭

∑∑

(12)

()

2 2

aax a

pCe e dx e

And the probability

that the noise is less than

(13)

where .

is chosen empirically for each specific image such

that

to optimize the estimate. The ALSAF

is iteratively applied to reduce any residual noise by

estimating the variables

and

from the statistics

of the neighbor differences at each iteration. The

algorithm stops when the residual error between the

current and the next estimate falls below some

threshold at each pixel, which is usually less than

Recall the Perona-Malik (PM) anisotropic diffusion

(Perona and Malik, 1990)

(14)

where ,

∇

denotes divergency and gradient,

respectively. Since the robust estimation can be

posed as:

(15)

where

is the domain of the image. Eq. (15) can

be solved using the gradient descent as follows:

(16)

Comparing Eq. (14) with Eq. (16), we can obtain the

relation

(17)

Thus, our denoising algorithm can be implemented

using PM anisotropic diffusion by selecting the edge

stopping function

(

)

given in Eq. (17) (Black, et

al., 1998).

−

()

∑

()

,..........

,......

gx ax x

=−≤≤

−<−

⎧

⎪

⎨

⎪

⎩

() ()

yfydy

∞

−∞

∞

−∞

′

⎡⎤

⎣⎦

∫

()

(

)

()

=−

′

()

........................

.............

ax x

≤

−>

⎧

⎪

⎨

⎪

⎩

()()

()

pC a a

=Φ−Φ−

()

edΦ=

∫

−

−∞

(

)

{

}

IhII

∇⋅ ∇ ∇

∇

δδδ δ

∞

−−− −

∫

(

)

min

∇

∫

()

∇

′

=∇⋅ ∇

∇

⎧

⎫

⎨

⎬

⎩⎭

()

..................

sgn

....

ak x

≤

′

⎧

⎪

⎨

⎪

⎩

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

426

3.2 Amplitude-Limited Myriad Filter

Similarly, the myriad filter which is the MLE of

location under heavier-tailed Cauchy distributed

noise is defined as

(18)

The behavior of the myriad filter is determined by

the value of

, which is called the linearity

parameter. Given a set of samples

,,,

⋅

⋅⋅

the sample myriad

in Eq. (18) converges to

sample mean

in Eq. (4), as

(Gonzalez,

Arce, 2001). It is proposed in this paper that outliers

which are samples outside the region

(

→∞

)

−

, are

limited, as shown in Eq. (9). That is, the sample

myriad is computed as

(19)

where

()

⋅

is as given in Eq. (9). This filter is

named an amplitude-limited myriad filter (ALMF).

Its sample myriad

results in amplitude-limited

sample average

depicted in Eq. (8), as

.This can be easily proved in the same way

as the myriad filter converges to a mean filter as

as given in (Gonzalez, Arce, 2001).

k →∞

3.3 Filtering Scheme

As mentioned above, if the given image pixel is

known to belong to one of the smooth regions

Gaussian noise can be reduced by a mean filter This

filter, however, tends to degrade the sharpness of the

boundaries between regions of an image if it belongs

to the boundary regions. This problem can be

reduced effectively by the ALSAF, which however,

produces visually grainy output as the amount of

impulsive noise increases. Thus, our proposed

approach utilizes the statistics of the samples in the

window. The parameter in Eq. (19).is determined

according to the presence of impulsive noise in the

window.

3.3.1 Processing of Impulsive Noise

Deciding which pixels in an image are replaced with

impulsive noise is not clearly defined yet. Especially

in cases they are also corrupted with Gaussian noise,

the problem will be very complicated. Fortunately,

image pixel values does not vary severely from its

surrounding pixels even in the boundary regions.

Thus, each pixel isolated with its neighbors is

detected as an impulse-like pixel.

()

In order to decide how impulse-like each pixel is,

the pixels within the window are arranged in the

ascending order for each pixel location, and it is

decided whether the pixel is located at some

predefined range as given in Eq. (20),

[

]

[

]

{

}

iii

xww

otherwise

∈

⎧

⎨

⎩

(20)

where

[

]

is the k th-order statistics of the

samples in the window of size , that is

2N +1

[

]

[

]

[

]

12 2

ii i

ww w

≤≤⋅⋅⋅≤

lN u N

(21)

and and are such that

l u

1121

≤

≤+≤≤ +

. If

corresponding

to the pixel

equals 0, then the ALSAF or ALMF

with a large value of is applied to the samples in

the window because it is more probable the pixel

belongs to smooth regions. However, when

the pixel is regarded as impulse-like if the mean of

absolute values of its neighbour differences (MAD),

as given in Eq. (22) is above the predefined

threshold,

AD x y

∈Ω

−

∑

(22)

where is the center pixel and

is the set of its

neighbors. It is verified experimentally to be a good

indicator of impulsive noise. Its idea is borrowed

from (Garnett, et al., 2005). Fig. 1 and Fig. 2 depict

mean MAD values on whole image pixels as

functions of types of noise and its amount. Impulsive

noise pixels have much larger mean MAD values

than the uncorrupted pixels or the pixels corrupted

with Gaussian noise. When impulsive noise exists at

some pixel in Lena image, its mean MAD value is

arg min

= +−

∏

()

(

)

arg min

kgx

= +−

∏

OPTIMAL NONLINEAR IMAGE DENOISING METHODS IN HEAVY-TAILED NOISE ENVIRONMENTS

427

127, which does not vary with the amount of

Gaussian noise. In our method, the image pixels

whose MAD values exceed 80 are classified as

impulsive noise. The pixels decided to be impulse-

like are separated to process with an ALMF, whose

parameter as given in Eq. (19) is small. In case

there is no impulse within the window,

is set to a

large value so that the ALMF may function as an

ALSAF.

Figure 1: Mean MAD values as a function of standard

deviation of Gaussian noise.

Figure 2: Mean MAD values as a function of probability

of impulsive noise.

4 EXPERIMENTAL RESULTS

The widely used gray-scale Lena image is selected

to test our proposed method. Impulsive noise as well

as Gaussian noise are injected to the test image. In

other words, the pixel corrupted with Gaussian noise

is replaced randomly with impulse, which has the

value of 0 (“black”) or 255(“white”) with equal

probability. Simulations are carried out for a wide

range of noise density levels. The performance of

our denoising filter is evaluated by way of mean-

square-error (MSE) metric and peak signal-to-noise

ratio (PSNR) given by

(23)

where

is the standard deviation of the residual

errors

(24)

()

∈Ω

=−

∑

In the above equation,

represents the number of

pixels in the image.

(a) (b)

Figure 3: (a) Corrupted Lena image degraded by Gaussian

noise of variance , with a measured

924

18.5

SNR dB

(b) PM anisotropic diffused image after

10 iterations with and

153.3

26.3

SNR dB=

and

137.6

26.8

SNR dB

(d) Output of the

ALMF with and

155.2

26.2

SNR dB=

Fig. 3 shows the simulation results when gray scale

image of size

256 256

is corrupted with additive

Gaussian noise of variance

924

)(20

SNR dB

. Obviously, both our methods

suppress additive Gaussian noise without severely

destroying the fine details compared with PM

equation in spite of the fact that there are no

significant differences in their PSNR values.

Simulation results are depicted in Fig. 4 when the

Lena image is corrupted with both Gaussian noise of

variance

and 10% of impulsive noise

900

(20PSNR dB)

. Simulation results show that the

ALSAF is not effective in removing impulsive noise,

while the myriad filter can be extended to reduce

255

20log

PSNR

⎛⎞

⎜⎟

⎝⎠

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

428

both Gaussian and impulsive noise by limiting the

amplitude of samples outside predefined range as

given in Eq. (9). This ALMF tends to preserve the

fine details while reducing both Gaussian and

impulsive noise.

(a) (b)

Figure 4: (a) Lena image corrupted with both Gaussian

noise of

and impulsive noise of

10%

with a measured residual variance and

2059.3

15.0

SNR dB=

, (b) Output of the ALSAF after 10

iterations with and

359.6

22.6

SNR dB=

(c)

Output of myriad filter with and

557.9

20.67

SNR dB=

(d) Output of ALMF with

and

234.7

24.42

SNR dB=

5 CONCLUSIONS

Optimal nonlinear filter which maximizes the

efficacy under mixed Gaussian noise environment is

derived. This filter effectively can be implemented

using PM anisotropic diffusion by selecting the

appropriate edge stopping function. However, it

produces visually grainy output as the amount of

impulsive noise increases. Thus, impulse-like signal

detection is introduced to process impulsive pixels

differently from the remaining pixels. For this

process, a myriad filter is selected, which is a

maximum log-likelihood estimator of the location

parameter for Cauchy density. The filter is known to

outperform median-based filters in removing

impulsive noise. By combining ALSAF which is a

MLE in mixed Gaussian noise with a myriad filter,

the resulting filter (ALMF) effectively removes both

Gaussian and impulsive noise, preserving the fine

details.

REFERENCES

Rabie, T., 2005. Robust Estimation Approach for Blind

Denoising. IEEE Trans. Image Process., vol. 14, No.

11, pp. 1755-1765.

Black, M. J., Sapiro, G., Marimont, D. H., 2005. Robust

Anisotropic Diffusion. IEEE Trans. Image Process.,

vol. 14, No. 11, pp.421-432.

Huber, P., 1981. Robust Statistics, New York: Wily.

Perona, P., Malik, J., 1990. Scale-Space and Edge

Detection Using Anisotropic Diffusion. IEEE Trans.

PAMI, vol. 12, No. 7, Jul., pp. 629-639.

Hamza, A. B., Krim, H., 2001. Image Denoising: A

Nonlinear Robust Statistical Approach. IEEE Trans.

Signal Process., vol.49, No.12, pp.3045-3054.

Garnett, R., Huegerich, T., Chui, C., He, W., 2005. A

Universal Noise Removal Algorithm With an Impulse

Detector. IEEE Trans. Image Process., vol. 14, No. 11,

pp. 1747-1754.

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Myriad Filter in Practical Impulsive-Noise

Environments. IEEE Trans. Signal Process., vol.49,

No.2, pp. 438-441.

Zurbach, P., Gonzalez, J. G., Arce, G. R., 1996. Weighted

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