Non Linear Image Restoration in Spatial Domain

doi:10.4236/jsip.2011.23029

Paper Menu >>

Journal Menu >>

Journal of Signal and Information Processing, 2011, 2, 211-217

doi:10.4236/jsip.2011.23029 Published Online August 2011 (http://www.SciRP.org/journal/jsip)

211

Non Linear Image Restoration in Spatial Domain

Bushra Jalil, Fauvet Eric, Laligant Olivier

Le2i Laboratory, Universite de Bourgogne, Le Creusot, France.

Email: [email protected]

Received May 10th, 2011; revised June 22nd, 2011; accepted July 30th, 2011.

ABSTRACT

In the present work, a novel image restoration method from noisy data samples is presented. The restoration was per-

formed by using some heuristic approach utilizing data samples and smoo thness criteria in spatial domain . Unlike most

existing techniques, this approach do es n ot require p rio r mode lling o f eith er the image or noise statistics. The p roposed

method works in an interactive mode to find the best compromise between the data (mean square error) and the

smoothing criteria. The method has been compared with the shrinkage approach, Wiener filter and Non Local Means

algorithm as well. Experimental results showed that the proposed method gives better signal to noise ratio as compared

to the previously proposed denoising solutions. Furthermore, in addition to the wh ite Gaussian noise, the effectiveness

of the proposed techn ique has also been proved in the presence of multiplicative noise.

Keywords: Restoration, Nonlinear Filtering, Mean Square Error, Signal Smoothness

1. Introduction

The recovery of a signal fro m observed no isy data, while

still preserving its important features, continues to re-

main a fundamentally elusive challenging problem in

signal and image processing. More importantly, the need

for an efficient image restoration method has grown with

the massive production of digital images of different

types. The two main limitations in any image accuracy

are categorized as blur and noise. The main objective of

any filtering method is to effectively suppress the noise

elements.

Not only that, it is of extreme importance to preserve

and enhance the edges at the same time. Several methods

have been proposed in the past to attain these objectives

and to recover the original (noise free) image. Most of

these techniques uses averaging filter e.g. the Gaussian

smoothing model has been used by Gabor [1], some of

these techniques uses anisotropic filtering [2,3] and the

neighbourhood filtering [4,5] and some works in fre-

quency domain e.g. Wiener filters [4]. In the past few

years, wavelet transform has also been used as a signifi-

cant tool to denoise the signal [6-8]. A brief survey of

some of these approaches is given by Buades et al. [9].

Since the scope of this work is limited to spatial domain,

therefore the proposed method has been compared with

the classical methods for denoising in spatial domain.

Traditionally, linear models have been used to extract

the noise elements e.g. Gaussian filter as they are com-

putationally less expensive. However, in most of the

cases the linear models are not able to preserve sharp

edges which are later recognized as discontinu ities in the

image. On the other hand, nonlinear models can effec-

tively handle this task (preserve edges) but more often,

non linear model are computationally expensive. In the

present work, we attempt to propose a non linear model

with the very less computational cost to restore image

from noisy data samples. The method utilizes data sam-

ples and find the best compromise between the data

samples and smoothness criteria which ultimately result

in giving the denoise signal at a very low computational

cost. We have also presented the comparative analysis of

the present technique with some of the previously pro-

posed method.

The paper is organized as follows. The principle of the

proposed technique is given in Section 2. Section 3 ex-

plains the overview of the restoration method. Applica-

tion on images and comparative analysis is given in Sec-

tion 4 and finally Section 5 concludes the work with

some future perspectives.

2. Principle of the Method

We assume that the given data specify the model:





where ,1,,

ijij ij

yfx ij n (1)

f is the noise free signal, uniformly sampled (e.g., an

image) and ij



is the white Gaussian noise





0,N



Non Linear Image Re st oration in Spatia l Domain

212

In the present work, the given data will always be an

matrix with nn

2

n

. The aim of the current work

is to estimate the function with respect



ij ij

Ffx 



to an estimator ,

 such that:



10log10

ij ij

SNR dBF











(2)

In order to estimate the function F, the method utilizes

the data samples and performs non linear functioning to

estimate the best fit. The filtering has been performed on

each row and column matrix individually and at the final

stage by utilizing filtering in x and y directions yield in

fully denoised image.



2xy

GGG

(3)

where 12 the normalizing is factor,

G is the fil-

tered image in horizontal direction and

G is the fil-

tered image in vertical direction.

3. Restoration Method

In this section, a brief explanation of the proposed de-

noising method (Mse-Smooth) is given. The restoration

method utilizes all sampled points and smoothness of the

signal to estimate the best fit by working in an iterative

mode.

The method is design for one dimensional signal;

therefore it performs non linear filtering on image ini-

tially row by row and then column by column. It means

if the size of the image is , then there would be 2n

one dimensional signals (correspond to the operation in

horizontal and vertical direction). At the final stage, by

using expressio n in given Equation (3) merging of filter-

ing operation in horizontal and vertical direction results

in giving the fully denoised image.

nn

Denoising Method

As explained before, in the present work filtering opera-

tion is performed on each row and column individually.

Therefore let’s define any one dimensional signal as:

SE kMSO k

kkk k

YY YY







 













(4)

Y denotes the one dimensional signal with

samples. K defines the iteration step. 1,l, 2N



is the mean square error estimation of the restored sub-

set with the original signal such that:





Cdefines the smoothness of the reconstructed subset

signal.

SE k





 (6)

In order to find the value of k



and k



in Equation

(4), consider the Taylor series expansion, and for the

given series, to find the minimum mean square error we

want





0;fxdx





 therefore, by simplifying the

Taylor series expansion,



xdf



 (7)

and we know that

1kk

xdx



 (8)

As 1k



 and ,

SE k

fC the variables in

Equation (8) can be replaced such that

SE k



 (9)

Thus,

SE k



 (10)

and we can simplify the equation to



MSE

kkkC

YY Y









 









(11)

Similarly, by using the Taylor series as smoothing cri-

teria, k



is as follows

SO k



 (12)

By using Equation (4), each row wise and column

wise vector were restored individually and result in giv-

ing two nn



matrices,

G and

G respectively. At

the last stage, by using mathematical expression given in

Equation (3), the final denoised image has been restored.

4. Results and Discussions

The test image use in this work is Lena 256 × 256pictur e.

We generated noisy data from clean image by adding

pseudorandom numbers (white gaussian noise) resulting

in signal to noise ration (SNR) of approximately 15 dB

(SNR is defined in Equation (2)). Figure 1 shows the

corrupted Lena image with white Gaussian noise (15 dB)

and the denoising result with the proposed technique as

shown in Figure 2(f). It can be seen from the figure that,

the new method denoised the image reasonably well



SEk k





y

(5)

Non Linear Image Re st oration in Spatia l Domain

213

while keeping the edges preserved.

In order to illustrate the amount of the noise in the data

and the effects of the denoising, we show in Figure 3, a

single row (100) of the image (15 dB white Gaussian

noise), plotted as a curve. It can be seen from the figure

that the method not only smooth the noisy part of the

signal but also preserve the sharp edges or transitions to

good extent. At the same time, the proposed method

performs the whole operations at a very low computa-

tional cost which makes it useful for many real time ap-

plications (maximum time taken for any experiment was

approx 16 sec).

4.1. Comparative Analysis

The human eye is able to decide if the quality of the im-

age has been improved by the denoising method. Figure

2 displays the comparative analysis of the proposed

method with other smoothing filters including Non-Locals

(a) (b)

Figure 1. (a) Original Lena image, (b) Lena image with “white Gaussian noise” SNR of 15 dB.

(a)

(b)

(c)

Figure 2. (a) Original Lena image, (b) Lena image with “white Gaussian noise” SNR of 15 dB, (c) Denoised Lena image with

Visu shrink method, (d) Denoising with Wiener filtering, e) Denoising with Non Local means, f) Denoising with proposed

Mse-Smooth method.

Non Linear Image Re st oration in Spatia l Domain

214

(a) (b) (c)

(d) (e) (f)

Figure 3. (a) Row 100 of original Lena image, (b) Row 100 of noisy Lena image with “white Gaussian noise” SNR of 15 dB, c)

Row 100 of denoised Lena image with proposed Mse-Smooth method.

[10], Wiener filter [4] and classical VISU shrink method

[7]. The experiment has been simulated by adding white

Gaussian noise. It can be seen from the figure that the

VISU shrink results in giving good smoothness but at the

cost of the edges. Not a single edge has been preserved;

this however is not, in the case of NL-means or Wiener

filtering. NL-means and Wiener filtering preserved the

edges reasonably well, but in this case the noise elements

are visible (clearly visible on the background of Lena)

and can be seen with the naked eye as well. The pro-

posed (Mse-Smooth) method gave the best compromise

of the two (blur and noise artefacts) as shown in Figure

4.2. Test with Multiplicative Noise

At the secon d stage, th e method has been tested with the

addition of multiplicative noise. In the case of multipli-

cative noise, variance of the noise is a function of the

signal amplitude. Figure 4 shows a comparison between

one dimensional step like signal corrupted with an addi-

tive white Gaussian noise and multiplicative noise. In the

case of multiplicative noise, the noise variance is higher

when amplitude of the signal is higher as shown in Fig-

ure 4. In relation to the images, noise in bright regions

have higher variations and could be interpreted wrongly

as features in the original image [11]. Thus, it is harder

and more complicated to smooth the noise without de-

grading true image features.

The proposed method has been tested with an addition

of multiplicative noise as well. Figure 5 illustrates the

amount of the multiplicative (speckle 15 dB) noise in the

data samples and the effects of the denoising with the

proposed method, a single row (100) of the image with

15 dB speckle noise, plotted as a curve. It can be seen

from the figure that in this case as well, the method

smooth the noisy part of the signal reasonably well and

also sharp edges are preserved. Figure 6 illustrate the

results on real Lena image. It can be seen from the figure

that the proposed method gave the best performance in

the presence of multiplicative noise. Table 1 summarizes

the numerical results in term of SNR (dB) of the new

method and some of the previously proposed techniques

with different types of noise. We can see that the new

method out performs VISU shrink, NL means and Wie-

ner filtering techniques in different cases.

5. Conclusions

In this work, we have presented a new denoising algo-

rithm, based on the actual noisy image samples. Unlike

other existing techniques, we have not considered mod-

elling of an image or noise characteristics in developing

the approach. Instead we have estimated the best fit of

the signal by utilizing actual noisy data samples and

smothness criteria. At the same time, the proposed non o

Non Linear Image Re st oration in Spatia l Domain215

(a)

(b)

(c)

Figure 4. (a) Synthetic “Step signal”, (b) Step signal with the addition of white Gaussian noise 15dB, (c) Step signal with an

addition of Speckle (multiplicative noise) 15dB.

(a)

(b)

(c)

Figure 5. (a) Row 100 of original Lena image, (b) Row 100 of noisy Lena image corrupted with “Speckle noise” SNR of 15dB,

Non Linear Image Re st oration in Spatia l Domain

216

(a) (b) (c)

(d) (e) (f)

Figure 6. (a) Original Lena image, (b) Lena image with “Speckle noise” SNR of 15 dB, (c) Denoised Lena image with Visu

shrink method, (d) Denoising with Wiener filtering, (e) Denoising with Non Local mean, (f) Denoising with proposed Mse-

Smooth method.

Table 1. Statistical Results in terms of SNR (dB) of “Lena Image” with different types of noise.

Noise Type Input NLM Weiner VISU Shrinkage Mse-Smooth

Gaussian 15.02 18.28 21.49 19.44 21.23

Speckle 17.55 18.75 23.76 20.73 22.32

Salt/Pepper 15.10 15.90 17.23 16.60 20.47

linear image filtering technique works equally well in the

presence of signal dependent nature of multiplicative

noise in spatial domain. The proposed non linear expres-

sion has effectively directed the algorithm in the

smoothing operation at a very low computational cost.

For images, it is important that edges data should be pre-

served. The denoising algorithm presented in this work,

to a large extent has satisfied the constraint that phase

should not be corrupted. The effectiveness of this tech-

nique encourages the possibility of improving this ap-

proach to preserve the edges to more extent.

REFERENCES

[1] A. Bruckstein M . Lind enbaum and M. Fischer, “On Gabor

Contribution to Imag e Enhancem ent. Pa ttern re cogniti on,”

Computer Methods and Programs in Biomedicine, Vol. 27,

No. 1, 1994, pp. 1-8.

[2] J. Malik and P. Perona, “Scale Space and Edge Detection

Using Anisotropic Diffusion,” IEEE Transactions on Pat-

tern Analysis, Vol. 12, No. 7, March 1990, pp. 629-639.

[3] J. M. Morel L. Alvarez and P. L. Lions, “Image Selective

Smoothing and Edge Detection by Nonlinear Diffusion,”

Journal of Numerical Analysis, Vol. 29, No. 1, 1992, pp.

182-193.

[4] L. Yaroslavsky, “Digital Picture Processing—An Intro-

duction,” Springer-Verlag, Berlin, 1985.

[5] R. Manduchi and C. Tomasi, “Bilateral Filtering for Gray

and Color Images,” Proceedings of the Sixth International

Conference on Computer Vision, Bombay, 4-7 January

1998, pp. 839-846.

[6] D. Donoho and R. Coifman, “Wavelets and Statistics,

Chapter Translation Invariant Denoising,” Springer-Ver-

lag, Berlin, 1995, pp. 125-150.

[7] D. Donoho, “Denoising by Soft-Thresholding,” IEEE

Transactions on Information Theory, Vol. 41, 1995, pp.

613-627.

[8] A. Alecu, A. Munteanu, L. Tessens and A. Pizurica,

“Context Adaptive Image Denoising through Modeling of

Curvelet Domain Statistics,” Journal of Electronic Imag-

Non Linear Image Re st oration in Spatia l Domain217

ing, Vol. 17, No. 3, 2008, p. 033021.

doi:10.1117/1.2987723

[9] J. Morel, A. Buades and B. Coll, “On Image Denoising

Methods,” Technical Report CMLA, 2004.

[10] J. M. Morel, A. Buades and B. Coll, “A Non-Local Algo-

rithm for Image Denoising,” IEEE Computer Society

Conference on Computer Vision and Pattern Recognition,

San Diego, 20-26 June 2005, pp. 60-65.

[11] A. Reza and Y. Hawwar, “Spatially Adaptive Multiplica-

tive Noise Image Denoising Technique,” IEEE Transac-

tions on Image Processing, Vol. 11, No. 12, 2002, pp.

1397-1404. doi:10.1109/TIP.2002.804526