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)

Copyright © 2011 SciRes. JSIP

211

Non Linear Image Restoration in Spatial Domain

Bushra Jalil, Fauvet Eric, Laligant Olivier

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

Email: bushra.jalil@u-bourgogne.fr

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

2

0,N

.