IMAGE RESTORATION

A New Explicit Approach in Filtering and Restoration of Digital Images

Pejman Rahmani, Beno

ıt Vozel and Kacem Chehdi

TSI2M, Universit

e de Rennes 1 - ENSSAT, BP 80518, 22305 Lannion Cedex, France

Keywords:

Inverse Problems, Image Restoration, Filtering, Deconvolution.

Abstract:

Image restoration, in presence of noise, is well known to be an ill-posed inverse problem. Deconvolution of

blurry and noisy digital images is a very active research area in image processing. This paper introduces a

novel approach composed of two optimized sequential stages of image processing: denoising followed by

deconvolution. In the ﬁrst stage, the denoising ﬁlter and the number of iteration are chosen in order to obtain

the best value of the usual criteria and the good recovering of the blurry image. We assume that the statistics

of the noise are previously estimated. In the second stage, a deconvolution method is applied on an almost

noise free version of the blurry image. Compared with the classical deconvolution methods, the numerical

experiments of proposed method, appear to give signiﬁcant improvement. The preliminary results of the new

cascade approach are very encouraging as well.

1 INTRODUCTION

It is well-known that the deconvolution of degraded

images is a difﬁcult and challenging problem. Be-

cause it is hard to recover the convolved original im-

age, uniquely from the observed data. A fundamental

issue in image restoration is blur removal from noisy

observation.

The mathematical discrete direct model of obser-

vation is represented by the following equation

g = h∗ f+ n (1)

where the PSF is presented by h and h∗ f means the

output of the convolutive linear system. The term n

in the equation (1) is non-correlated and independent

additive observation noise. It is known that estimating

the (unknown) true image f from observed image g is

an ill-posed problem even if the PSF is known. The

knowledge of the degradation model is, in general, in-

sufﬁcient to obtain satisfactory results. The blurry im-

ages are often disturbed and the process of restoration

is eminently unstable in the presence of noise.

Many methods have been reported for restoring

the degraded image under the assumption that the blur

operator is exactly known (Biemond et al., 1990). The

basic involved operation is simply a deconvolution

process that faces the usual difﬁculties related to the

noise and the ill-conditioning of the blur operator.

Many of the proposed methods are, therefore,

structured in the context of regularization procedures

to make the problem of inversion well-posed. The so-

lution is, in general, regularized by introducing con-

straints translating prior knowledge on the original

image and/or the PSF and/or the noise. Several ap-

proaches of regularization based on minimization of

least squares error are possible in order to integrate

a priori knowledge into the model of inversion. In

this case, the a priori knowledge is expressed through

the terms which are added to the basical a posteriori

term of the least squares error. These terms represent

a penalization, or a constraint on the restored solution.

The mathematical formulation in the regularized im-

age recovery problem, is stated as follows :

f = argmin

kg− (h∗ f)k

∑

(f)

(2)

where the hyperparameters λ

determine the weights

of the terms of regularization Φ

in the process of in-

version.

This regularized inversion presents some disad-

vantages. In most of the methods of regularization,

the knowledge of the noise is not directly translated in

the term of regularization Φ

. Taking into account of

noise is often weighed by the hyperparameters λ

. It is

not easy to ﬁnd a value of λ

high enough to limit the

inﬂuence of noise, and at the same time, weak enough

200

Rahmani P., Vozel B. and Chehdi K. (2007).

IMAGE RESTORATION - A New Explicit Approach in Filtering and Restoration of Digital Images.

In Proceedings of the Second International Conference on Signal Processing and Multimedia Applications, pages 196-199

DOI: 10.5220/0002140501960199

 SciTePress

to keep the maximum of details on original image.

In some methods, like the stochastic approaches, it

is possible to estimate the hyperparameters explicitly

(Mohammad-Djafari, 1996), (Jalobeanu et al., 2002),

but that requires an important computing time.

In the case of non-linear regularization, like the

variational regularization methods, the process of in-

version is, in fact, a spatial-variant deconvolution. In

other words, the response of the reconstructed inverse

ﬁlter depends on the local properties (edges, uniform

areas, etc.) of the restored image. This is contradic-

tory to the two-dimensional direct model of observa-

tion which occurs in linear optical system character-

ized by a spatial-invariant impulse response.

Recent research works (Bronstein et al., 2005),

(Nikolova et al., 1998), (Park and Kang, 2006),

(Molina et al., 2003), (Chantas et al., 2006), (Molina

et al., 2006), (Likas and Galatsanos, 2004), have been

developed to modify the terms of regularization in or-

der to attenuate the noise over the uniform areas and

to avoid the smoothing of the edges.

In the section 2 of this paper, we brieﬂy mention

the main idea of our approach. The numerical quan-

titative and visual experimental results of each stage

are presented in subsections 2.1 and 2.2. Finally, we

conclude the paper in section 3 by giving thoughts on

future research.

2 PROPOSED APPROACH AND

EXPERIMENTAL RESULTS

We split the restoration of blurry and noisy images

into two sequential stages: denoising and deconvolu-

tion. It consists of applying ﬁrst an algorithm of ﬁl-

tering adapted to the nature of the noise and then per-

forming a deconvolution process on the ﬁltered im-

age.

The critical stage of the proposed approach is

noise elimination. Because it has to accomplish the

preservation of edges and ﬁne details. But, it is also

important to estimate the blurry image h ∗ f without

altering it, nor the PSF. For these reasons, the choice

of the ﬁltering method is of the primary importance

for our approach. To select the best ﬁlter, we have

performed a comparative study of a set of represen-

tative ﬁlters requiring only the a priori knowledge on

the noise. After this ﬁrst ﬁltering stage, two decon-

volution processes (non-regularized and regularized)

are evaluated.

2.1 Denoising Process

Six iterative ﬁlters were retained: Koenderink’s ﬁlter

(Koenderink, 1984), ﬁlter of Rudin et al. (Rudin et al.,

1992), ﬁlter of Chan et al. (Chan et al., 2001), ﬁlter of

Perona-Malik (Perona and Malik, 1990), Lee’s ﬁlter

(Lee, 1980) and ﬁlter of Beaurepaire et al. (Klaine,

2004). For all these ﬁlters, we follow an iterative

denoising strategy for the purpose of evaluating the

PSNR of the ﬁltered images in each iteration. Invari-

ably, these ﬁlters require the determination of a sig-

niﬁcant parameter in the process. This parameter is

the optimal iteration where the PSNR of the ﬁltered

image is maximum.

The experiments were carried out on various se-

ries of images, synthetically degraded (defocusing

blur with different diameters: 5, 7 and 9 and addi-

tive gaussian noise with different levels of standard

deviation: 10, 14 and 16). Results are assessed ﬁrst

globally and then, locally. The global evaluation is

obtained by calculing the three usual criteria MAE,

MSE and PSNR for each ﬁltering iteration number.

These criteria are also considered to determine the in-

ﬂuence of the estimator of h∗f on the ﬁnal estimations

of PSF and f. For the local evaluation, the same crit-

era are selected but in three different zones: a zone

containing the pixels of contours, a zone around these

contours and the areas out of these two zones (cf. Fig-

ure 1).

(a) (b)

Figure 1: Local evaluation zones on degraded [LA-

CORNOU]. image (defocusing blur d = 7, σ

= 14). (a)

Detected edges; (b) the zones around the edges(white) and

the areas of out (black).

The PDE-based ﬁlters, Koenderick, Rudin et al.,

Chan et al., and Perona-Malik, give better results and

among them, the ﬁlter of Chan et al. is slightly better

with a low iteration number (cf. Table 1).

The results show that after a reasonable number

of ﬁltering iterations, the change in the PSF remains

weak independently of the used denoising method and

the selected initialization. These results remain valid

whatever the size of PSF and the level of noise. The

ﬁlter of Chan et al. gives appreciably better results.

This ﬁlter will be retained for the denoising stage.

IMAGE RESTORATION - A New Explicit Approach in Filtering and Restoration of Digital Images

201

Table 1: Global and local evaluations of ﬁlters on degraded [LACORNOU] image (defocusing blur d = 7, σ

= 14): PSNR

values of

h∗f corresponding to the optimal iteration number of ﬁltering, calculated on the whole of image, on the contours,

on the zones around contours and on the area out of them.

h∗f

contours zone of contours zone out of contours

ﬁlter iteration PSNR iteration PSNR iteration PSNR iteration PSNR

Koenderink 27 34.812 16 13.149 24 36.712 37 39.765

Rudin et al. 103 34.255 109 13.368 106 35.643 100 39.90

Chan et al. 4 35.124 3 13.109 3 36.875 5 40.341

Perona-Malik 20 33.635 23 13.223 18 34.887 24 39.789

Lee 17 31.439 4 12.877 15 32.611 26 38.559

Beaurepaire et al. 2 30.655 0 12.789 1 32.549 2 35.583

2.2 Deconvolution Process

In this stage, we assume that the PSF is known. Two

different deconvolution processes are examined. For

both of them, an algorithm of conjugate gradient (CG)

is considered to minimize the least squares criteria ac-

cording to equation (2). For the ﬁrst one, we minimize

the least squares criteria without any prior knowledge

on the original image. In this case, no regularization

is considered in order to study the inﬂuence of ﬁlter-

ing on the process of inversion and the quality of the

restored image. For the second method, we regularize

the process of deconvolution by penalizing highly os-

cillatory solutions. The motivation for using the reg-

ularization is due to the fact that the estimator of h∗ f

introduces a residual error. Thus, the inﬂuence of the

residuals of the error on the ﬁltered image can be re-

duced by slightly regularizing the process of inver-

sion.

From numerous simulations, the experimental re-

sults show that when the level of noise is low, the non-

regularized deconvolution gives slightly better results

(cf. Table 2). The regularized method gives better

results for higher level of noise. In both cases, the re-

sults of the proposed scheme,

h∗ f), are better ac-

cording to the retained objective criteria, and com-

pared to the results of the restoration obtained directly

from observed image,

f(g) (cf. Table 2, iteration num-

ber = 0, i.e. without ﬁltering).

These results are also conﬁrmed by the visual

quality of the estimated images (Figure 2). Edges and

details of the restored image are preserved.

It is clear that the optimal iteration number for

stopping of the ﬁltering depends strongly on the level

of the noise and also the size of the PSF. As observed

in Table 2, in the presence of noise (σ

6= 0), the re-

sults of restoration by using the ﬁlter of Chan et al. are

systematically better than those without ﬁltering (iter-

ation number= 0) for the different levels of the noise.

The quantitative results show that the ﬁlter of Chan et

al. associated with the regularized restoration method

gives better PSNR with high level of noise. In the

case of low level of noise, the non-regularized method

gives slightly better PSNR values. The numerical ex-

periments with other ﬁlters conﬁrm the same results.

(a) (b)

(e) (f)

Figure 2: (a) original [LACORNOU] image; (b) blurry im-

age y (defocusing blur d = 7); (c) blurry and noisy image

(defocusing blur d = 7, σ

= 14); (d)

f(g) estimated from

observed image; (e) ﬁltered image

h∗f obtained after 4 iter-

ations by the ﬁlter of Chan et al.; (f)

h∗f) obtained from

image (e).

SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

202

Table 2: Usual criteria values for non-regularized and reg-

ularized

f obtained from

h.∗f from degraded [LACORNOU]

image (defocusing blur d = 9, and additive gaussian noise

with various standard deviation levels σ

=0, 8, 12 and 16)

ﬁltered by ﬁlter of Chan et al.

h∗ f

non-regularized

f regularized

iteration PSNR PSNR PSNR

0 0 Inf 30.861 23.907

0 30.046 10.952 23.101

3 39.526 23.663 23.534

0 26.566 8.790 22.219

4 37.127 22.709 23.302

0 24.134 7.837 21.255

5 35.495 22.005 23.062

3 CONCLUSION AND FUTURE

WORK

Both the numerical performance and visual evalua-

tion of the results obtained by the proposed restora-

tion method are signiﬁcantly more favorable than that

of the classical method without preliminary noise re-

duction. The numerical results indicate that the pro-

posed scheme is quite robust and the image f can be

recovered under presence of the noise even if it is high

level. The quality of the restored image depends vig-

orously on the estimation of h∗ f, which means, the

choice of the ﬁlter and the number of iteration are

two important factors to get the satisfying results. We

have evaluated a ﬁnite set of representative ﬁltering

methods in order to select the optimal one. The ﬁlter

of Chan et al. achieves the best compromise between

the quality of the ﬁltering result and the number of

iterations involved.

Primary results based on the estimation of the

standard deviation of the ﬁltering residuals are en-

couraging to solve the problem of the determination

of the optimal iteration number of ﬁltering.

ACKNOWLEDGEMENTS

Supported by the European Union. Co-ﬁnanced

by the ERDF and the Regional Council of Brittany

through the Interreg3B project number 190 PIMHAI.

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