an image restoration by fusion

*Corresponding author. Tel.: #64-6-350-5799; ext: 2476;fax: #64-6-350-5604.E-mail address: [email protected] (T.D. Pham).�Formerly with School of Computing, University of

Canberra, ACT 2601, Australia.

Pattern Recognition 34 (2001) 2403}2411

An image restoration by fusion

Tuan D. Pham*��

Institute of Information Sciences and Technology, Massey University, Private Bag 11222, Palmerston North, New Zealand

Received 23 February 2000; accepted 16 October 2000

Abstract

To deal with the problem of restoring images degraded with Gaussian white noise, the mean and adaptive Wiener"lters are the most common methods to be implemented. Although these methods are both lowpass in character, theyyield di!erent results on the same problem. The mean "lter reduces more noise than the adaptive Wiener but also blursthe image edges, whereas the adaptive Wiener "lter can preserve edge sharpness but reduces less noise than the mean"lter. Instead of trying to design a single mathematical technique to have the advantages of both methods, which isusually theoretically di$cult, we propose an alternative solution to this image restoration by fusing multiple image "ltersusing the mean, Sobel, and adaptive Wiener "lters. Performance of the fusion algorithm is based on both redundant andcomplementary information provided by di!erent "lters. Several experimental results show the e!ective application ofthe proposed approach. � 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Image restoration; Spatial "lters; Fusion; Fuzzy integral; Gaussian white noise

1. Introduction

The goal of image restoration is to bring back theoriginal form of the noise-degraded image by studyingthe characteristics of the image signal and noise. Fromthis standpoint, image enhancement and image restora-tion are viewed di!erently. The enhancement of imagesdoes not require information how an image is degraded,and most algorithms for image enhancement are simpleand heuristic. Whereas the restoration of images arebased on more mathematical and complex models.Topics on image restoration have been extensively dis-cussed in literature [1}3]. In this paper, we will focus ourinvestigation on restoring gray-level images which aredegraded by Gaussian white (additive) noise. The mostcommon approaches for restoring such degraded imagesare the mean, and adaptiveWiener "lters. The mean "lteris the simplest spatial "lter whose performance is based

on lowpass "ltering to remove high spatial frequenciesfrom an image. The mean "lter is known to be the bestone for removing Gaussian noise from images. However,the disadvantage of the mean "lter is that it blurs sharpimage edges. The adaptive Wiener "lter performs onlocal image regions from which space-variant "lters aredesigned. Therefore the blurring e!ect on image edges isreduced but, under this condition, the adaptive Wiener"lter may only not reduce but highlight the noise presentat the high-frequency parts of an image. These twomethods can yield di!erent results with their own e!ec-tiveness. Instead of designing a single complex "lter [4]that is hoped to overcome shortcomings of individual"lters, we attempt to take into account the di!erentadvantages of the two methods for restoring degradedimages by developing a mathematical model for fusingthese processed image data where the Sobel image isimplicitly integrated to indicate the locations of imageedges. There are a number of methods for fusing multiplesources of data such as Bayesian inference, Dempster}Shafer rules, voting methods, neural networks, andfuzzy-set algorithms [5}8]. In this study we apply thenotion of fuzzy measures and the Choquet fuzzy integralto combine the sources of information obtained fromdi!erent image "lters, because it has been shown that

0031-3203/01/$20.00 � 2001 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.PII: S 0 0 3 1 - 3 2 0 3 ( 0 0 ) 0 0 1 6 4 - 3

fuzzymeasure is a generalization of belief and plausibility(Dempster}Shafer theory), possibility, and probabilitymeasures [9,10]. Fuzzy integrals are generalized meanoperators, range betweenmin and max. Therefore the useof fuzzy integrals are suitable for the information fusionand multicriteria decision problems. Fuzzy integrals arealso weighted operators whose weights are de"ned notonly on di!erent attributes, but also on all the subsets.This allows the representation of importance and interac-tion between attributes. Furthermore, it is believed thatfuzzy integrals are the only operators at present whichcan model this type of interaction [11].In the following sections we will brie#y describe the

mean, adaptive Wiener, and Sobel image "lters beforepresenting a fusion scheme for image restoration. Experi-ments are carried out to validate the application of theproposed approach. We also discuss further directionsfor future research.

2. Image 5lters

2.1. Mean xlter

Linear spatial "ltering is based on two-dimensionalconvolution of an image f (x, y) with the impulse responseh(x, y). The "ltered or processed image p(x, y) is de"ned as

p(x, y)"��

��

f (x!i, y!j) * h(i, j), (1)

where 0)x!i(N, 0)y!j(M, i, j3= which isa set of pixels, and N�M is the size of the image.For the 3�3 "lter mask, the spatial convolution for

the mean or average "lter as de"ned by Eq. (1) is reducedto

p(x, y)"1

9

��

��

f (x!1#j, y!1#i), (2)

which means that the processed pixel is the average of itseight neighboring pixels with center pixel at the coordi-nate x, y.

2.2. Adaptive Wiener xlter

The Wiener "lter is known as a space-invariant "lterbecause it assumes that the characteristics of the signaland noise do not change over di!erent regions of theimage. The adaptive Wiener "lter is a space-variant "lterthat changes its "ltering performance throughout di!er-ent local regions of a degraded image. In low-detail orhomogeneous image regions where the local variancesare small, the adaptive "lter performs a large amount oflowpass "ltering in order to reduce as much noise aspossible. In high-detail image regions where local vari-ances are large, it performs no or small amount of

lowpass "ltering in order to preserve the image edges.One speci"c algorithm to adaptively design the Wiener"lter is as follows [12]. Assuming that the n(x, y) iszero-mean white (additive) noise with variance ��

�. Then

its power spectrum P�(�

�,�

�) is

P�(�

�,�

�)"��

�. (3)

The local signal f (x, y) which is assumed stationary ismodeled by

f (x, y)"m�#�

�w(x, y), (4)

where m�and �

�are the mean and standard deviation of

f (x, y) and w(x, y) is zero-mean white noise with unitvariance.Within a local region, the frequency response of the

Wiener "lter H(��,�

�) is

H(��,�

�)"

P�(�

�,�

�)

P�(�

�,�

�)#P

�(�

�,�

�),

"

��

��#��

�

. (5)

The impulse response h(x, y) from the above equation isscaled by

h(x, y)"��

��#��

�

�(x, y), (6)

From Eq. (6), the local processed image p(x, y) can beexpressed as

p(x, y)"m�#[g(x,y)!m

�]

��

��#��

�

�(x, y),

"m�#

��

��#��

�

[g(x,y)!m�], (7)

where g(x, y) is the degraded image. If the noise variance��is not known, one possible approach is to approxim-

ate ��to the average of all the local estimated variance.

2.3. Sobel xlter

An edge within a gray-scale image can be indicated bychange in the gray levels from one region to another.Therefore, the derivative of the gray-level change asa function of the x, y position can be used to detect thepresence of an edge. This two-dimensional vector ofderivative known as the gradient is given by

�f (x, y)"�f�(x, y)

f�(x, y)�"�

�f (x, y)/�x

�f (x, y)/�y�. (8)

The magnitude of the gradient is

��f (x, y)�"�f�(x, y)�#f

�(x, y)�. (9)

2404 T.D. Pham / Pattern Recognition 34 (2001) 2403}2411

For a discrete approximation of Eq. (9) that can beused for non-directional edge detectors, f

�(x, y) and

f�(x, y) are given by

f�(x, y)"f (x, y) * h

�(x, y),

f�(x, y)"f (x, y) * h

�(x, y). (10)

For a 3�3 spatial "lter mask with pixel coordinates:f (x!1, y!1), f (x, y!1), f (x#1, y!1), f (x!1, y),f (x, y), f (x#1, y), f (x!1, y#1), f (x, y#1), f (x#1,y#1). The Sobel gradient edge detector based on Eqs.(9) and (11) with h

�(x, y) and h

�(x, y) for the 3�3 mask are

given by

h�(x, y)"(!1, 0, 1,!�2, 0,�2,!1, 0, 1)

and

h�(x, y)"(!1,!�2,!1, 0, 0, 0, 1,�2, 1).

3. Fusion model for image restoration

3.1. Fusion scheme

The fusion of image data given by the mean, Sobel, andadaptive Wiener "lters for noise reduction can be impli-citly expressed as

c(x, y)"F[m(x,y),w(x, y), s(x,y)], (11)

where c(x, y) is the fused image, F is a fusion operator,m(x,y), w(x, y), and s(x,y) are the mean, adaptive Wiener,and Sobel images, respectively.At local image regions with low variances, both mean

and adaptive Wiener "lters perform the same lowpass"ltering, and thus the fusion of these two sources ofinformation is rather redundant. The information thatcan be considered as complementary for fusing is atimage edges, where the adaptive Wiener "lter tries topreserve the edges and may highlight noise. Therefore theedge-image data provided by these two "lters will befused so that both the preservation of image edges andnoise reduction can be both obtained. Image edges,which can be obtained by applying the Sobel "lter, areused to guide the fusion process of the mean and adaptiveWiener images. In order to avoid much noise present inthe image, the Sobel edge detection is carried out on themean image.Let m(x, y), w(x, y) be in the range [0,1], and Z"

�m�(x, y),w

�(x, y)�"�z

�: i"1, 2� where m

�(x, y), and

w�(x, y) are the mean, and adaptive Wiener images at the

Sobel edges. We can now extend the fused image given inEq. (11) as follows:

c(x, y)"m(x, y)�z (x, y), (12)

where � is the override operator, i.e. m(x, y) be overrid-den by z (x, y), and z (x, y) is also the fusion of the mean

and adaptive Wiener image at the edges indicated by theSobel "lter. From Eq. (12) it can be explained in thatwhen the information sources become redundant, onlythe best single evidence, m(x, y), is taken into considera-tion; and when they are complementary, both pieces ofevidence, m

�(x, y) and w

�(x, y), are fused together.

Using the fusion operation of the fuzzy integral, thefused image z (x, y) is given by

z (x, y)"��

(z�!z

��) g(A

�), (13)

where z�"0, z

�)z

�, g(A

�) is fuzzy measure of A

�, which

will be subsequently explained, A�"�z

�, z

��, and

A�"�z

��.

Similarly the neighboring pixels of m�(x, y) and w

�(x, y),

denoted as m��(x, y) and w

��(x, y), respectively, can be also

included in the fusion using Eq. (13), that is we haveZ"�m

��(x, y),w

��(x, y)� and the fuzzy measures can be

the same as given for m��(x, y) and w

��(x, y).

3.2. Fuzzy measure

Let > be an arbitrary set, and B be a Borel "eld of >.A set function g de"ned on B is a fuzzy measure if itsatis"es the following conditions:

1. Boundary conditions: g(�)"0, g(>)"1.2. Monotonicity: g(A))g(B) if ALB, and A,B3B.

A g�-fuzzy measure is also proposed by Sugeno [13]which satis"es another condition known as the -rule('!1):

g(A�B)"g(A)#g(B)#g(A)g(B), (14)

where A,BL>, and A�B"�.Let >"�y

�,y

�,2, y

��, and the fuzzy density of the

g�-fuzzy measure be de"ned as a function g: y�3>P

[0,1] such that g�"g�(�y��), i"1, 2,2,m. With bound-

ary condition g(>)"1, the constant can be determinedby solving the following equation:

#1"

� ��

(1#g�). (15)

It was proved [10] that for a "xed set of g�, 0(g

�(1,

there exists a unique root of '!1, and O0, usingEq. (15). And also from Eq. (15) it can be seen that if thevalues of g

�are known, then can be calculated.

3.3. Fuzzy integral

Let (>,B, g) be a fuzzy measure space and f :>P[0,1]be a B-measurable function. The discrete Choquet inte-gral [14] of f with respect to a fuzzy measure g is de"ned

T.D. Pham / Pattern Recognition 34 (2001) 2403}2411 2405

as follows:

�

f (y)dg( ) )"��

[f (y�)!f (y

��)] g(A

�), (16)

where f (y�)"0, 0)f (y

�))f (y

�)2)f (y

�))1, (if

not, the elements of > are rearranged to make thisrelation hold), A

�"�y

�,y

��,2, y

��, and g(A

�) can be

recursively calculated in terms of the g�-fuzzy measure asfollows:

g(A�)"g

�#g(A

��)#g

�g(A

��), 1(i)m. (17)

From now it can be seen that the fused image z (x, y)de"ned by Eq. (13) is modeled using the Choquet integralgiven in Eq. (16). For the calculation of z (x, y), the priorinformation needs to be given is the fuzzy densities whichstand for the degrees of importance assigned to the "lters.That is the greater the value of the fuzzy density ofthe "lter is, the more important attribute the "lter has.Since only two sources of information (mean and adap-tive Wiener images) are dealt with in this fuzzy fusionmodel, the calculation of fuzzy measures for the Choquetintegral is straightforward. That is, for w

�(x, y))m

�(x, y),

we have z�"w

�(x, y), and z

�"m

�(x, y). For m

�(x, y))

w�(x, y), we have z

�"m

�(x, y), and z

�"w

�(x, y). Thus the

fuzzy measures are obtained by having g(A�)"

g(�z�, z

��)"1, and g(A

�)"g(�z

��) which is either the

fuzzy density of the mean or adaptive Wiener "lter.

3.4. Outline of fusion algorithm

The proposed fusion algorithm can be summarized asfollows:

1. Given degraded image f (x, y).2. Obtain mean image m (x, y).3. Obtain adaptive Wiener image w(x, y).4. Obtain Sobel image s(x, y) using m(x, y).5. Mark m(x, y) and w(x, y) at s(x,y) as m

�(x, y) and

w�(x, y), respectively.

6. Given fuzzy densities, compute the fused image z (x, y)using Eq. (13) with Z"�m

�(x, y),w

�(x, y)�.

7. Mark neighboring pixels of m(x, y) and w(x, y) asm

��(x, y) and w

��(x, y), respectively.

8. With the same given fuzzy densities for m(x, y) andw(x, y), compute the fused image z (x, y) using Eq. (13)with Z"�m

��(x, y),w

��(x, y)�.

9. Repeat steps (7) and (8) for all m��(x, y) and w

��(x, y).

10. Compute "nal fused image c(x, y) using Eq. (12).

4. Experiments

The fusion approach is tested on degraded gray-scaleimages. Fig. 1(a) is the original image of the Saturn. This

original image is added with Gaussian white noise of zeromean and 0.1 variance as shown in Fig. 1(b). Figs. 1(c), (d),and (f) are the images given by the mean, adaptiveWiener, and Sobel "lters respectively using a 3�3"lter mask. Finally, Fig. 1(f) shows the Saturn imagerestored by fusing the information provided by the three"lters, where the fuzzy densities for the mean, and adap-tive Wiener "lters are chosen to be 0.1, and 0.8, respec-tively. The values of the two fuzzy densities express thatthe adaptive "lter is considered to be much more impor-tant than the mean "lter when the two sources of in-formation are fused at the edge locations. It can be seenthat more noise is reduced in the degraded image byapplying the mean "lter than the adaptive Wiener "lter.However, the image edges are better preserved by thelatter "lter. The fused image, that obviously makes use ofthe di!erent e!ective features of the two "lters, yields thebest result by having less noise in the homogeneousregions of the image as given by the mean "lter, whereasimage edges are still sharpened by largely taking intoaccount the information given by the adaptive Wiener"lter. The use of the Choquet fuzzy integral for thiskind of image noise reduction can be seen in thesense of information fusion which nonlinearly searchesfor the degree of satisfaction between the two sourcesof evidence coming from the mean and the adaptiveWiener "lters. The fusion performance of the fuzzy inte-gral not only depends on the pixel intensity values butalso the fuzzy measures of the degrees of importance ofthe two "lters and their interaction when both attributesare taken together into account. In this case we considerthe information around the image edges given by theadaptive Wiener "lter is much more important thanthat given by the mean "lter and the combined valuetherefore tends to be much closer to Wiener-basedpixel value. As a result, the fused image in Fig. 1(f)is better restored than the two individual "lteredimages which are the mean and the adaptive Wiener asgiven in Figs. 1(c) and (d), respectively, in the sensethat noisy pixels are reduced as much as the mean "ltercan o!er, and the edge sharpness is preserved at thequality provided by the adaptive Wiener "lter. A com-parison with the weighted average rule is also made toillustrate the e!ectiveness of the proposed method. Usingthe weighted average, the fused image c(x, y) is obtainedas a linear combination of the mean and the Wienerimages with the associated weighting values, respectively,that is

c(x, y)"=�m(x, y)#=

�w(x, y), (18)

where =�

and =�

are the weights assigned to themean image m(x, y), and the Wiener image w(x, y), respec-tively.Fig. 1(g) shows the fused image given by the weighted

average combination rule as described in Eq. (18) with


Fig. 1. (a) Original image. (b) Degraded image. (c) Mean "ltered image. (d) Adaptive Wiener "ltered image. (e) Sobel "ltered image.(f) Proposed fused image. (g) Weighted average image.


Fig. 2. (a) Original image. (b) Degraded image. (c) Mean "ltered image. (d) Adaptive Wiener "ltered image. (e) Sobel "ltered image.(f) Fused image.


Fig. 3. (a) Original image. (b) Degraded image. (c) Mean "ltered image. (d) Adaptive Wiener "ltered image. (e) Sobel "ltered image.(f) Fused image.


=�

"0.1 and=�

"0.8, which are the same values of thecorresponding fuzzy densities for m(x, y) and w(x, y), re-spectively. It is obvious to see that the fused image asshown in Fig. 1(g) has edges that are less sharpened thanthe fused image given by the proposed fusion method,and has other homogeneous regions with more noisehighlighting than the fused image obtained from pro-posed method. This can be readily explained in that theweighted average combination rule largely takes intoaccount the information given by the Wiener "lter allover the entire image, which is not always neccessary inthe fusion process. Observation of the fused images at theedges between the two methods can show that the fuzzyintegral performs better than the weighted average com-bination rule in the sense of information fusion.The proposed fusion is tested on another image of

coins whose original version is shown in Fig. 2(a).Fig. 2(b) is the image degraded by Gaussian white noiseof zero mean and 0.05 variance. Figs. 2(c), (d), and (e) arethe images given by the mean, adaptive Wiener, andSobel "lters, respectively. The same values of the fuzzydensities as speci"ed for the Saturn image are also usedfor this image fusion, and the result of the fused image isshown in Fig. 2(f). Analysis of the results as given by themean "lter, adaptive Wiener "lter, and fusion are foundto be consistent with those in Figs. 1(c), (d), and (f),respectively. The fused image given in Fig. 2(f) gives thebest improvement on restoring the degraded image incomparison with the mean and the adaptive Wiener"ltered images given in Figs. 2(c) and (d), respectively.Although the information about the edges given by theSobel "ltered image is not quite su$cient, by taking intoaccount the neighboring pixels in the fusion, the fusedimage still gives the best restoration in noise reductionand edge sharpness.We further applied the "ltering to a third image of

a camera-man. Fig. 3(a) shows the noise-free image.Fig. 3(b) is the image degraded by Gaussian white noiseof zero mean and 0.2 variance. Figs. 3(c), (d), and (e) arethe images given by the mean, adaptive Wiener, andSobel "lters, respectively. With the same fuzzy densitiesas speci"ed for other images, the fusion is performed andits result is shown in Fig. 3(f). This image is the mostdegraded with noise. While it can be seen that both meanand adaptive Wiener "lters successfully reduce theGaussian noise, particularly the mean "lter, the Wiener"ltered image preserves slightly better sharpness than themean "ltered image. The fused image combines the di!er-ent advantages of the two "lters based on the Sobelimage, hence yields the best result over the two individualmean and Wiener "ltered images. It can be seen in a sim-ilar comparison as discussed in the above two experi-ments that the fused image has both noise reduction inthe homogeneous regions as given by the mean "lteredimage and edge sharpness as given by the Wiener "lteredimage.

5. Conclusions

A restoration of degraded images by fusion has beendiscussed in the foregoing sections. This work is hoped tobring into a new direction for dealing with an importantproblem in image analysis. Currently, reports on fusionin this domain of image analysis have been rarely made[15], and this proposed multi-"lter fusion based on fuzzyintegral for image restoration seems to be the "rst of itskind. Furthermore, there are a number of issues beingwell worth investigating. One important issue is the opti-mal identi"cation of the fuzzy densities and fuzzymeasures for the fusing operation of the fuzzy integral[11}17]. It is also necessary to consider the interaction ofdi!erent sources of information in which the redundancyand complementarity of multiple-"lter information needto be well studied so that the fusion performance cane!ectively improve the image restoration.

Acknowledgements

This work was supported by Y2000 ARC SmallResearch Grants Scheme.

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About the Author*TUAN D. PHAM received the B.E. degree (1990) from the University of Wollongong, and the Ph.D. degree (1995)from the University of New South Wales. His Ph.D. thesis was also published as a mini-monograph entitled Fuzzy Logic Applied toNumerical Modelling of Engineering Problems (Elsevier, 1995) co-authored with S. Valliappan. From 1994 to 1995, he was a seniorsystems analyst with Engineering Computer Services, where his work mainly involved with R&D in computer algorithms andsimulations for mining exploration. He then joined the Laboratory for Imaging Science and Engineering in the Department of ElectricalEngineering at the University of Sydney as a postdoctoral fellow during 1996}1997. He joined the Laboratory for Human-ComputerCommunication at the University of Canberra (UC) during 1997}1998 as a research fellow, and began researching in speech and speakerrecognition. He then became a lecturer in the School of Computing at UC until 2000 before joining the Information Engineering groupas a senior lecturer with the Institute of Information Sciences and Technology at Massey University in New Zealand. His research areasof interest include image and speech analysis, speaker recognition, signal processing, computational intelligence and AI methods (fuzzylogic, neural networks, genetic algorithms), pattern recognition (statistical and non-statistical), data fusion, applied geostatistics,optimization methods, hidden Markov models, information and uncertainty, fractals and chaos, software speci"cations, and know-ledge-based systems. He also coauthored with Z. Chi and H. Yan in the book Fuzzy Algorithms: With Applications to Image Processingand Pattern Recognition (World Scienti"c, Singapore, 1996). He has published over 60 papers in international journals and conferences,and served as reviewer, conference session chair/organizer and member of technical and organizing committees for several internationaljournals and conferences respectively. Dr. Pham is a member of the IEEE, and Pattern Recognition Society.


an image restoration by fusion

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