Image Restoration using Multilayer Neural Networks with Minimization of Total Variation Approach

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1 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): Image Restoraton sng Mltlaer Neral Networs wth Mnmzaton of Total Varaton Approach Mohammed Debala, Khalfa Djemal and Mohamed Benetto MOSIM Laborator, Unverst of USTO Oran Algera IBISC Laborator, Unverst of Evr Val d'essonne, France Abstract Nose redcton s a ver mportant tas n mage processng. In ths am, man approaches and methods have been developed and proposed n the lteratre. In ths paper, we present a new restoraton method for nos mages b mnmzng the Total Varaton (TV) nder constrants sng a mltlaer neral networ (MLP). Indeed, the obtaned Eler-Lagrange fnctonal s resolved b mnmzng an error fnctonal. The MLP parameters (weghts) n ths case are adjsted to mnmze approprate fnctonal and provdes optmal solton. The proposed method can restore degraded mages and preserves the dscontntes. The effectveness of or approach has been tested on snthetc and real mages, and compared wth nown restoraton methods Kewords: Image restoraton, Partal Dfferental Eqatons, Total varaton, Mltlaer neral networ.. Introdcton Image restoraton s an mportant operaton n man processes of mage analss. Indeed, restored mage can mprove the accrac of downstream processng operatons sch as segmentaton. Degraded mage restoraton problems are largel treated n lteratre n man applcatons []-[5]. Image restoraton was one of the frst problems to attract attenton. It sees to correct the dstortons that reslt n man degradatons sch redcton of contrast, blr and random or de to nose occrrng drng mage formaton. Observed mage modelng consderng nose and blr s generall assmed to mnmze establshed crtera n order to fnd the best orgnal mage. The man dffclt s that the restoraton methods shold preserve contors of objects and dscontntes contaned n the degraded mages. Ths needs the ntrodcton of varatonal methods [6] or stochastc [], whch can smooth the homogeneos regons and preserve dscontntes of mage. Methods based on partal dfferental eqatons (PDE) [8]-[] and ansotropc flterng technqes []-[3], well establshed, meet these reqrements and have been partclarl stded n recent ears. Inclde also some nonlnear methods that compte a weghted average of ntenst vales n a local neghborhood [4]-[6], [9] In recent ears, a new emergng technqe has grown consderabl. It helps to process nformaton more effcentl than conventonal sstems, to remove data prevosl navalable to assst n decson mang, t s the neral networ. The have the specfct of learnng b themselves to etract nformaton hdden n a mass of data, and provde powerfl models as well as to the nowledge of a gven problem. In other words, the represent a class of powerfl algorthms that are sed for classfcaton, predcton and aggregaton of data. These methods have been sccessfll ntrodced n mage processng and compter vson. Ther applcatons are nmeros, sch as edge detecton [7]-[8], [8], segmentaton [8]-[9], stereovson [], restoraton []-[4]. In ths am, the fncton mappng to be mnmzed nto energ of a gven networ s the commonl adopted strateg. Restoraton of a hgh qalt mage from a degraded recordng s a good applcaton area of neral nets. Zho et al. [4] are the frst who proposed the se of the Hopfeld neral networ (HNN) n mage restoraton and showed the nstablt of the orgnal (HNN) when appled to mage restoraton. The proposed an algorthm to ensre the stablt of the HNN. The also proposed the se of smlated annealng algorthm that allows energ ncrease wth a probablt decreasng n tme so as to converge to a better solton n stochastc sense. These two algorthms are tme-consmng becase energ change has to be checed step b step. In [5], Pa et al. proposed a Modfed Hopfeld neral networ (MHNN) model for solvng the restoraton problem whch mproves pon the algorthm proposed b Zho et al. n [4]. The algorthms based on the MHNN ensre networ stablt wthot checng energ change step b step, and two new pdatng schemes (one seqental and the other parallel) are ntrodced. However, Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

2 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): the convergence proof for the parallel scheme s based on an almost never satsfed condton. Sn Y, presents n [6], [] a Generalzed Updatng Rle (GUR) of the MHNN for gra mage recover. The stablt propertes of the GUR are gven. It s shown that the neral threshold set p n ths GUR s necessar and sffcent for energ decrease wth probablt one at each pdate. In [3] the athors present two novel mage restoraton algorthms based on a modfed Hopfeld neral networ and varatonal partal dfferental eqatons (PDE). The frst algorthm s based on harmonc model and the other s based on total varaton (TV) model [3]. Both algorthms can restore the degraded mages and preserve the edges. A novel neral networ based mltscale restoraton approach was proposed n [7] and mproved n [8]. The method ses a Mltlaer Perceptron (MLP) algorthm, traned wth snthetc an 8-bt gra level mage of artfcall degraded co-centered crcles, wth pels. Recentl n [9], athors provde an effectve algorthm wth Celllar Neral networ and Contor matchng that can be sed to the npantng dgtal mages or vdeo frames wth ver hgh nose rato. In ths paper, we present a new mage restoraton method based on TV mnmzaton proposed b Rdn et al. n [3]. In ths am, the obtaned PDE of the TV model mnmzaton nder constrants s consdered as an error fncton whch s resolved b mltlaer neral networ. Ths approach rel on the fncton appromaton capabltes of feedforward neral networs and reslts n the constrcton of a solton wrtten n a dfferentable, closed analtc form. In ths approach, the mltlaer neral networ s consdered as the bass of an appromaton, whose parameters (weghts) are adjsted to mnmze an approprate error fncton. To tran the networ we se optmzaton technqes, whch reqre the calclaton of the gradent of the error to set the networ settngs wth respect to ts parameters. The paper s organzed as follows: secton descrbes the proposed restoraton method. In Secton 3 we llstrate the method b presentng some eamples of snthetc and real mage restoraton. The reslts are compared wth now restoraton methods sch as Tchonov reglarzaton [], mnmzng the total varaton of ROF [3] methods and Mltscale Neral Networ [8]. Fnall, secton 4 brngs the conclson for the wor.. Proposed mage restoraton method We present frstl n ths secton, the formlaton of the mage restoraton problem b mnmzng the TV model nder constrants. The problem s characterzed b a PDE fnctonal, tpcall measrng some reconstrcton error, and the solton s defned as the mnmzaton of the consdered fnctonal. Secondl, we propose to mnmze ths error fncton sng MLP neral networ approach and fnall, a general algorthm s detaled.. Restoraton Problem Statement TV reglarzaton has been etremel sccessfl n a wde varet of restoraton problems, and remans one of the most actve areas of research n mathematcal mage processng and compter vson. B now, ther scope encompasses not onl the fndamental problem of mage denosng, bt also other restoraton tass sch as deblrrng, blnd deconvolton, and npantng [3], [3]- [33]. In all these approaches, a TV model s mnmzed n dfferent was. The tpcal problem n mage restoraton case were ntrodced b Rdn et al. n the poneerng wor [3] on edge pre-servng mage denosng wth the mnmzaton of the followng fnctonal : Ω F ( ) D d d () where Ω D represents the TV model of the mage. If mage s reglar, the eqaton () becomes onl Ω d. In [3] the athors consdered that the nose wtch corrpted mage s dstngshed from noseless one b the sze of total varaton, whch s defned as d d, where Ω denotes the mage Ω doman and denote the correspondng partal dfferentaton. Conseqentl, the propose to restore a nos and blrred mage b mnmzng total varaton: mn Ω d d Under constrants: ( (, ) (, )) dd Ω ( (, ) (, )) dd (3) Ω Where (, ) represents the gven observed mage, whch s consdered to be corrpted b a Gassan nose of varance and (, ) denotes the desred clean mage. To mnmze () Rdn et al. [3] have appled the Eler- Lagrange eqaton nder the two constrants (3) the obtan the followng eqaton: ( ) () (4) Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

3 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): Where λ the Lagrange mltpler s gven b: ( ) ( ) Over the ears, the TV model [3], has been etended to man other mage restoraton tass, and has been modfed n a varet of was to mprove ts performance. The classcal approach s then to se the assocated Eler- Lagrange eqaton to compte the solton. Fed step gradent descent [3], or later qas-newton methods [8], [34]-[36], have been proposed [37]-[39]. Iteratve methods have proved sccessfl [4]-[4]. Ideas from dalt can be fond n [43],[45]-[46]. In [45], [47], a graph cts based algorthms cold also be sed. A combnaton of the prmal and dal problems has been ntrodced n [48]. Recentl, t has been shown that Neral Networ for classfcaton of nose followed b classfcaton of flter s eplored n [49]. Notce that all these wors se an appromate nmercal method for solvng the assocated PDE wth the restoraton problem. An mage restoraton problem can be transformed to an optmzaton problem. In or assmpton and from (4) we can formlate the mage restoraton problem as mnmzng the followng error fncton: (6) E (, ) ( ) where and are dscretzatons of the horzontal and vertcal dervatves. A dffclt wth TV s that t has a dervatve snglart when s locall constant. To avod ths, some algorthms reglarze TV b ntrodcng a small parameter > wthn the sqare root. We propose to mnmze the error fncton E n eqaton (6) sng an MLP neral networ approach. We present a generalzaton of the problem and then we ntrodce a weghtng technqe of weght for a term based on the data fdelt.. Error Fncton Mnmzaton Based MLP Approach The general dea of or method s to mnmze an error fncton, tang n to accont the TV approach as llstrated n eqaton (6). In ths am, we se MLP whch shown b Lagars et al. [5], where the athors were able to solve partal dfferental eqatons b neral networs. To mnmze eqaton (6), a neral networ (NN) wth three laers s sed, accordng to the MLP archtectre (5) llstrated n Fgre. The npt laer of the MLP conssts of one neron whch s the pel to be restored. The otpt laer contans one neron correspond to the processed pel. Net research has been done n evalatng the nmber of nerons n the hdden laer bt no optmal nmber has been dscovered. So the hdden laer conssts of a varng nmber of nerons fed b the ser after several tests, n or epermentaton case we have sed ten nerons. The sgmod fncton s appled to each neron n the hdden laer and otpt neron. In all prevos wors sng neral networs n the feld of restoraton, the chosen networ tpe s related to the assmptons gven b the athors n the restoraton process. In or case we tae the assmpton that the otpt of the MLP s an mage correspondng to the desred mage (, ). So we provded as npts to the mltlaer neral networ the degraded mage (, ) and at the otpt of the networ, we have: (, ) N ( (, ), w ) o (7) where (, ) s the nos mage represented b each o ntenst of pel (, ) and w s the weghts vector of the MLP. Fgre.Mltlaer neral networ strctre of or approach We consder that all vales of the sgmod fncton are taen between and ; we mst normalze the npt and otpt vales of the networ. Indeed, each pel of nos mage s coded b a one bte gra-level, the npt vales wll be dvded b 55, and the otpt reslts wll be mltpled b 55. The vale of each neron n the hdden laer s gven b: h (, w ) ( ). (8) where w are the weghts from the npt laer to the hdden laer. The gra level of the otpt of each pel s obtaned b: Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

4 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): n N h w (9) 55 where w are the weghts from the hdden laer to the otpt laer, and n s the nmber of the nerons n the hdden laer. n () N(, ) 55 w ( w (, ) ) Once all pels of the nos mage are presented to the networ, we get an mage amed that satsfes the eqaton (4). Ths mage s replaced n eqaton (5) to gve, and n eqaton (6) for calclatng the error fncton E provded b the networ. The MLP s traned b the bacprobagaton algorthm [5] ntl convergence s obtaned. Convergence s obtaned when the fed teratons nmber s eceed or the mnmm vale of error E s reached whch corresponds to the convergence error E. To mae changes the weghts, we se the C steepest gradent descent method. The adaptaton of weghts s done b the followng eqaton: () w ( t ) w ( t) w ( t) Where ( t ) w and (t) w represent respectvel the new and the last vales of weghts, wth (weghts between npt laer and hdden laer) or (weghts between hdden laer and otpt laer), a learnng postve constant. The weght varaton w,.., n and s obtaned b mnmzaton of error E(, ) presented n eqaton (6) sng the followng eqaton: w To compte, E, ) w w ( (), and for the smplct of the calclaton, we denote the two terms of eqaton (6) as follow: Then we have: f, Eqaton (6) now became ( ) g (4) E(, ) f g (5) From these consderatons, the weght varaton term,, and,.., n can be formlated as follow: E(, ) f g ( ) w (6) All terms of the eqaton (6) are calclated from the vale of w, so t s necessar to provde frstl these dervatves. Under or assmpton and consderaton n (7) and as N s consdered n (), n these condtons the dervatves can be gven as follow: N (7) So, the dervatve of the desred mage wth respect to the weghts from the npt laer to the hdden laer s gven b: w 55 ' [ n ' [ (, ) w ( w (, )) ] w ( w (, ))] (8) In the same wa, dervng wth respect to the weghts from the hdden to the otpt laer s gven b: n ' ( w (, ) ) [ w ( w (, ))] 55 (9) From the obtaned dervatons and we can easl calclate w and w w w f (, ) and g (, ) (3) Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

5 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): General algorthm of the method Inpt: nos mage of sze (n,m), MLP neral net (see sb-secton.) Otpt: desred clean mage Begn - Random ntalzaton of networ weghts w and w. - Normalzaton of the nos mage. Itt ; Repeat For each pel (,) of do - Actvaton of each neron n the hdden laer: h (, w ) Algorthm : General algorthm of the proposed mage Restoraton method. 4. Epermental reslts ( ) - Calclate the neron vale n the otpt laer whch s the weghted sm of otpt nerons vales n the hdden laer: N (, ) ( w h ) End For - calclate desred clean mage N - calclate the error fncton E - calclate accordng to the delta rle: w w E w - Updatng the weghts In ths secton, we present some epermental reslts that evalate the performance of or approach. We also chose to compare the denosng performance of or approach wth other methods sng ther optmal parameters: Tchonov reglarzaton [], mnmzng TV model of ROF [3] and Mltscale Neral Networ (MNN) [8] for snthetc and real nos mages. For the prpose of objectvel testng the performance of mage restoraton algorthm, the mprovement ISNR s often sed. It represents the amont of nose removed from the degraded mage. If best. Ths metrc, sng the restored mage, s gven b: w w n w Itt Itt Untl (Itt > matt or E E ) /* Where E C Cs closer to zero, matt s the mamm nmber of teraton */ End. ISNR( f [ f (, ], (, j, ) log [ f (, (, ], j f (, j, (, ) and (, Where ) j () denote the orgnal, degraded and restored mages, respectvel. We also sed the Normalzed Mean Sqare Error (NMSE) as another measre of qalt. If the vale of NMSE decreases, the restoraton s better. NMSE s gven b: [ f (, (, ] (), j NMSE ( f, ) [ f (, ], j The frst denosng eperment s shown n Fgre. For ths eperment, sng a snthetc mage, we added whte Gassan nose wth three dfferent standard devatons (σ 5,, 5). The nos snthetc mages are n the frst colmn of Fgre, and the denosed mages b the Tchonov (wth: n, epslon. and alpha5), TV model (wth: n4, dt., alpha5 and epslon.), mltscale neral networ (MNN) methods and or approach (hdden laer wth nerons) are shown n the second, thrd, forth and ffth colmns, respectvel. The correspondng ISNR and NMSE vales are shown n Table. Fgre 3 and 4, respectvel, llstrates the evolton of the Lagrange mltpler and the error fncton E accordng to the teratons nmber. The consdered nose standard devaton s 5. We can see clearl that the Thonov method tends to blr the mage and can t preserve detals (Fgre second colmn). In Fgre at the thrd colmn, the restored mage obtaned b the ROF method, sppress some detals and blr s stll apparent. For the MNN method we can observe an ncrease n brghtness and contrast for the restored mage (Fgre forth colmn). The proposed method gves a good vsal qalt wth strong nose sppresson for all varatons and also more detals are preserved (Fgre ffth colmn). In Table, vales of ISNR and NMSE shows that or approach gves good performance compared to the other methods. Indeed, or approach wth nose standard devaton 5, obtans the best ISNR 3,8 n comparson wth other methods. Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

6 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): Nos Image Thonov method ROF method MNN method Or approach Fgre Eamples of whte Gassan nose redcton: The colmns from left to rght show the nos mage and the restored mages b Tchonov, the total varaton, mltscale neral networ and or approach. The rows from top to down are showng the eperments wth dfferent standard devatons (σ5,, and 5). E Iteratons nmber Iteratons nmber Fgre 3 Evolton of accordng to the teratons nmber Fgre 4 Evolton of E accordng to the teratons nmber Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

7 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): Table. The ISNR and NMSE vales of whte Gassan nose redcton. Image σ 5 σ σ 5 Thonov algorthm ISNR NMSE (ROF) approch ISNR Mltscale Neral Networ MLP wth 4 teratons NMSE.4..4 ISNR NMSE.5.8. ISNR NMSE.3.8. We appled or approach on real nown nos mages: Hose, Cameraman and Lena mages. The comparsons wth other methods are gven n fgre 5, 6 and 7 wth the optmal parameters of each method. The three mages are dstrbed b a Gassan addtve nose wth zero mean and standard devatons 5, restored mages, obtaned after 4 teratons of or approach are presented n Fgre 5f, 6f and 7f. a c e b d f a c b d Fgre 6 cameraman mage restoraton : (a) orgnal mage, (b) Gassan degraded mage wth 5, restoraton reslt sng: (c) Thonov s algorthm, ISNR.8, (d) (ROF) Total Varaton model, ISNR3.94, (e) Mltscale neral networ method, ISNR 4.59, (f) MLP approach wth 4 teraton, ISNR6.57 The reslts presented show the good performance of or algorthm, especall the preservaton of dscontntes. Moreover the geometrc characterstcs sch as corners and edges and orgnals contrast are well restored. For prposes of comparson, the reslts of restoraton b the for methods chosen are smmarzed n Table. Table. The ISNR and the NMSE Vales based on for methods Image Hose Cameramen Lena Thonov algorthm ISNR NMSE (ROF) approch ISNR NMSE e f Mltscale Neral Networ ISNR NMSE Fgre 5 hose garden mage restoraton : (a) orgnal mage, (b) Gassan degraded mage wth 5, restoraton reslt sng: (c) Thonov s algorthm, ISNR.5, (d) (ROF) Total Varaton model, ISNR5.35, (e) Mltscale neral networ method, ISNR5.79, (f) MLP approach wth 4 teraton, ISNR7.58 MLP wth 4 teratons ISNR NMSE Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

8 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): a c e Fgre 7 Lena mage restoraton : (a) orgnal mage, (b) Gassan degraded mage wth 5, restoraton reslt sng: (c) Thonov s algorthm, ISNR.6, (d) (ROF) Total Varaton model, ISNR5., (e) Mltscale neral networ method, ISNR4.79, (f) MLP approach wth 4 teratonisnr Conclsons We have presented n ths paper a new mage restoraton method based on TV model mnmzaton nder constrants. Ended, we have shows that the obtaned Eler-Lagrange fncton can be resolved b mnmzng an error fncton sng mlt-laer neral networ (MLP) approach. The developed algorthm mproves nose redcton and preserves the orgnal geometrc characterstcs and contrasts of the mage well. The reslts comparson demonstrates the performance of or approach. In ftre wor, we wll nvestgate dfferent neral networ archtectres sch as recrrent or celllar neral networs, and consder dfferent nose models. References [] A.N Thonov, V.Y Arsenn, Soltons of Ill-Posed Problems, Wnston and Sons, Washngton. 977, [] S. Geman, D. Geman, Stochastc relaaton, gbbs dstrbtons, and the baesan restoraton of mages, b d f IEEE Transactons on Pattern Analss and Machne Intellgence, Vol. 6, No. 6, 984, pp [3] L. Rdn, S. Osher,, E. Fatem, Nonlnear total varaton based nose removal algorthms, Phsca D 6, 99, [4] M.R. Banham, A.K. Katsaggelos, Dgtal mage restoraton, IEEE Sgnal Process. Mag. 4 (),997, pp 4-4. [5] A.K. Katsaggelos, J. Bemond, R.W. Schafer, R.M. Merserea, A reglarzed teratve mage restoraton algorthm, IEEE. Trans. Sgnal Processng, Vol. 39, 999, p [6] D. Mmford, J. Shah, Optmal appromatons b pecewse smooth fnctons and varatonal problems, Commncaton on Pre and appled Mathematcs, Vol. 4, No.5, 989, pp [7] F. Catté, P.L.Lons, J.M. Morel, T. Coll, Image selectvesmoothng and edge-detecton b nonlnear dffson, SIAM J. Nmercal Analss, Vol. 9, No., 99, pp [8] P. Charbonner, L. Blanc-Férad, G. Abert, M. Barlad, Determnstc edge-preservng reglarzaton n compter magng, IEEE Transactons on Image Processng, 6(), 997, pp [9] T. Chan, S. Osher, J. Shen, The dgtal TV flter and nonlnear denosng, IEEE Transactons on Image Processng, Vol., No.,, pp [] F. Dbos, G. Koepfler, P. Monasse, Total Varaton Mnmzaton for Scalar and Vector Image Reglarzaton, Geometrc Level Set Methods n Imagng, Vson and Graphcs, Sprnger Verlag., [] P. Perona, J. Mal, Scale space and edge detecton sng ansotropc dffson, IEEE Transactons on Pattern Analss and Machne Intellgence, Vol., No. 7, 99, pp [] M.J. Blac, G. Sapro, D.Marmont, D. Heeger, Robst ansotropc dffson, IEEE Transactons on Image Processng, Vol. 7, No. 3, 998, pp [3] M.J. Blac, G. Sapro, Edges as otlers: Ansotropc smoothng sng local mage statstcs In Scale-Space Theores n Compter Vson, Kerra,Greece. 999, pp. 59-7, [4] F. Godtlebsen, E. Spjotvoll, J.S. Marron, A nonlnear gassan flter appled to mages wth dscontntes, J. Nonparametrc Statstcs, Vol. 8, 997, pp [5] C. Tomas, R. Mandch, Blateral flterng for gra and color mages, In Internatonal Conference on Compter Vson, Bomba, Inda, 998, pp [6] D. Barash, A fndamental relatonshp between blateral flterng, adaptve smoothng and the nonlnear dffson eqaton, IEEE Transactons on Pattern Analss and Machne Intellgence, Vol. 4, No. 6,, pp [7] J.K Pa, Image restoraton and edge detecton sng neral networs, Ph.D.Dssertaton, Dep.Elec.Eng., Compter Sc, North-Western Unv. 99. [8] B Meftah. O.Lezora A. Benetto, Segmentaton and Edge Detecton Based on Spng Neron Networ, Neral Processng Letters, Vol. 3(),, pp [9] B. Meftah, A. Benetto, O. Lzora, M. Debala, Image Segmentaton wth Spng Neron Networ, st Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

9 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): Medterranean Conference on Intellgent Sstems and Atomaton (CISA 8), Annaba, Algera, AIP Conf. Proc. Volme 9, 8, pp. 5-9, [] N. Nasrabad, C. Choo, Hopfeld networ for stereovson correspondence, IEEE. Trans.Neral Networ, Vol. 3, 99, p [] J. Pa, A. Kastagellos, Edge detecton sng a neral networ, Proc of the Int.Conf.ASSP-ICASSP, Albresqe, NM, 99, p [] Y. Sn, Hopfeld neral networ based algorthms for mage restoraton and reconstrcton Part II: Algorthms and smlaton, IEEE Trans. Sgnal Processng, vol. 48,, pp [3] Y.D. W, Q.Z. Zh, S.X. Sn, H.Y. Zhang, Image restoraton sng varatonal PDE-based neral networ, Nerocomptng, 69, No. 6-8, 6, pp [4] Y.T. Zho, A. Vad, B.K. Jenns, Image restoraton sng a neral networ, IEEE Trans. Acost. Speech Sgnal Processng, vol.36, 988, p [5] J K. Pa, A. Katsaggelos, Image restoraton sng a modfed Hopfeld Networ, IEEE Transactons on mage processng, Vol, No., 99, pp [6] Y. Sn, A generalzed pdatng rle for modfed Hopfeld neral networ for qadratc optmzaton, Nerocompt, vol. 9, 998, pp [7] A.P.A. Castro, J.D.S. Slva, Neral Networ-Based Mltscale Image Restoraton Approach In: Proceedng on Electronc Imagng, Vol. 6497, San Jose, 7, pp [8] A.P.A. Castro, I.N. Drmmond, J.D.S. Slva, A Mltscale Neral Networ Method for Image Restoraton, TEMA Tend. Mat. Apl. Compt, 9, No., 8, pp [9] P. Elango, K. Mrgesan, Image Restoraton Usng Celllar Neral Networ Wth Contor Tracng Ideas,Internatonal Jornal of Compter Theor and Engneerng, Vol., No. 5.. [3] S.Karth et al, Drectonal Total Varaton Flterng Based Image Denosng Method, IJCSI Internatonal Jornal of Compter Scence Isses, Vol. 9, Isse, No, March. [3] G. Abert,, P. Kornprobst, Mathematcal Problems n Image Processng, Appled Mathematcal Scences, vol. 47. Sprnger, New Yor,. [3] T. Chan,, J. Shen, Image Processng and Analss Varatonal, PDE, Wavelet, and Stochastc Methods, SIAM, Phladelpha. 5, [33] K. Djemal, Specle Redcton n Ultrasond Images b Mnmzaton of Total Varaton, IEEE Internatonal Conference on Image Processng, Genova, Itala, Vol. 3, 5, p [34] A. Chambolle, P.L Lons, Image recover va total varaton mnmzaton and related problems. Nmer. Math. 76(3), 997, pp [35] M.K. Ng, L. Q, Y.F. Yang, Y. Hang, On semsmooth Newton methods for total varaton mnmzaton, J. Math. Imagng Vs. 7, 7, pp [36] M. Nolova, R. Chan, The eqvalence of half-qadratc mnmzaton and the gradent lnearzaton teraton, IEEE Trans. Image Process, 6(6), 7, pp [37] T.F. Chan, J. Shen,, L.Vese, Varatonal PDE models n mage processng, Notces Amer. Math. Soc., 5, 3, pp 4-6. [38] T.F. Chan, S.Esedogl, F.Par, A.Yp, Total varaton mage restoraton: overvew and recent developments, Handboo of mathematcal models n compter vson, 7-3, Sprnger, New Yor. 6, [39] J. F. Ajol, Some Frst-Order Algorthms for Total Varaton Based Image Restoraton, Math Imagng Vs, DOI.7/s , 9. [4] I. Dabeches, M. Defrse,, C. De Mol, An teratve thresholdng algorthm for lnear nverse problems wth a sparst constrant, Commn. Pre Appl. Math. 57, 4, [4] J. Bect,, L. Blanc-Frad, G. Abert, A. Chambolle,, A l-nfed varatonal framewor for mage restoraton, In ECCV 4. Lectre Notes n Compter Scences, vol. 34, Sprnger, Berln, 4 pp. -3. [4] J. Bocas-Das, M.,Fgeredo, Thresholdng algorthms for mage restoraton, IEEE Trans. Image Process, 6(), 7, pp [43] T. Chan, G. Golb,, P. Mlet,, A nonlnear prmal-dal method for total varaton-based mage restoraton, SIAM J. Sc. Compt. (6), 999.pp , [44] A. Chambolle,, An algorthm for total varaton mnmzaton and applcatons, J. Math. Imagng Vs., 4, pp [45] A. Chambolle, Total varaton mnmzaton and a class of bnar MRF models, In: EMMCVPR 5. Lectre Notes n Compter Scences, vol. 3757, Sprnger, Berln, 5, pp [46] P.L Combettes,. V. Wajs, Sgnal recover b promal forwardbacward splttng, SIAM J. Mltscale Model. Sml. 4(4), 5, pp68. [47] J. Darbon,, M. Sgelle,, Image restoraton wth dscrete constraned total varaton part I: Fast and eact optmzaton. J. Math. Imagng Vs. 6(3), 6, pp77 9. [48] M. Zh, T.F. Chan, An effcent prmal-dal hbrd gradent algorthm for total varaton mage restoraton, UCLA CAM Report , [49] T. Santhanam, S. Radha, Applcaton of Neral Networs for Nose and Flter Classfcaton to enhance the Image Qalt, IJCSI Internatonal Jornal of Compter Scence Isses, Vol. 8, Isse 5, No, September. [5] I Lagars, L Arstds, D. Fotads, Artfcal Neral Networs for Solvng Ordnar and Partal Dfferental Eqaton, IEEE Transacton on Neral Networ, Vol. 9, No [5] D. E. Rmelhart, G. E. Hnton and R. J. Wllams Learnng nternal representatons b error propagaton Parallel Dstrbted Processng vol, Cambrdge, MA: MIT Press, 986, pp Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.

10 IJCSI Internatonal Jornal of Compter Scence Isses, Vol., Isse, No, Janar 4 ISSN (Prnt): ISSN (Onlne): Debala Mohammed Is a PhD stdent n compter Scence Unverst of Scence and Technolog of Oran (USTO) - ALGERIA. He receved the dploma of engneerng n Compter Scence from the Unverst of Oran (USTO) - ALGERIA n 996. He receved the dploma of teachng n Compter Scence from the Unverst of Oran (USTO) - ALGERIA, n 5. Is a teacher at the Unverst of Mascara - Algera, snce 5. Hs research nterests are n the feld of mage processng and optmzaton methods. KhalfaDjemal receved hs dploma degree n Optcal, Image and Sgnal Processng n 999 from the Natonal School of Phscs at the Unverst of Marselle, France and hs Ph.D. n Image and Sgnal Processng,, from the Unverst of Tolon, France. Snce 3, he s an Assocate Professor at the Electrcal Engneerng Department of the Insttte of Technolog at the Unverst of Evr Val d Essonne, France. He wors now wthn the S.I.M.O.B team of the IBISC Laborator. Hs crrent research nterests are n the areas of mage and data processng (Restoraton, Segmentaton, Clsterngand CBIR). Dr. Djemal chared the Internatonal Conference on Image Processng Theor, Tools and Applcatons IPTA, n 8, and, and also Internatonal Worshop on Medcal Image Analss and Descrpton for Dagnoss Sstems, MIAD, n 9, and. He was the char of some specal sessons n a nmber of conferences. He acted as techncal charman for a nmber of conferences. He s a revewer for a nmber of nternatonal jornals and conferences. Mohamed Benetto s a professor n department of compter scence, Unverst of Scences and Technologes Mohamed Bodaf USTO, ORAN- ALGERIA. He s also a drector of laborator of modelng and optmzaton LAMOSI, hs man research nterests nclde: artfcal ntellgence, mage processng, optmzaton methods Coprght (c) 4 Internatonal Jornal of Compter Scence Isses. All Rghts Reserved.