A NONLINEAR FOURTH-ORDER DIFFUSION-BASED MODEL FOR IMAGE DENOISING AND RESTORATION

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volme 8, Nmber /7, pp. 8 5 A NONLINEAR FOURTH-ORDER DIFFUSION-BASED MODEL FOR IMAGE DENOISING AND RESTORATION Tdor BARBU, Ioţ MUNTEANU, Isie of Comper Sciece of he Romaia Academy Isie of Mahemaics Ocav Mayer of he Romaia Academy Correspodig ahor: dor.barb@ii.academiaromaa-is.ro Absrac. A oliear PDE-based image resoraio approach is described i his aricle. The proposed filerig echiqe is based o a ovel forh-order diffsio model. Ulike may oher forh-order PDE deoisig schemes, his oliear model provides a opimal rade-off bewee oise removal, image deail preservaio ad avoidig of desired effecs. A rigoros mahemaical ivesigaio of he well-posdedess of his differeial model is also performed. The, a cosise ad fas covergig merical approximaio scheme, based o he fiie-differece mehod, is cosrced for he forh-order PDE. The deoisig mehod proposed here operforms o oly he coveioal image filers ad hose based o liear PDEs, b also may sae of he ar oliear secod ad forh order diffsio-based echiqes, as reslig from he mehod compariso. Key words: image deoisig ad resoraio, oliear diffsio, forh-order PDE model, mahemaical reame, merical approximaio scheme, fiie-differece mehod.. INTRODUCTION The oliear parial differeial eqaio (PDE) based echiqes have bee represeig he mai isrme for digial image deoisig ad resoraio i he las cery qarer. Sice he coveioal wodimesio image filers ad he liear PDE-based deoisig schemes are geeraig he blrrig effec ha desroys he edges ad oher image deails [], he oliear PDE models represe a mch beer smoohig solio []. They prodce a direcioal diffsio ha is degeerae alog he gradie direcio, havig he effec of filerig he image alog b o across is edges. The bodaries ad oher impora feares are hs preserved drig he resoraio process []. Noliear diffsio-based image resoraio models ca be divided io secod- ad forh-order PDEbased approaches. The secod-order PDE models have bee widely sed for resoraio ad scale-space image aalysis sice Peroa ad Malik elaboraed heir ifleial aisoropic diffsio scheme [3]. Becase hese oliear differeial models ca be obaied from variaioal problems, meros variaioal deoisig schemes have bee developed sice Rdi, Osher ad Feami irodced heir TV Deoisig echiqe [4]. While he secod-order PDE-based echiqes provide a effecive deblrrig, hey cold also prodce he desired saircase (block effec, represeig creaig of fla zoes separaed by arifac bodaries [5]. So, a lo of improved PDE ad variaioal models derived from Peroa-Malik ad TV Deoisig schemes, sch as he Spli Bregma mehods, have bee proposed o solve his problem [, 6 8]. We also developed sch variaioal ad oliear secod-order aisoropic diffsio models ha redce cosiderably he image saircasig, which are dissemiaed i or pas papers [9 ]. However, he oliear PDE schemes based o forh-order diffsio represe a mch beer solio for saircase effec removig. Sch a poplar forh-order PDE resoraio model is ha developed by Y.L. Yo ad M. Kaveh [3], which approximaes he observed image wih a piecewise harmoic oe. Aoher ifleial resoraio approach based o forh order diffsios was proposed by M. Lasaker, A. Ldervold ad X. Tai [4]. While hese forh-order PDE deoisig mehods remove he Gassia oise sccessflly ad overcome he blocky effec, hey may also geerae image blrr ad speckle oise. May improved forh-order PDE schemes ha avoid hese ieded effecs have bee elaboraed i he las decade [5]. We also developed some forh-order diffsio-based resoraio schemes ha improve Yo-Kaveh model [6, 7].

2 A oliear forh-order diffsio-based model for image deoisig ad resoraio 9 Here we cosider a ovel forh-order oliear PDE-based image deoisig echiqe ha is oally differe from Yo-Kaveh mehod. I performs a effecive oise removal ad overcomes all desired effecs. Also, he proposed approach operforms some sae of he ar forh-order PDE resoraio models. Or oliear diffsio approach is deailed i he followig secio. A rigoros mahemaical ivesigaio is performed o he well-posedess of his forh-order PDE i he hird secio. The, a robs fiie differecebased merical approximaio scheme developed for his PDE model is described i he forh secio. The fifh secio of his paper describes or resoraio experimes ad mehod compariso. This aricle fializes wih a coclsio secio, some ackowledgemes ad a bibliography.. NOVEL FOURTH-ORDER DIFFUSION-BASED MODEL I his secio we propose a ovel differeial model for image deoisig ad resoraio. The proposed oliear diffsio-based resoraio model is composed of a forh-order parial differeial eqaio wih some bodary codiios. Ths, or PDE-based smoohig echiqe is expressed as followig: ξ + γ + Δ ( ϕ ( ) ) + λ ( ) (, x, y ) = y ) (, x, y ) = y ) (, x, y ) =,, y ) = where fcio represes a evolvig image, parameers, γ, λ (,] R, y ) () ξ, is he froier of domai Δ = is he Laplacia operaor, is he gradie magide, is he iiial image corrped by Gassia oise ad is a velociy modificaio of i. The diffsiviy fcio :[, ) (, ) ϕ has he form: where ψ ( ) ϕ ( s) = δ k α log ( s + ψ ( ) ) + β () ( ) ε coefficies δ, α, η (,.5 ), k, β, ε [,5 ) ad ( ) ψ ( ) = ημ, (3) μ rers he average vale of he argme. The diffsiviy fcio give by () (3) is properly chose for a effecive resoraio process. Obviosly, i is posiive: ϕ ( s ) >, s. Also, i is moooically decreasig, sice ψ ( ) ψ ( ) ϕ ( s) = δ ϕ k ( s ) = δ, s k s, α log ( s + ψ ( ) ) + β α log ( s + ψ ( )) + β ad i coverges o zero, sice lim ϕ ( s ) =, which is aoher deoisig-relaed codiio [, 3]. Also, s he fcio ϕ ( s ) is boded, sice b, b : b ϕ ( s) b, s. I also represes a Lipschiz fcio, becase is derivaive ϕ ' ( s) is boded. These codiios are very impora for he mahemaical reame of he proposed model ha is performed i he hird secio ad is relaed o is well-posedess. The opimally filered image is achieved by solvig he PDE model expressed by (), represeig is solio. Therefore, we have o ivesigae he exisece ad he iqeess of sch a solio for his forhorder PDE scheme. Ths, he well-posedess of he proposed oliear diffsio model will be reaed i he ex secio. So, we demosrae ha his parial differeial eqaio admis a iqe ad weak solio ha correspods o he resored image. Also, or diffsio scheme has he localizaio propery, is solio propagaig wih fiie speed [8]. I he forh secio, oe performs a discreizaio of he PDE model o approximae ha solio.

3 Tdor BARBU, Ioț MUNTEANU 3 3. MATHEMATICAL INVESTIGATION OF THE PDE SCHEME A mahemaical reame of he PDE model () is performed i his secio. We say ha a fcio L (, T ; H ( ) ) L (, T ; L ( ) ) = C(, T ; L ) ad is a weak solio o he eqaio () if ( ) T L L [, ]: ( ) ( ) ad ( ) ( ) xy (,, ) = xy (,, ) =Δ xy (,, ) =, [, T], ( xy, ) ξ [ ( ) ( ) ] fdxdy +γ [ ( ) ( ) ] fdxdy + ϕ ( ) ΔΔ fdxdy ds +λ ( ) fdxdy ds =, (4), [, T], f H ( ) H( ) (, xy, ) = ( xy, ), (, xy, ) = ( xy, ), ( xy, ) Le s cosider a boded posiive coios fcio a=a(, x, ad sdy he followig eqaio: ξ + γ + Δ( a Δ) + λ( ) = (, x, = (5) (, x, = (, x, = (, x, = Δ(, x, =,,, = λ, Le s deoe by A: D( A) L ( ) [ D( A) ]' he operaor A v aδ Δvdxdy+ avdxdy,, v D( A) where ( A) = { H ( ); = = Δ = o } D ad [ D ( A ) ]' is he dal space wih respec o he pivo space L ( ). We see ha A, v, D( A), so A is moooe. Also, by he Schwarz ieqaliy, we have: A( ) v a Δ ( ) v i D(A), he lim Δ ( ) = lim = Fially, we see ha A,,,, v D ( A). I is clear ha if C, D ( A) [ D ( A ) ]' D ( A ) ad. I follows ha A is demi-coios [9].. We coclde ha A is maximal moooe, he, via classical heory, he eqaio () has a iqe solio i L (, T; D( A) ) sch ha L (, T ; D( A) ) []. The we ge: ξ d d By + γ + a ( Δ) dxdy ds + λ ds = ξ dxdy + dxdy + γ λ dxdy ds we have deoed he orm of L ( ) ξ d d. The eqaliy (6) implies ha ( + ) + γ, + γ + λ ds C + λ ds ξ d for some posiive cosa C. We ge C, [, T ] where he cosa C is idepede of, d he solio, ad he fcio a. I depeds oly o he parameers ξ, γ, λ ad he L - orm of ad. We also ge ha ( ) C ad he cosa C depeds oly o he parameers of he sysem ad ad. Comig back o (6), we also ge ha (6) (7) a( Δ ) dxdy ds C 3, [, T ] (8)

4 4 A oliear forh-order diffsio-based model for image deoisig ad resoraio wih C depedig o ξ, γ, λ, 3 > iegraig over [, ] oe obais: ξ γ d d ad. Now, by mliplyig eqaio (5) by, ad + + aδ s ( Δ) ds + λ = ξ dxdy + γ dxdy + λ + λ ds Sice a Δ ( Δ ) ds= a( Δ) a( Δ ) as ( Δds ) s d ad ds = ds =, we ge s s ( ) C 4, [, T ], wih he cosa C 4 ds depedig o ξ, γ, λ, ad Δ = Δ. Now we are ready o cosrc, where we deoe ( ) s s he weak solio o (). To his ed, le s cosider he seqece ( ) L(, T ; D ( A) ) ( ( ) ) ( ), defied as: ξ + γ + Δ ϕ Δ + λ = (, x, = () ( ) (, x, = (, x, = (, x, = Δ (, x,,. Eqaio () has a solio for all N, by replacig a o ϕ ( ). This follows by sadard exisece of he resls for liear hyperbolic eqaio wih ime-depede coefficies. By sig (8), ( ) C ad he Poicare ieqaliy, we see ha he seqece is boded i L (, T ; H ( ) ) ad so, he exiss L (, T ; H ( ) ) sch ha i, T ; H ( ) from where we also ge ha i L (, T ; L ( )), i L (, T ; H ( ) ), () ( ) L ( ) ad ( )() () i L ( ), [,T ] H ( ) H ( ) o obai: f ad iegrae over [ ], ( ) ( ) ( ) ( ) ( ) ( ) ξ fdxdy+γ fdxdy+ ( ) ( ) s (9) ( ) L, i. Le s mliply he PDE i () by fdxdy ds fdxdy ds + ϕ Δ Δ +λ = Makig se of he meioed covergece relaios, he Lipschiz coiiy of ϕ, we may pass () o he limi ( ) o arrive o he fac ha saisfies he hird codiio i (4), cocldig so ha represes a weak solio o (). () 4. NUMERICAL APPROXIMATION ALGORITHM Now we ied o approximae ha iqe ad weak solio of he forh-order PDE, whose exisece has bee demosraed i he previos secio. Therefore, he coios oliear diffsio-based model is discreized by applyig he fiie-differece mehod []. A cosise merical approximaio scheme is hs cosrced for his mahemaical model. So, le s cosider a space grid size of h ad a ime sep Δ. The space ad ime coordiaes are qaized as follows: x = ih, y = h, =, i {,.., I}, {,., J}, {,.., N}. () The forh-order diffsio eqaio give by () ca be rewrie as:

5 Tdor BARBU, Ioț MUNTEANU 5 ξ + γ + Δ ϕ + + +λ = x y x y ( ). The, he eqaio (3) is discreized by sig he fiie differeces []. The compoe +Δ Δ +Δ Δ i, + i, i, i, i, ξ + γ + λ ( ) is approximaed firs as ξ + γ + λ( ) i,. i, Δ Δ The secod compoe is discreized ex by applyig he fiie differece-based discreizaio of he Laplacia operaor. Therefore, we compe = ϕ ( ),.., N, where ϕ for { } i, i, i, ( i + h, ) + ( i h, ) + ( i, + h) + ( i, h) 4 ( i, ) Δ i, = (4) h ad ( ) ( ) ( ) i + h, i h, i, + h i, h ϕ i, = ϕ + (5) 4 h 4 h is comped sig (). The, we apply he Laplacia ad ge he discreizaio of he secod compoe: ϕ +, +ϕ, +ϕ, + +ϕ, 4ϕ, ( ( ) ) i i i i i i, i, i, ϕ =Δ ϕ =. (6) h If we cosider h = ad Δ =, he ex implici merical approximaio scheme is obaied for (3): + + i, i, ( i, i, i, ) i, ( i, i, ) ξ + + γ + Δϕ + λ =. (7) The, (7) leads o he ex explici merical approximaio algorihm: + 4ξ λ ξ γ Δϕ i, λ ( i, ) = ( i, ) ( i, ) + + ( i, ), [, N ](8) ξ + γ ξ + γ ξ + γ ξ + γ where ( i, ) =. This ieraive scheme is applied o he evolvig image for each {,...,N }. The resoraio i, process sars wih = = represeig iiial [ I J ] image corrped by Gassia oise. Or explici merical approximaio scheme is cosise o he oliear diffsio model (). For he opimal vales of parameers ξ, γ, λ specified i (9), i also coverges fas o he approximaio of is weak solio, N + represeig he opimal image deoisig,, he mber of ieraios, N, becomig qie low. (3) 5. EXPERIMENTS AND METHOD COMPARISON We have sccessflly esed he proposed oliear diffsio-based smoohig echiqe o hdreds of images affeced by Gassia oise. Some impora image collecios, sch as he Volme 3 of he USC - SIPI daabase have bee sed i or deoisig experimes. We have deermied o a rial ad error basis, hrogh empirical observaio, he ex se of parameer vales ha provide a opimal image ehaceme: δ=.5, ξ=.8, γ=.7, λ=., α=.4, η=.3, k = 3, β= 4, ε= 3, N = 4. (9) The performed filerig experimes prove ha or PDE-based deoisig approach removes sccessflly he oise, while preservig some esseial image deails, sch as he edges. I overcomes he desired effecs, like blrrig, saircasig [5] ad speckle oise [3]. The proposed deoisig echiqe execes very fas, give he low vale of he ieraios mber, N. Is rig ime is less ha secod. The performace of he resoraio algorihm has bee assessed by sig performace measres like Peak Sigal-o-Noise Raio (PSNR), Norm of he Error (NE) ad Srcral Similariy Image Meric (SSIM) []. The described oliear forh-order PDE model operforms meros sae-of-he-ar resoraio mehods, prodcig mch beer vales for he performace parameers. Or resoraio approach performs

6 6 A oliear forh-order diffsio-based model for image deoisig ad resoraio 3 beer ha classic D image filers, sch as Average, Media, Gassia D, Wieer [] ad more effecive filers like LLMMSE Lee [3], ad he liear PDE-based smoohig approaches, becase i overcomes he ieded blrrig effec ad preserves esseial feares. Also, like hose filerig mehods, he proposed smoohig echiqe has he localizaio propery. Also, or diffsio-based deoisig mehod operforms some ifleial oliear secod-order PDE resoraio schemes, sch as boh versios of he Peroa-Malik aisoropic diffsio model ad oher derived algorihms [,3], ad he TV Deoisig [4]. Ths, i provides a more effecive Gassia oise removal ad execes mch faser. Also, like hese secod-order differeial models, i ca sccesflly overcome he saircase effec [5]. The described resoraio echiqe is also more effecive ha some sae of he ar oliear forh-order PDE deoisig models. Or PDE-based filerig model performs beer ha some poplar oliear forh order PDE schemes, sch as he isoropic diffsio model Yo-Kaveh [3] ad he LLT deoisig algorihm [4]. Besides avoidig he blrrig effec ad providig a beer edge preservaio ha hose echiqes, or filerig algorihm removes sccessflly he ieded speckle oise [4] ad operaes mch faser ha hem. The PSNR vales achieved by he proposed echiqe ad oher resoraio approaches meioed here are regisered i he followig able. Or forh-order diffsio model ges higher PSNRs ha he oher coveioal ad PDE-based approaches. 5 5 Lea Some resoraio resls geeraed by hese schemes are displayed i Fig.. Origial [ ] image is depiced i (a). I is corrped wih a amo of Gassia oise characerized by μ =. ad var D filers, sch =., he degraded image beig displayed i (b). The smoohig resls prodced by [ 3 3] as classic Gassia filer, Average ad Wieer, are displayed i (c) (e). The LLMMSE-Lee based filerig resl is depiced i (f). The resoraio resls provided by he PDE-based approaches are displayed i (g) (): Peroa-Malik scheme, TV Deoisig, Yo-Kaveh model ad he echiqe proposed here. Oe ca observe ha image (), marked i red ad represeig or resoraio resl looks beer ha oher ops. Also, he ieded deoisig effecs, sill visible i (c) o (i), are almos compleely removed i (). Table PSNR vales provided by several mehods This PDE scheme Average Gassia Wieer LLMMSE-Lee Peroa-Malik TV Deoisig Yo-Kaveh 8.5(dB) 5.43(dB) 5.7(dB) 6.5(dB) 7.63(dB) 6.84(dB) 6.79(dB) 7.3(dB) Fig. Resoraio resl achieved by varios filerig echiqes.

7 4 Tdor BARBU, Ioț MUNTEANU 7 6. CONCLUSIONS A ovel PDE-based image resoraio echiqe has bee described i his aricle. The proposed deoisig approach is based o a oliear forh-order diffsio-based model ha represes or mai coribio. This origial forh-order PDE scheme has a mch differe form ha well-kow Yo-Kaveh model ad oher derived forh-order PDEs, ses a ew diffsiviy fcio ad special bodary codiios. The rigoros mahemaical reame of he differeial model represes aoher coribio of or work. We demosrae ha he diffsiviy fcio is properly seleced ad he PDE scheme is well-posed, admiig a iqe ad weak solio. The, a cosise fas-covergig merical approximaio scheme is developed for his forh-order PDE model. We apply he fiie-differece mehod o ge a explici ieraive discreizaio scheme for he coios model. Is effeciveess is proved by he resls of or experimes ad mehod compariso. I achieves a opimal rade-off bewee oise redcio, feare preservaio ad ieded effec removal. Or echiqe operforms he coveioal filerig solios, sice i avoids he blrrig effec. I performs beer ha may secod-order PDE-based algorihms, by rig faser ad overcomig he saircasig, ad operforms he Yo-Kaveh like forh-order PDE deoisig models, by providig a beer deblrrig ad speckle removal. We will focs o frher improvig his deoisig echiqe drig or fre research. So, ovel varias of he diffsiviy fcio will be cosidered by s. Also we ied o iegrae his PDE-based resoraio scheme io more complex image ehaceme solios, sch as some effecive hybrid filerig models icorporaig boh secod- ad forh-order PDEs. ACNOWLEDGEMENTS This work was maily sppored by he proec PN-II-RU-TE , fiaced by UEFSCDI Romaia. I was sppored also by he Isie of Comper Sciece of he Romaia Academy. REFERENCES. R. GONZALES, R. WOODS, Digial Image Processig, Preice Hall, New York, NY, USA, d ediio,.. J. WEICKERT, Aisoropic Diffsio i Image Processig, Eropea Cosorim for Mahemaics i Idsry, B.G. Teber, Sgar, Germay, P. PERONA, J. MALIK, Scale-space ad edge deecio sig aisoropic diffsio, Proc. of IEEE Comper Sociey Workshop o Comper Visio, 987, pp L. RUDIN, S. OSHER, E. FATEMI, Noliear oal variaio based oise removal algorihms, Physica D: Noliear Pheomea, 6,, pp , A. BUADES, B. COLL, J.M. MOREL, The saircasig effec i eighborhood filers ad is solio, IEEE Trasacios o Image Processig, 5, 6, pp , Q. CHEN, P. MONTESINOS, Q. SUN, P. HENG, D. XIA, Adapive oal variaio deoisig based o differece crvare, Image Vis. Comp., 8, 3, pp ,. 7. J.F. CAI, S. OSHER, Z. SHEN, Spli Bregma mehods ad frame based image resoraio, Mliscale Model. Sim., 8,, pp , Y. HU, M. JACOB, Higher degree oal variaio (HDTV) reglarizaio for image recovery, IEEE Tras. Image Processig,, 5, pp ,. 9. T. BARBU, Robs aisoropic diffsio scheme for image oise removal, Procedia Comper Sciece (Proc. of KES 4, Sep. 5 7, Gdyia, Polad), 35, pp. 5 53, 4 (Elsevier).. T. Barb, A. FAVINI, Rigoros mahemaical ivesigaio of a oliear aisoropic diffsio-based image resoraio model, Elecroic Joral of Differeial Eqaios, 9, pp. 9, 4.. T. BARBU, Variaioal Image Deoisig Approach wih Diffsio Poros Media Flow, Absrac ad Applied Aalysis, 3.. T. BARBU, A Novel Variaioal PDE Techiqe for Image Deoisig, Lecre Noes i Comper Sciece (Proc. of ICONIP 3, Par III, Daeg, Korea, Nov. 3 7), 88, pp. 5 58, Spriger-Verlag, Berli, Heidelberg, M. Lee e al. (Eds.), 3.

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