An Iterative Conjugate Gradient Regularization Method for Image Restoration

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1 SSN , Eglad, UK Joual of foatio ad Coputig Siee Vol. 5, No., 00, pp A teative Cojugate Gadiet Regulaiatio Method fo age Restoatio Jiaju Zhag +, Qi Wag Dept. of Math., Shaghai Uivesity, Shaghai Reeived July 30, 009, aepted August 6, 009 Abstat: age estoatio is a ill-posed ivese poble, whih has bee itodued the egulaiatio ethod to suppess ove-aplifiatio. this pape, we popose to apply the iteative egulaiatio ethod to the iage estoatio poble ad peset a ested iteative ethod, alled iteative ojugate gadiet egulaiatio ethod. Covegee popeties ae established i detail. ased o [6], we also siultaeously detee the egulaiatio paaete based o the estoed iage at eah step. Siulatio esults show that the poposed iteative egulaiatio ethod is feasible ad effetive fo iage estoatio. Key Wods: ihoov egulaiatio, iage estoatio, blu.. todutio age estoatio poble has bee etesively studied ad used i seveal aeas of siee ad egieeig [][6][8-]. t alls fo the eovey of a oigial see fo a degaded obsevatio. Fo eaple, stella iages obseved by goud-base telesopes ae degaded due to atosphei tubulee, while thee ae also appliatios whee the stella iages eed to be estoed eve if they ae ot obseved though the atosphee. ost ases, the iage degadatio poess a be odeled by a liea blu ad a additive white Gaussia oise poess, that is y. whee y,, ae MN vetos ad epeset espetively the leiogaphially odeed M N piel obseved degaded iage, oigial iage, ad additive oise. he ati epesets the degadatio ati of sie MN MN, whih ay epeset a spatially ivaiat o a spatially vayig degadatio. he iage estoatio poble alls fo applyig a ivese poedue to obtai a appoiatio of the oigial iage based o the iage degadatio odel. t is a ill-posed poble, whih eas that a sall petubatio i the data leads to a lage petubatio i the solutio. heefoe, a egulaiatio ethod has to be used i ode to detee a useful appoiatio of the tue iage. Oe of the ost popula egulaiatio tehiques is ihoov egulaiatio. his ethod appoiately solves. by solvig the uostaied iiatio poble M,. With M, y.3 whee is a positive egulaiatio paaete. A solutio of. is oputed by solvig its fist-ode oditios y.4 his ethod has bee etesively studied i iage estoatio. oweve fo ill-posed poble, the ovegee ate ay be ipoved [4] i a iteated vesio of.4 give by y.5 + Coespodig autho. E-ail addess: jjhag@staff.shu.edu. Published by Wold Aadei Pess, Wold Aadei Uio

2 56 Jiaju Zhag, et al: A teative Cojugate Gadiet Regulaiatio Method fo age Restoatio We all it iteative egulaiatio. this pape, we popose to apply the iteative egulaiatio ethod to the iage estoatio poble ad peset a ested iteative ethod, alled iteative ojugate gadiet egulaiatio CGR ethod. Covegee popeties ae established i detail. ased o [6], we also siultaeously detee the egulaiatio paaete based o the estoed iage at eah step. Siulatio esults show that the poposed iteative egulaiatio ethod is feasible ad effetive fo iage estoatio. he est of the pape is ogaied as follows. Setio, CGR ethod is itodued, alog with the hoie of egulaiatio paaete. Setio 3, ovegee popeties ae established. Epeietal esults ae peseted i Setio 4 ad olusios ae eahed i Setio 5.. teative Regulaiatio Fo ill-posed pobles ivolvig losed, desely defied liea opeatos, M. ae ad C. W. Goetsh have studied the iteative egulaiatio ethod [4]. age degadatio is a ill-posed poble, ad the iteative egulaiatio ethod ould be atually applied to this poble. Fo the iage degadatio odel., iteative egulaiatio is y. whih is equivalet to the followig iiatio poble. L,. whee.. Choie of egulaiatio paaete L, y.3 ode fo the oliea ost futio L, to have a global iu, the egulaiatio paaete should be hose i a pope way. t is oted that hoosig a suitable egulaiatio paaete a pioi is diffiult, though thee ae ay eaigful hoies of the egulaiatio paaete. this pape, the followig popeties ae eeded whih is adapted fo [6]. Popety. should be a futio of the soothig futioal: We hoose to be popotioal to L, whih epesets the egulaied oise powe. Popety. Etee iies of L,. he iie of L, should epeset a solutio betwee two etee solutios: oe epesetig the geealied ivese solutio of. whe the data ae oiseless, ad the othe epesetig the soothest possible solutio, whe the oise powe beoes ifiite. Popety 3. he futioal L, should be ove fo all hoies of. his equieet o oveity is obviously vey ipotat, sie a loal eteu of a oliea futioal beoes a global eteu, if the futioal is ove. heefoe, the iteative algoith that will be eployed fo obtaiig a iie of L, will ot deped o the iitial oditio. ased o the above popeties, a tae the followig fo, Also aodig to [6], we set y. y.. CGR ethod he basi idea of ou poposed CGR ethod is as follows. Give a statig veto that we have got appoiatios,..., to the solutio of the oal equatio 0, R 0, suppose y.4 he the et appoiatio to is obtaied by solvig the followig equatio iteatively, JC eail fo otibutio: edito@ji.og.u

3 Joual of foatio ad Coputig Siee, Vol No., pp y.5 with the CG ethod, to etai aitheti peisio. Moe peisely, this iteative CG egulaiatio ethod a be desibed as follows. Algoith.. he CGR ethod. put the lagest adissible ube of oute iteatio a ad the oute iteatio stopig toleae. put the lagest adissible ube of ie iteatio la ad the ie iteatio stopig toleae 3. put the statig veto ad the egulaiatio paaete, 0 4. Do the followig steps y, s, s 4.., l l 4.3. Do while ad l la a f l the 0 ad p s else l l ad p s p b q p w d e p f s s w l g h l l 4.4. EdDo 4.5. es q p l p w Update Util es o a Notie that, at th oute iteates, is hose as the iitial appoiatio, whih eas that is eo fo, theefoe, we tae at the th oute iteates the followig fo, y y 3. Covegee Aalysis efoe povig the ovegee of the CGR ethod, we fist itodue soe otatios ad leas. Fo a syeti positive defiite SPD ati R, we use to epeset its spetu set, ad ad its sallest ad lagest eigevalues espetively. Fo ay a R, its -o is defied by. Fo a osyeti ati R, we use to epeset its sigula value, ad ad a its sallest ad lagest sigula values espetively. Clealy, fo ay osigula ati,. a JC eail fo subsiptio: publishig@wau.og.u

4 58 Jiaju Zhag, et al: A teative Cojugate Gadiet Regulaiatio Method fo age Restoatio ad Lea 3.. Let Lea 3.. Let R be a SPD ati. he fo ay a R, R be a SPD ati. f the CG ethod is stated fo a iitial iteate 0 R, the af te -steps of iteates, it geeates a appoiatio to the solutio of the liea equatio y, whih satisfies a p P p a 0 0 a whee P p p is a polyoial of degee ad p 0 is the set of th degee esidual polyoials. Coollay 3.. f equatio.5 is stated fo a iitial iteate R is a osigula ad osyeti ati. f the CG ethod fo 0 R, the afte -steps of iteates, it geeates a appoiatio to the solut io of the liea equatio.5, whih satisfies,, a a 0, heoe 3.. Let R be a osigula ati. f the CGR ethod is stated fo a iitial iteate 0 R, ad applies steps of CG iteatio to get the et appoiatio to the solutio of the oal equatio y. he whee q 3. q a Poof. Deote y, it satisfies, ad the fial esult of the ie ethod. he fo Coollay 3. we have, a a a - 3 a the eat solutio of the liea equatio.5, i.e.,, CG iteatio at the th, 3. oute iteate of the egulaied CGNR Notie that,,0,,, a a, the above estiate iediately leads to,0, K, Defie the veto he we have y ad a a., K JC eail fo otibutio: edito@ji.og.u

5 Joual of foatio ad Coputig Siee, Vol No., pp y y So, = heefoe, 3.3 O the othe had, a a a, a a a, a a a, a, a a a 3.4 y usig 3.4 ad taig -o o both sides of 3.3, we get a a a a a a a a is poved. 4. Epeietal Results this setio we peset the esults of two iage estoatio test pobles, i ode to illustate the JC eail fo subsiptio: publishig@wau.og.u

6 60 Jiaju Zhag, et al: A teative Cojugate Gadiet Regulaiatio Method fo age Restoatio pefoae of the poposed iteative egulaiatio ethod. he fist is Satellite, whih is a iage estoatio test poble that was developed at the US Ai Foe Phillips Laboatoy, Lases ad agig Dietoate, Kitlad Ai Foe ase, New Meio. he iage is a opute siulatio of a field epeiet showig a satellite as tae fo a goud based telesope, ad theefoe epesets a eaple of atosphei bluig. he tue ad blued, oisy iages have piels, ad ae show i Figue. Figue 3a displays the estoed iage. t has a elative eo of 0.697, ad its oputatio equies 45 iteatios. the seod eaple, we use the Gai test iage whih is otaied i the Restoeools paage [9]. he tue ad blued, oisy iages have piels, ad ae show i Figue. Figue 3b displays the estoed iage. t has a elative eo of , ad its oputatio equies 7 iteatios. oth the ueial epeiets have bee aied out o a Petiu V PC usig Matlab 7.. both the epeiets, the iitial iteate has bee hose as the bluy ad oisy iage. he oute iteatio 0 stoppig toleae.0e 3. hat is the iteatio is teated whe.0e 3. he ie iteatio is set as l a 0 ad is set as eps, that is.04e 6. able shows the ueial esults of the poposed algoith applied to the test pobles. he seod olu of the table gives the elative eo oputed oe. he thid olu epots the ube of the oute iteates. eal eal able. Nueial esults fo the test pobles est poble Relative eo eatios Satellite Gai betwee the oputed solutio ad the eal a.oigial iage b. lued iage Figue : Oigial ad blued satellite iage JC eail fo otibutio: edito@ji.og.u

7 Joual of foatio ad Coputig Siee, Vol No., pp a. Oigial iage b. lued iage Figue : Oigial ad blued Gai iage a b 5. Colusio Figue 3: Restoed Satellite iage a ad estoed Gai iage b We popose a iteative ojugate gadiet egulaiatio ethod to the iage estoatio poble. Epeiets have bee doe fo two iage estoatio pobles, the esults illustate that the poposed ethod has good pefoae. 6. Refeees [] Z. Z. ai ad S. L. Zhag. A egulaied ojugate gadiet ethod fo syeti positive defiite syste of liea equatios. JCM. 00, 0: [] M. ei ad M. K. Ng. Reoditioed iteative ethods fo weighted oeplit least squaes pobles. SAM J. Mati Aal. Appl. 006, 7: [3] R.. Cha ad M. K. Ng. Cojugate gadiet ethods fo oeplit systes. SAM Review. 996, 38: [4] M. ae ad C. W. Goetsh. Nostatioay iteated ihoov egulaiatio. JOA. 998, 98: JC eail fo subsiptio: publishig@wau.og.u

8 6 Jiaju Zhag, et al: A teative Cojugate Gadiet Regulaiatio Method fo age Restoatio [5] P. C. ase ad D. P. O Leay. he use of the L-uve i the egulaiatio of disete ill-posed pobles. SAM J. Si. Stat. Coput. 993, 4: [6] M. G. Kag ad A. K. Katsaggelos. Geeal hoie of the egulaiatio futioal i egulaied iage estoatio, EEE as. age Poess. 995, 4: [7] C.. Kelley. teative Methods fo Liea ad Noliea Equatios. SAM Philadelphia [8] G. Ladi. A fast tuated Lagage ethod fo lage-sale iage estoatio pobles. Applied Matheatis ad Coputatio. 007, 86: [9] K. P. Lee, J. G. Nagy ad L. Peoe. teative ethods fo iage estoatio: a Matlab objet oieted appoah. 00. [0] J. G. Nagy ad K. M. Pale. Steepest deset, CG, ad iteative egulaiatio of ill-posed pobles.. 004, 43: [] J.G. Nagy, R.J. Pleos ad.c. ogese. teative iage estoatio usig appoiate ivese peoditioes. EEE as. age Po. 996, 5: 5-6. JC eail fo otibutio: edito@ji.og.u