Inverse Problems in Imaging Systems and General Bayesian Inversion Framework

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1 Invrs roblms n Iman Sysms and Gnral Baysan Invrson Framwor Al Mohammad-Daar Snal and Sysm lab. Mxd Branch o sarch No.856 CNS Suélc Unvrsy o Souh ars Suélc laau d Moulon 3 Jolo-Cur s. 99 GIF-SU-YVETTE Cdx Franc Absrac: In hs ar rs a ra numbr o nvrs roblms whch ars n nsrumnaon n comur man sysms and n comur vson ar rsnd. Thn a common nral orward modln or hm s vn and h corrsondn nvrson roblm s rsnd. Thn ar shown h nadquacy o h classcal analycal and las squar mhods or hs ll osd nvrs roblms a Baysan smaon ramwor s rsnd whch can handl n a cohrn way all hs roblms. On o h man ss n Baysan nvrson ramwor s h ror modln o h unnowns. For hs rason a ra numbr o such modls and n arcular h comound hddn Marov modls ar rsnd. Thn h man comuaonal ools o h Baysan smaon ar brly rsnd. Fnally som arcular cass ar sudd n dal and nw rsuls ar rsnd. y words: Invrs roblm Baysan ramwor Lar sysm Marov modl. INTODUCTION Invrs roblms ars n many alcaons n scnc and nnrn. Th man rason s ha vry on w wan o masur h dsrbuon o an non-obsrvabl quany r rom h obsrvaon o anohr quany s whch s rlad o and accssbl o h masurmn. Th mahmacal rlaon whch vs s whn r s nown s calld orward roblm: 3 s [ r] s s whr s h orward modl. In hs rlaon r and s may rrsn hr m oson on a ln x oson on a surac r x y oson n sac r x y or any combnaons o hm. Ths orward modl s on non lnar bu can b lnard. So n hs ar w only consdr h lnar modl whch n s nral orm can b wrn as s h r s r dr s Whr h r s rrsns h masurn sysm rsons and s all h rrors modln lnaraon and h ohr unmodlld rrors on calld nos. In hs ar w assum ha h orward modl s nown rcly or a las nown xcd a w numbr o aramrs. Th nvrs roblm s hn h as o on bac rom h obsrvd quany s o r. Th man dculy s ha vry on hs roblms ar ll-osd n oson o h orward roblms whch ar wllosd as dnd by adamard []. A roblm s mahmacally wll-osd h roblm has a soluon xsnc h soluon xss unqunss and h soluon s sabl sably. مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6

2 A roblm s hn calld ll-osd any o hs condons ar no sasd []. In hs ar w wll only consdr h albrac mhods o nvrson whr n a rs s h orward roblm s dscrd.. h nral quaon s aroxmad by a sum and h nu h ouu and h rrors ar assumd o b wll rrsnd by h n dmnsonal vcors and such ha: n... n 3 Whr s s r and h r s or n a mor nral cas whr h h nsrumn s rsons. I hs rsons s nvaran n m hn w hav a convoluon orward modl: h d 7 and h corrsondn nvrs roblm s calld dconvoluon. F.: Dconvoluon o ID snals. φ s s φ s s ψ r s ψ φ s s ds φ s s ds r r dr 4 whr φ s and ψ s ar arora bass uncon n hr corrsondn sacs whch mans ha w assum Th convoluon quaon 7 can also b wrn h τ τ dτ 8 whch s oband by chan o varabl τ. Assumn h samln nrval o h and o b qual o h dscrd vrson o h dconvoluon quaon can hn b wrn: Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 s s r m m φ s φ s ψ r φ s ψ s n ψ r φ s dr ds مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان Bu bor on urhr n dals o h nvrson mhods w ar on o rsn a w xamls... D Snals Any nsrumn such as a hrmomr whch rs o masur a non-drcly masurabl quany hr h m varaon o h mraur ransorms o h m varaon o a masurabl quany hr h lnh o h lqud n h hrmomr. A rc nsrumn has b a las lnar. Thn h rlaon bwn h ouu and h nu s: h d 6 4 h... T 9 whch can b wrn n h nral vcor-marx orm: whr and conans samls o h ouu and h nu and h marx n hs cas s a Tol marx wh a nrc ln comosd o h samls o h muls rsons h. Th Tol rory s hus dnd o h m nvaranc rory o h sysm rsons convoluon orward roblm... Ima soraon In hs ar w consdr mor h cas o bvara snals or mas. As an xaml whn h unnown and masurd quans ar mas w hav: r h r r r dr r and h sysm rsons s saally nvaran w hav

3 r h r r r r Th cas o dnosn s h arcular cas whr h on srad uncon s h r s h r δ r : r r r 3 Th dscrd vrson o hs orward quaon can also b wrn as Q whr [ L ] conans samls o rocon daa r φ L φ or drn anls { r r } conans h ma xls u n a vcor and h lmns o h marx n hs cas rrsns h lnh o h -h ray n h -h xl. Ths marx s a vry sars marx wh ra numbr o ro valud lmns [7 8]. F.: Ima rsoraon as an nvrs roblm Th dscrd vrson o h D dconvoluon quaon can also b wrn as whr and conans rscvly h rasrd samls o h ouu r and h nu r and h marx n hs cas s a hu dmnsonal Tol-Bloc-Tol TBT marx wh a nrc bloc-ln comosd o h samls o h on srad uncon SF h r. Th TBC rory s hus dnd o h sac nvaranc rory o h sysm rsons D convoluon orward roblm. For mor dals on h srucur o hs marx rr o h boo [3] and h ars [4 5 6]..3. Ima consrucon n Comud Tomorahy In rvous xamls s and r whr dnd n h sam sac. Th cas o ma rconsrucon n X ray comud omorahy CT s nrsn bcaus h obsrvd daa s and h unnown ma r ar dnd n drn sacs. Th usual orward modl n CT s shown n Fur.3. In D cas h rlaon bwn h ma o b rconsrucd x y and h rocon daa r φ r s vn by h adon ransorm: r φ φ x y dl r φ x y δ r xcosφ ysnφ dxy L r φ r φ 4 F.3: D and 3D ray comud omorahy F.4: Dscrd D X ray coud omorahy F.5:Invrs roblm o ma rconsrucon n x ray comud omorahy Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6 5 مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385

4 .4 Tm varyn man sysms Whn h obsrvd and unnown quans dnd on sac r and m w hav r h r r r dr d r 5 I h on srad uncon o h man sysm dos no dnd on m hn w hav In hs cas can also b consdrd as an ndx: r h r r r dr r 7 r h r r r dr r 6 On xaml o such roblm s h vdo ma rsoraon shown n Fur MIMO Sourcs Localaon and Esmaon On such xaml s h cas whr n rado sourcs L n mn n h sam m ar { } rcvd by m rcvrs { L m} ach on rcvn a lnar combnaon o dlayd and dradd vrsons o ornal wavs: N h τ d... N Whr h s h muls rsons o h channl bwn h -h rcvr and h -h sourc. Th dscrd vrson o hs nvrs roblm can b wrn as Whr and and h nu conans samls o h ouu and h marcs ar Tol marcs dscrbd by h muls F.6: nvrs roblm o vdo ma rsoraon Th dscrd vrson o hs nvrs roblm can b wrn as r h r r r dr r h. rsonss Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 Whr and 8 conans samls o h ouu r and h nu r and h marx n hs cas s aan a Tol-Bloc-Tol TBT marx wh a nrc bloc-ln comosd o h samls o h on srad uncon SF h r..5. Mul Inus Mul Ouus Invrs roblms Mul Inus Mul Ouus MIMO man sysms can b modld as: s N h s r r dr... N مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 r MIMO Dconvoluon A MIMO ma rsoraon roblm s : and on such xaml s h cas o color ma rsoraon whr ach color comonn can b consdrd as an nu. F.7: Color ma rsoraon as an xaml o MMO nvrs roblm.6. Sourc Saraon A arcular cas o a MIMO nvrs roblm s h blnd sourc saraon BSS: r A h r r r dr r 3 and a mor arcular on s h cas o nsananous mxn:

5 r A r r 4 Th arculary o hs roblms s ha h h A s also unnown. A mxn marx { }.7.. Ima Sur-rsoluon as a MISO Invrs roblm Anohr MISO sysm s h cas o Sur- soluon S man usn a w Low soluon L mas oband by low cos camras: s h s r r dr r 6 whr ar h L mas and s h dsrd h soluon ma. Th uncons h rrsn a combnaon o a las hr oraons: a low ass lrn c a movmn ranslaonal or wh roaon and oomn cs o h camra and a sub-samln. Th ollown ur shows on such suaon. F.8: Blnd ma saraon.7. Mul Inus Snl Ouu Invrs roblms A Mul Inus Snl Ouu MISO sysm s a arcular cas o MIMO whn w hav only on nu: s h s r r dr r MIMO sourcs localaon and Esmaon On xaml o MISO nvrs roblm s a non dsrucv sn NDT or dcon and valuaon o h dc crad du o an mac on a surac o an obc usn mcrowav man whr wo mas ar oband whn a rcanular wavud scans hs surac wo ms. In h rs scan h rcanular wavud s ornd n shorr sd and n h scond cas n lonr sd. By hs way wo mas r ar oband ach has o b consdrd as h ouu o a lnar sysm wh h sam nu r and wo drn channls. Ths s a MISO lnar and nvaran sysms. 7 F.9: S roblm whr a srs o L ma ar usd o consruc a ma Th dscrd vrson o hs nvrs roblm can b wrn as 7 Whr and conans samls o h ouu and h nu and h marcs ar Tol marcs dscrbd by h muls rsonss h..8. Mul Modaly n CT Iman Sysms Usn drn modals has bcom a man ool n man sysms whr o xlor h nrnal rory o a body on can us X rays ulrasounds mcrowavs nra-rd manc rsonanc c. As an xaml n X ray man h obsrvd radorahs v som normaon on h volumnal dsrbuon o h maral dnsy nsd h obc whl h ulrasound chorahy vs normaon on h chann osons conours o ulrasound rors nsd h obc. On can hn wan o us boh chnqus and us a nd o daa مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6

6 uson o oban a hhr qualy o mas o h body. An xaml o such suaon s vn n.8. 8 Th da can b asly xndd o h cas o MISO or MIMO. For an xnd dals o hs mhods rr o [3 4]. Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 F.: mul modaly n CT man sysms a Ornal obc b Conours o h drn homonous rons c Daa omry n X ray omorahy d Daa acquson omry n ulrasound chorahy Obsrvd daa sonoram n X ray omorahy Obsrvd daa n ulrasound chorahy..9. Fuson o X-ay and Ulrasound Echorahy An xaml o mulmodaly and daa uson n CT s h us o X ray radorahc daa and h ulrasound chorahc daa s shown n Fur and or mor dals on hs alcaon s [9 ]. F.: Invrs roblm o X ray and ulrasound daa uson. Bascs o Drmnsc Invrson Mhods To llusra h bascs o h nvrson mhods w sar by consdrn h cas o a Snl Inu Snl Ouu SISO lnar sysm: 8 مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان Mach Flrn Frs assum ha h rrors and masurmn nos ar nlbl and ha w could choos h bass uncons φ and ψ could b chosn n such a way ha h marx s squar n m and sl-don un unralsc hyohss. Thn h soluon o h roblm would b: ˆ 9 Ths soluon has bn usd n many cass. For xaml n dconvoluon hs soluon s calld Machn lrn. Th man rason s ha n a dconvoluon roblm h marx s a Tol marx so s s ransos. Th orward marx oraon corrsonds o a convoluon conv h. Th adon marx oraon hn also corrsonds o a convoluon conv h whr ~ h hˆ. Anohr xaml s n comud omorahy CT whr h rocon daa n ach anl drcon s rlad o h ma hrouh a rocn marx n ha drcon such ha w can wr: M M M And h adon oraon: ˆ 3 3 corrsonds o wha s calld bac-rocon. owvr as s mnond h hyohss mad hr ar unralsc.

7 .. Drc Invrson Th nx s s us o assum ha h orward marx s nvrbl. Thn on can ry o dn h soluon as: ˆ 3 Bu n racc hs also s an lluson bcaus vn h marx s mahmacally nvrbl s vry on vry ll-condond. Ths mans ha small rrors on h daa δ wll nra ra rrors δ on h soluon. Ths mhod n dconvoluon corrsonds o h analycal mhod o nvrs lrn whch s n nral unsabl. In ohr alcaons h man dculy s ha vry on h marx s vn no squar m n bcaus h numbr o h masurd daa m may no b qual o h numbr o aramrs n dscrbn h unnown uncon.. n Las Squar and Gnrald Invrson For h cas whr squar LS dnd as: m n a soluon wll b h las { } 33 ˆ ar mn whch rsuls o h normal quaon: [ ] ˆ 34 s nvrsabl ran and h marx hn h soluon s vn by: n [ ] 35 ˆ n Whn m h roblm may hav an nn numbr o soluons. So w may choos on o hm by rqusn som arcular a ror rory or xaml o hav mnmum norm. Th mahmacal roblm s hn: ˆ ar mn { } or wrn drnly { } 36 mn m subc o 37 9 Th soluon s oband va h Laran mullr mhod whch n hs cas rsuls o I whch vs λ ˆ s nvrbl. Th man dculy n hs mhods s ha h soluon n nral s oo snsv o h rror n h daa du o h ll condonn o h marcs o b nvrd..4. ularaon Mhods Th man da n rularaon hory s ha a sabl soluon o an ll-osd nvrs roblm can no b oband only by mnmn a dsanc bwn h obsrvd daa and h ouu o h modl as s or xaml n LS mhods. A nral ramwor s hn o dn h soluon o h roblm as h mnmr o a comound crron such as: ˆ ar mn wh { J } 4 J λ 4 whr and ar wo dsancs h rs dnd n h obsrvd quany sac and h scond n h unnown quany sac. λ s h rularaon aramr whch rulas h comroms wh h wo rms and s an a ror soluon. An xaml o such crron s: λ 4 J whch rsuls o ˆ مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 [ λi ] 43 W may no ha h condon numbr o h marx o b nvrd hr can b conrolld by aroraly choosn h valu o h rularaon aramr λ. Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6

8 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 Evn h mhods basd on rularaon aroach hav bn usd wh succss n many alcaons hr man on roblms sll rmans: Drmnaon o h rularaon aramr Th arumns or choosn h wo dsancs and and Quancaon o h uncrans assocad o h oband soluons. Evn hr hav bn a lo o wors ryn o answr o hs roblms and hr ar cv soluons such as h L-curv or h Croos Valdaon or h rs h wo ohrs ar sll on roblms. Th Baysan smaon ramwor as w wll s can v answrs o hm [5]. 3. Baysan Esmaon Framwor To llusra h bascs o h Baysan smaon ramwor l rs consdr h sml cas o SISO sysm whr w assum ha s nown. In a nral Baysan smaon ramwor h orward modl s usd o dn h llhood uncon θ and w hav o ransla our ror nowld abou h unnowns hrouh a ror robably law θ and hn us h θ. Bays rul o nd an xrsson or θ whr θ θ θ 44 θ s h llhood whos xrsson s oband rom h orward modl and θ θ θ rrsns assumon on h rrors all h hyr-aramrs aramrs o h llhood and rors o h roblm and θ θ θ d 45 s calld h vdnc o h modl. θ s oband w Whn h xrsson o can us o dn any smas or. Two usual smaors ar h maxmum a osror MA ˆ armax { θ } 46 مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 and h Man Squar Error MSE smaor whch corrsonds o h osror man ˆ θ d 47 Unorunaly only or h lnar roblms and h θ s also Gaussan w Gaussan laws whr hav analycal soluons or hs wo smaors. For almos all ohr cass h rs on nds an omaon alorhm and h scond an nraon on. For xaml h rlaxaon mhods can b usd or h omaon and h MCMC alorhms can b usd or xcaon comuaons. Anohr dcul on s ha h xrssons o θ and θ and hus h xrsson o θ dnd on h hyr-aramrs θ whch n raccal alcaons hav also o b smad hr n a survsd way usn h rann daa or n an unsurvsd way. In boh cass w nd also o ransla our ror nowld on hm hrouh a ror robably θ. Thus on o h man ss n any nvrson mhod or any nvrs roblm s modln h unnowns. In robablsc mhods and n arcular n h Baysan aroach hs s bcoms h assnmn o h robably law θ. Ths on as wll as h assnmn o θ ar dscussd h nx wo subscons. 3. Sml cas o Gaussan modls L consdr as a rs xaml h sml cas whr ǫ and ar assumd o b Gaussan: δ N I x N x Thn s asy o show ha: 48

9 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6 ناتسمز و زيياپ مود هرامش -موس لاس -ناريا كينورتكلا و قرب نيسدنهم نمجنا هلجم 385 x 49 x N I N and 5 ˆ ˆ N Wh 5 ˆ ˆ ˆ Whn and non by λ hs rlaons wr: 5 ˆ ˆ ˆ λ λ I s nod ha n hs cas all h on smaors such as h MA h osror man or osror mdan ar h sam and can b oband by: { } { } { } 53 ar mn ln ar mn ar max ˆ J wh 54 J λ Thr arcular cass ar o nrs: I. Ths s h cas whr ar assumd cnrd Gaussan and..d.: 55 x x CC. Ths s h cas whr ar assumd cnrd Gaussan bu corrlad. Th vcor s hn consdrd o b oband by: 56 Cξ wh C corrsonds o a movn avra MA lrn and N ξ. In hs cas w hav: [ ] 57 x x C C A I D D. Ths s h cas whr ar assumd cnrd Gaussan and auorrssv: 58 ξ A wh A a marx oband rom h A cocns and N ξ. In hs cas w hav 59 x D A arcular cas o A modl s h rs ordr Marov chan 6 N wh corrsondn A and A I D marcs 6 A D whch v h ossbly o wr 6 x x D

10 Ths arcular cass v us h ossbly o xnd h ror modl o ohr mor sohscad non-gaussan modls whch can b classd n hr rous: Sarabl: x α φ Modln Usn ddn Varabls 3.. Snal and Imas wh Enry Modulaon A sml modl whch can caur h varanc modulad snal or mas s [9 7 ]. d λ N d d λ ς 3 / λ 67 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 whr φ s any osv valud uncon. Sml Marovan: x φ whr φ s any osv valud uncon calld onal uncon o h Marovan modl. Comound Marovan: c xα φ c whr φ s any osv valud uncon whos xrsson dnds on h hddn varabl c. Som xamls o h φ xrssons usd n many alcaons ar: β φ ; β ; ln > ; mn ; Ths quaons can asly b xndd or h cas o mul-snsor cas. owvr vn a Gaussan modl or h nos s accabl hs modl s rarly ralsc or mos ral word snals or mas. Indd vry on a snal or an ma can b modld locally by a Gaussan bu s nry or amlud can b modulad..; cws homonous and Gaussan [6 7 8]. To nd an arora modl or such cass w nroduc hddn varabls and n arcular hddn Marov modln MM. In h ollown w rs v a summary dscron o hs modls and hn w wll consdr h nral cas o MIMO sysms wh ror MM modln. مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان whr G s a Gamma dsrbuon. I s hn asy o show h ollown rlaons: d λ x λ x and 4d [ J d ] d d x 69 J d λ d 4d I w ry o nd h on MA sma o h unnowns d by omaon succssvly wh rsc o whn d s xd and wh rsc o d whn s xd w oban h ollown rav alorhm: ˆ λd [ / 4d...n] D da 7 dˆ / 3... Amlud Modulad Snals To llusra hs wh alcaons n lcommuncaon snal and ma rocssn w consdr h cas o a Gaussan snal modulad wh a wo lvl or bnary snal. A sml modl whch can caur h varanc modulad snal or mas s λ N / λ 68

11 { m m } 7 m /... I s hn asy o show h ollown: [ N / λ N / λ ] λ / / N m / λ λ x [ λ ] [ λ ] λ x [ λ ] [ ] λ x 7 λ λ x and λ x [ J ] J λ 73 ln / δ m whr [ L ]. N Aan ryn o oban h JMA sma ˆ omn succssvly and w oban: ˆ by J wh rsc o [ λ] ˆ λi > ẑ 74 < a whr h hrshold a s a uncon o λ Gaussans Mxur Modl Th rvous modl can b nrald o h nral mxur o Gaussans. W hn hav h ollown rlaons: m υ N m υ / λ π {...} 3 π m π N m υ υ υ 75 Nm m λ x[ { } λ ] : m [ λ m ] x δ m x and hus m λ x and m λ δ C π [ [ λ ] δ m ln π m λ [ λ m ln π ] m λ x [ J ] 76 m λ x wh J m ln π δ مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 [ J ] { } λ λ m n ln π 77 whr m { m } λ { λ } n L m m L λ δ s h numbr o samls whch ar n h class and { } :. For mor dals and alcaons o such modln s [ 3 4] Mxur o Gauss-Marov Modl In h rvous modl w assumd ha h samls n ach class ar ndndn. r w xnd hs o a Marovan modl: m υ N m υ m υ N υ {...} π 78 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6

12 whch can b wrn n a mor comac way w q δ by nroduc m υ Nq m q whch rsuls n: υ 79 m λ x[ λ [ ] ] δ q m [ δ [ q q m ] 8 x[ λ and whn combnd wh J q [ q q ] λ 83 λ QD n ln α Whr n q s h numbr o dsconnus lnh o h conours n h cas o an ma q α q. α and In all hs mxur modls w assumd. owvr ndndn wh Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 x vs: m λ x[ J ] wh J λ [ ] δ q m q n ln π 8 ~ ~ q مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 n ln π ~ GD whr ˆ λ m n ln π D s h rs ordr n drnc marx and Q s a marx wh q as s daonal lmns. A arcular cas o hs modl s o ra nrs: m and. Thn w hav: q λ q m λ x λ m υ N [ λ [ q ] ] [ [ q ] q υ x λ 8 and q m λ x wh [ J q ] 4 corrsonds o h labl o h saml. I s hn br o u a Marovan srucur on o caur h ac ha n nral whn h nhborn samls o hav all h sam labl hn mus b mor robabl ha hs saml has h sam labl. Ths aur can b modld va h os-marov modln o h classcaon labls. In h nx scon w us hs modl and a h sam m w xnd all h rvous modls o D cas or alcaons n ma rocssn and o MIMO alcaons Mxur and ddn Marov Modls or Imas In ma rocssn alcaons h noons o conours and rons ar vry moran. In h ollown w no by r x y h oson o a xl and by r s ray lvl or by r { r... N r } s color or scral comonns. In classcal GB color rrsnaon N 3 bu n hyr-scral man N may b mor han on hundrd. Whn h obsrvd daa ar also mas w no hm by r { r... M r }. For any ma r w no by q r a bnary valud hddn varabl s conours and by r a dscr valu hddn varabl rrsnn s ron labls. W ocus hr on mas wh homonous rons and us h mxur modls o h rvous scon wh an addonal Marov modl or h hddn varabl r.

13 3.3. omonous rons modln In nral any ma r r s comosd o a n s o homonous rons wh vn labls... such ha { r : } U and h corrsondn xl valus { r : r } and U. Th ddn Marov modln MM s a vry nral and cn way o modl aroraly such mas. Th man da s o assum ha all h xl valus r : r o a homonous ron { } ollow a vn robably law or xaml a Gaussan N m whr s a nrc vcor o ons o h s n h numbr o xls n ron. In h ollown w consdr wo cass: Th xls n a vn ron ar assumd d: r r Nm and hus r r r Nm I 85 Ths corrsonds o h classcal sarabl and monovara mxur modls. Th xls n a vn ron ar assumd o b locally dndn: r rr Nm 86 whr s an arora covaranc marx. Ths corrsonds o h classcal sarabl bu mulvara mxur modls. In boh cass h xls n drn rons ar assumd o b ndndn: N m 87 5 F.: mxur and hddn Marov modls or mas Modln h Labls Non ha all h modls and 86 ar condond on h valu o r hy can b rwrn n h ollown nral orm r N m 88 whr hr s a daonal marx I or no. Now w nd also o modl h vcor varabls { r r }. r also w can consdr wo cass: Indndn Gaussan Mxur modl IGM whr { r r } ar assumd o b ndndn and مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 r wh 89 Conxual Gaussan Mxur modl CGM whr { r r } ar assumd o b Marovan: x δ r s r rv r α 9 whch s h os Marov random ld MF. Th aramr α conrols h man valu o h rons ss. Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6

14 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr yr-aramrs ror Law Th nal on bor obann an xrsson or h osror robably law o all h unnowns.. θ θ s o assn a ror robably law o h hyr-aramrsθ. Evn hs on has bn on o h man dscussn ons bwn Baysan and classcal sascal rsarch communy and sll hr ar many on roblms w choos hr o us h conua rors or smlcy. Th conua rors hav a las wo advanas: hy can b consdrd as a arcular amly o a drnal omry basd amly o rors [5 6 7] and hy ar asy o us bcaus h ror and h osror robably laws say n h sam amly. In our cas w nd o assn ror robably laws o h mans m o h varancs or o h covaranc marcs and also o h covaranc marcs o h noss o h llhood uncons. Th conua rors or h mans m ar n nral h Gaussans varancs N hos o m o ar h nvrs Gammas IG α β and hos or h covaranc marcs ar h nvrs Wshar s Iw α Λ Exrssons o Llhood ror and osror Laws W now hav all h lmns or wrn h xrssons o h osror laws. W ar on o summars hm hr: Llhood: whr w assumd ha h noss ar ndndn cnrd and Gaussan wh covaranc marcs whch hrar ar also assumd o b daonal. M M θ N I MM or h mas: N m usd {... N} ha θ whr w and whr w assumd ar ndndn. مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان MF or h labls: [ ] N x α δ r s v r r s whr w usd h smld noaon Z r r r and whr w assumd {... N} ar ndndn. Conua rors or h hyr-aramrs: m N m τς α β τ W α Λ τς α β Jon osror law o and θ. θ θ θ θ θ 3.4. Baysan Esmaors and Comuaonal Mhods Th xrsson o hs on osror law s n nral nown u o a normalaon acor. Ths mans ha w consdr h Jon Maxmum A osror JMA sma: { θ } ˆ ˆ ˆ θ ar max 9 θ w nd a lobal omaon alorhm bu w consdr h Mnmum Man Squar Esmaor MMSE or quvalnly h osror Man M smas hn w nd o comu hs acor whch nds hu dmnsonal nraons. Thr ar howvr hr man aroachs o do Baysan comuaon: Lalas aroxmaon: Whn h osror law s unmodal s rasonabl o aroxma wh an quvaln Gaussan whch allows hn o do all comuaons analycally. Unorunaly vry on as a uncon o only may b θ Gaussan bu as a uncon o or θ s no. So n nral hs aroxmaon mhod can no b usd or all varabls. Varaonal and man ld aroxmaon: Th man da bhnd hs aroach s o aroxma h wh anohr smlr on osror θ dsrbuon q θ or whch h comuaons can b don. A rs s smlr dsrbuon q s a sarabl ons: θ

15 q θ q q q3 θ 9 In hs way a las rducs h nraon comuaons o h roduc o hr sara ons. Ths rocss can aan b ald o any o hs hr dsrbuons or xaml q q. Wh h Gaussan mxur modln w roosd q can b chosn o b Gaussan q o b sarad o wo ars q B and q W whr h xls o h mas ar sarad n wo classs B and W as n a chcr board. Ths s hans h rors o h roosd os-marov modl wh h our nars nhborhood whch vs h ossbly o us q B and q W saraly. For q 3 θ vry on w also choos a sarabl dsrbuon whch us h conua rors o h ror dsrbuons. Marov Chan Mon Carlo MCMC samln whch vs h ossbly o xlor h on osror law and comu h ncssary osror man smas. In our cas w roos h nral MCMC Gbbs samln alorhm o sma and θ by rs saran h unnowns n wo ss θ and θ. Thn w sara aan h rs s n wo subss θ and θ. Fnally whn ossbl usn h sarably alon h channls θ sara hs wo las rms n and θ. Th nral schm s hn usn n n hs xrssons o nras samls n θ rom h on osror law θ and ar h convrnc o h Gbbs samlrs o comu hr man and o us hm as h osror smas. In hs ar w ar no on o dal hs mhods. owvr n h ollown w roos o xamn som arcular cass hrouh a w cas suds n rlaon o ma rsoraon ma uson and on smnaon blnd ma saraon.] 7 4. Cas Suds 4.. Snl Channl Ima Dnosn and soraon Th smls xaml o nvrson s a snl channl ma dnosn and rsoraon whn h SF o h man sysm s vn. Th orward modl or hs roblm s r hr * r r r or 93 whr h dnosn cas corrsonds o h cas whr h r r and. Assumn h nos o b cnrd wh and Gaussan wh nown varanc w hav N Σ wh Σ I 94 Th rors or hs cas can b summard as ollows: r r N m مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 r r α δ r s 96 r rv r whr { r : r } { r : r } r N m Σ wh I N m N m Σ wh [ m...m ] da[ ] m Σ... m N m 97 τς α β τς α β and h osror robably laws w nd o mlmn an MCMC l alorhm ar: θ N ˆ ˆ Σ Wh ˆ Σ Σ Σ 98 And ˆ ˆ Σ Σ Σ m θ θ 99 θ N m Σ wh Σ Σ Σ and h osror robabls o h hyraramrs ar: n m n m N µ υ wh υ µ υ Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6

16 n n s τς α β wh α and β β whr r s r m n r r n τς α β wh α α and β β n numbr o xls n and n oal numbr o xls. r w show wo xamls o smulaons: h rs n rlaon wh ma dnosn and h scond n rlaon wh ma dconvoluon. In boh cass w hav chosn h sam nu ma r. In h rs cas w only has addd a Gaussan nos and n h scond cas w rs blurrd wh box car SF o s 7 7 xls and addd a Gaussan nos. F. 3 shows h ornal ma s conours and s rons. F. 4 shows h obsrvd nosy ma and h rsuls oband by h roosd mhod. mmbr ha n hs mhod w hav also h smad conours and ron labls as byroducs. F. 5 shows h obsrvd blurrd and nosy ma and h rsuls oband by h roosd rsoraon mhod. For ohr nvrs roblms whch can b modld as a SISO modl and whr such Baysan aroach has bn usd rr o [8]. 4.. srd Imas Fuson and Jon Smnaon r ach obsrvd ma r or quvalnly s assumd o b a nosy vrson o h unobsrvd ral ma r or quvalnly r r r r or...m whch vs and N Σ wh Σ I wh and all h unobsrvd ral mas r... M ar assumd o hav a common smnaon r or quvalnly whch s modld by a dscr valu os andom Marov Fld MF. Thn usn h sam noaons as n rvous cas w hav h ollown rlaons: Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 F.3: Ornal ma s conour and s ron labls usd or ma dnosn and ma rsoraon F.4: Obsrvd nosy ma and h rsuls o h roosd dnosn mhod. F.5: Obsrvd nosy ma and h rsuls o h roosd rsoraon mhod. 8 مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 r r N m... r : r r : r { } { } r N m Σ wh Σ I r r x α Σ rσ sv r [ δ r s ] N m Σ wh m [ m...m ] Σ da[ Σ... Σ ] m N m τς α β τς α β and all h condonal and osror robably laws w nd o mlmn h roosd Baysan mhods ar summard hr: N ˆ ˆ θ Σ Wh ˆ ˆ ˆ Σ Σ Σ Σ Σ Σ m

17 m µ θ r r wh θ θ N m Σ wh Σ Σ Σ υ N µ υ m n n s τς α β wh α β β whr Σ r r s Σ r r m n n τς α β wh α α β β For mor dals on hs modl and s alcaon n mdcal ma uson as wll as n ma uson or scury sysms s [9 3]. υ n 5. Concluson In hs ar w rs showd ha many ma rocssn roblms can b rsnd as nvrs roblms by modln h rlaon o h obsrvd ma o h unnown dsrd aurs xlcly. Thn w rsnd a vry nral orward modln or h obsrvaons and a vry nral robablsc modln o mas hrouh a hddn Marov modln MM whch can b usd as h man bass or many ma rocssn roblms such as: sml or mul channl ma rsoraon sml or on ma smnaon 3 mul-snsor daa and ma uson 4 on smnaon o color or hyrscral mas and 5 on blnd sourc saraon BSS and smnaon. Fnally w rsnd dald orward modls ror and osror robably law xrssons or h mlmnaon o MCMC alorhms or a w cass o hos roblms shown ycal rsuls whch can b oband usn hs mhods. F.6: Ima uson and smnaon o wo mas rom a scury sysm masurmn Jon Smnaon o yr-scral Imas Th roosd modl s h sam as h modl o h rvous scon xc or h las quaon o h orward modl whch assums ha h xls n smlar rons o drn mas ar ndndn. For hyr-scral mas hs hyohss s no vald and w hav o accoun or hr corrlaons. Ths wor s undr consdraon Smnaon o a Vdo Squnc o Imas r w can no assum ha all h mas n h vdo squnc hav h sam smnaon labls. owvr w may us h smnaon oband n an ma as an nalaon or h smnaon o nx ma. For mor dals on hs modl and o s a ycal rsul s Jon Smnaon and Saraon o Insananous Mxd Imas r h addonal dculy s ha w also hav o sma h mxn marx A. For mor dals on hs modl and o s som ycal rsul n on smnaon and saraon o mas s [ ]. 9 EFEENCES [] J. adamard Sur ls roblms aux drvs arlls lur sncaon hysqu rncon Unv. Bull. vol [] G. Dmomn D convoluon ds snaux Cours d l Ecol su rur d lcr [3]. C. Andrws and B.. un Dal Ima soraon rnc-all Enlwood Cls n 977. [4] B.. un A marx hory roo o h dscr convoluon horm IEEE Trans. Auoma. Conr. vol. AC [5] B.. un A horm on h dculy o numrcal dconvoluon IEEE Trans. Auoma. Conr. vol. AC [6] B.. un Dconvoluon o lnar sysms by consrand rrsson and s rlaonsh o h Wnr hory IEEE Trans. Auoma. Conr.vol. AC [7] A. Mohammad-Daar Bnary olyonal sha ma rconsrucon rom a small numbr o rocons Elr vol. 5 no [8] A. Mohammad-Daar and C. Soussn Comac obc rconsrucon n Dscr مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6

18 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.- Fall and wnr 6 Tomorahy: Foundaons Alorhms and Alcaons G. T. rman and A. uba Eds. char Brhausr Boson MA 999. [9] A. Mohammad-Daar Baysan aroach wh hrarchcal marov modln or daa uson n ma rconsrucon alcaons n Fuson 7- Jul. Annaols Maryland USA July. [] A. Mohammad-Daar Fuson o x ray and omrcal daa n comud omorahy or non dsrucv sn alcaons n Fuson 7- Jul. Annaols Maryland USA July. [] A. Mohammad-Daar rarchcal marov modln or uson o x ray radorahc daa and anaomcal daa n comud omorahy n In. Symosum on Bomdcal Iman ISBI 7- Jul. Washnon DC USA July. [] A. Mohammad-Daar Fuson bay snn d donn s n mar x ulrasonor n GETSI 3 Franc S. 3. [3] A. Mohammad-Daar Solvn nvrss roblms: From drmnsc o robablsc aroachs n Smnar n Elcrcal En. D. o urdu Unvrsy n Dc [4] A. Mohammad-Daar N. Qaddoum and. Zouh A blnd dconvoluon aroach or rsoluon nhancmn o nar-ld mcrowav mas n Mahmacal modln Baysan smaon and Invrs rob-lms SIE 99 Dnvr Colorado USA F. rˆux A. Mohammad-Daar and E. Douhry Eds. 999 vol [5] A. Mohammad-Daar J.-F. Govannll G. Dmomn and J. Idr ularaon maxmum nroy and robablsc mhods n mass scromry daa rocssn roblms In. Journal o Mass Scromry vol. 5 no Ar.. [6] M. Nolova J. Idr and A. Mohammadaar Invrson o larsuor ll-osd lnar oraors usn a cws Gaussan mr IEEE Trans. Ima rocssn vol. 7 no Ar [7] J. Idr A. Mohammad-Daar and G. Dmomn ularaon mhods and nvrs roblms: an normaon hory sandon n nd In-rnaonal Conrnc on Invrs مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 roblms n Ennrn L CroscFranc Jun [8] J. Idr Ed. Aroch baysnn our ls roblms nvrss Tra IC Sr ramn du snal d l ma rms ars. [9] J. Idr Convx hal-quadrac crra and nracn auxlary varabls or ma rsoraon IEEE Trans. Ima rocssn vol. no July. [] J. Idr robl`ms nvrss n rsauraon d snaux d mas ablaon drr ds rchrchs Unvrs d ars-sud Orsay Franc July. []. Snouss and A. Mohammad-Daar Baysan sourc saraon wh mxur o Gaussans ror or sourcs and Gaussan ror or mxur cocns n Baysan Inrnc and Maxmum Enroy Mhods A. Mohammad-Daar Ed. G-sur-Yv Franc July roc. o MaxEn Amr. Ins. hyscs. [] chm Snouss AND Al Mohammad- Daar Fas on saraon and smnaon o mxd mas Journal o Elcronc Iman vol. 3 no Arl 4. [3] chm Snouss AND Al Mohammad- Daar Baysan unsurvsd larnn or sourc saraon wh mxur o aussans ror Journal o VLSI Snal rocssn Sysms vol. 37 no. / Jun/July 4. [4] Mahddn Ichr AND Al Mohammad- Daar ddn marov modls or blnd sourc saraon IEEE Trans. on Snal rocssn vol. 5 no Jul 6. [5]. Snouss and A. Mohammad-Daar Inormaon Gomry and ror Slcon. n Baysan Inrnc and Maxmum Enroy Mhods C. Wllams Ed. MaxEn Worshos Au Amr. Ins. hyscs. [6]. Snouss Baysan aroach o sourc saraon. Alcaons n m-ary h.d. hss Unvrsy o ars Sud Orsay Franc smbr 3.

19 [7]. Snouss and A. Mohammad-Daar Fas on saraon and smnaon o mxd mas Journal o Elcronc Iman vol. 3 no Ar. 4. [8] A. Mohammad-Daar Baysan aroach or nvrs roblms n ocs n SIE3 USA S. 3. [9] O. F ron and A. Mohammad-Daar Ima uson and on smnaon usn an MCMC alorhm Journal o Elcronc Iman vol. 4 no.. ar no. 34 Ar 5. [3] O. F ron D. B. and A. Mohammad- Daar Mcrowav man o nhomonous obcs mad o a n numbr o dlcrc and conducv marals rom xrmnal daa Invrs roblms vol. no Dc 5. [3] A. Mohammadour O. Fron and A. Mohammad-Daar Baysan smnaon o hyrscral mas n BAYESIAN INFEENCE AND MAXIMUM ENTOY METODS IN SCIENCE AND ENGINEE- ING: 4h Inrnaonal Worsho on Baysan Inrnc and Maxmum Enroy Mhods n Scnc and Ennrn. 4 vol AI. [3] A. Mohammad-Daar and A. Mohammadour yrscral ma rocssn usn a baysan classcaon aroach n rocdns o SI 5 hyscs n Snal and Ima rocssn SI 5 hyscs n Snal and Ima rocssn. [33] Nada Bal AND Al Mohammad-Daar Jon dmnsonaly rducon classcaon and smnaon o hyrscral mas n ICI 6. Oc. 6 ICI6 Ocobr 8- Alana GA USA. [34] Nada Bal AND Al Mohammad-Daar rarchcal marovan modls or on classcaon smnaon and daa rducon o hyrscral mas n ESANN 6. S. 6 ESANN 6 Smbr 4-8 Blum. [35] Nada Bal AND Al Mohammad-Daar rarchcal marovan modls or hyrscral ma smnaon n IC 6. Au. 6 IC6 Au. -4 on Gon. مجله انجمن مهندسين برق و الكترونيك ايران- سال سوم- شماره دوم پاييز و زمستان 385 Journal o Iranan Assocaon o Elcrcal and Elcroncs Ennrs - Vol.3- No.Fall and Wnr 6