Pobablsc mage Pocessng b Eended Gauss-Makov Random Felds Kauuk anaka Munek asuda Ncolas Mon Gaduae School of nfomaon Scences ohoku Unves Japan and D. M. engon Depamen of Sascs Unves of Glasgow UK 3 Sepembe 009 SSP009 adff UK
mage Resoaon b Baesan Sascs Nose Assumpon : Degaded mages ae andoml geneaed fom he ognal mage b accodng o he condonal pobabl of degadaon pocess. Baes Fomula Poseo 644444474444444 8 P{Ognal mage Degaded mage Degadao n Pocess Ognal mage ansmsson Poseo Esmae Assumpon : Ognal mages ae andoml geneaed b accodng o a po pobabl. Po Degaded mage 644444474444444 86444 74448 P{Degaded mage Ognal magep{ognal mage 3 Sepembe 009 SSP009 adff UK
Baesan mage Analss Assumpon : Po Pobabl consss of a poduc of funcons defned on he neghboung pels. Po Pobabl P{ Poseo 6444444 74444444 8 P{Ognal mage Degaded mage Lkelhood 6444444 74444444 8 P{Degaded mage Ognal mage 6444 7444 8 P{Ognal mage ep ep ep { E { E 4 4 > 0 { E { E 0 ohewse Gbbs Sample Po 0.000 0.0005 0. 0030 0 0.45 3 Sepembe 009 SSP009 adff UK 3
Baesan mage Analss Poseo 6444444 74444444 8 P{Ognal mage Degaded mage Lkelhood 6444444 74444444 8 P{Degaded mage Ognal mage P{ Po 6444 7444 8 P{Ognal mage ep Assumpon : Degaded mage s geneaed fom he ognal mage b Addve Whe Gaussan Nose. ep :Se of all he pels > 0 3 Sepembe 009 SSP009 adff UK 4
3 Sepembe 009 SSP009 adff UK 5 Baesan mage Analss g P{ P{ Ognal mage Degaded mage Po Pobabl Poseo Pobabl Degadaon Pocess { { ep ep P{ P{ P{ P{ E E d P{ ˆ Makov Random Feld Model Gauss Smoohng Daa Domnan Baesan Newok Esmae ohewse 0 { { 4 4 E E
Aveage of Poseo Pobabl Aveage of he poseo pobabl can be calculaed b usng he mul-dmensonal Gauss negal Fomula ˆ L P{ L ep L ep L ep μ μ μ L L Gaussan negal fomula L L 64748 0 0 L0 3 Sepembe 009 SSP009 adff UK 6
g Sascal Esmaon of Hpepaamees Hpepaamees ae deemned so as o mame he magnal lkelhood P{ wh espec o. ˆ ˆ P{ ag map{ P{ P{ P{ P{ Ognal mage P{ Magnal Lkelhood Degaded mage Magnaled wh espec o 3 Sepembe 009 SSP009 adff UK 7
3 Sepembe 009 SSP009 adff UK 8 Sascal Esmaon of Hpepaamees P{ P{ P{ PR / POS π Z Z A A de ep π μ μ L L POS ep de ep ep ep Z L L L L π Gaussan negal fomula de ep π PR Z L L
3 Sepembe 009 SSP009 adff UK 9 Eac Epesson of Magnal Lkelhood n Gaussan Gaphcal Model ep de de P{ π 4 Eemum ondons fo and 4 ag map{ ˆ ˆ Q eaed Algohm EM Algohm 0 P{ 0 P{
Baesan mage Analss b Gaussan Gaphcal Model eaon Pocedue of EM Algohm n Gaussan Gaphcal Model ˆ ˆ ag map{ Q EM ˆ 3 Sepembe 009 SSP009 adff UK 0 ˆ
mage Resoaon b Gaussan Makov Random Feld GMRF Model and onvenonal Fles Ognal mage Degaded mage onvenonal GMRF 0 Eended GMRF Smulaneous AR MSE 0. 0.45 ˆ MSE 309 73 3 5 Resoed mage ˆ ˆ onvenonal GMRF 0 0.00053 Eended GMRF 0. Eended GMRF 0.45 Smulaneous AR 0.0008833 0.0046536 0.00738 3 Sepembe 009 SSP009 adff UK
mage Resoaon b Gaussan Makov Random Feld GMRF Model and onvenonal Fles Ognal mage Degaded mage onvenonal GMRF 0 Eended GMRF Smulaneous AR MSE 0. 0.45 ˆ MSE 33 30 34 383 Resoed mage ˆ ˆ onvenonal GMRF 0 0.000733 8 Eended GMRF 0. Eended GMRF 0.45 Smulaneous AR 0.00339 0.006874 0.0040043 3 Sepembe 009 SSP009 adff UK
Sascal Pefomance b Sample Aveage of Numecal Epemens Sample Aveage of Mean Squae Eo E 5 5 n Ognal mages h n Nose 3 4 5 Degaded mages Poseo Pobabl h h h 3 h 4 h 5 Resoed mages 3 Sepembe 009 SSP009 adff UK 3
3 Sepembe 009 SSP009 adff UK 4 Sascal Pefomance Esmaon d h E P{ h g Addve Whe Gaussan Nose P{ P{ Poseo Pobabl Resoed mage Ognal mage Degaded mage P{ Addve Whe Gaussan Nose
3 Sepembe 009 SSP009 adff UK 5 Sascal Pefomance Esmaon fo Gauss Makov Random Felds 4 4 ep ep ep ep ep ep ep ep P{ d d d d d d d d d h E π π π π π π π π 0
3 Sepembe 009 SSP009 adff UK 6 00 50 300 350 400 0 0.005 0.0 0.05 00 50 300 350 400 0 0.005 0.0 0.05 Sascal Pefomance Esmaon fo Gauss Makov Random Felds 4 ep P{ d d h E π 40 40 E E ohewse 0 { { 4 4 E E 0 0 0. 0.45 0. 0.45
Summa We popose an eenson of he Gauss-Makov andom feld models b noducng ne-neaes neghbou neacons. alues fo he hpepaamees n he poposed model ae deemned b usng he EM algohm n ode o mame he magnal lkelhood. n addon a measue of mean squaed eo whch quanfes he sascal pefomance of ou poposed model s deved analcall as an eac eplc epesson b means of he mul-dmensonal Gaussan negal fomulas. Sascal pefomance analss of pobablsc mage pocessng fo ou eended Gauss Makov Random Felds has been shown. 3 Sepembe 009 SSP009 adff UK 7
Refeences. K. anaka: : Sascal-mechancal appoach o mage pocessng opcal Revew Jounal of Phscs A: Mahemacal and Geneal vol.35 no.37 pp.r8-r50 R50 00.. K. anaka and J. noue: Mamum Lkelhood Hpepaamee Esmaon fo Solvable Makov Random Feld Model n mage Resoaon EE ansacons on nfomaon and Ssems vol.e85-d no.3 pp.546-557 557 00. 3. K. anaka J. noue and D. M. engon: : Pobablsc mage Pocessng b Means of Behe Appomaon fo Q-sngQ Model Jounal of Phscs A: Mahemacal and Geneal vol. 36 no. 43 pp.03-036 036 003. 4. K. anaka H. Shouno M. Okada and D. M. engon: : Accuac of he Behe Appomaon fo Hpepaamee Esmaon n Pobablsc mage Pocessng Jounal of Phscs A: Mahemacal and Geneal vol.37 no.36 pp.8675-8696 8696 004. 5. K. anaka and D. M. engon: : Sascal aeco of Appomae EM Algohm fo Pobablsc mage Pocessng Jounal of Phscs A: Mahemacal and heoecal vol.40 no.37 pp.85-300 300 007. 3 Sepembe 009 SSP009 adff UK 8