Statistical performance analysis by loopy belief propagation in probabilistic image processing
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1 Statstcal perormance analyss by loopy bele propaaton n probablstc mae processn Kazuyuk Tanaka raduate School o Inormaton Scences Tohoku Unversty Japan Collaborators D. M. Ttternton Unversty o lasow UK M. Yasuda Tohoku Unversty Japan S. Kataoka Tohoku Unversty Japan 4 October 00 LRI Semnar 00 Unv. Pars-Sud
2 Introducton Bayesan network s ornally one o the methods or probablstc nerences n artcal ntellence. Some probablstc models or normaton processn are also rearded as Bayesan networks. Bayesan networks are expressed n terms o products o unctons wth a couple o random varables and can be assocated wth raphcal representatons. Such probablstc models or Bayesan networks are reerred to as raphcal Model. 4 October 00 LRI Semnar 00 Unv. Pars-Sud
3 Probablstc Imae Processn by Bayesan Network Probablstc mae processn systems are ormulated on square rd raphs. Averaes varances and covarances o the Bayesan network are approxmately computed by usn the bele propaaton on the square rd raph. 4 October 00 LRI Semnar 00 Unv. Pars-Sud 3
4 MR Bele Propaaton and Statstcal Perormance eman and eman 986: IEEE Transactons on PAMI Imae Processn by Markov Random elds MR Tanaka and Morta 995: Physcs Letters A Cluster Varaton Method or MR n Imae Processn CVM eneralzed Bele Propaaton BP Nshmor and Won 999: Physcal Revew E Statstcal Perormance Estmaton or MR Innte Rane Isn Model and Replca Theory Is t possble to estmate the perormance o bele propaaton statstcally? 0 March 00 IW-SMI00 Kyoto 4
5 Outlne. Introducton. Bayesan Imae Analyss by auss Markov Random elds 3. Statstcal Perormance Analyss or auss Markov Random elds 4. Statstcal Perormance Analyss n Bnary Markov Random elds by Loopy Bele Propaaton 5. Concludn Remarks 4 October 00 LRI Semnar 00 Unv. Pars-Sud 5
6 Outlne. Introducton. Bayesan Imae Analyss by auss Markov Random elds 3. Statstcal Perormance Analyss or auss Markov Random elds 4. Statstcal Perormance Analyss n Bnary Markov Random elds by Loopy Bele Propaaton 5. Concludn Remarks 4 October 00 LRI Semnar 00 Unv. Pars-Sud 6
7 Imae Restoraton by Bayesan Statstcs Ornal Imae 4 October 00 LRI Semnar 00 Unv. Pars-Sud 7
8 Imae Restoraton by Bayesan Statstcs Nose Transmsson Ornal Imae Deraded Imae 4 October 00 LRI Semnar 00 Unv. Pars-Sud 8
9 Imae Restoraton by Bayesan Statstcs Nose Transmsson Ornal Imae Estmate Deraded Imae 4 October 00 LRI Semnar 00 Unv. Pars-Sud 9
10 Imae Restoraton by Bayesan Statstcs Nose Ornal Imae Transmsson Posteror Estmate Deraded Imae Bayes ormula Posteror Ornal Imae Deraded Imae Deradaton Process Pror Deraded Imae Ornal ImaeOrnal Imae 4 October 00 LRI Semnar 00 Unv. Pars-Sud 0
11 Imae Restoraton by Bayesan Statstcs Nose Bayes ormula Posteror Ornal Imae Deraded Imae Deradaton Process Ornal Imae Transmsson Posteror Estmate Assumpton : Ornal maes are randomly enerated by accordn to a pror probablty. Pror Deraded Imae Deraded Imae Ornal ImaeOrnal Imae 4 October 00 LRI Semnar 00 Unv. Pars-Sud
12 Imae Restoraton by Bayesan Statstcs Nose Assumpton : Deraded maes are randomly enerated rom the ornal mae by accordn to a condtonal probablty o deradaton process. Ornal Imae Transmsson Posteror Estmate Deraded Imae Bayes ormula Posteror Ornal Imae Deraded Imae Deradaton Process Pror Deraded Imae Ornal ImaeOrnal Imae 74 July October LRI Semnar 00 Unv. Pars-Sud
13 Bayesan Imae Analyss Assumpton: Pror Probablty conssts o a product o unctons dened on the nehbourn pxels. Pror Probablty Posteror Ornal Imae Deraded Imae Lkelhood Deraded Imae Ornal Imae exp exp { E { E > 0 Ornal Imae Pror 4 October 00 LRI Semnar 00 Unv. Pars-Sud 3
14 Bayesan Imae Analyss Assumpton: Pror Probablty conssts o a product o unctons dened on the nehbourn pxels. Pror Probablty Posteror Ornal Imae Deraded Imae Lkelhood Deraded Imae Ornal Imae exp exp { E { E > 0 Ornal Imae Pror 4 October 00 LRI Semnar 00 Unv. Pars-Sud 4
15 Bayesan Imae Analyss Assumpton: Pror Probablty conssts o a product o unctons dened on the nehbourn pxels. Pror Probablty Posteror Ornal Imae Deraded Imae Lkelhood Deraded Imae Ornal Imae exp exp { E { E > 0 Ornal Imae Patterns by MCMC Pror 4 October 00 LRI Semnar 00 Unv. Pars-Sud 5
16 Bayesan Imae Analyss Posteror Ornal Imae Deraded Imae Lkelhood Deraded Imae Ornal Imae Pror Ornal Imae Assumpton: Deraded mae s enerated rom the ornal mae by Addtve Whte aussan Nose. V exp V:Set o all the pxels > 0 4 October 00 LRI Semnar 00 Unv. Pars-Sud 6
17 Bayesan Imae Analyss Posteror Ornal Imae Deraded Imae Lkelhood Deraded Imae Ornal Imae Pror Ornal Imae Assumpton: Deraded mae s enerated rom the ornal mae by Addtve Whte aussan Nose. V exp 4 October 00 LRI Semnar 00 Unv. Pars-Sud 7
18 Bayesan Imae Analyss Posteror Ornal Imae Deraded Imae Lkelhood Deraded Imae Ornal Imae Pror Ornal Imae Assumpton: Deraded mae s enerated rom the ornal mae by Addtve Whte aussan Nose. > 0 4 October 00 LRI Semnar 00 Unv. Pars-Sud V exp 8
19 4 October 00 LRI Semnar 00 Unv. Pars-Sud 9 Bayesan Imae Analyss Ornal Imae Deraded Imae Pror Probablty Posteror Probablty Deradaton Process E V { exp Estmate
20 0 Bayesan Imae Analyss Ornal Imae Deraded Imae Pror Probablty Posteror Probablty Deradaton Process E V { exp Smoothn Data Domnant Estmate 4 October 00 0 LRI Semnar 00 Unv. Pars-Sud
21 March 00 IW-SMI00 Kyoto Bayesan Imae Analyss Ornal Imae Deraded Imae Pror Probablty Posteror Probablty Deradaton Process E V { exp Smoothn Data Domnant Bayesan Network Estmate 4 October 00 LRI Semnar 00 Unv. Pars-Sud
22 dz z z h ˆ + C I Bayesan Imae Analyss Ornal Imae Deraded Imae Pror Probablty Posteror Probablty Deradaton Process E V { exp Markov Random eld auss + Smoothn Data Domnant Bayesan Network Estmate 4 October 00 LRI Semnar 00 Unv. Pars-Sud otherwse 0 { E V C
23 Imae Restoratons by aussan Markov Random elds and Conventonal lters Restored Imae Ornal Imae Deraded Imae auss Markov Random eld Lowpass lter Medan lter MSE V V MSE 35 3x x5 43 3x x5 445 ˆ V: Set o all the pxels auss Markov Random eld 3x3 Lowpass 5x5 Medan + 4 October 00 LRI Semnar 00 Unv. Pars-Sud 3 auss Markov Random eld
24 Outlne. Introducton. Bayesan Imae Analyss by auss Markov Random elds 3. Statstcal Perormance Analyss or auss Markov Random elds 4. Statstcal Perormance Analyss n Bnary Markov Random elds by Loopy Bele Propaaton 5. Concludn Remarks 4 October 00 LRI Semnar 00 Unv. Pars-Sud 4
25 Statstcal Perormance by Sample Averae o Numercal Experments Ornal Imaes 4 October 00 LRI Semnar 00 Unv. Pars-Sud 5
26 Statstcal Perormance by Sample Averae o Numercal Experments Ornal Imaes Nose Observed Data 4 October 00 LRI Semnar 00 Unv. Pars-Sud 6
27 Statstcal Perormance by Sample Averae o Numercal Experments Ornal Imaes Nose Posteror Probablty h h h 3 h 4 h 4 Observed Data Estmated Results 4 March October 00 LRI Semnar IW-SMI00 Unv. Kyoto Pars-Sud 7
28 Statstcal Perormance by Sample Averae o Numercal Experments Sample Averae o Mean Square Error D 5 Ornal Imaes 5 n Nose h n Observed Data Posteror Probablty h h h 3 h 4 h 4 Estmated Results 4 October 00 LRI Semnar 00 Unv. Pars-Sud 8
29 4 October 00 LRI Semnar 00 Unv. Pars-Sud 9 Statstcal Perormance Estmaton d h V D h + C I Addtve Whte aussan Nose Posteror Probablty Restored Imae Ornal Imae Deraded Imae Addtve Whte aussan Nose otherwse 0 { E V C
30 4 October 00 LRI Semnar 00 Unv. Pars-Sud 30 Statstcal Perormance Estmaton or auss Markov Random elds T 4 Tr exp I V V d V d h V D V C C C I I C I I π otherwse E V 0 { C D D
31 Outlne. Introducton. Bayesan Imae Analyss by auss Markov Random elds 3. Statstcal Perormance Analyss or auss Markov Random elds 4. Statstcal Perormance Analyss n Bnary Markov Random elds by Loopy Bele Propaaton 5. Concludn Remarks 4 October 00 LRI Semnar 00 Unv. Pars-Sud 3
32 Marnal Probablty n Bele Propaaton Pr 3 4 { { Pr 3 4 In order to compute the marnal probablty we take summatons over all the pxels except the pxel. N N 4 October 00 LRI Semnar 00 Unv. Pars-Sud 3
33 Marnal Probablty n Bele Propaaton Pr 3 4 { { Pr 3 4 N N 3 4 N 4 October 00 LRI Semnar 00 Unv. Pars-Sud 33
34 Marnal Probablty n Bele Propaaton Pr 3 4 { { Pr 3 4 N N 3 4 N In the bele propaaton the marnal probablty s approxmately expressed n terms o the messaes rom the nehbourn reon o the pxel. 4 October 00 LRI Semnar 00 Unv. Pars-Sud 34
35 Marnal Probablty n Bele Propaaton Pr 3 4 { { Pr 3 4 N N 3 4 N In order to compute the marnal probablty we take summatons over all the pxels except the pxels and. 4 October 00 LRI Semnar 00 Unv. Pars-Sud 35
36 Marnal Probablty n Bele Propaaton Pr 3 4 { { Pr 3 4 N N 3 4 N In the bele propaaton the marnal probablty s approxmately expressed n terms o the messaes rom the nehbourn reon o the pxels and. 4 October 00 LRI Semnar 00 Unv. Pars-Sud 36
37 Bele Propaaton n Probablstc Imae Processn 4 October 00 LRI Semnar 00 Unv. Pars-Sud 37
38 Bele Propaaton n Probablstc Imae Processn 4 October 00 LRI Semnar 00 Unv. Pars-Sud 38
39 Imae Restoratons by aussan Markov Random elds and Conventonal lters Ornal Imae Deraded Imae MSE auss MR Exact 35 auss MR Bele Propaaton Lowpass lter Medan lter 37 3x x5 43 3x x5 445 Restored Imae Exact Bele Propaaton V: Set o all the pxels + auss Markov Random eld 4 October 00 LRI Semnar 00 Unv. Pars-Sud 39 MSE V V ˆ
40 ray-level Imae Restoraton Spke Nose Deraded Imae Ornal Imae Bele Propaaton Lowpass lter Medan lter MSE: 075 MSE: 44 MSE: 7 MSE:35 MSE: 3469 MSE: 37 MSE: 53 MSE: October 00 LRI Semnar 00 Unv. Pars-Sud 40
41 Bnary Imae Restoraton by Loopy Bele Propaaton October 00 LRI Semnar 00 Unv. Pars-Sud 4
42 Statstcal Perormance by Sample Averae o Numercal Experments Sample Averae o Mean Square Error D 5 Ornal Imaes 5 n Nose h n Observed Data Posteror Probablty h h h 3 h 4 h 4 Estmated Results 4 October 00 LRI Semnar 00 Unv. Pars-Sud 4
43 4 October 00 LRI Semnar 00 Unv. Pars-Sud 43 Statstcal Perormance Estmaton or Bnary Markov Random elds d h V D { exp lo exp V z z z E V V d d d z z z V ± ± ± It can be reduced to the calculaton o the averae o ree enery wth respect to locally non-unorm external elds V. ree Enery o Isn Model wth Random External elds ± Lht ntenstes o the ornal mae can be rearded as spn states o erromanetc system. > > E { E V { exp
44 Statstcal Perormance Estmaton or Bnary Markov Random elds 4 October 00 LRI Semnar 00 Unv. Pars-Sud 44 λ ρ λ V h d d V d h V D + + tanh tanh tanh * /{ /{ \{ λ ρ λ λ δ λ λ ρ k k k l l V k k d d h λ sn
45 Statstcal Perormance Estmaton or Markov Random elds D h V d Spn lass Theory n Statstcal Mechancs Loopy Bele Propaaton Mult-dmensonal auss Interal ormulas D D 4 October 00 LRI Semnar 00 Unv. Pars-Sud
46 4 October 00 LRI Semnar 00 Unv. Pars-Sud 46 Statstcal Perormance Estmaton or Markov Random elds d h V D
47 Outlne. Introducton. Bayesan Imae Analyss by auss Markov Random elds 3. Statstcal Perormance Analyss or auss Markov Random elds 4. Statstcal Perormance Analyss n Bnary Markov Random elds by Loopy Bele Propaaton 5. Concludn Remarks 4 October 00 LRI Semnar 00 Unv. Pars-Sud 47
48 Summary ormulaton o probablstc model or mae processn by means o conventonal statstcal schemes has been summarzed. Statstcal perormance analyss o probablstc mae processn by usn auss Markov Random elds has been shown. One o extensons o statstcal perormance estmaton to probablstc mae processn wth dscrete states has been demonstrated. 4 October 00 LRI Semnar 00 Unv. Pars-Sud 48
49 Imae Impantn by auss MR and LBP Our ramework can be extended to erase a scrbbln. auss MR and LBP 4 October 00 LRI Semnar 00 Unv. Pars-Sud 49
50 Reerences. K. Tanaka and D. M. Ttternton: Statstcal Traectory o Approxmate EM Alorthm or Probablstc Imae Processn Journal o Physcs A: Mathematcal and Theoretcal vol.40 no.37 pp M. Yasuda and K. Tanaka: The Mathematcal Structure o the Approxmate Lnear Response Relaton Journal o Physcs A: Mathematcal and Theoretcal vol.40 no.33 pp K. Tanaka and K. Tsuda: A Quantum-Statstcal-Mechancal Extenson o aussan Mxture Model Journal o Physcs: Conerence Seres vol.95 artcle no.003 pp.-9 January K. Tanaka: Mathematcal Structures o Loopy Bele Propaaton and Cluster Varaton Method Journal o Physcs: Conerence Seres vol.43 artcle no.003 pp M. Yasuda and K. Tanaka: Approxmate Learnn Alorthm n Boltzmann Machnes Neural Computaton vol. no. pp S. Kataoka M. Yasuda and K. Tanaka: Statstcal Perormance Analyss n Probablstc Imae Processn Journal o the Physcal Socety o Japan vol.79 no. artcle no October 00 LRI Semnar 00 Unv. Pars-Sud 50
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