em algorithm with markov chain monte carlo method for bayesian image analysis
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10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 11
EM Algorithm withEM Algorithm withMarkov Chain Monte Carlo Method forMarkov Chain Monte Carlo Method for
Bayesian Image AnalysisBayesian Image Analysis
EM Algorithm withEM Algorithm withMarkov Chain Monte Carlo Method forMarkov Chain Monte Carlo Method for
Bayesian Image AnalysisBayesian Image Analysis
Kazuyuki TanakaKazuyuki TanakaGraduate School of Information Sciences,Graduate School of Information Sciences,
Tohoku UniversityTohoku Universityhttp://www.smapip.is.tohoku.ac.jp/~kazu/http://www.smapip.is.tohoku.ac.jp/~kazu/
Collaborators: D. M. Titterington (Department of Statistics, University of Glasgow)
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 22
ContentContentss
1.1. IntroductionIntroduction
2.2. Gaussian Graphical Model Gaussian Graphical Model and EM Algorithmand EM Algorithm
3.3. Markov Chain Monte Carlo Markov Chain Monte Carlo MethodMethod
4.4. Concluding RemarksConcluding Remarks
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 33
ContentContentss
1.1. IntroductionIntroduction
2.2. Gaussian Graphical Model Gaussian Graphical Model and EM Algorithmand EM Algorithm
3.3. Markov Chain Monte Carlo Markov Chain Monte Carlo MethodMethod
4.4. Concluding RemarksConcluding Remarks
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 44
MRF and Statistical InferenceMRF and Statistical Inference
Geman and Geman (1986): IEEE Transactions on PAMIGeman and Geman (1986): IEEE Transactions on PAMIImage Processing for Image Processing for Markov Random Fields (MRF)Markov Random Fields (MRF) (Simulated Annealing, Line Fields) (Simulated Annealing, Line Fields)
How can we estimate hyperparameters in the degradation process and in the prior model only from observed data?
•EM Algorithm
In the EM algorithm, we have to calculate some statistical quantities in the posterior and the prior models. •Belief Propagation
•Markov Chain Monte Carlo Method
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 55
Statistical Analysis in EM AlgorithmStatistical Analysis in EM Algorithm
J. Inoue and K. Tanaka: Phys. Rev. E 2002, J. Phys. A 2003J. Inoue and K. Tanaka: Phys. Rev. E 2002, J. Phys. A 2003Statistical Behaviour of EM Algorithm for MRFStatistical Behaviour of EM Algorithm for MRF
(Graphical Models on Complete Graph)(Graphical Models on Complete Graph)
K. Tanaka, H. Shouno, M. Okada and D. M. Titterington: K. Tanaka, H. Shouno, M. Okada and D. M. Titterington: J. Phys. A 2004J. Phys. A 2004
Hyperparameter Estimation by using Belief Propagation Hyperparameter Estimation by using Belief Propagation (BP) for Gaussian Graphical Model in Image Processing(BP) for Gaussian Graphical Model in Image Processing
It is possible to estimate statistical behaviour of EM algoritIt is possible to estimate statistical behaviour of EM algorithm with belief propagation analytically.hm with belief propagation analytically.
K. Tanaka and D. M. Titterington: J. Phys. A 2007K. Tanaka and D. M. Titterington: J. Phys. A 2007Statistical Trajectory of Approximate EM Algorithm for Statistical Trajectory of Approximate EM Algorithm for Probabilistic Image ProcessingProbabilistic Image Processing
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 66
ContentContentss
1.1. IntroductionIntroduction
2.2. Gaussian Graphical Model Gaussian Graphical Model and EM Algorithmand EM Algorithm
3.3. Markov Chain Monte Carlo Markov Chain Monte Carlo MethodMethod
4.4. Concluding RemarksConcluding Remarks
10 October, 2007 University of Glasgow 7
Bayesian Image Restoration
Original Image Degraded Image
transmission
Noise
Likelihood Marginal
yProbabilit PrioriA Process nDegradatio
yProbabilit PosterioriA
Image Degraded
Image OriginalImage OriginalImage Degraded
Image DegradedImage Original
Pr
PrPr
Pr
10 October, 2007 University of Glasgow 8
Bayes Formula and Probabilistic Image Processing
,,2,1
g fP ,σfgP
g
g
g
g
2
1
Original Image Degraded Image
α,σgP
αfP,σfgP,gfP
,
Prior Probability
PosteriorProbability
Degradation Process
Pixel
f
f
f
f
2
1
links the all of Set:N
10 October, 2007 University of Glasgow 9
Prior Probability in Probabilistic Image Processing
Bijji ff
Z2
Prior)(
2
1exp
)(
1Pr
fF
0005.0 0030.00001.0
Samples are generated by MCMC.
B: Set of All the Nearest Neighbour pairs of Pixels
: Set of all the nodes
Markov Chain Monte Carlo Method
10 October, 2007 University of Glasgow 10
Degradation Process
Additive White Gaussian Noise
2,0~ Nfg ii
iii gf 2
22 2
1exp
2
1Pr
fFgG
Histogram of Gaussian Random Numbers
n NoiseGaussian f Image Original g Image Degraded
10 October, 2007 University of Glasgow 11
gzd,gzPz
m
m
m
gm
CI
I2
2
1
,,,
Degradation Process and Prior
Degradation Process
Prior Probability Density Function
,, ii gf
Bijji ff
ZfP 2
PR 2exp
1
iii gffgP 2
22
1exp
2
1,
,
,,,
gP
fPfgPgfP
Posterior Probability Density Function
2,0~ Nn
nfg
i
iii
,,2,1 links theall ofSet :B
otherwise0
1
4
Bij
ji
ji C
Multi-Dimensional Gaussian Integral Formula
10 October, 2007 University of Glasgow 12
Maximization of Marginal Likelihood by EM Algorithm
zdzPzgPgP
,,Marginal Likelihood zdgzPgzPgQ
,,ln',',,',',
.,,maxarg1,1
.,,ln,,,,
,ttQtt
zdgzPttgzPttQ
:Step-M
:Step-E
Iterate the following EM-steps until convergence:
EM Algorithm
Q-Function
A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximum likelihood from incomplete data
via the EM algorithm,” J. Roy. Stat. Soc. B, 39 (1977).
10 October, 2007 University of Glasgow 13
Maximization of Marginal Likelihood by EM (Expectation Maximization) Algorithm
i
ii ttPgftf
gf ))(),(,|()()1(|| 22
.,)(),(,maxarg)1(),1(,
gttQtt
Bij
jiBij
ji ttPfftPffff
gff ))(),(,|()())1(|()( 22
)1(
||)(ln2
)1(PR
t
Zt
=
Extremum Condisions of Q(,|(t),(t),g) w.r.t. and
10 October, 2007 University of Glasgow 14
Maximization of Marginal Likelihood by EM (Expectation Maximization) Algorithm
g
g
g
2
1
g
gCI
Cg
CI
C2
T2
21 1
Tr1
1tttt
tt
gCI
Cg
CI
I22
242T
2
22 1
Tr1
1tt
tt
tt
tt
.,)(),(,maxarg)1(),1(,
gttQtt
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100
t
t
g
gmf
,ˆ,ˆˆ
10 October, 2007 University of Glasgow 15
Statistical Behaviour of EM (Expectation Maximization) Algorithm
g
g
g
2
1
g1
22*
2**
2
2
))()((
)(Tr
1
)()(
)(Tr
1)1(
CI
CC
CI
C
tt
I
tt
tt
22*
2**42
2
2
))()((
)()()(Tr
1
))()((
)(Tr
1)1(
CI
CIC
CI
I
tt
tt
tt
tt
ggg dPttQ
tt
**
,,,)(),(,maxarg
)1(),1(
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100
t
t t
t
1000
0001.00
40
001.0*
*
Numerical Experiments for Standard Image
1000
0001.00
40*
Statistical Behaviour of EM Algorithm
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 1616
ContentContentss
1.1. IntroductionIntroduction
2.2. Gaussian Graphical Model Gaussian Graphical Model and EM Algorithmand EM Algorithm
3.3. Markov Chain Monte Carlo Markov Chain Monte Carlo MethodMethod
4.4. Concluding RemarksConcluding Remarks
10 October, 2007 University of Glasgow 17
Maximization of Marginal Likelihood by EM (Expectation Maximization) Algorithm
i
ii ttPgftf
gf ))(),(,|()()1(|| 22
.,)(),(,maxarg)1(),1(,
gttQtt
Bij
jiBij
ji ttPfftPffff
gff ))(),(,|()())1(|()( 22
)1(
||)(ln2
)1(PR
t
Zt
=
Markov Chain Monte Carlo
10 October, 2007 University of Glasgow 18
Markov Chain Monte Carlo Method
Bij
jiij ffP ),(),,|( gf
ii
ikk
cjjiij
cjjiij
cjjiij
ikffi ffff
ff
w),(),(
),(
)|'(\
,ff
fi(t) fi(t+1)
wi(f(t+1)|f(t))
),,|()|(
),,|()|(
gfff
gfff
Pw
Pw
i
i
ijff jj \
i
i
ci
Basic Step
10 October, 2007 University of Glasgow 19
)()1()0(
)()1()0(
)()1()0(
)()()(
)2()2()2(
)1()1()1(
LLL fff
fff
fff
Frequency
fi
L
lff
fiii
ili
i
L
ffffPfP
1,
\21
)(1
),,|,,,,,()(
f
g
Marginal Probabilities can be estimated from histograms.
Markov Chain Monte Carlo Method
10 October, 2007 University of Glasgow 20
)()1()0(
)()1()0(
)()1()0(
)()()(
)2()2()2(
)1()1()1(
LLL fff
fff
fff
Markov Chain Monte Carlo Method
i
ii ttPgftf
gf ))(),(,|()(||
1)1( 2
1
2 ))(),(,|()(||
2)1(
Bijji ttPff
Bt
f
gf
1
)()0(,),0()0(),()0( )()()2()2()1()1(
tt
LL ffffff
EM
MCMC
10 October, 2007 University of Glasgow 21
Markov Chain Monte Carlo Method
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100
Non-Synchronized Updateg
.,,,maxarg1,1,
gttQtt
1000
0001.00
40*
Numerical Experiments for Standard Image
t
t
20 Samples
Input
Output
MCMC =50
EM
MCMC (=50)
MCMC (=1)
Exact
EMInput
Output
MCMC
10 October, 2007 University of Glasgow 22
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0 20 40 60 80 100
Markov Chain Monte Carlo Method
Non-Synchronized Updateg
.,,,maxarg1,1,
gttQtt
1000
0001.00
40
0010.0*
*
Numerical Experiments for Standard Image
t
t
20 Samples
Input
Output
MCMC =50
EM
MCMC (=50)
MCMC (=1)
Exact
EMInput
Output
MCMC
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 2323
ContentContentss
1.1. IntroductionIntroduction
2.2. Gaussian Graphical Model Gaussian Graphical Model and EM Algorithmand EM Algorithm
3.3. Markov Chain Monte Carlo Markov Chain Monte Carlo MethodMethod
4.4. Concluding RemarksConcluding Remarks
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 2424
SummarySummary
We construct EM algorithms by means of We construct EM algorithms by means of Markov Chain Monte Carlo method and Markov Chain Monte Carlo method and compare them with some exact calculations.compare them with some exact calculations.
Input
Output
Exact EM
Input
Output
MCMC =50
EM EMInput
Output
MCMC
10 October, 2007 University of Glasgow 25
New Project 1
fi(t) fi(t+1)
wi(f(t+1)|f(t))
ii
ci
Basic Step
EMInput
Output
MCMC
Can we derive the trajectory of EM algorithm by solving the master equations for any step t in the case of ?
i
ii ttPgftf
gf ))(),(,|()(||
1)1( 2
1
2 ))(),(,|()(||
2)1(
Bijji ttPff
Bt
f
gf
EM
10 October, 2007 University of Glasgow 26
New Project 1
i
ci
iitit
tt
dwpwp
pp
fffffff
ff
)|()()|()(
)()(1
Bij
ijiBij
ji PffttPffff
fgf )()())(),(,|()( 22
Transition Probability
i
tiii
ii PgfttPgfff
fgf )()())(),(,|()( 22
From the solution of master equation, we calculate
These are included in the EM update rules.
i
kkcj
jiijik
ffi ffw ),()|'(\
,ff
10 October, 2007 University of Glasgow 27
New Project 2
Can we replace the calculation of statistical quantities in the prior probability by the MCMC?
EM
i
ii ttPgftf
gf ))(),(,|()()1(|| 22
Bij
jiBij
ji ttPfftPffff
gff ))(),(,|()())1(|()( 22
Input
Output
MCMC
EM
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 2828
New Project 3New Project 3
K. Tanaka, H. Shouno, M. Okada and D. M. Titterington: K. Tanaka, H. Shouno, M. Okada and D. M. Titterington: J. Phys. A 2004J. Phys. A 2004
Hyperparameter Estimation by using Belief Propagation Hyperparameter Estimation by using Belief Propagation (BP) for Gaussian Graphical Model in Image Processing(BP) for Gaussian Graphical Model in Image Processing
K. Tanaka and D. M. Titterington: J. Phys. A 2007K. Tanaka and D. M. Titterington: J. Phys. A 2007Statistical Trajectory of Approximate EM Algorithm for Statistical Trajectory of Approximate EM Algorithm for Probabilistic Image ProcessingProbabilistic Image Processing
Our previous works in EM algorithm and Loopy Belief Propagation
10 October, 2007 University of Glasgow 29
New Project 3
.,,,maxarg1,1,
gttQtt
g
f̂
Loopy Belief Propagation
Exact
0006000ˆ
335.36ˆ
.
LBP
LBP
0007130ˆ
624.37ˆ
.
Exact
Exact
MSE:327
MSE:315
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100 t
tLoopy BP
Exact
10 October, 2007 University of Glasgow 30
New Project 3
g
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100
t
t
0
0.0002
0.0004
0.0006
0.0008
0.001
0 20 40 60 80 100
gdgPgttQtt
**
,,,,,maxarg1,1
.,,,maxarg1,1,
gttQtt
t
t
1000
0001.00
40
0007.0*
*
1000
0001.00
40*
0005640ˆ
542.38ˆ
.
LBP
LBP
Statistical Behaviour of EM Algorithm
Numerical Experiments for Standard Image
Loopy BP
Exact
Loopy BP
Exact
10 October, 200710 October, 2007 University of GlasgowUniversity of Glasgow 3131
New Project 3New Project 3
Can we update both messages and Can we update both messages and hyperparameters in the same step?hyperparameters in the same step?Can we calculate the statistical trajectory?Can we calculate the statistical trajectory?
Input
Output
BP EM EM
Input
Output
BP
More Practical Algorithm
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