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Presentation on theme: "1 Bayesian Image Modeling"— Presentation transcript:
Slide1
1
Bayesian Image Modeling by Generalized Sparse Markov Random Fields and Loopy Belief Propagation
)Sun Kataoka (GSIS, Tohoku University, Japan)D. M. Titterington (Department of Statistics, University of Glasgow, UK)
20 March, 2013
SPDSA2013, Sendai, Japan
Slide2
2
Outline
Supervised Learning of Pairwise Markov Random Fields by Loopy Belief PropagationBayesian Image Modeling by Generalized Sparse PriorNoise Reductions by Generalized Sparse PriorConcluding Remarks
20 March, 2013
SPDSA2013, Sendai, Japan
Slide3
3
Probabilistic Model and Belief Propagation
Probabilistic Information Processing
Probabilistic Models
Bayes Formulas
Belief Propagation
=Bethe Approximation
Bayesian
Networks
Markov Random Fields
20 March, 2013
SPDSA2013, Sendai, Japan
V
: Set of all the nodes (vertices) in graph
G
E
: Set of all the links (edges) in graph
G
j
i
Message
=Effective Field
Slide4
4
Supervised Learning of Pairwise Markov Random Fields by Loopy Belief
Propagation
Prior
Probability
of natural images is assumed to be the
following pairwise Markov random fields:
4
20 March, 2013
SPDSA2013, Sendai, Japan
Slide5
Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields:
5
Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation
5
Histogram from
Supervised Data
20 March, 2013
SPDSA2013, Sendai, Japan
M
. Yasuda, S.
Kataoka
and
K.Tanaka
, J. Phys. Soc.
Jpn
, Vol.81, No.4, Article
No.044801,
2012.
Slide6
6
Supervised Learning of Pairwise Markov Random Fields by Loopy Belief Propagation
Supervised Learning Scheme by Loopy Belief Propagation in Pairwise Markov random fields:
6
20 March, 2013
SPDSA2013, Sendai, Japan
Slide7
7
Bayesian Image Modeling by Generalized Sparse
Prior
Assumption
: Prior Probability is given as the following Gibbs distribution with the interaction
a
between every nearest neighbour pair of pixels:
7
p
=0:
q-state Potts modelp=2: Discrete Gaussian Graphical Model
20 March, 2013
SPDSA2013, Sendai, Japan
Slide8
8
Prior in Bayesian Image Modeling
8
In the region of 0<
p
<0.3504…, the first order phase transition appears and the solution
a
*
does not exist.
20 March, 2013
SPDSA2013, Sendai, Japan
Loopy Belief Propagation
q
=16
Slide9
9
Bayesian Image Modeling by Generalized Sparse Prior: Conditional Maximization of Entropy
9
Lagrange Multiplier
Loopy Belief Propagation
20 March, 2013
SPDSA2013, Sendai, Japan
K. Tanaka
,
M.
Yasuda and D. M.
Titterington
:
J. Phys. Soc.
Jpn
,
81
,
114802
, 2012
.
Slide10
10
Prior Analysis by LBP and Conditional Maximization of Entropy in Bayesian Image Modeling
10
Prior Probability
q
=16
p
=0.2
q
=16
p
=0.5
Repeat until
A
converges
20 March, 2013
SPDSA2013, Sendai, Japan
LBP
Slide11
11
Prior Analysis by LBP and Conditional Maximization of Entropy in Generalized Sparse Prior
11
Log-Likelihood for
p
when the original image
f
*
is given
q
=16
LBP
Free Energy of Prior
20 March, 2013
SPDSA2013, Sendai, Japan
K. Tanaka
,
M.
Yasuda and D. M.
Titterington
:
J. Phys. Soc.
Jpn
,
81
,
114802
, 2012
.
Slide12
12
Degradation Process in Bayesian Image Modeling
Assumption
: Degraded image is generated from the original image by Additive White Gaussian Noise.
12
20 March, 2013
SPDSA2013, Sendai, Japan
Slide13
13
Noise Reductions by Generalized Sparse Prior
Posterior Probability
13
Lagrange Multipliers
20 March, 2013
SPDSA2013, Sendai, Japan
K. Tanaka
,
M.
Yasuda and D. M.
Titterington
:
J. Phys. Soc.
Jpn
,
81
,
114802
, 2012
.
Slide14
14
Noise Reduction Procedures Based on LBP and Conditional Maximization of Entropy
14
Repeat until
C
converge
Repeat
until
u
,
B
and
L converge
20 March, 2013
SPDSA2013, Sendai, Japan
Marginals and Free Energy in LBP
Repeat until
C
and
M
converge
Input
p
and data
g
Slide15
15
Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation
p
=0.5
p=0.2
Original Image
Degraded Image
Restored
Image
p=1
20 March, 2013
SPDSA2013, Sendai, Japan
K. Tanaka
,
M.
Yasuda and
D
. M.
Titterington
:
J. Phys. Soc.
Jpn
,
81
,
114802
, 2012
.
Slide16
16
Noise Reductions by Generalized Sparse Priors and Loopy Belief Propagation
p
=0.5
p=0.2
Original Image
Degraded Image
Restored
Image
p
=1
20 March, 2013
SPDSA2013, Sendai, Japan
K. Tanaka
,
M.
Yasuda and
D
. M.
Titterington
:
J. Phys. Soc.
Jpn
,
81
,
114802
, 2012
.
Slide17
17
Summary
Formulation of Bayesian image modeling for image processing by means of generalized sparse priors and loopy belief propagation are proposed.Our formulation is based on the conditional maximization of entropy with some constraints.In our sparse priors, although the first order phase transitions often appear, our algorithm works well also in such cases.
20 March, 2013
SPDSA2013, Sendai, Japan
Slide18
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References
S. Kataoka, M. Yasuda, K. Tanaka and D. M. Titterington: Statistical Analysis of the Expectation-Maximization Algorithm with Loopy Belief Propagation in Bayesian Image Modeling, Philosophical Magazine: The Study of Condensed Matter, Vol.92, Nos.1-3, pp.50-63,2012.M. Yasuda and K. Tanaka: TAP Equation for Nonnegative Boltzmann Machine: Philosophical Magazine: The Study of Condensed Matter, Vol.92, Nos.1-3, pp.192-209, 2012. S. Kataoka, M. Yasuda and K. Tanaka: Statistical Analysis of Gaussian Image Inpainting Problems, Journal of the Physical Society of Japan, Vol.81, No.2, Article No.025001, 2012. M. Yasuda, S. Kataoka and K.Tanaka: Inverse Problem in Pairwise Markov Random Fields using Loopy Belief Propagation, Journal of the Physical Society of Japan, Vol.81, No.4, Article No.044801, pp.1-8, 2012.M. Yasuda, Y. Kabashima and K. Tanaka: Replica Plefka Expansion of Ising systems, Journal of Statistical Mechanics: Theory and Experiment, Vol.2012, No.4, Article No.P04002, pp.1-16, 2012.K. Tanaka, M. Yasuda and D. M. Titterington: Bayesian image modeling by means of generalized sparse prior and loopy belief propagation, Journal of the Physical Society of Japan, Vol.81, No.11, Article No.114802, 2012.M. Yasuda and K. Tanaka: Susceptibility Propagation by Using Diagonal Consistency, Physical Review E, Vol.87, No.1, Article No.012134, 2013.
