Fields of Experts: A Framework for Learning Image Priors 2006. 7. 10 (Mon) Young Ki Baik, Computer...
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Transcript of Fields of Experts: A Framework for Learning Image Priors 2006. 7. 10 (Mon) Young Ki Baik, Computer...
Fields of Experts:A Framework for LearningImage Priors
2006. 7. 10 (Mon)Young Ki Baik, Computer Vision Lab.
2
Fields of Experts
• References
• On the Spatial Statistics of Optical Flow• Stefan Roth and Michael J. Black (ICCV 2005)
• Fields of Experts: A Framework for Learning Image Priors• Stefan Roth, Michael J. Black (CVPR 2005)
• Products of Experts• G. Hinton (ICANN 1999)
• Training products of experts by minimizing contrastive divergence• G. Hinton (Neural Comp. 2002)
• Sparse coding with an over-complete basis set • B. Olshausen and D. Field (VR1997)
3
Fields of Experts
• Contents• Introduction
• Products of Experts
• Fields of Experts
• Application : Image denoising
• Summary
4
Fields of Experts
• Introduction (Image denoising)
• Spatial filter• Gaussian, Mean, Median … .
5
Fields of Experts
• Introduction (Image denosing)
)()|()|( XPXYPYXP
Y X
6
Fields of Experts
• Introduction• Target
• Developing a framework for learning rich, generic image priors (potential function) that capture the statistics of natural scenes.
• Special features• Sparse Coding methods and Products of Experts• Extended version of Products of Experts.• MRF(Markov Random Field) model with learning
potential function in order to solving conventional PoE problems.
7
Fields of Experts
• Sparse Coding• Sparse coding represent an image patch in terms of a
linear combination of learned filters( or bases).
• To express the image probability with small parameters
• An example of mixture model
2
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),(min j i
ijij
JaaaE JxJ
basesor filters :
tscoefficien :
filters ofindex :
vectorsofindex :
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i
jia
i
j
J
8
Fields of Experts
• Products of Experts• Mixture model
• Build a model of a complicated data distribution by combining several simple models.
• Mixture models take a weighted sum of the distributions.
Mixture model: Scale each distribution down and add them together
)()( xx m
mm pp propotion mixture:
9
Fields of Experts
• Products of Experts• Mixture model
• Mixture models are very inefficient in high-dimensional spaces.
10
Fields of Experts
• Products of Experts• PoE model
• Build a model of a complicated data distribution by combining several simple models.
• multiply the distributions together and renormalize. • The product is much sharper than the individual
distributions.
Product model: Multiply the two densities together at every point and then renormalize.
11
Fields of Experts
• Products of Experts• PoE model
• PoE’s work well on high dimensional distributions.• A normalization term is needed to convert the
product of the individual densities into a combined density.
Z
pp m
m
)()(
xx
12
Fields of Experts
• Products of Experts• Geoffrey E. Hinton : Products of Exports
• Most of perceptual systems produce a sharp posterior distribution on high-dimensional manifold.
• PoE model is very efficient to solve vision problem.
13
Fields of Experts
• Products of Experts• PoE framework for vision problem
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ii
TiiZ
p1
;1
)( xJx
Learning sparse topographic representation with products of Student-t distributions
-M. Welling, G. Hinton, and S. Osindero(NIPS 2003)
iii
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14
Fields of Experts
• Products of Experts• PoE framework for vision problem
• Experts : Student-t distribution Responses of linear filters applied to natural images
typically resemble Studient-t experts
i
Tii
Tii
2
2
11; xJxJ
Learning sparse topographic representation with products of Student-t distributions
-M. Welling, G. Hinton, and S. Osindero(NIPS 2003)
15
Fields of Experts
• Products of Experts• PoE framework for vision problem
• Probability density in Gibbs form
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ii
TiiPoEE
1
;log, xJx
,exp1
xx PoEEZ
p
N
ii
TiiZ
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16
Fields of Experts
• Products of Experts• Problems
• Patch based method• Patch can be set to whole image or collection of
patch with specific location in order to treat whole image region.
17
Fields of Experts
• Products of Experts• Problems
• The number of parameters to learn would be too large.
• The model would only work for one specific image size and would not generalize to other image size.
• The model would not be translation invariant, which is a desirable property for generic image priors.
18
Fields of Experts
• Fields of Experts• Key idea
• Combining MRF models
EVG ,
nodes connecting edges the:
image)an in pixels (or the nodes:
E
V
VE
19
Fields of Experts
• Fields of Experts• Key idea
• Define a neighborhood system that connects all nodes in an m x m rectangular region.
• Defines a maximal clique in the graph
VE
kx Kk ,...,1
20
Fields of Experts
• Fields of Experts• The Hammersley-Clifford theorem
• Set the probability density of graphical model as a Gibbs distribution.
• Translation-invariance of an MRF model • assume that potential function is same for all
cliques.
kkkVZ
p xx exp1
kkkV xx
x
cliquefor function potential the:
imagean :
kkk VV xx
21
Fields of Experts
• Fields of Experts• Potential function
• Learn from training images
• Probability density of a full image under the FoE
kV x
,kPoEk EV xx
k
ikTi
N
iiFoE xJE ;log,
1
x
,exp1
xx FoEEZ
p
22
Fields of Experts
• Learning• Parameter and filter can be learned from a set
of training images by maximizing its likelihood.
• Maximizing the likelihood for the PoE and the FoE model is equivalent.
• Perform a gradient ascent method on the log-likelihood
i iJ
Xi
FoE
pi
FoEi
EE
X data trainingover the average the:
p(x)on distributi model therespect to with n valueexpectatio the:
rate learning defined-user a :
X
p
23
Fields of Experts
• Application• Image denoising
• Given an observed noisy image y,• Find the true image x that maximizes the
posterior probability.
• Assumption The true image has been corrupted by additive,
i.i.d Gaussian noise with zero mean and known standard deviation.
xxyyx ppp ||
2
22
1exp| jj
j
p xyxy
24
Fields of Experts
• Application• Image denoising
• In order to maximize the posterior probability, gradient ascent method on the logarithm of the posterior probability is used.
• The gradient of the log-likelihood
• The gradient of the log-prior
xyxy 2
1|log
px
N
iiiix p
1
**log xJJx
iii
i
i
yy ;log/ :
center its around J mirroringby obtainedfilter the:
filter with image ofn convolutio the:*
y
J
JxxJ
25
Fields of Experts
• Application• Image denoising
• The gradient ascent denoising algorithm
tN
i
tiii
tt xyxJJxx2
1
1 **
weightoptionalan :
rate update :
indexiteration :
t
xyx |loglog pp xx
26
Fields of Experts
• Applications• Image denoising
a) Original image b) Noisy image c) Denoising image
27
Fields of Experts
• Summary• Contribution
• Point out limitation of conventional Product of Experts.
PoE focus on the modeling of small image patches rather than defining a prior model over an entire image.
• Propose FoE which models the prior probability of an entire image in term of random field with overlapping cliques, whose potentials are represented as a PoE.