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物理フラクチュオマティクス論Physical Fluctuomatics
応用確率過程論Applied Stochastic Process
第 5 回グラフィカルモデルによる確率的情報処理 5th Probabilistic information processing by means of
S. Geman and D. Geman (1986): IEEE Transactions on PAMIS. Geman and D. 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)
J. Zhang (1992): IEEE Transactions on Signal ProcessingJ. Zhang (1992): IEEE Transactions on Signal ProcessingImage Processing in EM algorithm for Image Processing in EM algorithm for Markov Markov Random Fields (MRF)Random Fields (MRF) (Mean Field Methods) (Mean Field Methods)
Markov Random Fields are One of Probabilistic Methods for Image processing.
物理フラクチュオマティクス論 ( 東北大 ) 6
Markov Random Fields for Image Processing
In Markov Random Fields, we have to consider not only the states with high probabilities but also ones with low probabilities.In Markov Random Fields, we have to estimate not only the image but also hyperparameters in the probabilistic model.We have to perform the calculations of statistical quantities repeatedly.
Hyperparameter Estimation
Statistical Quantities
Estimation of Image
We can calculate statistical quantities by adopting the Gaussian graphical model as a prior probabilistic model and by using Gaussian integral formulas.
物理フラクチュオマティクス論 ( 東北大 ) 7
Purpose of My TalkReview of formulation of probabilistic model for image processing by means of conventional statistical schemes.Review of probabilistic image processing by using Gaussian graphical model (Gaussian Markov Random Fields) as the most basic example.
K. Tanaka: Statistical-Mechanical Approach K. Tanaka: Statistical-Mechanical Approach to Image Processing (Topical Review), J. to Image Processing (Topical Review), J. Phys. A: Math. Gen., vol.35, pp.R81-R150, Phys. A: Math. Gen., vol.35, pp.R81-R150, 2002.2002.
Section 2 and Section 4 are summarized in the present talk.
Digital image is defined on the set of points arranged on a square lattice.The elements of such a digital array are called pixels.We have to treat more than 100,000 pixels even in the digital cameras and the mobile phones.
xx
y y
)1,1( )1,2( )1,3(
)2,1( )2,2( )2,3(
)3,1( )3,2( )3,3(
),( yxPixels 200,307480640
物理フラクチュオマティクス論 ( 東北大 ) 10
Image Representation in Computer Vision
Pixels 65536256256
x
yAt each point, the intensity of light is represented as an integer number or a real number in the digital image data.A monochrome digital image is then expressed as a two-dimensional light intensity function and the value is proportional to the brightness of the image at the pixel.
yxfyx ,),(
0, yxf 255, yxf
物理フラクチュオマティクス論 ( 東北大 ) 11
Noise Reduction by Conventional FiltersNoise Reduction by Conventional Filters
173
110218100
120219202
190202192
Average
192 202 190
202 219 120
100 218 110
192 202 190
202 173 120
100 218 110
It is expected that probabilistic algorithms for image processing can be constructed from such aspects in the conventional signal processing.
Markov Random Fields Probabilistic Image ProcessingAlgorithm
Smoothing Filters
The function of a linear filter is to take the sum of the product of the mask coefficients and the intensities of the pixels.
物理フラクチュオマティクス論 ( 東北大 ) 12
Bayes Formula and Bayesian Network
Posterior Probability
}Pr{
}Pr{}|Pr{}|Pr{
B
AABBA
Bayes Rule
Prior Probability
Event A is given as the observed data.Event B corresponds to the original information to estimate. Thus the Bayes formula can be applied to the estimation of the original information from the given data.
Assumption 1: The degraded image is randomly generated from the original image by according to the degradation process. Assumption 2: The original image is randomly generated by according to the prior probability.
Bayes Formula
物理フラクチュオマティクス論 ( 東北大 ) 14
Image Restoration by Probabilistic Model
Degraded
Image
i
fi: Light Intensity of Pixel iin Original Image
),( iii yxr
Position Vector
of Pixel i
gi: Light Intensity of Pixel iin Degraded Image
i
Original
Image
The original images and degraded images are represented by f = {fi} and g = {gi}, respectively.
Assumption 1: A given degraded image is obtained from the original image by changing the state of each pixel to another state by the same probability, independently of the other pixels.
Assumption 2: The original image is generated according to a prior probability. Prior Probability consists of a product of functions defined on the neighbouring pixels.
i j
Product over All the Nearest Neighbour Pairs of Pixels
物理フラクチュオマティクス論 ( 東北大 ) 17
Prior Probability for Binary Image
== >p p p
2
1p
2
1i j Probability of Neigbouring Pixel
Eij
ji ff ),(}Pr{ fFi j
It is important how we should assume the function (fi,fj) in the prior probability.
)0,1()1,0()0,0()1,1(
We assume that every nearest-neighbour pair of pixels take the same state of each other in the prior probability.
1,0if
物理フラクチュオマティクス論 ( 東北大 ) 18
Prior Probability for Binary Image
Prior probability prefers to the configuration with the least number of red lines.
Which state should the center pixel be taken when the states of neighbouring pixels are fixed to the white states?
?
>
== >p p p
2
1p
2
1i j Probability of Nearest Neigbour Pair of Pixels
物理フラクチュオマティクス論 ( 東北大 ) 19
Prior Probability for Binary ImagePrior Probability for Binary Image
Which state should the center pixel be taken when the states of neighbouring pixels are fixed as this figure?
? - ?== >
p p
> >=
Prior probability prefers to the configuration with the least number of red lines.
物理フラクチュオマティクス論 ( 東北大 ) 20
What happens for the case of large umber of pixels?
p 0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
lnp
Disordered StateCritical Point
(Large fluctuation)
small p large p
Covariance between the nearest neghbour pairs of pixels
Sampling by Marko chain Monte Carlo
Ordered State
Patterns with both ordered statesand disordered states are often generated near the critical point.
物理フラクチュオマティクス論 ( 東北大 ) 21
Pattern near Critical Point of Prior Probability
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0ln p
similar
small p large p
Covariance between the nearest neghbour pairs of pixels
We regard that patterns generated near the critical point are similar to the local patterns in real world images.
Formulation of probabilistic model for image processing by means of conventional statistical schemes has been summarized.
Probabilistic image processing by using Gaussian graphical model has been shown as the most basic example.
ReferenceReferencessReferenceReferencess
K. Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) .
K. Tanaka: Statistical-Mechanical Approach to Image Processing (Topical Review), J. Phys. A, 35 (2002).
A. S. Willsky: Multiresolution Markov Models for Signal and Image Processing, Proceedings of IEEE, 90 (2002).
K. Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) .
K. Tanaka: Statistical-Mechanical Approach to Image Processing (Topical Review), J. Phys. A, 35 (2002).
A. S. Willsky: Multiresolution Markov Models for Signal and Image Processing, Proceedings of IEEE, 90 (2002).
物理フラクチュオマティクス論 ( 東北大 ) 44
物理フラクチュオマティクス論 ( 東北大 ) 45
Problem 5-1: Derive the expression of the posterior probability Pr{F=f|G=g,,} by using Bayes formulas Pr{F=f|G=g,,}=Pr{G=g|F=f,}Pr{F=f,}/Pr{G=g|,}. Here Pr{G=g|F=f,} and Pr{F=f,} are assumed to be as follows:
Viii gf 2
22 2
1exp
2
1,Pr
fFgG
Ejiji ff
Z },{
2
PR 2
1exp
)(
1|Pr
fF
||21},{
2PR 2
1exp)( V
Ejiji dzdzdzzzZ
Ejiji
Viii ffgf
Z },{
22
2POS 2
1
2
1exp
,,
1},,|Pr{
ggGfF
||21
},{
22
2PR 2
1
2
1exp V
Ejiji
Viii dzdzdzzzgzZ
[Answer]
物理フラクチュオマティクス論 ( 東北大 ) 46
Problem 5-2: Show the following equality.
TT2
},{
22
2
2
1))((
2
1
2
1
2
1
fCfgfgf
Eji
jiVi
ii ffgf
otherwise0
},{1
4
Eji
ji
ji C
物理フラクチュオマティクス論 ( 東北大 ) 47
Problem 5-3: Show the following equality.
T
2
T
22
22
TT2
2
1
2
1
2
1))((
2
1
gCI
Cg
gCI
IzCIg
CI
Iz
zCzgzgz
物理フラクチュオマティクス論 ( 東北大 ) 48
Problem 5-4: Show the following equalities by using the multi-dimensional Gaussian integral formulas.
T22
||2
||21},{
22
2POS
2
1exp
)det(
)2(
2
1
2
1exp,,
gCI
Cg
CI
g
V
VEji
jiVi
ii dzdzdzffgfZ
Cdet
)2(
2
1exp)(
||
||
||21},{
2
V
V
VEji
jiPR dzdzdzffZ
物理フラクチュオマティクス論 ( 東北大 ) 49
Problem 5-5: Derive the extremum conditions for the following marginal likelihood Pr{G=g} with respect to the hyperparameters and .
T
22 2
1exp
det2
det},|Pr{ g
CI
Cg
CI
CgG
V
T
22
2
||
1Tr
||
11g
CI
Cg
CI
C
VV
T
22
242
2
22
||
1Tr
||
1g
CI
Cg
CI
I
VV
[Answer]
物理フラクチュオマティクス論 ( 東北大 ) 50
Problem 5-6: Derive the extremum conditions for the following marginal likelihood Pr{G=g} with respect to the hyperparameters and .
T
22 2
1exp
det2
det},|Pr{ g
CI
Cg
CI
CgG
V
T
22
2
||
1Tr
||
11g
CI
Cg
CI
C
VV
T
22
242
2
22
||
1Tr
||
1g
CI
Cg
CI
I
VV
[Answer]
物理フラクチュオマティクス論 ( 東北大 ) 51
Problem 5-7: Make a program that generate a degraded image by the additive white Gaussian noise. Generate some degraded images from a given standard images by setting =10,20,30,40 numerically. Calculate the mean square error (MSE) between the original image and the degraded image.
Histogram of Gaussian Random Numbers Fi Gi~N(0,402)
K.Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 .
物理フラクチュオマティクス論 ( 東北大 ) 52
Problem 5-8: Make a program of the following procedure in probabilistic image processing by using the Gaussian graphical model and the additive white Gaussian noise.
1T
22
2
11||
1
11
1Tr
||
1
g
CI
Cg
CI
C
ttVtt
t
Vt
T22
242
2
2
11
11
||
1
11
1Tr
||
1g
CI
Cg
CI
I
tt
tt
Vtt
t
Vt
gCI
Im
2)()()(
ttt
2)(minarg)(ˆ tmztf ii
zi
i
Algorithm: Repeat the following procedure until convergence