Ising model and its simulation

1
Ising model and its simulation

• Liming Hu
• liminghu_at_cc.usu.edu

2
The Origin of the Ising Model

• The Ising Model was motivated by model for
  magnetism
• Ising Model is the most important and simple
  model in MRF.
• The status at s,

3
The Ising Model

• MRF with 4 point neighborhood
• Let , 4 point neighborhood, then
• If , the poles are aligned, they (i.e. r,
  s ) are in low energy status
• If , the poles are opposite, they are in
  high energy status

4
The Ising Model (continued)

• Total energy
• x is one detail image configuration, x represents
  the status of the whole specific configuration
• C is the set of all cliques
• J is a physical constant
• r, s are any two sites.

5
Analysis the Ising Model

• The probability of the whole image,
• The entropy of the image
• The expected energy
• If the system is in thermodynamic equilibrium
  then,
• And i.e.
• Maximize entropy while containing energy
  (Lagrange multiplier.)

6
Solution- Gibbs distribution (1)

• Ka universal constant
• TTemperature.
• Z(T)Partition function , normalizing constant.

7
Solution- Gibbs distribution (2)
8
Solution- Gibbs distribution (3)
9
What is the noncausual dependence ? Markov
Condition (1)

• Define
• those cliques that
  include s.
• , those cliques that exclude s

10
Markov Condition (2)
11
Markov Condition (3)

• Define
• number of neighbors
• in not equal to

12
Markov Condition (4)

• Notice that,

13
Markov Condition (5)

• is a function of ss
• neighborhood (i.e. ,
• number of neighbors in not equal to
  )

14
Markov Condition (6)

• Figure is as a function
  of ss neighborhood

15
Markov Condition (7)

• Simulation
• The following figure are the Simulation images
  that use Gibbs sampler to draw the samples from
  the above probability, using the parameter
  . I will demonstrate it.

16
Simulation(1)
(a)
(b)
(c)
(d)
(e)
(f)
Typical configurations of an Ising field on a
150150 torus, (a) initialzed image using the
uniform random generator (b) Using Gibbs sampler
after 10 iteration(c) after 50 iteration (d)
after 100 iteration (e) after 200 iteration (f)
after 500 iteration
17
Simulation(2)
(g)
(h) (i)

• (g) after 700 iteration (h) after 900 iteration
  (i) after 1000 iteration.

18
Application to microcalcification detection

• y
• x

• MAP estimation
• P(xy)?
• Suppose it is Gibbs
• Distribution

19
Whats the difficulties?

• Parameters estimation
• Sampling from the posterior distribution
• Combine with other approaches
• Frequency domain, i.e. fft, dwt
• Fuzzy domain

20

• Questions Comments?

21
References

• Zheng, L. and A.K. Chan, An artificial
  intelligent algorithm for tumor detection in
  screening mammogram. IEEE Transactions on Medical
  Imaging, 2001. 20(7) p. 559-567.
• Winkler, G., Image Analysis, Random Fields and
  markov Chain Monte Carlo Methods A Mathematical
  Introduction. 2 ed. 2003 Springer.
• Charles A. Bouman Markov Random Fields and
  Stochastic Image Models, Presented at 1995 IEEE
  International Conference on Image Processing
  23-26 October 1995 Washington, D.C.