1. Markov random field Institute of Electronics, NCTU
指導教授: 王聖智 S. J. Wang
學生: 羅介暐 Jie-Wei Luo

2. Prior Knowledge Sites Label: An event happen to a site 𝑆={1,…,𝑚}
Ex: pixel, feature(line, surface patch)
Label: An event happen to a site
EX: L ={edge,nonedge}, L={0, , 255}

3. Labeling Problem f = {f1, . . . , fm}
Each fi labeling sites in term of Labels f : S →L
Labeling is called configuration in random field

4. Prior knowledge(conti)
In order to explain the concept of the MRF, we first introduce following definition:
1. i: Site (Pixel)
2. Ni: The neighboring point of i
3. S: Set of sites (Image)
4. fi: The value at site i (Intensity) f1 f2 f3 f4 fi f6 f7 f8 f9
A 3x3 imagined image

5. Neighborhood system f1 f2 f3 f4 fi f6 f7 f8 f9
The sites in S are related to one another via a neighborhood system. Its definition for S is defined as:
where Ni is the set of sites neighboring i.
The neighboring relationship has the following properties:
(1) A site is not neighboring to itself
(2) The neighboring relationship is mutual f1 f2 f3 f4 fi f6 f7 f8 f9

6. Example(Regular sites)
First order neighborhood system
Second order neighborhood system
Nth order neighborhood system

7. Example(Irregular sites)
The neighboring sites of the site i are m, n, and f.
The neighboring sites of the site j are r and x

8. Clique A clique C is defined as a subset of sites in S.
Following are some examples
Single-site
pair-site
triple-site

9. Clique: Example
Take first order neighborhood system and second order neighborhood for example:
Neighborhood system
Clique types

10. Random field
Random field is a list of random numbers whose indices are mapped onto a space (of n dimensions)
F = {F1, , Fm} be a family of random variables defined on the set S in which each random variable Fi takes a value fi in L. The family F is called a random field.

11. Markov Random field
 View the 2D image f as the collection of the random variables (Random field)
Markov Random field is a set of random variables having a Markov property

12. Gibbs random field (GRF) and Gibbs distribution
A random field is said to be a Gibbs random field if and only if its configuration f obeys Gibbs distribution, that is:
Design U for different applications f1 f2 f3 f4 fi f6 f7 f8 f9
Image configuration f
U(f): Energy function; T: Temperature Vi(f): Clique potential

13. Role of Energy Function
(1) As the quantitative measure of the global quality of the solution and
(2) As a guide to the search for a minimal solution.
By MRF Modeling to find 𝑓 ∗ = arg 𝑚𝑖𝑛𝑓 𝐸(𝑓|𝑑)

14. Role of Temperature
The temperature T controls the sharpness of the distribution.
When Temperature is high, all configurations tend to be equally distributed.

15. Markov-Gibbs equivalence
Hammersley-Clifford theorem: A random field F is an MRF if and only if F is a GRF
Proof: Let P(f) be a Gibbs distribution on S with the neighborhood system N. f1 f2 f3 f4 fi f6 f7 f8 f9
A 3x3 imagined image

16. Markov-Gibbs equivalence
Divide C into two set A and B with A consisting of cliques containing i and B cliques not containing i: f1 f2 f3 f4 fi f6 f7 f8 f9
A 3x3 imagined image

17. Optimization-based vision problem

18. Denoising
Noisy signal d
denoised signal f

19. The MAP-MRF Framework When both prior and likelihood is known
MAP-MRF Labeling
𝑓 ∗ = arg 𝑚𝑎𝑥𝑓 𝑃(𝑓|𝑑)
𝑃 𝑓 𝑑 ∝𝑃 𝑓,𝑑 =𝑝 𝑑 𝑓 𝑃 𝑓
𝑓 ∗ = arg 𝑚𝑎𝑥𝑓 𝑝 𝑑 𝑓 𝑃 𝑓

20. MAP formulation for denoising problem
Assume the observation is the true signal plus the independent Gaussian noise, that is
𝑑𝑖=𝑓𝑖+𝑒𝑖 , 𝑒𝑖~𝑁(0,𝜎2)
Under above circumstance, the observation model could be expressed as
U(d|f): Likelihood energy

21. MAP formulation for denoising problem
Assume the unknown data f is MRF, the prior model is:
Based on above information, the posteriori probability becomes

22. MAP formulation for denoising problem
The MAP estimator for the problem is: ?

23. The Smoothness Prior
U(f)=[f(n)(x)]2 the order n determines the number of sites in the cliques involved
N=1 (constant gray level)
𝑈 𝑓 = 𝑓 ′ 𝑥 2𝑑𝑥 𝑈 𝑓 = 𝑖 𝑓𝑖− 𝑓 𝑖−1 2
N=2 (constant gradient)
𝑈 𝑓 = 𝑓 ′ ′ 𝑥 2𝑑𝑥 𝑈 𝑓 = 𝑖 𝑓 𝑖+1 −2𝑓𝑖+ 𝑓 𝑖−1 2
N=3 (constant curvature)
𝑈 𝑓 = 𝑓 ′′′ 𝑥 2𝑑𝑥 𝑈 𝑓 = 𝑖 𝑓 𝑖+1 −3 𝑓 𝑖1 +3 𝑓 𝑖−1 − 𝑓 𝑖− 𝑖 𝑓 𝑖+1 −3 𝑓 𝑖1 +3 𝑓 𝑖−1 − 𝑓 𝑖−2 2

24. MAP formulation for denoising problem
Define the smoothness prior:
Substitute above information into the MAP estimator, we could get:
Call posterior
Energy function
Observation model (Similarity measure)
Prior model (Reconstruction constrain)

25. Piecewise Continuous Restoration
𝐸 𝑓 = 𝑖=1 𝑚 𝑓 𝑖 − 𝑑 𝑖 2+ 2λ 𝑖=1 𝑚 𝑔 𝑓 𝑖 − 𝑓 𝑖−1
If g(x)=x2, at discontinuities (𝑓 𝑖 − 𝑓 𝑖−1 )2 tend to be very large , giving an oversmoothed result.
To encode piecewise smoothness , g should be saturate at its asymptotic upper bound to allow discontinuities
𝑔 𝑥 =min⁡{𝑥2,𝐶}

26. Result