1. Hidden Markov Models 戴玉書
L.R Rabiner, B. H. Juang, An Introduction to Hidden Markov Models
Ara V. Nefian and Monson H. Hayeslll, Face detection and recognition using Hidden Markov Models
戴玉書

2. Outline Markov Chain & Markov Models Hidden Markov Models HMM Problem
-Evaluation
-Decoding
-Learning
Application

3. Outline Markov Chain & Markov Models Hidden Markov Models HMM Problem
-Evaluation
-Decoding
-Learning
Application

4. Markov chain property:
Probability of each subsequent state depends only on what was the previous state

5. Markov Models
State
State
State

6. Outline Markov Chain & Markov Models Hidden Markov Models HMM Problem
-Evaluation
-Decoding
-Learning
Application

7. Hidden Markov Models …… q1 q2 q3 q4
If you don’t have complete state information, but some
observations at each state
N - number of states :
M - the number of observables: …… q1 q2 q3 q4

8. Hidden Markov Models State:{ , , } Observable:{ , } 0.1 0.3 0.9 0.7
0.8
0.2

9. Hidden Markov Models M=(A, B, )
 = initial probabilities : =(i) , i = P(si)

10. Outline Markov Chain & Markov Models Hidden Markov Models HMM Problem
-Evaluation
-Decoding
-Learning
Application

11. Evaluation
Determine the probability that a particular sequence of symbols O was generated by that model

12. Forward recursion
Initialization:
Forward recursion:
Termination:

13. Backward recursion
Initialization:
Backward recursion:
Termination:

14. Outline Markov Chain & Markov Models Hidden Markov Models HMM Problem
-Evaluation
-Decoding
-Learning
Application

15. Decoding Given a set of symbols O determine the most likely
sequence of hidden states Q that led to the
observations
We want to find the state sequence Q which
maximizes P(Q|o1,o2,...,oT)

16. Viterbi algorithm s1 si sj sN a1j aij aNj qt-1 qt General idea:
if best path ending in qt= sj goes through qt-1= si then it should coincide with best path ending in qt-1= si

17. Viterbi algorithm
Initialization:
Forward recursion:
Termination:

18. Viterbi algorithm

19. Outline Markov Chain & Markov Models Hidden Markov Models HMM Problem
-Evaluation
-Decoding
-Learning
Application

20. Learning problem
Given a coarse structure of the model, determine HMM parameters M=(A, B, ) that best fit training
data
determine these parameters

21. Baum-Welch algorithm
Define variable t(i,j) as the probability of being in state si at time t and in state sj at time t+1, given the observation sequence o1, o2, ... ,oT

22. Baum-Welch algorithm
Define variable k(i) as the probability of being in state si at time t, given the observation sequence
o1,o2 ,...,oT

23. Outline Markov Chain & Markov Models Hidden Markov Models HMM Problem
-Evaluation problem
-Decoding problem
-Learning problem
Application

24. Example 1 -character recognitions1 s2 s3
The structure of hidden states:
Observation = number of islands in the vertical slice

25. Example 1 -character recognition
After character image segmentation the following sequence
of island numbers in 4 slices was observed :
{1,3,2,1}

26. Example 2- face detection & recognition
The structure of hidden states:

27. Example 2- face detection
A set of face images is used in the training of one HMM model
N =6 states
Image:48, Training:9, Correct detection:90%,Pixels:60X90

28. Example 2- face recognition
Each individual in the database is represent by an HMM face model
A set of images representing different instances of same face are used to train each HMM
N =6 states

29. Example 2- face recognition
Image:400, Training :Half, Individual:40, Pixels:92X112