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

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

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

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

Markov Models State State State

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

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

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

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

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

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

Forward recursion Initialization: Forward recursion: Termination:

Backward recursion Initialization: Backward recursion: Termination:

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

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)

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

Viterbi algorithm Initialization: Forward recursion: Termination:

Viterbi algorithm

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

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

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

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

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

Example 1 -character recognition s1 s2 s3 The structure of hidden states: Observation = number of islands in the vertical slice

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

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

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

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

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