timothy and rahule6886 project1 statistically recognize faces based on hidden markov models...

Timothy and Rahul E6886 Project 1

Statistically Recognize Faces Based on Hidden Markov Models

Presented by

Timothy Hsiao-Yi Chin

Rahul Mody


What is Hidden Markov Model?

•Its underlying is a Markov Chain.

•An HMM, at each unit of time, a single observation is generated from the current state according to the probability distribution, which is dependent on this state.

i jnnnn PiXiXiXiXjXP ),,...,,|( 0011111

1 n32

O :O bs e rva tio n P : pro ba bility o f m o ving to ne xt s ta te

O

P PP P

F ig ure 1 H M M


Mathematical Notation of HMM

• Suppose that there are T states {S1, …, ST} and the probability between state i and j is Pij.

Observation of system can be defined as ot at time t. Let bSi(oi) be the probability function of ot at time t. Lastly, we have the initial probability , i = 1, …, n of Markov chain. Then the likelihood of the observing the sequence o is

i

S TSS

TS TS TS TSSSSSobPobPoboP

,. . . ,2,1,1222,1111

)() ...()()(


Which probability function of ot?

• In HMM framework, observation o is assumed to be governed by the density of a Gaussian mixture distribution.

• Where k is the dimension of ot , and where oi and are the mean vector and covariance matrix,

respectivelyi

))()(2

1e x p (

)d e t()2(

1)( 1'

itiit

ik

ti ooooob


Re-estimation of mean, covariances, and the transition probabilities

T

ti

T

tti

tL

utLu

1

1

)(

)(

T

ti

T

t

ititi

tL

uuuutL

1

1

'

)(

))()((

T

ti

T

tij

ij

tL

tHP

1

1

1

)(

)(

)(tLi d eno tes the c o nd itio nal p ro b ab ility o f b eing in s tate i at tim e t

)(, tH jid eno tes the c o nd itio nal p ro b ab ility fro m s tate i at tim e t to s tate j at tim e t+ 1


Example: A Markov Model*

Sunny

Rainy

Snowy

70%

25%

5%

60%

12%

28%

20%

70% 10%

* C o urte s y o f D r. D o a n, U IU C


Represent it as a Markov Model*

• States:

• State transition probabilities:

• Initial state distribution:

},,{ snowyrainysunny SSS

2.1.7.

12.6.28.

05.25.7.

P

)2.2.6.(



What is sequence o in this example?*

• Sequence o:

• The probability could be computed by the conditional probability:

)|(*)|(*)|(*)( R ain yS n ow yR ain yR a in yS u n n yR ain yS u n n y SSPSSPSSPSP

1S 2,11 SSP2,1 SSP 3,2 SSP



Example: A HMM*

Sunny

Rainy

Snowy

80%

15%

5%

60%

2%

38%

20%

75% 5%

70%

10%

20%

75%

5%

20%

50%5%45%


Insid e not ob servab le


What other parameters will be needed?

• If we can not see what is inside blue circle, what can we actually see?

• Observations:

• Observation probabilities:

noooo ...,, ,321

),|()( iiiSi SstateitoPob


Forward-Backward Algorithm: Forward

• If Observation probability is

• then),|()( iiiSi SstateitoPob

T

i

T

iT iiOPOP

1 1

)(),()(

),,...,,()( 21 itt SstateOOOPi

)()( 11 Obi iSi

)(*

*)()(*),|,(),(

*),|,,(),,()(

12,1

221121

121212

iP

ObiSstateOSstateOPSstateOP

SstateOSstateOOPSstateOOPj

SS

Siji

ijj


Forward-Backward Algorithm: Backward

• If there is a

• Then

• The Forward-Backward Algorithm tells us that

• for any time t

),,..,()( 21 istateOOOPi Tttt

1)( iT

T

jttjSjSit jObPi

111, )()()(

t

itt iiOP

1

)(*)()(


Face identification using HMM

• An Observation sequence is extracted from the unknown face, the likelihood of each HMM generating this face could be computed.

• In theory, the likelihood is

• The maximum P(O) can identifies the unknown faces.

• However, it takes too much time to compute.

S TSS

TS TS TS TSSSSSobPobPoboP

,. . . ,2,1,1222,1111

)() ...()()(


Face identification using HMM• In practice, we only need one S sequence

which maximizes

• This is a dynamic programming optimization procedure.

1,1

,1 *)(),|,(

STSTtSt

T

tStSt PObPbPSOP


Viterbi Algorithm

• Given a S sequence, a dynamic programming approach to solve this problem

• where • By induction, the max Probability in state i+1 at

time t+1 is based on the max probability in state I at time t.

)|,,...,,(max)( 21 istateOOOPi tt ),( bP

)(])([max)( 1,1 tStStSttt ObPij


Algorithm itself

• Initialization

where denotes the collection of that sequence which is based on max

• Recursion:

Ni

Obi Sii

1

)()( 11 0)(1 i

)(])(max[)( 1,1

1 tStStSttNi

t ObPij

])([maxarg)( ,11

jitNi

t Pij

NjTt 1,2


Algorithm itself (2)

• Termination

• Sequence constructing from T to t

)]([max1

iP TNi

)]([maxarg1

iq TNi

1,...,2,1,11 TTtqq ttt


So far we have this block diagram

feature extrac tio nB ulk p ro c es s ing

fac ep ic ture

O b s ervatio ns eq uenc es

S eq uenc e w ithm ax

p ro b ab ility

V iterb i A lgo rithm

fac eid entific atio n

p erfo rm anc eevaluatio n

F a c e R e c o gn itio n B lo c k D ia g ra m


Face Detection

• In simple face recognition framework, the picture is assumed to be a frontal view of a single person and the background is monochrome.

• This project assumes that with the techniques of face detection, the performance of face recognition may be better than the approach presented above.


Acknowledgement

• The authors of this presentation slides would like to give thanks to Dr. Doan, UIUC.


Reference

• [1] Ferdinando Samaria, and Steve Young, HMM-based architecture for face identification.

• [2] Jia, Li, Amir Najmi, and Robert M. Gray, Image Classification by a Two-Dimensional Hidden Markov Model

• [3] Ming-Hsuan Yang, David J. Kriegman, Narendra Ahuja, Detecting Faces In Images: A survey

• [4] T.K. Leung, M. C. Burl, and P. Perona, Finding Faces in Cluttered Scenes using Random Labeled Graph Matching

• [5] James Wayman, Anil Jain, Davide Maltoni, and Dario Maio, Biometric Systems, Springer, 2005

timothy and rahule6886 project1 statistically recognize faces based on hidden markov models...

Documents