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Timothy and Rahul E6886 Project 1
Statistically Recognize Faces Based on Hidden Markov Models
Presented by
Timothy Hsiao-Yi Chin
Rahul Mody
Timothy and Rahul E6886 Project 2
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
Timothy and Rahul E6886 Project 3
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
)() ...()()(
Timothy and Rahul E6886 Project 4
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
Timothy and Rahul E6886 Project 5
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
Timothy and Rahul E6886 Project 6
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
Timothy and Rahul E6886 Project 7
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.(
* C o urte s y o f D r. D o a n, U IU C
Timothy and Rahul E6886 Project 8
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
* C o urte s y o f D r. D o a n, U IU C
Timothy and Rahul E6886 Project 9
Example: A HMM*
Sunny
Rainy
Snowy
80%
15%
5%
60%
2%
38%
20%
75% 5%
70%
10%
20%
75%
5%
20%
50%5%45%
* C o urte s y o f D r. D o a n, U IU C
Insid e not ob servab le
Timothy and Rahul E6886 Project 10
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
Timothy and Rahul E6886 Project 11
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
Timothy and Rahul E6886 Project 12
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
)(*)()(
Timothy and Rahul E6886 Project 13
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
)() ...()()(
Timothy and Rahul E6886 Project 14
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
Timothy and Rahul E6886 Project 15
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
Timothy and Rahul E6886 Project 16
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
Timothy and Rahul E6886 Project 17
Algorithm itself (2)
• Termination
• Sequence constructing from T to t
)]([max1
iP TNi
)]([maxarg1
iq TNi
1,...,2,1,11 TTtqq ttt
Timothy and Rahul E6886 Project 18
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
Timothy and Rahul E6886 Project 19
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.
Timothy and Rahul E6886 Project 20
Acknowledgement
• The authors of this presentation slides would like to give thanks to Dr. Doan, UIUC.
Timothy and Rahul E6886 Project 21
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