FACE RECOGNITION BASED ON SEPARABLE LATTICE 2-D HMMS
USING VARIATIONAL BAYESIAN METHOD

Kei Sawada, Akira Tamamori, Kei Hashimoto, Yoshihiko Nankaku, Keiichi Tokuda (Nagoya Institute of Technology, Japan)	 
4. Separable lattice 2-D HMMs
using variational Bayesian method

1. Introduction	 ¢ Image recognition based on statistical approaches¢ Maximum a posteriori (MAP) method [Gauvain, et al.; ’94]

 Eigen-image and subspace methods based on PCA
 Heuristic normalization techniques for each task are required

5. Experiments	 ¢ Experimental conditions

Database	 XM2VTS	 Image size	 

64×64, grayscale	 

Training data	 

6, 5, 4, 3, 2 images per person × 100 subjects	 

Test data	 

2 images per person × 100 subjects	 

Number of states	 

8×8, 16×16, 24×24, 32×32, 40×40, 48×48, 56×56, 64×64	 

Tuning parameterss	 

50, 100, 500, 1000, 2000, 3000, 4000, 5000,6000, 7000, 8000, 9000, 10000, 50000, 100000

 Estimation of model parameters by maximizing posterior probability¢ Separable lattice 2-D HMMs (SL2D-HMMs) [Kurata, et al.; ’06]

 Training and normalization are integrated
 ML criterion produces point estimation of model parameters⇒ Estimation accuracy may be decreased due to the over-fitting Use of prior distribution
 Over-fitting problem because of point estimates¢ Variational Bayesian (VB) method [Attias; ’99]

¢ Bayesian criterion

 Use of prior distribution and marginalization of model parameters

Apply Bayesian criterion to separable lattice 2-D HMMs

 Estimation of approximated posterior distribution
 Define a low bound of log marginal likelihood

2. Separable lattice 2-D HMMs	 ¢ Examples of training images and mean vectors

Jensen’s inequality¢ Separable lattice 2-D hidden Markov models: State sequence	: Arbitrary dist.	

 SL2D-HMMs with horizontal and vertical Markov chains⇒ An elastic matching in both horizontal and vertical directions

 Relation between the log marginal likelihood and the lower boundHorizontal state sequence	⇒	 Assume that random variables are conditionally independent

Location andsize variations⇒ Normalize bychanging theduration of the state	Mean vectorof UBM

All training images of one subject	 : Variational posterior dist.	

 Estimation of posterior distribution based on maximizing
VB E-step 	Alternately update	VB M-step 	

Images are divided intoa rectangular region in the state	 Derive variational posterior distribution

Each pixel is emitted froma corresponding output
Horizontal state transition	
probability distribution	

¢ Results

Training data are 6 images	

Output probability distribution	 Use of prior distribution and marginalization of model parameters¢ Maximum likelihood (ML) criterion

 ML criterion produces point estimation ⇒ Over-fitting problem¢ Bayesian criterion

 Use of prior distribution and marginalization of model parameters
 Complex integral and expectation calculations⇒ Effective approximation techniques are requiredPrior information	  Prior dist.	 

Bayesian criterion	

Training data	  Posterior dist.	 

Predictive dist.	 

Test data	 

Predictive dist.	 

: Set of model parameters	

: Training data	: Test data	Training data are 6 to 2 images	

MLMAP
VB70

3. Bayesian criterion

ML criterion	

Mean vector of posterior distribution (VB)	 

¢ Prior distribution

 Conjugate prior distributionu Posterior dist. belongs to the same dist. family as the prior dist.
Initial state probability	 Dirichlet distribution	 

State transition probability	 Dirichlet distribution	 

Output probability distribution	 

Gauss-Wishart distribution	  Universal background model (UBM)u UBM is trained from all training data for all subjects⇒ UBM roughly represents a training data Tuning parameteru Representation of the reliability of the UBMu
is small ⇒ Prior distribution has a larger impact on posterior distributionuis large ⇒ Prior distribution has a smaller impact on posterior distribution

70
Recognition rate (%)

Vertical state transition	

Mean vectorof model (ML)Recognition rate (%)

Vertical state sequence	All training images for all subjects

6050406050
4030201030

08x8 16x16 24x24 32x32 40x40 48x48 56x56 64x64
Number of states

ML : ML criterion (conventional)

MLMAPVB
65432
Number of training images
40 x 40 states

MAP and VB : Bayesian criterion (proposed)	

 Bayesian criterion achieved significantly higher recognition rates thanML criterion
 The difference between ML criterion and Bayesian criterion became
larger when small numbers of training images were used The use of a prior distribution was more effective than themarginalization of model parameters