Master Thesis - Department of Computer Science

common vectors. These discriminative common vectors are used for classifica-
tion of new faces.
• Probabilistic Eigenspace Method: Probabilistic subspace method models
intra-personal and extra-personal variations to classify the face intensity differ-
ence ∆ as intra-personal variation (ΩI) for the same class and extra-personal
variation (ΩE) for different classes. The MAP similarity between two images
is defined as the intra-personal a posterior probability:
S(I1, I2) = P (ΩI|∆) =
P (∆|ΩI)P (ΩI)
P (∆|ΩI)P (ΩI) + P (∆|ΩE)P (ΩE)
(2.8)
To estimate P (∆|ΩI) and P (∆|ΩE), the eigenvectors of intra-personal and
extra-personal subspaces are computed from the difference set {(xi−xj)|L(xi) =
L(xj)} and {(xi − xj)|L(xi) �= L(xj)}, respectively. The covariance matrices
for intra-personal and extra-personal difference sets are defined as:
SI =
SE =
�
L(xi)=L(xj)
�
L(xi)�=L(xj)
(xi − xj)(xi − xj) T , (2.9)
(xi − xj)(xi − xj) T . (2.10)
To estimate P (∆|ΩI), the eigenspace of SI is decomposed into intra-personal
principal subspace F, spanned by the L largest eigenvectors, and its orthogonal
complementary subspace ¯ F , with dimension M − L. Then P (∆|ΩI) can be
obtained as the product of two independent marginal Gaussian densities in F
and ¯ F ,
P (∆|ΩI) =
Here, dF (∆) = � L i=1
y 2 i
λi
=
�
1 exp(− 2dF (∆))
(2Π) L/2 �L i=1 λ 1/2
� �
2 exp(−ε (∆)/2ρ)
(2Πρ)
i
(N−L)/2
�
exp �
− 1
2 (dF (∆) + ε2 (∆)/ρ) �
�
(2Π) L/2 �L i=1 λ 1/2
�
i [(2Πρ) (N−L)/2 . (2.11)
]
is a Mahalanobis distance in F and referred as “distance-
in-feature-space” (DIFS). yi is the principal component of ∆ projecting to the
i th intra-personal eigenvector, and λi is the corresponding eigenvalue. ε 2 (∆),
defined as “distance-from-feature-space” (DFFS), is the PCA residual (recon-
struction error) in ¯ F . ρ is the average eigenvalue in ¯ F . P (∆|ΩE) can be
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