Face recognition using PCA
DANG THE HUONG
VINH UNIVERSITY


CONTENTS
•
•
•
•
•

IDEA
OPERATIONS
MERITS
DEMERITS
APPLICATIONS


IDEA
PCA

Eigenfaces: the idea

Eigenvectors and Eigenvalues

Learning Eigenfaces from training sets of faces

Co-variance

Recognition and reconstruction

Where A is a matrix ,

 2 3
A=

2
1



e.g.

is a scalar (called the eigenvalue)

 3since

ν = 
2

one eigenvector of is

 2 3  3  12 
 3
 2 1  2  =  8  = 4 ×  2 

   
 
so for this eigenvector of this matrix the eigenvalue is 4



a
 N2 

 d1 

÷
d
2 ÷
=
M ÷

÷
÷
d
2
 N 

 b1 

÷
b
2 ÷
=
M÷

÷
÷
b
 N2 




=




f1 
÷
f2 ÷
M ÷
÷
fN2 ÷



We compute the average face

 a1 + b1 + L + h1 

÷
r 1  a2 + b2 + L + h2 ÷
m=
,
M ÷
MM M

÷
÷
a

M
M
M
M ÷
M ÷
M ÷
M ÷

÷

÷

÷

÷
÷
÷
÷
÷
a
m
b
m
c
m
d
m
2 −
2
2 −

2 −
2
N 
 N


r 
fm =




f1 − m1 
 g1 − m1 
 h1 − m1 
÷

÷ r 
÷
f 2 − m2 ÷ r
g
−
m
h
−
m
2
2 ÷
2
2 ÷


2
Now we build the matrix which is N by M

r r r r r r r r
A =  am bm cm d m em f m g m hm 
2
2
The covariance matrix which is N by N

Cov = AA

Τ


The covariance matrix has eigenvectors
covariance matrix

eigenvectors

eigenvalues

Eigenvectors with larger eigenvectors correspond to

directions in which the data varies more

Finding the eigenvectors and eigenvalues of the
covariance matrix for a set of data is termed
principle components analysis


)(
x
∑ 1 1 2 −x2 )
i =1

n −1


Recognition
A face image can be projected into this face space by
T
pk = U (xk – m) where k=1,…,m

To recognize a face

Subtract the average face from it

 r1 
 ÷
r2
= ÷
M ÷
 ÷
÷
 rN 2 

 r1 − m1 

÷
r

θ = max Ωi − Ω j
2

}

2

for i = 1.. M

for i, j = 1.. M


Distinguish between

•

If

ξ ≥then
θ it’s not a face; the distance between the face and its reconstruction is

larger than threshold

•
•

If
If

then it’s a new face

- Change in feature location and shape.


DEMERITS
Variations in lighting conditions
Different lighting conditions for enrolment and query.
Bright light causing image saturation.


APPLICATIONS:
Various potential applications, such as

•
•
•

Person identification.
Human-computer interaction.
Security systems.


Thank You