1. 1
Part 2 ICA for Face Recognition
１. ICA –general framework-
e.g. 2-source/2-sensor case
AS X
Mixing
matrix
Independent Sources
Observation
(Data matrix)
to be obtained
   
   
   
   
1 1 1 111 12
21 22 2 2 2 2
1 , 2 1 , 2
1 , 2 1 , 2
s s x xa a
a a s s x x
    
     
        

X
AS X

2. 2
 
 
 
 
 
 
 
 
 
1
2
1 1 1
2 2 2
Data Matrix
1 2
1 2
, ,
1 2
T
T
T
N
N N N
x x x M
x x x M
x x x M
 
 
 
 
 
  
 
 
 
 
 
  
x
x
X =
x
1st image
2nd image
1st pixel 2nd pixel
N-th image
The last pixel
N face images

3. as a solution which has two ambiguities on the order and
the scale of the real source images
ICA gives
,
where
Permutation
scale



S
U = WX
S = PDU
P
D
Reference [2]

4. 4
2. Face Recognition by ICA
[Training set of face images: FERET database]
For a given ensemble of N (425) training face images with M(=3000)
dimensional vectors
with zero mean and its data matrix X as in the first part.
{ , 1 }n n Nx
1
N
 
    
  
u
U WX
u
1
2
Face image matrix
T
T
T
N
X
 
 
 
 
 
  
x
x
=
x
face 1
face 2
ICA
3000-dim.
425-dim.

5. 5
Remarks
- Row vectors ui (i=1~N) would be as statistically independent as
possible, the obtained these row vectors are the basis images to
represent faces.
- One problem is the number of independent images will become quite
large because it is equal to the number of faces of training database.
- One solution is apply PCA prior to ICA for dimensionality reduction.
[Feature vector of training faces]
 
1
11 12
2
. . 21 22
1 1 2 2
1 2
th face:
of -th face image
train train
N
T
i i i iN N
train
i i i iN
b b
b b
i b b b
i
b b b
 
   
        
    
 
   

Feature vector
u
u
X B U
u
x u u u
b

6. Reference [2]
Statistically Independent
Component Basis Images
(25 images) which
provide local features.
As shown in this Figure,
ICA images are local.

7. 7

8. PCA Basis Images
from the same training
faces as in [2].
The order of the principal
components starts from
left to right, top to bottom.
Reference [2]

9. 9
1train T 
b z U
Representation of test images z: feature vectors of test face
,
length-normalized evaluation
test train
i
i test train
i
c 
b b
b b
test image: z
1test T
U 
b z
Identification of the test face: pattern recognition
Define the similarity measure (cosine of the angle) between two faces
.-th row oftrain
i trainib B
The best fit face image = arg Max i
i
c
similarity measure (cosine of the angle)
training images

10. 10
Reference [2]

11. 11
References (Part 2)
[1] A. J. Bell and T J. Srjnowski, The “Independent Components” of Natural Scene are Edge
Filters: Vision Research, Vol. 37, No. 23, pp. 3327-3338, 1997.
[2] M. S. Bartlett et al. “ Face recognition by Independent Component Analysis,” IEEE. Trans
on Neural Networks, Vol. 13, No. 6, Nov., 2002
[3] A. Hyvarinen et al. “Independent Component Analysis” , Wiley-InterScience, 2001
[4] B. A. Draper et al. , “Recognizing faces with PCA and ICA,” Computer Vision and Image
Understanding, vol. 91, pp. 115-137, 2003. 1, Jan. 2004

12. 12