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Part 2 ICA for Face Recognition １. ICA –general framework- e.g. 2-source/2-sensor case

AS XMixing

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

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

xX =

x

1st image

2nd image

1st pixel 2nd pixel

N-th image

The last pixel

N face images

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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]

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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 Nx

1

N

u

U WX

u

1

2Face image matrix

T

T

T

N

X

x

x=

x

face 1

face 2

ICA 3000-dim.

425-dim.

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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

uX B U

u

x u u u

b

Reference [2]

Statistically Independent

Component Basis Images

(25 images) which

provide local features.

As shown in this Figure,

ICA images are local.

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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

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 TU b z

Identification of the test face: pattern recognition

Define the similarity measure (cosine of the angle) between two faces

.-th row of train

i trainib B

The best fit face image = arg Max ii

c

similarity measure (cosine of the angle)

training images

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Reference [2]

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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

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