08 biometrics lecture 8 part3 2009-11-09

7/31/2019 08 Biometrics Lecture 8 Part3 2009-11-09

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2Face Recognition

Face detection

Face tracking

Face recognition (appearance-based) Local features

DCT-based methods

Global features (holistic approach)

Principal Component Analysis (PCA)

Linear Discriminant Analysis (LDA)

Performance evaluation Advantages and disadvantages


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3Principal Component Analysis (PCA)

Principal component analysis (PCA), orKarhunen-Loevetransformation, is a data-representation method that findsan alternative set of parameters for a set of raw data (or

features) such that most of the variability in the data iscompressed down to the first few parameters

The transformed PCA parameters are orthogonal The PCA, diagonalizes the

covariance matrix, and the

resulting diagonalelements are the variances

of the transformed PCA

parameters


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5PCA


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6PCA

The covariance (scatter) matrix of the data , which encodes thevariance and covariance of the data, is used in PCA to find the optimalrotation of the parameter space

PCA finds the eigenvectors and eigenvalues of the covariance matrix.

These have the property that

where:

- covariance (scatter) matrix

- eigenvectors

- transformed covariance matrix (diagonal scatter matrix of eigenvalues)

- eigenvalues

Example:

T =W SW V

S

1 2[ , , , ]

Dw w w=W r r rK

1 2diag( ) [ , , , ]

Dv v v= =V v K

0.26 0.96 14492.28 20760.14 0.26 0.96 302.84 0

0.96 0.26 20760.14 14492.28 0.96 0.26 0 94.40

=

ST

W W V

x

r

V


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7PCA

Having found the eigenvectors and eigenvalues,the principal components are found by the followingtransformation:

Example:

T

PCAx x= Wr r

,1 1 2 1

,2 1 2 2

0.26 0.96 0.26 0.96

0.96 0.26 0.96 0.26

PCA

PCA

x x x x

x x x x

= = +

The eigenvectors give an idea of the importance of each of

the original parameters in accounting for the variance in the data


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8PCA

A face image defines a pointin the high-dimensional imagespace

Different face images share anumber of similarities witheach other

They can be described by arelatively low-dimensional subspace

They can be projected into anappropriately chosen subspace ofeigenfaces and classification can beperformed by similarity computation(distance)

1x

2x

,1PCAx

,2PCAx


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92D DCT and PCA

Graphs from: C. Sanderson, On Local Features for Face Verification, IDIAPRR 04-36

Feature VectorV= [C0

, C1

, C2

, ... , Cuv

];

Feature vector with first few

local PCA basis functions


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10PCA

Suppose data consists ofMfaces withD feature values

1) Place data inD xMmatrix x

2) Mean-center the data

ComputeD-dimensional (mean). x0 = x -

3) ComputeD xD covariance matrix ( c = x0 x0T)

4) Compute eigenvectors and eigenvalues of covariancematrix

5) ChooseKlargest eigenvalues (K


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11PCA

PCA seeks directions that are efficient forrepresenting the data

efficientnot efficient

Class A

Class B

Class A

Class B


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12Eigenfaces, the algorithm

The database

2

1

2

N

b

b

b

=

M

2

1

2

N

c

c

c

=

M

2

1

2

N

d

d

d

=

M

2

1

2

N

e

e

e

=

M

2

1

2

N

f

f

f

=

M

2

1

2

N

g

g

g

=

M

2

1

2

N

h

h

h

=

M

2

1

2

N

a

a

a

=

M


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We compute the average face

2 2 2 2

1 1 11

2 2 2 21, where 8

N N N N

a b hm

m a b hm M

Mm a b h

+ + + + + + = =

+ + +

L

Lr

M M M ML


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Then subtract it from the training faces

2 2 2 2 2 2 2 2

2 2

1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2

1 1 1 1

2 2

, , , ,

,

m m m m

N N N N N N N N

m m

N N

a m b m c m d m

a m b m c m d ma b c d

a m b m c m d m

e m f m

e m fe f

e m

= = = =

= =

r rr r

M M M M M M M M

rr

M M

2 2 2 2 2 2

1 1 1 1

2 2 2 2 2 2, ,m m

N N N N N N

g m h m

m g m h mg h

f m g m h m

= =

rr

M M M M M M


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Now we build the matrix which is N2by M

The covariance matrix which is N2by N2

Find eigenvalues of the covariance matrix

The matrix is very large The computational effort is very big

We are interested in at mostMeigenvalues We can reduce the dimension of the covariance (scatter) matrix

Find theMeigenvalues and eigenvectors Eigenvectors ofC and S are equivalent

Build transform matrix W from the eigenvectors ofS

m m m m m m m ma b c d e f g h = xr r r rr r r r

T= C x x

T= S x x


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Compute for each face its projection onto the facespace

Compute the threshold

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

,1 ,2 ,3 ,4

,5 ,6 ,7 ,8

, , , ,

, , ,

T T T T

PCA m PCA m PCA m PCA m

T T T T

PCA m PCA m PCA m PCA m

x a x b x c x d

x e x f x g x h

= = = =

= = = =

W W W W

W W W W

r rr r r r r r

r rr r r r r r

{ }, ,1

max , 1,2, ,2

PCA i PCA jx x for i j M = =r r

K


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To recognize a face

Subtract the average face from it

2

1

2

N

r

r

r

=

M

2 2

1 1

2 2m

N N

r m

r mr

r m

=

rM M


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Compute its projection onto the face space

Compute the distance in the face space between the faceand all known faces

( )PCA mx r= W

r r

22, 1,2, ,i PCA PCA ix x for i M = =r r K


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Reconstruct the face from eigenfaces

Compute the distance between the face and itsreconstruction

Distinguish between If then it is not a face

If then it is a new face

If then it is a known face

PCA PCAr x= Wr r

22

m PCAr r = r r

, ( 1,2, , )iand i M < = K{ }min iand

08 biometrics lecture 8 part3 2009-11-09

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