eigenfisherfaces 2

8/18/2019 eigenfisherfaces 2

1/20

Face Recognition

using

PCA (Eigenfaces) and LDA (Fisherfaces)

Slides adapted from Pradeep Buddharaju


2/20

Principal Component Analsis

• A ! x N pi"el image of a face#represented as a $ector occupies asingle point in N 2 %dimensional imagespace&

• 'mages of faces eing similar in o$erallconfiguration# ill not e randomldistriuted in this huge image space&

• *herefore# the can e descried alo dimensional suspace&

• +ain idea of PCA for faces, – *o find $ectors that est account for

$ariation of face images in entireimage space&

– *hese $ectors are called eigen$ectors&

– Construct a face space and project theimages into this face space(eigenfaces)&


3/20

'mage Representation

• *raining set of m images of si-e N*N are

represented $ectors of si-e N .

"/#".#"0#1#"+

E"ample

33154

213

321

×

−

191

5

4

21

3

3

2

1

×

−


4/20

A$erage 'mage and Difference 'mages

• *he a$erage training set is defined

µ2 (/3m) 4mi2/ "i

• Each face differs from the a$erage $ector

r i 2 "i 5 µ


5/20

Co$ariance +atri"

•

*he co$ariance matri" is constructed as

C 2 AA* here A26r /#1#r m7

• Finding eigen$ectors of N 2 " N 2 matri" is intractale& 8ence# use the

matri" A* A of si-e m " m and find eigen$ectors of this small matri"&

Si-e of this matri" is N 2 " N 2


6/20

Eigen$alues and Eigen$ectors % Definition

• 'f $ is a non-ero $ector and 9 is a numer such that

Av = λv# then

$ is said to e an eigenvector of A ith eigenvalue 9&

E"ample

A

λ

$ (eigenvectors)

(eigenvalues)

×=

×

1

1

31

1

21

12


7/20

Eigen$ectors of Co$ariance +atri"

•

*he eigen$ectors $i of A

*

A are,

• Consider the eigen$ectors $i of A* A such that

A* A$i 2 µi$i

• Premultipling oth sides A# e ha$e

AA*( A$i) 2 µi( A$i)


8/20

Face Space

• *he eigen$ectors of co$ariance matri" are

ui 2 A$i

• ui resemle facial images hich loo: ghostl# hence called Eigenfaces

Face Space


9/20

Projection into Face Space

• A face image can e projected into this face space

p: 2 ;*(": 5 µ) here :2/#1#m


10/20

Recognition

• *he test image " is projected into the face space tootain a $ector p,

p 2 ;*(" 5 µ)

• *he distance of p to each face class is defined

: 2 /#1#m

• A distance threshold ?c# is half the largest distance

eteen an to face images,

?c 2 @ ma" j#: ==p j%p:==> j#: 2 /#1#m


11/20

Recognition

• Find the distance < eteen the original image " and itsreconstructed image from the eigenface space# "f #


12/20

Limitations of Eigenfaces Approach

• Variations in lighting conditions – Different lighting conditions for

enrolment and Guer&

– Bright light causing image saturation&

• Differences in pose – Head orientation

% .D feature distances appear to distort&

• Expression

% Change in feature location and shape&


13/20

Linear Discriminant Analsis

• PCA does not use class information – PCA projections are optimal for reconstruction from

a lo dimensional asis# the ma not e optimal

from a discrimination standpoint&

• LDA is an enhancement to PCA – Constructs a discriminant suspace that minimi-es

the scatter eteen images of same class and

ma"imi-es the scatter eteen different class

images


14/20

+ean 'mages

• Let H/# H.#1# Hc e the face classes in the dataase and let

each face class Hi# i 2 /#.#1#c has : facial images " j# j2/#.#

1#:&

• Ie compute the mean image µi of each class H i as,

• !o# the mean image µ of all the classes in the dataase cane calculated as,

∑==k

j

ji xk 11 µ

∑=

=c

i

i

c 1

1 µ µ


15/20

Scatter +atrices

• Ie calculate ithin%class scatter matri" as,

• Ie calculate the eteen%class scatter matri" as,

T

ik

c

i X x

ik W x xS ik

)()(1

µ µ −−= ∑ ∑= ∈

T

ii

c

i

i B N S ))((

1

µ µ µ µ −−= ∑=


16/20

+ultiple Discriminant Analsis

Ie find the projection directions as the matri" I that ma"imi-es

*his is a generali-ed Eigen$alue prolem here the

columns of I are gi$en the $ectors i that sol$e

W

^

= argmax J (W ) = |W T S

BW |

|W T S W W |

S Bw

i = λ

iS W w

i


17/20

Fisherface Projection

• Ie find the product of SI%/ and SB and then compute the Eigen$ectors

of this product (SI%/ SB) % AF*ER RED;C'!J *8E D'+E!S'K! KF

*8E FEA*;RE SPACE&

• ;se same techniGue as Eigenfaces approach to reduce the

dimensionalit of scatter matri" to compute eigen$ectors&

• Form a matri" I that represents all eigen$ectors of SI%/ SB placing

each eigen$ector i as a column in I&

• Each face image " j ∈ Hi can e projected into this face space the

operation

pi 2 I*(" j 5 µ)


18/20


19/20

*esting

• Same as Eigenfaces Approach


20/20

References

• *ur:# + Pentland# A&, Eigenfaces for recognition& & Cogniti$e

!euroscience 3 (/MM/) N/5O&

• Belhumeur# Pespanha# Qriegman# D&, Eigenfaces vs. Fisherfaces:

recognition using class specific linear projection& 'EEE *ransactions on

Pattern Analsis and +achine 'ntelligence 19 (/MMN) N//5N.&

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