Linear Discriminant Analysis

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2 LDA comes with concept of class. PCA don’t use concept of classes. LDA is an enhancement to PCA Class in face recognition means a specific person, and elements of

class are his/her face images. Suppose there two class, then class 1 will have images of 1st

person and class 2 will have images of 2nd person.

3• Multiple classes and PCA

− Suppose there are C classes in the training data.− PCA is based on the sample covariance which characterizes the

scatter of the entire data set, irrespective of class-membership.− The projection axes chosen by PCA might not provide good

discrimination power.

• What is the goal of LDA?

− Perform dimensionality reduction while saving as much of the class discriminatory information as possible.

− Search to find directions along which the classes are best separated.− Takes into consideration the scatter within-classes but also the

scatter between-classes.

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LDA maximizes the between-class scatter LDA minimizes the within-class scatter

Class A

Class B

5 Algorithm

Assumptions Square images with Width=Height=N M is the number of images in the database P is the number of persons in the database

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The database

2

1

2

N

bb

b

2

1

2

N

cc

c

2

1

2

N

dd

d

2

1

2

N

ee

e

2

1

2

N

aa

a

2

1

2

N

ff

f

2

1

2

N

gg

g

2

1

2

N

hh

h

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Fisherfaces, the algorithm We compute the average of all faces

2 2 2

1 1 1

2 2 21 , 8

N N N

a b ha b h

m where MM

a b h

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Fisherfaces, the algorithmCompute the average face of each person

2 2 2 2

2 2 2 2

1 1 1 1

2 2 2 2

1 1 1 1

2 2 2 2

1 1, ,2 2

1 1,2 2

N N N N

N N N N

a b c da b c d

x y

a b c d

e f g he f g h

z w

e f g h

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Fisherfaces, the algorithmAnd subtract them 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 x b x c y d ya x b x c y d y

a b c d

a x b x c y d y

e z f ze z f

e f

e z

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 w h wz g w h w

g h

f z g w h w

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Fisherfaces, the algorithm

We build scatter matrices S1, S2, S3, S4

And the within-class scatter matrix SW

1 2

3 4

, ,

,

m m m m m m m m

m m m m m m m m

S a a b b S c c d d

S e e f f S g g h h

1 2 3 4WS S S S S

Fisherfaces, the algorithmThe between-class scatter matrix

We are searching the matrix W maximizing

2 2 2 2BS x m x m y m y m z m z m w m w m

B

W

W S WJ W

W S W

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Fisherfaces, the algorithm

Columns of W are eigenvectors ofWe have to compute the inverse of SW

We have to multiply the matricesWe have to compute the eigenvectors

1W BS S

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RecognitionProject faces onto the LDA-spaceTo classify the face

Project it onto the LDA-spaceRun a nearest-neighbor classifier Nearest is our answer