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Discriminant Analysis

1. Fisher Linear Discriminant2. Multiple Discriminant Analysis

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Motivation

Projection that best separates the data in a least-squares sense– PCA finds components that are useful for representing

data – However no reason to assume that components are useful

for discriminating between data in different classes– Pooling data may discard directions that are essential

• OCR example: Discriminating between O and Q may ignore tail

8

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Fisher Linear Discriminant

Projecting data from d dimensions onto a line

,.. samples of set ingcorrespond a and

as of components theofn combinatiolinear a form wish toWe labelled subset in the labelled subset in the

,.. samples ldimensiona- ofSet

1

222

111

1

n

n

yyny

DnDn

d n

xwx

xx

t=

ωω

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Two-Dimensional ExampleFisher Liner Discriminant: two-dimensional example

Better SeparationClasses mixed

Projection of same set of two-class samples onto two different linesin the direction marked w.

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Finding best direction w

|)|||is means projectedbetween Distance

11:points projected ofMean Sample

1:space ldimensiona-din Mean Sample

121 2

x

x

m(mw

mwxw

xm

−=−

===

=

∑∑

∑

∈∈

∈

t

it

D

t

iYyii

Dii

mm

ny

nm

n

ii

i

))

)

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Criterion for Fisher Linear Discriminant

Rather than forming sample variances, define scatter for the projected samples

22)( i

Yyi mys

i

∑ −=ε

Thus ))(/1(22

21 ssn + is an estimate of the variance of the pooled data

)(22

21 ss +Total within class scatter is

)(

||)( 22

21

221

ss

mmwJ+

−=Find that linear function wtx for which

is maximum and independent of ||w||.While maximizing J(.) leads to best separation between the two projected sets, we will need a threshold criterion to have a true classifier.

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Scatter Matrices

Within ClassScatter Matrix

Between ClassScatter Matrix

To obtain J(.) as an explicit function of w, we define scatter matrices Si and SW

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Scatter MatricesTo obtain J(.) as an explicit function of w, we define scatter matrices Si and SW

Within ClassScatter Matrix

21

))((

SSS

mxmxS

W

Dx

tiii

i

+=

−−= ∑ε

Between ClassScatter Matrix

tB mmmmS ))(( 2121 −−=

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Criterion Function in terms of Scatter Matrices

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Final form of Fisher Discriminant

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Case of Multivariate Normal pdfs

Bayes Decision rule is to compute Fisher LD and decide ω1 if it exceeds a threshold and ω2 otherwise

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Fisher’s Linear DiscriminantExample

Discriminating between machine-print and handwriting

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Cropped signature image

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hmw1

wm

h1

x1 = ( h1+w1) / (hm+wm) = 0.4034x2 = h1 / w1 = 0.9355

w1

Connected components and their features

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Feature Values of Labelled Components(from 20 images)

Print components Handwriting componentsComponent x1 x2 Compon

ent x1 x2

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Feature distribution in 2 dimensional space

T

T

T

)0040.0,0338.0()(

3043.730076.00076.01720.0

3043.732349.32349.39554.5

)2652.1,2725.0(

)0782.1,0843.0(

1

1

−−=−=

⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡−

−=

=

=

−

−

21w

w

w

2

1

mmSw

S

S

m

m

After projectionSamples are 1-DA Bayes classifieris designed for theprojected samples

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hmw1

wm

h1

x1 = ( h1+w1) / (hm+wm) = 0.4034x2 = h1 / w1 = 0.9355

w1

Discrete component analysis

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The obtained printed text component candidates

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Framework of printed text filtering

Step 1: Extract shape features of bounding box for each discrete component. Detect candidates by Fisher’s Linear Discriminant and Bayesian Classification

Step 2: Remove candidates that satisfy the spatial relation defined for printed text components

Step 3: For candidates surviving from step2, remove isolated and small pieces.

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Processed image after ( a ): Step 2, ( b ): Step 3 (final)

( a )

( b )

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Sample images before and after enhancement

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Multiple Discriminant Analysis

• c-class problem• Natural generalization of Fisher’s Linear

Discriminant function involves c-1 discriminant functions

• Projection is from a d-dimensional space to a c-1 dimensional space

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Mapping from d-dimensional space to c-dimensional space d=3, c=3