PCA vs ICA vs LDA. How to represent images? Why representation methods are needed?? –Curse of dimensionality – width x height x channels –Noise reduction.

PCA vs ICA vs LDA

How to represent images?• Why representation methods are needed??

– Curse of dimensionality – width x height x channels– Noise reduction– Signal analysis & Visualization

• Representation methods– Representation in frequency domain : linear transform

• DFT, DCT, DST, DWT, …• Used as compression methods

– Subspace derivation• PCA, ICA, LDA• Linear transform derived from training data

– Feature extraction methods• Edge(Line) Detection • Feature map obtained by filtering• Gabor transform• Active contours (Snakes)• …

1 1 2 2

1 2

ˆ ...

where , ,..., is a base in the -dimensionalsub-space (K<N)K K

K

x b u b u b u

u u u K

x̂ x

1 1 2 2

1 2

...

where , ,..., is a base in the original N-dimensionalspaceN N

n

x a v a v a v

v v v

• Find a basis in a low dimensional sub-space:

− Approximate vectors by projecting them in a low dimensional

sub-space:

(1) Original space representation:

(2) Lower-dimensional sub-space representation:

• Note: if K=N, then

What is subspace? (1/2)

What is subspace? (2/2)

• Example (K=N):

PRINCIPAL COMPONENT ANALYSIS (PCA)

Why Principal Component Analysis?

• Motive– Find bases which has high variance in data– Encode data with small number of bases with low MSE

Derivation of PCs

[ ]E x 0 T Ta x q q x

2 2 2 2[ ] [ ] [ ]

[( )( )] [ ]T T T T T

E a E a E a

E E

q x x q q xx q q Rq

Rq q

1 2 1 2, [ , ,..., ,..., ], [ , ,..., ,..., ]

1, 2,...,

Tj m j m

j j j

diag

j m

R QΛQ Q q q q q Λ

Rq q

Find q’s maximizing this!!

12|| || ( ) 1T q q q

Principal component q can be obtainedby Eigenvector decomposition such as SVD!

Assume that

0

5

10

15

20

25

PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10

Vari

an

ce (

%)

Dimensionality Reduction (1/2)

Can ignore the components of less significance.

You do lose some information, but if the eigenvalues are small, you don’t lose much– n dimensions in original data – calculate n eigenvectors and eigenvalues– choose only the first p eigenvectors, based on their eigenvalues– final data set has only p dimensions

Dimensionality Reduction (2/2)

Variance

Dimensionality

Reconstruction from PCs

q=1 q=2 q=4 q=8

q=16 q=32 q=64 q=100…Original Image

LINEAR DISCRIMINANT ANALYSIS (LDA)

Limitations of PCA

Are the maximal variance dimensions the relevant dimensions

for preservation?

Linear Discriminant Analysis (1/6)

• What is the goal of LDA?

− Perform dimensionality reduction “while preserving as much of the class discriminatory information as possible”.

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

scatter between-classes.− For example of face recognition, more capable of distinguishing

image variation due to identity from variation due to other sources such as illumination and expression.

Linear Discriminant Analysis (2/6)

1 1

1

( )( )

( )( )

incT

w j i j ii j

cT

b i ii

S Y M Y M

S M M M M

Within-class scatter matrix

Between-class scatter matrix

TUy xprojection matrix

− LDA computes a transformation that maximizes the between-class scatter while minimizing the within-class scatter:

| | | |max max

| || |

Tb b

Tww

S U S U

U S US

products of eigenvalues !

,b wS S : scatter matrices of the projected data y

1w

TbS S U U

Linear Discriminant Analysis (3/6)

− c.f. Since Sb has at most rank C-1, the max number of eigenvectors with non-zero eigenvalues is C-1 (i.e., max dimensionality of sub-space is C-1)

• Does Sw-1 always exist?

− If Sw is non-singular, we can obtain a conventional eigenvalue problem by writing:

− In practice, Sw is often singular since the data are image vectors with large dimensionality while the size of the data set is much smaller (M << N )

1w

TbS S U U

Linear Discriminant Analysis (4/6)

• Does Sw-1 always exist? – cont.

− To alleviate this problem, we can use PCA first:

1) PCA is first applied to the data set to reduce its dimensionality.

2) LDA is then applied to find the most discriminative directions:

Linear Discriminant Analysis (5/6)

D. Swets, J. Weng, "Using Discriminant Eigenfeatures for Image Retrieval", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 8, pp. 831-836, 1996

PCA LDA

Linear Discriminant Analysis (6/6)• Factors unrelated to classification

− MEF vectors show the tendency of PCA to capture major variations in the training set such as lighting direction.

− MDF vectors discount those factors unrelated to classification.

D. Swets, J. Weng, "Using Discriminant Eigenfeatures for Image Retrieval", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 8, pp. 831-836, 1996

INDEPENDENT COMPONENT ANALYSIS

PCA vs ICA• PCA

– Focus on uncorrelated and Gaussian components

– Second-order statistics

– Orthogonal transformation

• ICA– Focus on independent and non-Gaussian components

– Higher-order statistics

– Non-orthogonal transformation

Independent Component Analysis (1/5)

• Concept of ICA– A given signal(x) is generated by linear mixing(A) of independent

components(s)

– ICA is a statistical analysis method to estimate those independent components(z) and Mixing rule(W)

Aijs1

s2

s3

sM s A x

x1

x2

x3

xM

Wij

W z

z1

z2

z3

zM

WAsWxz

We do not knowBoth unknowns Some optimizationFunction is required!!

Independent Component Analysis (2/5)

AW

UAXUXW

1

Independent Component Analysis(3/5)

• What is independent component??– If one variable can not be estimated from other

variables, it is independent.– By Central Limit Theorem, a sum of two

independent random variables is more gaussian than original variables distribution of independent components are nongaussian

– To estimate ICs, z should have nongaussian distribution, i.e. we should maximize nonguassianity.

Independent Component Analysis(4/5)

• What is nongaussianity?– Supergaussian– Subgaussian– Low entropy

Gaussian Supergaussian Subgaussian

Independent Component Analysis(5/5)

• Measuring nongaussianity by Kurtosis– Kurtosis : 4th order cumulant of randomvariable

• If kurt(z) is zero, gaussian• If kurt(z) is positive, supergaussian• If kurt(z) is negative, subgaussian

• Maximzation of |kurt(z)| by gradient method

224 }){(3}{)( zEzEzkurt

]3})({))[((4|)(| 23 ||w||wxwxxw

w

xw

TT

T

Ekurtsignkurt

||w||w/w

x)x(wxww TT

}{))(( 3Ekurtsign

wx)x(ww T 3}{ 3 E

Fast-fixed point algorithm

Simply changeThe norm of w

PCA vs LDA vs ICA

• PCA : Proper to dimension reduction

• LDA : Proper to pattern classification if the number of training samples of each class are large

• ICA : Proper to blind source separation or classification using ICs when class id of training data is not available

References

• Simon Haykin, “Neural Networks – A Comprehensive Foundation- 2nd Edition,” Prentice Hall

• Marian Stewart Bartlett, “Face Image Analysis by Unsupervised Learning,” Kluwer academic publishers

• A. Hyvärinen, J. Karhunen and E. Oja, “Independent Component Analysis,”, John Willy & Sons, Inc.

• D. L. Swets and J. Weng, “Using Discriminant Eigenfeatures for Image Retrieval”, IEEE Trasaction on Pattern Analysis and and Machine Intelligence, Vol. 18, No. 8, August 1996