eigen value analysis in pattern recognition by dr. m. asmat ullah khan comsats institute of...

Eigen Value Analysis in Pattern Eigen Value Analysis in Pattern RecognitionRecognition

By

Dr. M. Asmat Ullah KhanCOMSATS Institute of Information Technology, Abbottabad

MULTI SPECTRAL IMAGE COMPRESSIONMULTI SPECTRAL IMAGE COMPRESSION

Principal Component Analysis Principal Component Analysis (PCA)(PCA)

Pattern recognition in high-dimensional spaces– Problems arise when performing recognition in a high-dimensional space

(curse of dimensionality).– Significant improvements can be achieved by first mapping the data into a

lower-dimensional sub-space.

– The goal of PCA is to reduce the dimensionality of the data while retaining as much as possible of the variation present in the original dataset.


Dimensionality reduction– PCA allows us to compute a linear transformation that maps data from a high

dimensional space to a lower dimensional sub-space.


Lower dimensionality basis– Approximate vectors by finding a basis in an appropriate lower dimensional

space.

(1) Higher-dimensional space representation:

(2) Lower-dimensional space representation:


Example


Information loss– Dimensionality reduction implies information loss !!– Want to preserve as much information as possible, that is:

How to determine the best lower dimensional sub-space?


Methodology– Suppose x1, x2, ..., xM are N x 1 vectors


Methodology – cont.


Linear transformation implied by PCA– The linear transformation RN RK that performs the dimensionality reduction is:


Geometric interpretation– PCA projects the data along the directions where the data varies the most.

– These directions are determined by the eigenvectors of the covariance matrix corresponding to the largest eigenvalues.

– The magnitude of the eigenvalues corresponds to the variance of the data along the eigenvector directions.


How to choose the principal components?

– To choose K, use the following criterion:


What is the error due to dimensionality reduction?– We saw above that an original vector x can be reconstructed using its

principal components:

– It can be shown that the low-dimensional basis based on principal components minimizes the reconstruction error:

– It can be shown that the error is equal to:


Standardization

– The principal components are dependent on the units used to measure the original variables as well as on the range of values they assume.

– We should always standardize the data prior to using PCA.– A common standardization method is to transform all the data to have

zero mean and unit standard deviation:


PCA and classification– PCA is not always an optimal dimensionality-reduction procedure for

classification purposes:

Principal Component Analysis (PCA)Principal Component Analysis (PCA) Case Study: Eigenfaces for Face Detection/Recognition

– M. Turk, A. Pentland, "Eigenfaces for Recognition", Journal of Cognitive Neuroscience, vol. 3, no. 1, pp. 71-86, 1991.

Face Recognition

– The simplest approach is to think of it as a template matching problem

– Problems arise when performing recognition in a high-dimensional space.

– Significant improvements can be achieved by first mapping the data into a lower dimensionality space.

– How to find this lower-dimensional space?


Main idea behind eigenfaces

Principal Component Analysis (PCA)Principal Component Analysis (PCA) Computation of the eigenfaces


Computation of the eigenfaces – cont.

Principal Component Analysis (PCA)Principal Component Analysis (PCA) Computation of the eigenfaces – cont.


Computation of the eigenfaces – cont.

Principal Component Analysis (PCA)Principal Component Analysis (PCA) Representing faces onto this basis


Representing faces onto this basis – cont.

Principal Component Analysis (PCA)Principal Component Analysis (PCA) Face Recognition Using Eigenfaces


Face Recognition Using Eigenfaces – cont.

– The distance er is called distance within the face space (difs)

– Comment: we can use the common Euclidean distance to compute er, however, it has been reported that the Mahalanobis distance performs better:


Face Detection Using Eigenfaces


Face Detection Using Eigenfaces – cont.

Principal Component Analysis (PCA)Principal Component Analysis (PCA) Problems

– Background (de-emphasize the outside of the face – e.g., by multiplying the input image by a 2D Gaussian window centered on the face)

– Lighting conditions (performance degrades with light changes)– Scale (performance decreases quickly with changes to head size)

multi-scale eigenspaces scale input image to multiple sizes

– Orientation (performance decreases but not as fast as with scale changes)

plane rotations can be handled out-of-plane rotations are more difficult to handle

Linear Discriminant Analysis (LDA)Linear Discriminant Analysis (LDA) 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 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.

– More capable of distinguishing image variation due to identity from variation due to other sources such as illumination and expression.

Linear Discriminant Analysis (LDA)Linear Discriminant Analysis (LDA) Methodology

Linear Discriminant Analysis Linear Discriminant Analysis (LDA)(LDA)

Methodology – cont.– LDA computes a transformation that maximizes the between-class scatter

while minimizing the within-class scatter:

– Such a transformation should retain class separability while reducing the variation due to sources other than identity (e.g., illumination).


Linear transformation implied by LDA

– The linear transformation is given by a matrix U whose columns are the eigenvectors of Sw

-1 Sb (called Fisherfaces).

– The eigenvectors are solutions of the generalized eigenvector problem:


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 )


Does Sw-1 always exist? – cont.

– To alleviate this problem, we can perform two projections:

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

2) LDA is then applied to further reduce the dimensionality.


Case Study: Using Discriminant Eigenfeatures for Image Retrieval– 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.

Content-based image retrieval– The application being studied here is query-by-example image retrieval.– The paper deals with the problem of selecting a good set of image

features for content-based image retrieval.


Assumptions– "Well-framed" images are required as input for training and query-by-

example test probes.– Only a small variation in the size, position, and orientation of the objects

in the images is allowed.

Linear Discriminant Analysis (LDA)Linear Discriminant Analysis (LDA) Some terminology

– Most Expressive Features (MEF): the features (projections) obtained using PCA.

– Most Discriminating Features (MDF): the features (projections) obtained using LDA.

Computational considerations– When computing the eigenvalues/eigenvectors of Sw

-1SBuk = kuk numerically, the computations can be unstable since Sw

-1SB is not always symmetric.

– See paper for a way to find the eigenvalues/eigenvectors in a stable way.

– Important: Dimensionality of LDA is bounded by C-1 --- this is the rank of Sw-1SB

Linear Discriminant Analysis (LDA)Linear Discriminant Analysis (LDA) Case Study: PCA versus LDA

– A. Martinez, A. Kak, "PCA versus LDA", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 2, pp. 228-233, 2001.

Is LDA always better than PCA?– There has been a tendency in the computer vision community to prefer

LDA over PCA.– This is mainly because LDA deals directly with discrimination between

classes while PCA does not pay attention to the underlying class structure.– This paper shows that when the training set is small, PCA can outperform

LDA.– When the number of samples is large and representative for each class,

LDA outperforms PCA.

Linear Discriminant Analysis (LDA)Linear Discriminant Analysis (LDA) Is LDA always better than PCA? – cont.


Is LDA always better than PCA? – cont.


Critique of LDA– Only linearly separable classes will remain separable after applying LDA.– It does not seem to be superior to PCA when the training data set is small.

Appearance-based RecognitionAppearance-based Recognition

• Directly represent appearance (image brightness), not geometry.

• Why?

Avoids modeling geometry, complex interactions between geometry, lighting and reflectance.

• Why not?

Too many possible appearances!

m “visual degrees of freedom” (eg., pose, lighting, etc)R discrete samples for each DOF

How to discretely sample the DOFs?

How to PREDICT/SYNTHESIS/MATCH with novel views?

Appearance-based RecognitionAppearance-based Recognition

• Example:

• Visual DOFs: Object type P, Lighting Direction L, Pose R

• Set of R * P * L possible images:

• Image as a point in high dimensional space:

}ˆ{ PRLx

x is an image of N pixels andA point in N-dimensional space

x

Pixel 1 gray value

Pix

el 2

gra

y va

lue

The Space of FacesThe Space of Faces

An image is a point in a high dimensional space– An N x M image is a point in RNM

– We can define vectors in this space as we did in the 2D case

+=

[Thanks to Chuck Dyer, Steve Seitz, Nishino]

Key IdeaKey Idea

}ˆ{ PRLx• Images in the possible set are highly correlated.

• So, compress them to a low-dimensional subspace that captures key appearance characteristics of the visual DOFs.

• EIGENFACES: [Turk and Pentland]

USE PCA!

EigenfacesEigenfaces

Eigenfaces look somewhat like generic faces.

Linear SubspacesLinear Subspaces

Classification can be expensive– Must either search (e.g., nearest neighbors) or store large probability density functions.

Suppose the data points are arranged as above– Idea—fit a line, classifier measures distance to line

convert x into v1, v2 coordinates

What does the v2 coordinate measure?

What does the v1 coordinate measure?

- distance to line- use it for classification—near 0 for orange pts

- position along line- use it to specify which orange point it is

Dimensionality ReductionDimensionality Reduction

Dimensionality reduction– We can represent the orange points with only their v1 coordinates

since v2 coordinates are all essentially 0

– This makes it much cheaper to store and compare points– A bigger deal for higher dimensional problems

Linear SubspacesLinear Subspaces

Consider the variation along direction v among all of the orange points:

What unit vector v minimizes var?

What unit vector v maximizes var?

Solution: v1 is eigenvector of A with largest eigenvalue v2 is eigenvector of A with smallest eigenvalue

Higher DimensionsHigher Dimensions

Suppose each data point is N-dimensional– Same procedure applies:

– The eigenvectors of A define a new coordinate system eigenvector with largest eigenvalue captures the most variation among

training vectors x eigenvector with smallest eigenvalue has least variation

– We can compress the data by only using the top few eigenvectors corresponds to choosing a “linear subspace”

– represent points on a line, plane, or “hyper-plane” these eigenvectors are known as the principal components

Problem: Size of Covariance Matrix AProblem: Size of Covariance Matrix A

Suppose each data point is N-dimensional (N pixels)

– The size of covariance matrix A is N x N– The number of eigenfaces is N

– Example: For N = 256 x 256 pixels, Size of A will be 65536 x 65536 ! Number of eigenvectors will be 65536 !

Typically, only 20-30 eigenvectors suffice. So, this method is very inefficient!

2 2

If B is MxN and M<<N then A=BTB is NxN >> MxM

– M number of images, N number of pixels

– use BBT instead, eigenvector of BBT is easily

converted to that of BTB

(BBT) y = e y

=> BT(BBT) y = e (BTy)

=> (BTB)(BTy) = e (BTy)

=> BTy is the eigenvector of BTB

Efficient Computation of EigenvectorsEfficient Computation of Eigenvectors

Eigenfaces – summary in wordsEigenfaces – summary in words

Eigenfaces are the eigenvectors of the covariance matrix of the probability distribution of the vector space of human faces

Eigenfaces are the ‘standardized face ingredients’ derived from the statistical analysis of many pictures of human faces

A human face may be considered to be a combination of these standardized faces

Generating Eigenfaces – in wordsGenerating Eigenfaces – in words

1. Large set of images of human faces is taken.2. The images are normalized to line up the eyes,

mouths and other features. 3. The eigenvectors of the covariance matrix of the

face image vectors are then extracted.4. These eigenvectors are called eigenfaces.

Eigenfaces for Face RecognitionEigenfaces for Face Recognition

When properly weighted, eigenfaces can be summed together to create an approximate gray-scale rendering of a human face.

Remarkably few eigenvector terms are needed to give a fair likeness of most people's faces.

Hence eigenfaces provide a means of applying data compression to faces for identification purposes.

Dimensionality ReductionDimensionality Reduction

The set of faces is a “subspace” of the set of images

– Suppose it is K dimensional

– We can find the best subspace using PCA

– This is like fitting a “hyper-plane” to the set of faces

spanned by vectors v1, v2, ..., vK

Any face:

EigenfacesEigenfaces

PCA extracts the eigenvectors of A– Gives a set of vectors v1, v2, v3, ...

– Each one of these vectors is a direction in face space what do these look like?

Projecting onto the EigenfacesProjecting onto the Eigenfaces

The eigenfaces v1, ..., vK span the space of faces

– A face is converted to eigenface coordinates by

Is this a face or not?Is this a face or not?

Recognition with EigenfacesRecognition with Eigenfaces Algorithm

1. Process the image database (set of images with labels)

• Run PCA—compute eigenfaces• Calculate the K coefficients for each image

2. Given a new image (to be recognized) x, calculate K coefficients

3. Detect if x is a face

4. If it is a face, who is it?

• Find closest labeled face in database• nearest-neighbor in K-dimensional space

Key Property of Eigenspace RepresentationKey Property of Eigenspace Representation

Given

• 2 images that are used to construct the Eigenspace

• is the eigenspace projection of image

• is the eigenspace projection of image

Then,

That is, distance in Eigenspace is approximately equal to the correlation between two images.

21 ˆ,ˆ xx

1x

2x1g

2g

||ˆˆ||||ˆˆ|| 1212 xxgg

Choosing the Dimension KChoosing the Dimension K

K NMi =

eigenvalues

How many eigenfaces to use?

Look at the decay of the eigenvalues

– the eigenvalue tells you the amount of variance “in the direction” of that eigenface

– ignore eigenfaces with low variance

PapersPapers

More Problems: OutliersMore Problems: Outliers

Need to explicitly reject outliers before or during computing PCA.

Sample Outliers

Intra-sample outliers

[De la Torre and Black]

PCAPCA

RPCARPCA

RPCA: Robust PCA, [De la Torre and Black]

Robustness to Intra-sample outliersRobustness to Intra-sample outliers

OriginalOriginal PCAPCA RPCARPCA OutliersOutliers

Robustness to Sample OutliersRobustness to Sample Outliers

Finding outliers = Tracking moving objects

Research QuestionsResearch Questions Does PCA encode information related to gender, ethnicity, age, and identity

efficiently?

What information do PCA encode?

Are there components (features) of PCA that encode multiples properties?

PCAPCA The aim of the PCA is a linear reduction of D dimensional data to d dimensional

data (d<D), while preserving as much information, in the data, as possible. Linear functions

y1= w1 Xy2= w2 X***yd= wd X

Y= W X X – inputs; Y – outputs, components; W – eigenvectors, eigenfaces, basis vectors

x1

w1

w2

x2

How many components?How many components? Usual choice consider the first d PC’s which account for some percentage,

usually above 90 %, of the cumulative variance of the data. This is disadvantageous if the last components are interesting

W2

W1

x1

x2

DatasetDataset

A subset of FERET dataset 2670 grey scale frontal face

images Rich in variety: face images

vary in pose, background lighting, presence or absence of glasses, slight change in expression

Property No.Categorie

s

Categories No. Face

s

Gender 2Male 1603

Female 1067

Ethnicity 3

Caucasian 1758

African 320

East Asian 363

Age 5

20 – 29 665

30 – 39 1264

40 – 49 429

50 – 59 206

60+ 106

Identity 358Individuals with 3

or more examples

1161

DatasetDataset

Each image is pre-processed to a 65 X 75 resolution. Aligned based on eye locations Cropped such that little or no hair information is available Histogram equalisation is applied to reduce lighting effects

Does PCA efficiently represents information in face Does PCA efficiently represents information in face images?images?

Images of 65 × 75 resolution leads to a dimensionality of 4875. The first 350 components accounted for 90% variance of the data. Each face is thus represented using 350 components instead of 4875

dimensions

Classification employing 5-fold cross validation, with 80% of faces in each category for training and 20% of faces in each category for testing

for identity recognition leave-one-out method is used. LDA is performed on the PCA data Euclidean measure is used for classification

Property Classification

Gender 86.4%

Ethnicity 81.6%

Age 91.5%

Identity 90%

What information does PCA encode? – GenderWhat information does PCA encode? – Gender

Gender encoding power estimated using the LDA 3rd component carries highest gender encoding power followed by the 4th components All important components are among the first 50 components

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

Components

Gen

der

En

cod

ing

Po

wer

What information does PCA encode? – GenderWhat information does PCA encode? – Gender

Reconstructed images from the altered components (a) third and (b) fourth components. The components are progressively added by quantities of -6 S.D (extreme left) to +6 S.D (extreme right) in steps of 2 S.D.

Third component encodes information related to the complexion, length of the nose, presence or absence of hair on the forehead, and texture around the mouth region.

Fourth component encodes information related to the eyebrow thickness, presence or absence of smiling expression

-6 SD -4 SD -2 SD Mean +2 SD +4 SD +6 SD

GenderGender

(a) Face examples with the first two being female and the next two being male faces. (b) Reconstructed faces of (a) using the top 20 gender important components. (c) Reconstructed faces of (a) using all components, except the top 20 gender important components.

What information does PCA encode? – EthnicityWhat information does PCA encode? – Ethnicity

6th component carries highest ethnicity encoding power followed by the 15 th components

All ethnicity important components are among the first 50 components

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10

Components

Eth

nic

ity

En

cod

ing

Po

wer

EthnicityEthnicity

Reconstructed images from the altered components (a) 6 th and (b) 4th components. The components are progressively added by quantities of -6 S.D (extreme left) to +6 S.D (extreme right) in steps of 2 S.D.

6th component encodes information related to complexion, broadness and length of the nose

15th component encodes information related to length of the nose, complexion, and presence or absence of smiling expression


What information does PCA encode? – AgeWhat information does PCA encode? – Age

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10

Components

Ag

e E

nco

din

g P

ow

er

Age – 20-39 and 50-60+ age groups termed as young and old)

10th component is found to be the most important for age

Reconstructed images from the altered tenth component. The component is progressively added by quantities of -6 S.D (extreme left) to +6 S.D (extreme right) in steps of 2 S.D


What information does PCA encode? – IdentityWhat information does PCA encode? – Identity

Many components are found to be important for identity. However, their importance magnitude is small.

These components are widely distributed and not restricted to the first 50 components

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Components

Iden

tity

En

cod

ing

Po

wer

Can a single component encode multiple properties?Can a single component encode multiple properties? A grey beard informs that the person is a male and also, most probably, old. As all important components of gender, ethnicity, and age are among the first 50

components there are overlapping components. One example is the 3rd component which is found to be the most important for

gender and second most important for age

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

Components

Gen

der

En

cod

ing

Po

wer

0 10 20 30 40 500

1

2

3

4

5

6

7

8

9

10

Components

Ag

e E

nco

din

g P

ow

er

Can a single component encode multiple properties?Can a single component encode multiple properties?

Normal distribution plots of the (a) third (b) and fourth components for male and female classes of young and old age groups.

-10 -5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Component 3

Young maleYoung femaleOld maleOld female

-10 -5 0 5 10 150

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Component 4

Young maleYoung femaleOld maleOld female

SummarySummary PCA encodes face image properties such as gender, ethnicity, age, and

identity efficiently. Very few components are required to encode properties such as

gender, ethnicity and age and these components are amongst the first few components which capture large part of the variance of the data. Large number of components are required to encode identity and these components are widely distributed.

There may be components which encode multiple properties.

Principal Component Analysis (PCA)Principal Component Analysis (PCA) PCA and classification

– PCA is not always an optimal dimensionality-reduction procedure for classification purposes.

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 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.

– More capable of distinguishing image variation due to identity from variation due to other sources such as illumination and expression.

Angiograph Image Enhancement Angiograph Image Enhancement

Webcamera Calibration Webcamera Calibration

QUESTIONSQUESTIONS

THANKSTHANKS

eigen value analysis in pattern recognition by dr. m. asmat ullah khan comsats institute of...

Documents

abbottabad slide

classification pca

goal of pca

principal components

high dimensional space

lowerdimensional subspace

lower dimensional subspace

eigen value analysis