face recognition & biometric systems support vector machines (part 2)

Face Recognition & Biometric Systems

Support Vector Machines (part 2)


Plan of the lecture

SVM – main issues repeatedSoft marginMulti-class problemsApplications to face recognitionTraining set optimization


SVM – main issues

Aim: data classificationTwo stages: learning (training) classification


SVM – main issues

Solves linearly separable problemsInput data are transformed mapping into higher dimensions

Training: find optimal hyperplane margin maximisation


SVM – main issues

A function:Data mapping: x(x)Dot product used in all calculationsDot product -> kernel of convolution

No need to know the function

Mn RR :

)()(),( vuvuK

nM


Convolution kernels

Linear

Polynomial

RBF (radial basis functions)

2

2||

),( vu

vu

eK

vuvu ),(K

dK )1(),( vuvu


SVM – main issues

Optimal hyperplane:w0 • x + b0 = 0

for 2D data it is a line

Optimal margin width:

00000

2

||

2

www),bρ(w


SVM – main issues

Optimal hyperplane:

yi – class label i – Lagrange multipliers

(obtained during optimisation)

l

iiii xyw

1

00


SVM – main issues

Lagrange coefficients (): calculated for every vector from the

training set non-zero for support vectors equal zero for the majority of vectors

Training set after the optimisation: support vectors coefficients for every vector number of vectors reduced


Training 1

...n


SVM – main issues

Classification of a vector:

xr, xs – support vectors from opposite classes

bxxKyxfl

iiii

1),()(

l

isiriii xxKxxKyb

1)],(),([

2

1


Soft margin

Error allowed during the training:

Number of errors minimisedOptimised function must be modified

iii bxwy 1)( 0i


Soft margin

Margin maximisationMinimisation of functional (F – monotonic, convex function):

C – penalty parameter presentation

Constraints:

l

iiCF

1

2

2

1 w

iii bxwy 1)( 0i


Soft margin

Optimisation without the soft margin:

DW TT

2

11)(

),( jijiij xxKyyD

),...,( 1 lT

0Λ 0YΛT ),...,( 1 lT yyY


Soft margin

Optimisation with the soft margin (for F(u) = u2):

)(2

11),(

2

CDW TT

),( jijiij xxKyyD

),...,( 1 lT

0YΛT ),...,( 1 lT yyY10 0


Multi-class problem

Example


Multi-class problem

Based on two-class problem solved by the SVM

N classes in the training setPossible solutions: base-class approach 1 – N comparisons 1 – 1 comparisons


The base-class approach

The base-class approach one class selected as a base class each class compared with the base

class the strongest response decides

Classification of a single vector: (N – 1) two-class classifications



Advantages: high speed effective when non-base classes are

easily separable

Disadvantages: problems with separating

non-base classes


1 – N comparisons

Each class compared with the restThe strongest response decidesClassification of a single vector: N two-class classifications

Compared to the base-class approach: more universal (symmetry) comparable speed


1 – N comparisons


1 – 1 comparisons

Each class compared with each otherThe highest precisionClassification of a single vector: N(N – 1)/2 two-class classifications

Very slow method


SVM for face recognition

Detection and verificationFeature vectors comparisonMulti-method fusionOther applications


Face detection

Ellipse detection Generalised Hough Transform a set of face candidates

Normalisation of the candidatesVerification image (as a vector) classified by the

SVM multi-class approach


Feature vectors comparison

Aim: measure similarity between feature vectorsDistance-based similarity: Euclidean distance Mahalanobis distance

Similarity measured by the SVM: two vectors subtracted from each

other create a difference vector difference vector classified

K11

K12

K1n

...

K21

K22

K2n

...


SVM

The sameclass

Differentclasses

K11 - K21

...

K12 - K22

K1n - K2n




Good improvement for EBGMEigenfaces not improved similar results to other metrics


Multi-method fusion

Many feature extraction methods

S1

S2

Sn

... S

K1

K2

Kn

...

Two images Feature vectors Similarities

K1

K2

Kn

...


Multi-method fusion

Vector of similarities classified linear kernel polynomial kernel time-consuming classification

SVM applied only for the training linear kernel – weights for the

methods (dimensions stand for methods)

average mean based on the weights


Other applications

Assessment of recognition accuracy vector of sorted similarities

to the elements in the gallery can be used for many images

of the same person

Image quality estimation e.g. elimination of blurred images


SVM – limitations

Constant and small number of classes too much time-consuming for many

classes

Training set: must be representative optimal number of elements

The parameters must be tuned Relevance Vector Machines


Training set optimization

Representative training set: similarity to the classified data universal classification rules difficult to acquire

Real training sets: data acquired automatically low quality, faulty data large number of data



Selection of available data subset drawn randomly genetic algorithms

Genetic algorithms heuristic optimization technique evolutional strategy population of individuals fitness genetic operators:

selection mutation crossover



Population drawn

Effectiveness test

Population of training sets

Evolutional operations

Class + Class –

N elements N elements

Individual

+ –

Individual

SVM training

Effectiveness test

Fittness


SVM compared to ANN

Support Vector Machines: more transparent calculations more controllable than neural networks implements various types of ANN useful for image processing

Artificial Neural Networks: more applications (e.g. compression) dedicated hardware solutions


Summary

Soft margin – automatic selectionMulti-class problems: can be solved basing on two-class

problems various approaches

Many possible applications in the area of face recognitionTraining set optimization


Thank you for your attention!

Next time:

Elastic Bunch Graph Matching

face recognition & biometric systems support vector machines (part 2)

Documents

baseclass approach slide

optimisation slide

function slide

nn slide

training classification

baseclass approach advantages

class classifications

lecture svm main issues