Face Recognition

Sumitha Balasuriya

Computer Vision

• Image processing is a precursor to Computer Vision – making a computer understand and interpret what’s in an image or video.

Recognition

3D shape

Tracking

SegmentationCategorisation / Retrieval

Robotics

Innate Face Recognition Ability

• Face recognition almost instantaneous• Highly invariant to pose, scale, rotation, lighting

changes• Can handle partial occasions and changes in age • And we can do all this for faces of several

thousand individuals

Who is this?

Fully Automated Face Recognition

Face Detection

Colour

Motion

Template

Fiduciary points

Face Detection RecognitionImage

/ VideoIdentity

Face Detection

Detect face and facial landmarks

Register (align) face using an image transform

Australian National University

Face Recognition approaches

Geometric features Recognition possible at low resolution and at

high noise levels because geometrical features such as nose width and eye separation are used

Automated extraction of facial geometric features is very hard

Template matching Extract facial regions (matrix of pixels) and

compare with that of known individuals Need templates for different face poses/views Very high dimensional data

High Dimensional Correlated Data• Images as a high dimensional vector

• A typical image used for image processing will be 512x512= 262144 dimension vector!

• (Registered) face images are highly correlated

Image space (high dimensional space of all possible images)

Face images

I1,1

I1,2

I … .

Transform Face images to a ‘Face Space’

• The basis vectors (Ui) of facespace are the Principal Components of face images.

• These Principal Components (PCs) correspond to the directions of greatest variation of the face dataset.

• The first PC corresponds to the direction (and scale) of greatest variation, the 2nd PC corresponds to the orthogonal direction with the second greatest variation and so on …

• Embedding a face image vector into face space is done by calculating the vector’s projection onto the Principal Components.

Face space (low dimensional space of face images)

U1

U2

U3

Face images

f = U * (I - A)

Embedded face Eigenfaces

Mean face image

Face image Change dimensionality of f by changing that of U

Principal Component Analysis

• PCA – find the Principal Components of a dataset• Finding the Eigenvectors and eigenvalues of the covariance matrix (Eigen decomposition)• The covariance matrix contains the relationships (correlations) between the variables of the vectors. Variances along the diagonals and covariance between each pair of variables in other positions

(Karhunen-Loeve transform)

, j ji j i i XX

PCA (continued)

*Tc X X

For images will run out of memory because for a dataset of say 512x512 images X*XT will be a 262144x262144 matrix. If it has double precision (64bit) values C will be 512GB!!!

Covariance matrix for images

Therefore for image we instead compute the reduced covariance matrix

* TX X

If we have n datapoints c will be nxn. As n is normally much smaller than the dimensionality of the images the matrix c is computable

Find the Eigenvectors and eigenvalues of c, solve the following linear equations

cV VEigenvalues (degree of variation)Eigenvectors

(Principal Compoments!!)

Reducedcovariancematrix

>> [V,lamda]=eig(c);

You can get Matlab to solve equations to give the eigenvectors and eigenvalues

PCA (continued)

Eigenvectors (Eigenfaces) of covariance matirx

*U X V

Another way to compute PCA – Singular Value Decomposition method

SVD decomposes an mxn matrix X to a product

X = U * S * VT

Don’t need to compute covariance matrix

Diagonal matrix with eigenvalues along diagonal

Right singular vectors (eigenvectors of XT*X)

Left singular vectors (eigenvectors of X*XT)

You can get Matlab to solve these equations as well to get U, S and V

>> [U,S,V]=svd(X);

Use eigen decomposition with the reduced covariance matrix if you run out of memory with svd(seems to work upto 100x100 images)

High eigenvalue eigenfaces seem to capture lighting and other global intensity changes

Eig

enfa

ces

1 to

9Average face (data not mean subtracted!)

Eig

enfa

ces

10 to

18

Lower eigenvalue eigenfaces are better for recognition

Transform into ‘Face Space’

• Projection f = U * (I - A)

U1

U2

U3

Transform known faces to face space

Face space

ReconstructionF = UT * U * (I - A)

Mean subtracted imageTransform to face space

Transform back to image space

Can know whether image contains a face by calculating reconstruction error

Face detection method!

Very high compression!

Face space with 28 eigenvectors

100x100x8bits = 10000 bytes compress to 28*32 bits=112 bytes

Only compress faces

Classifiers

Classify an unknown face into one of the known face classes

Important concepts

• Inter-class (between-class) variance

• Intra-class (within-class) variance Need a metric to measure distance for class

membership

?

Metric

A non-negative function g(x,y) describing the "distance" between neighbouring points for a given set and satisfies

(1) g(x,y)+g(y,z)≥g(x,z) (triangle inequality)

(2) g(x,y)=g(y,x) (symmetry)

(3) g(x,x)=0, as well as g(x,y)=0 x=y

Euclidean distance (~300BC)

Mahalanobis distance (1936)Superior to Euclidean distance because it takes distribution of the points (correlations) into account

The distance between two N dimensional points scaled by the statistical variation in each component of the point.

2

1

( , )n

i ii

d a b a b

T 1( , )d a b a b a b

Decision surface

Mahalanobis classifier

Need many representative examples from each class to compute covariance matrix

1

T( , )known

d unknown known unknown known unknown known

Recap

• Image processing precursor to Computer Vision• Face Detection• Images are high dimensional correlated data• Principal Component Analysis

– Eigen decomposition– Singular Value Decomposition

• Eigenfaces for recognition, compression, face detection• Classifiers and Metrics

Questions

• Will performance improve if we used orientated edge responses of the face image for recognition? Why?

• A popular approach uses orientated responses only from certain specific points on the face image (say edge of nose, sides of eyes). What are the advantages/disadvantages of this approach?

• PCA maximises the variance between vectors embedded in the eigenspace. What does LDA (Linear Discriminant Analysis) do?