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EFFECTS OF IMAGE R ESOLUTION ON FACE R ECOGNITION

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CONTENTS

ABSTRACT ................................ ...... 4

Chapter 1.Introduction ................................ ................................ ................................ ..................... 5

1.1.Face recognition ................................ ...... 6

1.1.1.Face Recognition Methods ............ 8

1.1.2.Face Recognition Robustness ............ 9

1.2.Resolution ................................ .... 11

1.2.1.Factors Affecting the performance of Face Recognition Systems .......... 12

1.2.2.Image Resolution Enhancement Techniques: .......... 12

1.2.3SUMMARIZING: .......... 13

Chapter 2.Face Image Resolution Analysis .............. ............. ..... ............ .............. ...... .............. ...... 14

2.1.Image Pyramids ................................ .... 14

2.2.Gaussian Pyramid ................................ .... 15

2.3.Bicubic Interpolation ................................ .... 18

2.3.1.Conventional Bicubic Interpolation .......... 19

Chapter 3.Principal Component Analysis ................................ ................................ ....................... 21

3.1.Details ................................ .... 22

3.2.Computing PCA Using The Covariance Method ................................ .... 24

3.2.1.Organize the data set .......... 24

3.2.2.Calculate the empirical mean .......... 24

3.2.3.Calculate the deviations from the mean .......... 25

3.2.4.Find the covariance matrix .......... 25

3.2.5.Find the eigenvectors and eigenvalues of the covariance matrix .......... 25

3.2.6.Rearrange the eigenvectors and eigenvalues .......... 26

3.2.7.Select a subset of the eigenvectors as basis vectors .......... 26

3.2.8.Convert the source data to z-scores .......... 27

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3.2.9.Project the z-scores of the data onto the new basis .......... 27

3.2.10.Derivation of PCA using the covariance method .......... 27

3.2.11.Computing principal components iteratively .......... 28

3.3.Summarizing: ................................ .... 29

3.3.1.Mathematics of PCA .......... 29

3.3.2.Eigenfaces in Face Recognition . .......... 31

3.3.3.Face Recognition using PCA .......... 33

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ACKNOWLEDGMENTS

I hope from Allah to success in discussion with the department staff. I would like

to exprsss my deep gratitude and thanks to my supervisor:

Dr / Mohamed Elsayed Ghoneim

For his valuable discussion with me guidance continuous encouragement and

providing many facilities through this work.

Special thanks to my parents for supporting me to complete this work.

Last but not least thanks for every one help thorough this work to emerge on this

way.

Mohamed Moneir El-Beidak

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ABSTR ACT

Images containing faces are essential to intelligent vision-based

human computer interaction, and research efforts in face processing

include face recognition, face tracking, pose estimation, and expression

recognition [1].To build fully automated systems that analyze the

information contained in face images, we require robust and efficientface detection algorithms. Researchers have proposed different

recognition methods under the various conditions such as different

pose, illumination and expression. The goal of face detection is to

identify all image regions which contain a face regardless of its three-

dimensional position, orientation, and lighting conditions. Such a

problem is challenging because faces are non stable and have a high

degree of variability in size, shape, color, and texture.

In this paper, we conjecture that the face recognition rate will level

off when face image resolution arrives at one certain resolution

threshold. We presents PCA based face recognition experiments and

show statistical results to depict how face recognition rate changes with

face image resolution. After analyzing this method and identifying its

limitations, we conclude with several promising directions for future

research.

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Chapter 2. INTRODUCTION

If you¶ve ever watched hi-tech spy movies, you¶ve most likely

seen biometric technology. Several movies have depicted biometric

technologies based on one or more of the following unique identifiers:

(keystroke - Face ± Fingerprint ± Handprint ± Iris- Retina ± Signature ±

Voice) They refers to authentication techniques that rely on measurable

physiological and individual characteristics that can be automatically

verified. Many forms of biometric systems exist for identification and

verification purposes; each has a different price range with associated

crossover error rates and user-acceptance levels.

Figure 2.1 Comparison of various biometric features:

(a) based on zephyr analysis [2]; (b) based on MRTD compatibility [3]

With the spreading of new information technology and media,

more effective and friendly methods for human computer interaction

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(HCI) are being developed which do not rely on traditional devices

such as keyboards, mice, and displays.

Face recognition is an important research problem spanning

numerous fields [

4

] because in additional to having many practicalapplications it is a fundamental human behavior that is essential for

effective communications and interactions among people . The rapidly

expanding research in face processing is based on the assumption that

information about a user¶s identity, state, and intent can be extracted

from images, and that computers can then react accordingly (e.g., by

observing a person¶s facial expression.) [5

] [6

] [7

].

FACE RECOGNITION

It is a popular topic in computer vision and object recognition. It has a

number of real-world applications such as surveillance, secure access and

human/computer interface, access control, security monitoring [8

].

Figure 2.2 Access Control System Based on Face Authentication Model

Recognize a person's identity is important to obtain quick access

to any type of records. Solving this problem is important because it

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could allow personnel to take preventive action, provide better service -

in the case of a doctors appointment, or allow a person access to a

secure area. Also, many methods successfully applied in face

recognition can be eventually transferred to other object recognition problems. Face detection from a single image is a challenging task

because of variability in scale, location, orientation (up-right, rotated),

and pose (frontal, profile). Facial expression, occlusion, and lighting

conditions also change the overall appearance of faces. We now give a

definition of face detection: the goal of face detection is to determine whether

or not there are any faces in the image and, if present, return the image locationand extent of each face.

The challenges associated with face recognition can be attributed

to the following factors: [9][

10]

y Pose: The images of a face vary due to the relative camera-face

pose (frontal, 45 degree, profile, upside down).

y Presence or absence of structural components: Facial features

such as mustaches, and glasses may or may not be present.

y Facial expression. The appearance of faces are directly affected

by a person¶s facial expression.

y Occlusion. Faces may be partially occluded by other objects

y Imaging conditions. When the image is formed, factors such as

lighting (spectra, source distribution and intensity) and camera

characteristics (sensor response, lenses) affect the appearance of a

face.

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F IM¢ £

E R ESOLUTION ON FACE R ECOGNITION

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2.0.1. Face Recogni ion Methods

Face recognition techni es are di ided roughl into t o categor ies:

a) t l al approach

b) t he component- ased approach.

The input to a classi ier in t he g l obal approach is a single feature

vector that represents the whole face i age. The classif ier is minimum

distance classif ication in the Eigen-space based on PCA [11][12] " a shown

in chapt er 3 " , This glo bal techni ue work well for classifying frontal

views of face. The correspondences between two face images are f irst built by la beling some key points and then an aff ine transform is

computed to warp an input image into a reference face image [13]. Ac-

tive morpha ble model is adopted to match the input face with the

reference face [14]. Active shape models are used in to align the input

face with the model face[15].

Figure 2.3 2D Facial Scanners Record Identities through Recognizing Facial Features

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While for the component based algorithms, the main idea is to

compensate for pose changes by allowing a flexible geometrical

relation between the components in the classification stage. In [16

] face

recognition was performed by independently matching templates of three facial regions (eyes, nose and mouth). The configuration of the

components during classification was unstrained since the system did

not include a geometrical model of the face.

2.0.2. Face Recognition Robustness

The general task of face recognition still poses a number of chal-

lenges with respect to the changes in illumination, facial expression

and pose. Therefore currently researchers pay more attention to the

study of the robustness against the changes in pose, illumination and

expression.

Bernd H eisele et al. [

17

] presented a component based and twoglobal recognition methods with multi-class support vector classifier

and evaluated them with respect to the robustness against the pose

changes. T akeshi Shakunaga and Kaz-uma Shigenari proposed a

decomposed eigenface based face recognition method for realizing

robust face recognition under various lighting conditions when too few

images are available for registration [18

]. G eof G ivens [19

] and his

collaborators considered 11 factors that influence PCA-based face

recognition performance including race, gender, age, glasses, facial

hair, bangs, mouth, eyes, complexion, makeup and expression . They

built a system to analyze the relation between these subject covariates

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and the distance of the image pairs of the same subject. Some

researchers are also concerned about how to optimize some specific

face recognition methods, e.g. W endy et al. [20

] analyzed PCA-based

face recognition algorithms and studied the relation betweenrecognition performance and the selection of eigenvector and distance

measures. When studying face recognition, researchers all run across

one problem: what resolution face image is proper for face

recognition. Fortunately, some researchers have done some related

work. Simon Baker et al. thought that the enhanced information of the

high resolution over the low resolution face could been decided by the

low resolution face and built the corresponding face hallucination

algorithm [21

]. C e Liu et al. [22

] first constructed a "recognized" global

model to the individual global face, then built a local model to enhance

the local face feature. Motivated by their work, we divide face imag e

information into two kinds of information: the discriminative

information & the structure information. The former represents the

individual information compared with other face images, the latter is

the common information of all face images under the same resolution.

Then one conjecture is given that face recognition rate will level

off when the face image resolution arrives at one certain value. Finally,

we perform PCA based recognition algorithms on the face database.

The relation curves between face recognition rate and face resolution

validate that the recognition rate will not increase until the face

resolution arrives at some certain value.

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R ESOLUTION

Resolution can be defined as " the ability of an imaging system to

record fine details in a distinguishable manner ".

The size of an image is measured in inches or centimeters [23

] . In

a digital device, like a monitor or a camera, the size of an image is

measured in pixels. Pixels are the smallest basic units that compose a

digital image. In fact, the term pixel is an abbreviation from " picture

element ". So, a lot of small pixels put together make up an image.

Obviously, the more pixels an image has, the more resolution it has,

and thus, more detail that can be seen. For instance, the average 14

inch monitor has an 11 inches wide x 8 inches high screen

(approximately). When configured to display 72 ppi or " pixels per

inch", it creates 800 pixels wide x 600 pixels high images on screen. A

6" by 4" photograph scanned at 300 ppi will generate 1800 pixels on

the wide side and 1200 pixels on the high side. An image shot with a 2

megapixel (MP) camera will usually have 1600 pixels wide x 1200

pixels high, making up 1.92 million pixels in total, or approximately "2

MP". Higher resolution images can let you crop in on part of an image

and blow it up, and still get a good definition. In digital terms, each

pixel is simply a piece of information regarding the specific color and

brightness of that particular dot. For each pixel, this information is

contained in three bytes representing each one these the particular

shade of Red, Green, and Blue (RGB) that combined together make up

the specific color and brightness of that pixel. Each RGB component or

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byte is represented by 8 bits , so it can have 256 possible values

ranging from 0 to 255 . T hat means that 24 bit RGB color image pixels

are capable of displaying 16.7 million different color combinations

(256x256x256). This is also important because pixel quantitydetermines computer file sizes (some file formats such as JPEG allow

for increasing levels of compression, and thus, decreasing file sizes).

2.1.1. Factors Affecting the performance of Face

Recognition Systems

Considerable progress has been made in face recognition research

over the last Decade, especially with the development of powerful

models of face appearance (e.g., eigen spaces). Despite the variety of

approaches and tools studied, however, face recognition has shown to

perform satisfactorily in controlled environments, but it is not accurate

or robust enough to be deployed in uncontrolled environments.

Illumination variation is one of the critical factors affecting face

recognition rate which depend on capture device physical properties

(e.g. resolution and contrast).

2.1.2. Image Resolution Enhancement Techniques:

Depending on the presence of anti-aliasing filter, there are two

ways of formulating the resolution enhancement problem for still

images, that is, how to obtain a high-resolution (HR) image from its

low-resolution (LR) version? When no anti-aliasing filter is used, we

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might use classical linear interpolation [24

] , edge-sensitive filter [25

] ,

directional interpolation [26

] , POCS-based interpolation [27

] , or edge

directed interpolation schemes [28

][29

] . When anti-aliasing filter takes

the form of low-pass filter in wavelet transforms (WT) [

30

] , there are aflurry of works [

31][

32] which transform the problem of resolution

enhancement in the spatial domain to the problem of high-band

extrapolation in the wavelet space.

2.1.3. SUMMARIZING:

High image quality means high resolution, to achieve that we need

high quality camera which is more expensive and then we need large

storage space. But if we know the threshold of the resolution needed to

implement the PCA algorithm we shall reduce cost for both storage

space and camera quality.

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Chapter 3. FACE IMAGE R ESOLUTION ANALYSIS

In this section, we first introduce Image Pyramids, the generalized

Gaussian Pyramid, which is used to generate the multi-resolution face

images, and bicubic interpolation. Then, we analyze the face image

information constitution and conjecture the relation between face

recognition performance and face image resolution.

IMAGE PYR AMIDS

Goal: Develop filter-based representations to decompose images

into information at multiple scales, to extract features/structures of

interest, and to attenuate noise. [33

]

Motivation:

y redundancy reduction and image modeling for efficient coding.

y image enhancement/restoration.

y image analysis/synthesis.

Linear Transform Framework

P rojection Vectors: Let denote a 1D signal, or a vectorized

representation of an image (so

), and let the transform be

= (1)

Here,

y are the transform coefficients.

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y The columns of P = are the projection

vectors; the coefficient, is the inner product

.y When P is complex-valued, we should replace

by the

conjugate transpose .

Sam pling : The transform is said to be critically

sampled when , Otherwise it is over-sampled(when ) or

under-sampled (when ).

Basis Vectors: For many transforms of interest there is a

corresponding basis matrix B satisfying

=

.

The columns B = are called basis vectors as

they form a linear basis for

GAUSSIAN PYR AMID

It [34

]is a technique used in image processing. The technique

involves creating a series of images which are weighted down using a

Gaussian average (blur) and scaled down.

Figure 3.1 Levels of Gaussian Pyramid

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When this techni ue is used multi ple times, it creates a stack of

successively smaller images, with each pi el containing a local average

that corresponds to a pi el neigh borhood on a lower level of the

pyramid.

Sequence of low-pass, down-sampled images, .

Usually constructed with a separa ble 1D kernel ,

and a down-sampling factor of 2 (in each direction):

In matr i notation (for 1D) the mapping from one level to the next

has the form:

Typical weights for the impulse response from binomial weights

convol ti on i s a mat hematical operation on two f nctions f and

g, producing a t hird f unction t hat i s t picall viewed as a mod ified

version o f one o f t he or i g inal f unctions.

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3.2 Example of image and next four pyramid levels:

Figure 3.3 First three levels scaled to be the same si e:

Properties of Gaussian pyramid:

y Used for multi-scale edge estimation.

y Efficient to compute coarse scale images. Only 5-tap 1D

filter kernels are used.

y Highly redundant, coarse scales provide much of the

information in the finer scales.

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3.2.1. Conventional Bicubic Interpolation

Figure 3.5 Original Pic«

The conventional bicu bic interpolation needs an up-sampling

distance S to estimate the unknown pixels for the interpolation

processing. At the position which is shown :

Figure 3.7 Photo was enlarge

without using any interpolatio

3.6 Photo was enlarged using

interpolation

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the bicu bic interpolation calculates the interpolated pixel as equ:

where and means the pixel value at

the position (i, j).

The weights (S), (S), (S), (S) in conventional

bicu bic interpolation are given as

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Chapter 4. PRINCIPAL COMPONENT ANALYSIS

The Principal Component Analysis (PCA) is one of the most

successful techniques that have been used in image recognition and

compression. PCA was invented by Karl Pearson in 1901 [37

] . PCA is

a statistical method under the broad title of factor analysis. T he

purpose of PCA is to reduce the large dimensionality of the data space

(observed variables) to the smaller intrinsic dimensionality of feature

space (independent variables), which are needed to describe the data

economically. The jobs which PCA can do are prediction, redundancy

removal, feature extraction, data compression, etc. PCA is a classical

technique which can do something in the linear domain, applications

having suitable linear models, such as signal processing, image

processing, system and control theory, communications, and so on.

The main idea of using PCA for face recognition is to express the

large 1-D vector of pixels constructed from 2-D facial image into the

compact principal components of the feature space. This can be called

eigenspace projection. Eigenspace is calculated by identifying the

eigenvectors of the covariance matrix derived from a set of facial

images(vectors). PCA involves a mathematical procedure that

transforms a number of possibly correlated variables into a number of

uncorrelated variables called principal components.

PCA can be done by eigenvalue decomposition of a data

covariance matrix or singular value decomposition of a data matrix, it

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is the simplest of the true eigenvector based multivariate analyses.

Often, its operation can be thought of as revealing the internal structure

of the data in a way which best explains the variance in the data.

DETAILS

PCA is mathematically defined [38

] as an orthogonal linear

transformation that transforms the data to a new coordinate system

[such that the greatest variance by any projection of the data comes to

lie on the first coordinate called the first principal component, the

second greatest variance on the second coordinate, and so on]

Define a data matrix, , with zero empiricalwhere each of :

y the n rows represents a different repetition of the experiment.

y the m columns gives a particular kind of datum.

The PCA transformation is then given by:

where the matrices W, , and V are given by a singular valuedecomposition (SVD) of X as W

. is an mn diagonal matrix

with nonnegative real numbers on the diagonal. Since W (by definition

of the SVD of a real matrix) is an orthogonal matrix, each row of is

simply a rotation of the corresponding row of .

If we want a reduced dimensionality representation, we can

project X down into the reduced space defined by only the first L

singular vectors, :

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The matrix W of singular vectors of X is equivalently the matrix

W of eigenvectors of the matrix of observed covariance ,

, given a set of points in Euclidean space.

Each eigenvalue is proportional to the portion of the "variance"

(more correctly of the sum of the squared distances of the points from

their multidimensional mean) that is correlated with each eigenvector,

so the sum of all the eigenvalues is equal to the sum of the squared

distances of the points from their multidimensional mean.

Mean subtraction is necessary for performing PCA to ensure that

the first principal component describes the direction of maximum

variance. If mean subtraction is not performed, the first principal

component might instead correspond more or less to the mean of the

data. A mean of zero is needed for finding a basis that minimizes the

mean square error of the approximation of the data [39

] . Assuming zero

empirical mean (the empirical mean of the distribution has been

subtracted from the data set), the principal component w1 of a data set

X can be defined as:

The

component can be found by subtracting the first

principal components from X:

and by substituting this as the new data set to find a principal

component in

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COMPUTING PCA USING THE COVARIANCE METHOD

The goal is to transform a given data set X of dimension M to an

alternative data set Y of smaller dimension

. Equivalently, we areseeking to find the matrix Y, where Y is the Karhunen±Loève

transform (KLT) of matrix X:

4.1.1. Organi e the data set

Suppose you have data comprising a set of observations of M

variables, and you want to reduce the data so that each observation can

be described with only L variables, L < M . Suppose further, that the

data are arranged as a set of N data vectors with each

representing a single grouped observation of the M variables.

Write as column vectors, each of which has M rows.

Place the column vectors into a single matrix X of dimensions M × N .

4.1.2. Calculate the empirical mean

Find the empirical mean along each dimension Place the calculated mean values into an empirical mean vector u of

dimensions M × 1,

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4.1.3. Calculate the deviations from the mean

Mean subtraction is an integral part of the solution towards finding

a principal component basis that minimizes the mean square error of

approximating the data. Hence we proceed by centering the data as

follows:

Subtract the empirical mean vector u from each column of the data

matrix X.

Store mean-subtracted data in the M × N matrix B :

4.1.4. Find the covariance matrix

Find the M × M empirical covariance matrix C from the outer

product of matrix B with itself:

where

The covariance matrix in PCA, is a sum of outer products between

its sample vectors, indeed it could be represented as B.B*.

4.1.5. Find the eigenvectors and eigenvalues of the

covariance matrix

Compute the matrix V of eigenvectors which diagonalizes the

covariance matrix C: where D is the diagonal matrix of

is the expected value operator.

is the outer product operator.

is the conjugate transpose operator.

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eigenvalues of C. T his step will typically involve the use of a computer

based algorithm for computing eigenvectors and eigenvalues. Matrix D

will take the form of an M × M diagonal matrix, where

for is the

eigenvalue of the covariance matrix C, and for . Matrix V, also of dimension M × M , contains M

column vectors, each of length M , which represent the M eigenvectors

of the covariance matrix C. The eigenvalues and eigenvectors are

ordered and paired. The eigenvalue corresponds to the

eigenvector.

4.1.6. Rearrange the eigenvectors and eigenvalues

o Sort the columns of the eigenvector matrix V and eigenvalue

matrix D in order of decreasing eigenvalue.

o Make sure to maintain the correct pairings between the

columns in each matrix.

4.1.7. Select a subset of the eigenvectors as basis vectors

y Save the first L columns of V as the M × L matrix W:

for p=1,«, M and q=1,«,L where .

y Use the vector g as a guide in choosing an appropriate value for L.

T he goal is to choose a value of L as small as possible while

achieving a reasonably high value of g on a percentage basis.

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4.1.8. Convert the source data to -scores

y Create an M × 1 empirical standard deviation vector s from the square

root of each element along the main diagonal of the cova riance matrix

C: for .

y Calculate the M × N z-score matrix: (divide element-by-

element)

4.1.9. Project the -scores of the data onto the new basis

y The projected vectors are the columns of the matrix

. y W* is the conjugate transpose of the eigenvector matrix.

y The columns of matrix Y represent the K arhunen±Loève

transforms (KLT) of the data vectors in the columns of

matrix X.

4.1.10. Derivation of PCA using the covariance method

Let X be a d -dimensional random vector expressed as column

vector. Without loss of generality, assume X has zero mean. We want

to find a orthonormal transformation matrix P such that

with the constraint that�� is a diagonal matrix and . By

substitution, and matrix algebra, we obtain:

��

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��

We now have: ��

��

R ewr ite P as column vectors, so

and ��as:

Su bstituting into equation a bove, we o btain:

Notice that in , P i is an eigenvector of the covar iance

matr ix of X. Therefore, by f inding the eigenvectors of the covar iance

matr ix of X, we f ind a pro jection matr ix P that satisf ies the or iginal

constraints.

4.1.11. Computing principal components iteratively

In practical implementations especially with high dimensional data

(large m), the covar iance method is rarely used because it is not

eff icient. One way to compute the f irst pr inci pal component eff iciently

[40

] is shown in the following pseudo-code, for a data matr ix XT

with

zero mean, without ever computing its covar iance matr ix. Note that

here a zero mean data matr ix means that the columns of XT

should

each have zero mean.

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--------------------------------------------------------------------------------------

P = a random vector do c times:

T =0 (a vector of length m) for each row

t = t +(x.p)x

return P

--------------------------------------------------------------------------------------

This algorithm is simply an efficient way of calculating .

One way to compute the eigenvalue that corresponds with each

principal component is to measure the difference in sum-squared-

distance between the rows and the mean, before and after subtracting

out the principal component. The eigenvalue that corresponds with the

component that was removed is equal to this difference.

SUMMARIZING:

4.2.1. Mathematics of PCA

A 2-D facial image can be represented [41

] as 1-D vector by

concatenating each row (or column) into a long thin vector. Suppose

we have M vectors of size N (= rows of image columns of image)

representing a set of sampled images. 's represent the pixel values.

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The images are mean centered by subtracting the mean image from

each image vector. Let m represent the mean image. and

let be defined as mean centered image , our goal is to

find a set of ¶ which have the largest possible projection onto each

of the 's. We wish to find a set of M orthonormal vectors for

which the quantity

is maximized with the

orthonormality constraint . It has been shown that the ¶s

and ¸i¶s are given by the eigenvectors and eigenvalues of the

covariance matrix

, where W is a matrix composed of the

column vectors placed side by side. The size of C is which

could be enormous. A common theorem in linear algebra states that

the vectors and scalars ¸ can be obtained by solving for the

eigenvectors and eigenvalues of the matrix . Let and

be the eigenvectors and eigenvalues of , respectively.

By multiplying left to both sides by W .

which means that the first

eigenvectors and eigenvalues ¸ of are given by W and ,

respectively. W needs to be normalized in order to be equal to .

The eigenvectors are sorted from high to low according to their

corresponding eigenvalues. T he eigenvector associated with the largest

eigenvalue is one that reflects the greatest variance in the image , that

is, the smallest eigenvalue is associated with the eigenvector that finds

the least variance. A facial image can be projected onto ( M )

dimensions by computing where = , is

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the coordinate of the facial image in the new space, which came to

be the principal component. The vectors are also images, so called,

eigenimages, or eigenfaces [42

] .

They can be viewed as images and indeed look like faces. So,

describes the contribution of each eigenface in representing the facial

image by treating the eigenfaces as a basis set for facial images. T he

simplest method for determining which face class provides the best

description of an input facial image is to find the face class k that

minimizes the Euclidean distance .

4.2.2. Eigenfaces in Face Recognition .

Eigenface is one of the most thoroughly investigated approaches to

face recognition. It is also known as K arhunen Loève [43

] expansion. Steps [

44]

1) Obtain a set S with M face images. Each image is

transformed into a vector of size N and placed into the set.

2) After you have obtained your set, you will obtain the mean

image

3) Then you will find the difference between the input image

and the mean image 4) Next we seek a set of M orthonormal vectors, un, which best

describes the distribution of the data. The k th

vector, uk , is

chosen such that

is a maximum,subject to

N ote: and are the eigenvectors and eigenvalues of the covariance

matrix .

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5) We obtain the covariance matrix C in the following manner

,

6)

7) Once we have found the eigenvectors, v l, ul:

,

4.1 set of images used to create eigen space

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4.2 the eigenfaces of our set of original images

4.2.3. Face Recognition using PCA

Once the eigenfaces [45

] have been computed, several types of

decision can be made depending on the application. What we call face

recognition is a broad term which may be further specified to one of

following tasks:

o Identification where the labels of individuals must be obtained .

o Recognition of a person, where it must be decided if the individual has

already been seen.

o Categorization where the face must be assigned to a certain class.

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Figure 4.3 Using PCA in Face Recognition Systems

PCA computes the basis of a space which is represented by its

training vectors. These basis vectors, actually eigenvectors, computed

by PCA are in the direction of the largest variance of the training

vectors. As it has been said earlier, we call them eigenfaces. Each

eigenface can be viewed a feature. When a particular face is projected

onto the face space, its vector into the face space describe the

importance of each of those features in the face. The face is expressed

in the face space by its eigenface coefficients (or weights). Each face in

the training set is transformed into the face space and its components

are stored in memory. The face space has to be populated with these

known faces. An input face is given to the system, and then it is

projected onto the face space. The system computes its distance from

all the stored faces.

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However, two issues should be carefully considered:

1. What if the image presented to the system is not a face?

2. What if the face presented to the system has not already

learned, i.e., not stored as a known face?

The first defect is easily avoided since the first eigenface is a good

face filter which can test whether each image is highly correlated with

itself. The images with a low correlation can be rejected. Or these two

issues are altogether addressed by categorizing following four different

regions:

1. Near face space and near stored face known faces

2. Near face space but not near a known face unknown faces

3. Distant from face space and near a face class non-faces

4. Distant from face space and not near a known class non-faces

Since a face is well represented by the face space, its reconstruction

should be similar to the original, hence the reconstruction error will be

small. Non-face images will have a large reconstruction error which is

larger than some threshold . The distance determines whether the

input face is near a known face.

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