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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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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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