PCA_FACE_RECOGNITION_REPORT

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FaceRecognition

A Report on Face Recognition Using

Principal Component Analysis.

Final Report

Preparedfor:

Submission of Assignment work(3)on Ap plied M athematics

Preparedby:

Darshan Venkatrayappa

[email protected]

Sharib Ali

[email protected]

Submitted to

Desire [email protected]

17th

November 20
mailto:[email protected]:[email protected]:[email protected]


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Contents

Acronyms .................................................................................................................. 1

Chapter 1 Introduction to Face Recognition Using PCA

1.1 Background ........................................................................................ 11.2 Objective............................................................................................. 2

1.3 Problem Statement............................................................................. 2

1.4 Stages in Face Recognition

Chapter 2 Normalization

2.1 Why Normalization of images?....................................................... 32.1.1 Flow Chart and Algorithm of Normalization 3-5

2.1.2 Algorithm for Mapping of the image to 64x64 window... 5-6

2.2 Limitations... .................................................................. 6

Chapter 3 Eigen Faces and Eigen Space

3.1 What does they mean?........................................................................ 7

3.2 Algorithm

..... 7-9

Chapter 4 Recognition of the Face

4.1 Analysis............................................................................................... 104.2 Algorithm .................................................. ......................................... 10

4.3 Result4.3.1 Accuracy 10

4.3.2 Limitation... 10

4.3.3 Scope of Improvement.. 11

Chapter 5 Conclusions

... 11

REFERENCES................................................................................................................ R-1


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Acronyms

SVD = Singular Value Decomposition

PCA = Principal Component Analysis

EV = Eigen Vector

Chapter 1.Introduction to PCA in Face

Recognition

1.1 BackgroundThe Face Recognition system has various potential applications, such as

person identification.

human-computer interaction.

security systems.

The history of it goes back to the start of computer vision. Several other approach like irish,

fingerprint etc has been used in the application but however this has been used more widely

as the research. Face recognition has always remain a major focus of research because of its

non-invasive nature and because it is people's primary method of person identification. The

most famous early example of a face recognition system is due to Kohonen. Kohonen'ssystem was not a practical success, however, because of the need for precise alignment and

normalization.

In Principal Component Analysis for the face recognition, we train the faces and create a

database of the trained sample images of the person. We then find the covariance matrix from

this tra ined set. We get the eigen faces which corresponds to the principal components in the

eigen vector obtained. This will give an eigen facewhich is ghost like in appearance. Now,

each face in the training set is the linear combination of the eigen vectors. Now, when we

take a test image for the recognition, we follow the same normalization steps and then project

the test image to the same eigen space containing the eigen faces. Finally we calculate theminimum Euclidean distance between the eigen faces in the trained set of images with the

eigen face of the test image. This is explained in the further discussions.

The main 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. This is the case here is a strong

correlation between observed variables.


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1.2 Objective

i. To Study and implement the concept of SVD and PCA.

ii. To decrease the dimension of the feature space

iii. To make the computation fast

iv. To built a strong correlation between observed variables.

v. To see how the Eigen vectors gives the principal components of the image to be

recognized from a trained set of images.

vi. To make effort towards improvement of the result obtained.

1.3 Problem Statement :

Given an image, to identify it as a face and/or extract face images from it.

To retrieve the similar images (based on a PCA) from the given database of trained face

images.

1.4 Stages In Face Recognition

Training of the Images

(to create database)

Eigen Face using PCA

(eigen space)

Test Face Location

Identification


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Chapter 2. Normalization

2.1 Why Normalization required?

The normalization steps are done for the sacling, orientation and location variation

adjustment to all the images with some predefined features and the feature of images. Basically,

all the images are mapped to a window 64x64(in our case) taking some of the important facial

features. In our case, we have taken, 1. Left Eye Center 2. Right Eye Center 3. Tip of Nose 4.

Left Mouth Corner and 5. Right mouth corner.

2.1.1 Flow chart and Algorithm for Normalization

1


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Fig. Flow Chart to get the Convergence FBAR

*which gives the affine transformation A and b used to map the images to 64x64 window

Algorithm:

1. We take the predefined co-ordinates ,p

if

14 20

50 20

34 34

16 50

48 50

2. We take all the feature if files starting from the first feature file and compute the equation

bfAf ip

i * , where, A and b are 6 unkonwns which gives the affine

transformation.

3. We update FBAR each time by using SVD

FBar=Singular_Value_Decomposition(FBar,fp);

1


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4. We take the average of all the FBAR calculated and update FBAR again with thisaverage value

5. Now, we compare the previous result with the current and if its greater than thethreshold error 10^-6 then we come out of the loop and the final converged FBAR gives

the affine transformation matrix A and vectoe b.

2.1.2 Algorithm for Mapping 384x384 image into 64x64 window

Since we have got the matrix A and vector b we can easily put the pixels of 384x384 into

64x64 window. This follows the following algorithm.

1. We now use FBAR to get the values ofA and b. The first 4 values gives the values ofmatrix A and the last two gives the vector b.

x=(V*S*U')*F_BAR; b=x(5:6,1);

a=x(1:4,1); a=a(1:1:4,1); A=zeros(2,2);

A(:,1)=a(1:2,1); A(:,2)=a(3:4,1);

2. Since we know the the transformation matrix so we will plot each pixel of 64x64 windowinto the 384x384 image using.

)( 40961

384384 bFAF x

3. We extract this window image i.e. the transformed image of 64x64.

Normalize image1 Normalize image2

Normalize image3 Normalize image4


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Fig. Normalized and Mapped Images in 64x64window

2.2 Limitations

Non-uniform illumination

Images do not show the correct positions of the features taken for the affine

transform may result in convergence or some bad result.

Chapter 3 Eigen Faces and Eigen Space

3.1. What do they mean?

PCA computes the basis of a space which is represented by its training vectors. Thesebasis vectors, actually eigenvectors, computed by PCA are in the direction of the largestvariance of the training vectors. Each eigenface can be viewed a feature. these are

Eigenvectors and have a face like appearance, they are called Eigenfaces. Sometimes,they are also called as Ghost Images because of their weird appearance. When aparticular 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 spaceby its eigenface coefficients (or weights).

3.2 Algorithm for Creating Eigen Face DataBase

1. We have taken the mean of all the 57 training faces. We substract these values from each

face. We concatenate all the substracted faces in a a single matrix which we will call D.

)64,64(...........................).........1,1(

.

.

.

.

.

)64,64(................................).........1,1(

5757

11

II

II

D


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2. We must use the covariace formula to compute PCA

DDN

CT

1

1

and compute the eigen values and eigen vector whose principal

components will give the eigen faces and hence eigen space. But, if the database has

many images then the situation will become worse and the computation may be very

vague. Even in our case it will give 4096x4096 dimentional matrix. Now, as the number

of non-zero covariance matrx is limited to N (57), we calculate the other way out which

will reduce the dimention but still give the eigen vectors which correspond to the eigen

vectors obtained from this covariance matrix.

3. So, we doT

DDN

C1

1'

. This reduces the dimention to 57x57 in our case. The

Eigenvector computed from this matrix will be of size 57x57

4. Now, we find the eigen faces which is given by multiplying rEigenvectoDT * obtained by

C.

5. Each face in the training set, i ,can be represented as a linear combination of these

Eigenvectors.

.ii X , where, is the principal components

iX is the training images

Subtracted Image1 Subtracted Image2

Subtracted Image3 Subtracted Image4


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Graph: Representing the Eigen Trained Eigen Faces

Fig. Eigen Faces

0 10 20 30 40 50 60-0.5

0

0.5

1

1.5

2

2.5

3x 10

7

1st Eigen Face 8th Eigen Face 10th Eigen Face

12th Eigen Face 20th Eigen Face 30th Eigen Face

35th Eigen Face 40th Eigen Face 57th Eigen Face


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Chapter 4 Recognition of the Face

4.1 Analysis

The face is expressed in the face space by its eigenface coefficients (or weights). We can handlea large input vector now, facial image, only by taking its small weight vector in the face space.

As seen in the previous chapter, we have already found 57 eigen faces of the trained images.Now, when user enters the face, (to be detected), which is unknown and we are taking it as a testset of faces. We follow the following algorithm to find the eigen face related to it.

4.2 Algorithm:1. We normalize the incoming image and map it as done in normalization chapter.2. We now project this normalized image onto the eigen space to get a corresponding

feature vector j as,

jj X

3. We finally find the Euclidean distance betweenj

andi

.

Euclidean Distance: The Euclidean Distance is probably the most widely used distancemetric. It is a special case of a general class of norms and is given as:

2

ii yxyx

4. The minimum distance position between them will give the nearly identical face in theEigen space.

5. We read the corresponding image which will be identical to the test face.

4.3 Result

The result with the 30 test images was not 100% accurate but it gave some good

matching with almost 28 images.

4.3.1 Accuracy

Implementing PCA in the Face recognition on MATLAB, we got nearly 87. 09 %.

%Accuracy= 100*dim

count

agenoofmatche

4.3.2 Limitaitions

The images of a face, and in particular the faces in the training set shouldlie near the face space.

Each image should be highly correlated with itself.


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4.3.2 Scope and Improvement

Further changes in the algorithm may lead to better accuracy. Inaddition, we cantake some more distinct features to the training set of faces like length of forehead, chin

positon etc. Facial recognition is still an ongoing research topic for computer vision

scientists.

Fig. Showing One sample of Matched Output

Fig. Showing One sample of Unmatched Output

Test Image

Matched Image

20 40 60

20

40

60

Image-Matched(:>

20 40 60

20

40

60

Test Image

Matched Image

20 40 60

20

40

60

!!!Miss-Matched(:

20 40 60

20

40

60


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Chapter-5 Conclusion:

1. We must choose some features of the sample face and create a database of the images. In

our case, we have taken 57 face images.

2. Use of the Affine transform in finding the variables responsible for the same orientation,scaling and other feature variations for all the images.

3. The special features taken should be mapped in the window we are taking the face, itshould include most part of the face rather than body.

4. Trained images should be mapped to smaller window.

5. Principal component Analysis can be used to both decrease the computational complexityand measure of the covariance between the images.

6. How PCA reduces the complexity of computation when large number of images aretaken?

7. The principal components of the Eigen vector of this covariance matrix whenconcatenated and converted gives the Eigen faces.

8. These eigenfaces are the ghostly faces of the trained set of faces forms a face space.

9. For each new face(test face), 30 in our case, we need to calculate its pattern vector.

10.The distance of it to the eigen faces in the eigen space must be minimum.

11.This distance gives the location of the image in the eigen space which is taken as theoutput matched image.

REFERENCES[1] Matthew Turk and Alex Pentland, Eigen Faces For Recognition, MIT , The MediaLaboratory[2] Desire Sidebe, Face Recognition Using PCA,Assignment-3 sheet,UB[3] Wikipedia

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PCA_FACE_RECOGNITION_REPORT