Face Detection Using PCA

8/3/2019 Face Detection Using PCA

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Face Detection using Principal

Component Analysis

Abhishek Roy Anuj Jain Arpit Gupta

2K3/EC/603 2K3/EC/613 2K3/EC/614

Guide: Ms S Indu, ECE Dept, DCE


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To do face detection using Principal Component Analysis (PCA).

Objective


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Why face detection ?

Various potential applications, such as

Applications in biometrics

Video surveillance

Human-computer interaction.

Image database management systems


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Principal Components Analysis

It is a way of identifying patterns in data, and expressing the

data in such a way as to highlight their similarities and

differences.

Eigenvector approach Data Compression


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Some important terms

Standard Deviation -- a measure of how spread out the

data is.

Variance -- is the square of standard deviation.

These two measures are purely 1-dimensional.


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Covariance -- is always measured between 2 dimensions.

Covariance is the measure of how much two random variables

vary together. If the value is +ve then both dimensions increase

together & if ve, they are inversely proportional.


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Covariance Matrix

where S is the covariance matrix, sjk

is the covariance of

variables Xj

and Xk

when j

k and the diagonal element

sjj

is

the variance of variable Xj

when j = k.


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Method

1. Obtain the data in the form of a matrix.

2. Obtain the covariance matrix for the data.

3. Calculate the eigenvectors and eigenvalues of the covariance

matrix. The eigenvector with the highesteigenvalue is the

principle componentof the data set.

4. Select the eigenvectors with the highest eigenvalues and

form a matrix which is smaller in size compared to the data

matrix.


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PCA projects the data along the directions

where the data varies the most.

Directions are determined by the eigenvectors of the

covariance matrix corresponding to the largest eigenvalues


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EIGENFACES

Set of eigenvectors

Develpoed by Matthew Turk and Alex Pentland

Eigenfaces can be extracted out of the image

data by means of the mathematical tool called

Principal Component Analysis (PCA)
http://en.wikipedia.org/wiki/Image:Eigenfaces.png


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Face detection based on eigenface approach

Acquire a set of training images of same size.

Calculate the covariance matrix for the training set.

Calculate the eigenfaces from the covariance matrix,

keeping only the bestMimages with the highest eigenvalues.

TheseMimages define the face space.


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Given an image, calculate a set of weights of theM

eigenfaces by projecting it onto each of the eigenfaces

Determine if the image is a face at all by checking tosee if the image is sufficiently close to the face space


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Image Representation

A square, N by N image can be expressed as an N2dimensional vector.

Say we have 20 images. Each image is N

pixels high by

N pixels wide. Write each image as an image vectorand put all the images together in one big image-matrixlike this:


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The Space of Faces

An image is a point in a high dimensional

space

x is an image of N pixels anda point in N-dimensional space

x

Pixel2

gray

value

Pixel 1 gray value


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Training Set


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Experiment

JAFFE image database 20 images(M) 256 x 256 (N x N)

Each image converted into a column vector of N2 x 1

Column vectors stacked together to get a matrix (N2 x M)

Covariance matrix needed (N2 x N2) over 4 billionentries

Calculate eigenvectors of the covariance matrix


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Problem: Size of Covariance Matrix C

Each data point is N2 -dimensional (N2 pixels)

The size of covariance matrix A is N2 x N2

The number of eigenfaces is N2

Example: For N2 = 256 x 256 pixels,

Size of C will be 65536 x 65536 !

Number of eigenvectors will be 65536 !

Typically, only a few eigenvectors suffice. So, this methodis very inefficient!


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Efficient Computation of Eigenvectors

If B is N2xM and MMxM

M number of images, N2 number of pixels

use BTB instead, eigenvector of BTB is easily

converted to that of BBT

(BTB) y = e y

=> B(BTB) y = e (By)

=> (BBT)(By) = e (By)

=> By is the eigenvector of BBT


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Eigenfaces PCA extracts the Eigenfaces of the set of images

Gives a set of vectors v1,v2,v3

Each vector represents a dimension in the face space

What do they look like?


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Projecting an image onto face space

The eigenfaces v1vk span the face space.

A face is projected onto eigen coordinates by

)

a9v9 a10v10


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Projections


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Procedure

1. Process the image database (training set of images)

Run PCAcompute eigenfaces

2. Given a new image (to be detected) x, calculate K coefficients

3. Detect if x is a face


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Detection

Result


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Result

Image name Distance

Images part of the training set

KA.HA1.29 3.0430e+012

MK.NE3.115 1.4271e+012

Images not part of training set but belonging to people in the

set

YM.HA2.53 4.4622e+012

KA.NE2.27 1.3121e+012


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Images with no relation to the training set

Containing a face

lena 2.6338e+012

Not containing a face

nens 7.9097e+012

wheel 8.8347e+012grey 3.4745e+012

2356R 8.9020e+012

CT_scan 5.9903e+012

Lone 3.8256e+012


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Threshold value

Threshold value = 3.4745e+12 (corresponding to image grey)

fails to detect a face in the image -- YM.HA2.53.

If the threshold value is increased to the distance corresponding

to the image -- YM.HA2.53, then 100% detection takes place but

accuracy goes down as the program will detect faces in images

with no human face in them.

Also, the image nens consists of a number of faces, but they are

not detected.


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Implementation

MATLAB

Imread

Reshape

DoubleClear

Eig

Zeros

Pdist

Uint8


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Eigenfaces summary in words

Eigenfaces are the eigenvectors of the covariance matrixobtained from the training set.

Eigenfaces are the standardized face ingredients derivedfrom the statistical analysis of many pictures of humanfaces

A human face may be considered to be a combination ofthese standardized faces


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Conclusion

Project images onto a low-dimensional linear subspace

face space, defined by eigenfaces.

The distance between an image and its projection in face

space is compared to a threshold value determined

experimentally.

This approach was tested on a number of images giving gooddetection results.


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Advantage

Ease of implementation

Simplicity of the maths behind the concept.

No knowledge of geometry or specific feature of the

face is required.


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Limitation

Applicable only to front views

Input images should be of the same type(size, color,

etc) as the images in the trainig set

Refrences


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Refrences

1.Eigenfaces for recognition, M. Turk and A. Pentland

2. Face recognition using eigenfaces, M. Turk and A. Pentland

3. Stellar Spectral Classification using Principal Component

Analysis and artificial neural networks, Harinder P Singh, Ravi KGulati and Ranjan Gupta

4. http://en.wikipedia.org/wiki/Eigenface

5. http://en.wikipedia.org/wiki/Principal_Component_Analysis

6. Advanced Engineering Mathematics by Erwin Kreyszig

7. JAFFE image database "Coding Facial Expressions with Gabor

Wavelets, Michael J. Lyons, Shigeru Akamatsu, Miyuki Kamachi,

Jiro Gyoba

Date post:	07-Apr-2018
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Face Detection Using PCA

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