PCA Based Face

Recognition SystemMD. ATIQUR RAHMAN

Face Recognition using PCA Algorithm

PCA-

Principal Component Analysis

Goal-

Reduce the dimensionality of the data by retaining as much as variation

possible in our original data set.

The best low-dimensional space can be determined by best principal-

components.

Eigenface Approach

Pioneered by Kirby and Sirivich in 1988

There are two steps of Eigenface Approach

Initialization Operations in Face Recognition

Recognizing New Face Images

Steps

Initialization Operations in Face Recognition

Prepare the Training Set to Face Vector

Normalize the Face Vectors

Calculate the Eigen Vectors

Reduce Dimensionality

Back to original dimensionality

Represent Each Face Image a Linear Combination of all K Eigenvectors

Recognizing An Unknown Face

Prepare the Training

Set to Face Vector

………..

112 × 92

10304 × 1𝜞𝒊

Face vector space

Images converted to vector

Each Image size

column vector

𝑀= 16 images in the training set Convert each of face images in

Training set to face vectors

Normalize the Face

Vectors

Average face vector/Mean image (𝜳)

𝑀= 16 images in the training set

……….. 𝜳

Converted

Face vector space

Mean Image 𝜳

𝜞𝒊

Calculate Average face vector

Save it into face vector space

Subtract the Mean from each Face Vector

………..

Ф𝒊

𝜳

Converted

Face vector space

𝑀= 16 images in the training set

− =

𝜞𝟏 𝚿 Ф𝟏

Normalized Face vector

Result of Normalization

Figure: Normalized Data set

Calculate the Eigen

Vectors

Calculate the Covariance Matrix (𝑪)

C = 𝑛=116 Ф𝑛Ф𝑛

𝑇

= 𝐴𝐴𝑇

= {(𝑁2×𝑀). (𝑀 × 𝑁2)}

= 𝑁2× 𝑁2

= (10304 × 10304)

Where 𝐴 = {Ф1, Ф2, Ф3, ……… ., Ф16}

[𝐀 = 𝐍𝟐 ×𝐌]……….. 𝜳

Ф𝒊

Face vector space

Converted

𝑀= 16 images in the training set

Converted

C = 10304 × 10304

10304 eigenvectors

………

Each 10304×1 dimensional

……….. 𝜳

Ф𝒊

Face vector space

𝒖𝒊

Converted

𝑀= 16 images in the training set

In 𝑪, 𝑵𝟐 is creating 𝟏𝟎𝟑𝟎𝟒 eigenvectors

Each of eigenvector size is 𝟏𝟎𝟑𝟎𝟒 × 𝟏 dimensional

Calculate Eigenvector (𝒖𝒊)

C = 10304 × 10304

10304 eigenvectors

Each 10304×1 dimensional

……….. 𝜳

Ф𝒊

Face vector space

Converted

………

𝒖𝒊

𝑀= 16 images in the training set

Find the Significant 𝑲𝒕𝒉 eigenfaces

Where, 𝑲 < 𝑴

C = 10304 × 10304

10304 eigenvectors

Each 10304×1 dimensional

……….. 𝜳

Ф𝒊

Face vector space

Converted

………

𝒖𝒊

𝑀= 16 images in the training set

Make system slow

Required huge calculation

Reduce

Dimensionality

Consider lower dimensional subspace

……….. 𝜳

Ф𝒊

Lower dimensional Sub-space

Face vector space

Converted

𝑀= 16 images in the training set

𝑳 = 𝑨𝑻𝑨

= 𝑴×𝑵2 𝑵2 ×𝑴

= 𝑴×𝑴

= 16 × 16…… . .

16 eigenvectors

Each 16 ×1 dimensional

Calculate eigenvectors 𝒗𝒊

𝒗𝒊

……….. 𝜳

Ф𝒊

Lower dimensional Sub-space

Face vector space

Converted

𝑀= 16 images in the training set

Calculate Co-variance matrix(𝑳)

of lower dimensional

𝑳 = 𝑨𝑻𝑨

= 𝑴×𝑵2 𝑵2 ×𝑴

= 𝑴×𝑴

= 16 × 16…… . .

16 eigenvectors

Each 16 ×1 dimensional

𝒖𝒊 V/S 𝒗𝒊

𝒗𝒊

……….. 𝜳

Ф𝒊

Lower dimensional Sub-space

Face vector space

Converted

10304 eigenvectors

………

Each 10304×1 dimensional

𝒖𝒊

C = 10304 × 10304

v/s

𝑀 images in the training set

𝑳 = 𝑨𝑻𝑨

= 𝑴×𝑵2 𝑵2 ×𝑴

= 𝑴×𝑴

= 16 × 16

Select K best eigenvectors

……….. 𝜳

Ф𝒊

Lower dimensional Sub-space

Face vector space

Converted

…… . .

16 eigenvectors

Each 16 ×1 dimensional

𝒗𝒊

Selected K eigenfaces MUST be inThe ORIGINAL dimensionality of theFace vector space

Back to Original

Dimensionality

𝑳 = 𝑨𝑻𝑨

= 𝑴×𝑵2 𝑵2 ×𝑴

= 𝑴×𝑴

= 16 × 16

……….. 𝜳

Ф𝒊

Lower dimensional Sub-space

Face vector space

Converted

…… . .

16 eigenvectors

Each 16 ×1 dimensional

𝒗𝒊

A=

𝒖𝒊 = 𝑨𝒗𝒊

10304 eigenvectors

………

Each 10304×1 dimensional

𝒖𝒊

𝑀= 16 images in the training set

𝑳 = 𝑨𝑻𝑨

= 𝑴×𝑵2 𝑵2 ×𝑴

= 𝑴×𝑴

= 16 × 16

……….. 𝜳

Ф𝒊

Lower dimensional Sub-space

Face vector space

Converted

…… . .

16 eigenvectors

Each 16 ×1 dimensional

𝒗𝒊

A=

𝒖𝒊 = 𝑨𝒗𝒊

10304 eigenvectors

………

Each 10304×1 dimensional

𝒖𝒊

𝑀 images in the training set

C = 𝐴𝐴𝑇

10304 eigenvectors

………

Each 10304×1 dimensional

𝒖𝒊

The K selected eigenface

……….. 𝜳

Ф𝒊

Face vector space

Converted

𝑀= 16 images in the training set

Result of Eigenfaces Calculation

Figure: The selected K eigenfaces of our set of original images

Represent Each Face Image

a Linear Combination of all

K Eigenvectors

𝛚𝟏 𝛚𝟐 𝛚𝟑 𝛚𝟒 𝛚𝟓 𝛚𝐊⋯⋯⋯

+ 𝜳 (Mean Image)

Each face from Training set can be represented a weighted sum of the K Eigenfaces + the Mean face

𝛚𝟏 𝛚𝟐 𝛚𝟑 𝛚𝟒 𝛚𝟓 𝛚𝐊⋯

+ 𝜳 (Mean Image)

The K selected eigenface

Each face from Training set can be represented aweighted sum of the K Eigenfaces + the Mean face

………..

Ф𝒊

𝜳

Converted

Face vector space

𝑀= 16 images in the training set

Weight Vector (𝜴𝒊)

𝛚𝟏 𝛚𝟐 𝛚𝟑 𝛚𝟒 𝛚𝟓 𝛚𝐊⋯

+ 𝜳 (Mean Image)

= 𝜴𝒊 =

𝝎1𝒊

𝝎2𝒊

𝝎3𝒊

.

.

.𝝎𝑲𝒊

Each face from Training set can be represented aweighted sum of the K Eigenfaces + the Mean face

A weight vector 𝛀𝐢 which is the eigenfaces representation of

the 𝒊𝒕𝒉 face. We calculated each faces weight vector.

Recognizing An Unknown Face

Convert the

Input to Face

Vector

Normalize the

Face Vector

Project Normalize Face Vector onto the Eigenspace

Get the Weight

Vector

𝜴𝒏𝒆𝒘 =

𝝎𝟏𝝎𝟐𝝎𝟑...𝝎𝑲

Euclidian Distance

(E) = (𝛀𝒏𝒆𝒘 −𝛀𝒊)

If

𝑬 < 𝜽𝒕

No

Unknown

Yes

Input of a unknown Image

Recognized as

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