HOW Face Recognition Works

Face Recognition 

Face recognition of known person is easy job for a human brain but for a computer system it is bit difficult. Human brain has special nerves that see the world as a whole image not in scatter pieces. Face has some inner features

                                I.            Nose

                              II.            Lips

                            III.            Eyes

And some outer features

                                I.            Head

                              II.            Shape

                            III.            Hairline

Face recognition is all about extracting those feature and recognize the image. In this project two techniques are used to reduce the dimensionality of the image. After reducing the dimensionality fro classification k-nearest neighbor is used.

Face Database

In this project take five pictures of ten people is used while they were talking. Every image is 180×200. Different images of these ten people are taken for the testing purpose. This data base of images is maintained by Dr Libor Spacek.

Reading the images

First step is reading the images from the folder. Give direct path of the training folder. Read all the images, names and label them. After this break the image into a vector and make a matrix which has all images vector. Structure of the matrix is given below.

X=

IMAGE VECTOR 1

IMAGE VECTOR 2

…………...

IMAGE VECTOR n

One vector has dimension 3600×1 and matrix has dimension 3600×50.

Principle Component Analysis

Now every pixel of image is a feature so it is 3600 dimensional space. To reduce the dimension PCA feature extraction is used. Algorithmic steps are given below

1.    Compute the mean of every row.

2.    Compute the Covariance Matrix S

S= X^T X

3.    Compute the Eigen values λi and eigenvectors vi of S

4.    Order the eigenvectors descending by their Eigen value. Find the eigenvectors corresponding to the largest Eigen values. They give the largest change.

5.    Multiply data matrix ‘A’ with matrix of Eigen vector to get ‘W’.

6.    Output matrix calculate by

  Y = W^T (x - µ)

7.    The reconstruction from the PCA is given by

X = W(y + µ)

We know that S= X X^T but if dimension of feature are greater than then sample ‘S’ has dimension 3600×3600. So to reduce the memory requirement, use S= X^T X it has dimension 3600×50. To get the original eigenvectors of S = XX^T with a left multiplication of the data matrix.

    X^TX (Xvi) = λi(Xvi)

Eigen faces are given below.

Figure 1(a)                                          Figure 1(b)

Eigen faces

Fisher discernment analysis

The PCA finds a linear combination of features that maximizes the total variance in data. While this is clearly a powerful way to represent data, it doesn't consider any classes and so a lot of discriminative information may be lost when throwing components away. Imagine a situation where the variance is generated by an external source. The components identified by a PCA do not necessarily contain any discriminative information at all. In order to find the combination of features that separates best between classes FDA is used. The Linear Discriminate Analysis maximizes the ratio of between classes to within classes scatter. In this analysis same class cluster remain together while different classes are as far away as possible from each other.

Algorithic discription is given below.

1.       The scatter matrices SB and SW are calculated as

2.       Fisher's classic algorithm now looks for a projection W, that maximizes the class reparability criterion

There's one problem left to solve: The rank of SW is at most (N-c), with N samples and c classes. In pattern recognition problems the number of samples N is almost always smaller than the dimension of the input data (the number of pixels), so the scatter matrix SW becomes singular. In this was solved by performing a Principal Component Analysis on the data and projecting the samples into the (N-c) dimensional space. A Linear Discriminant Analysis was then performed on the reduced data, because SW isn't singular anymore. The optimization problem can be rewritten as:

The transformation matrix W, that projects a sample into the (c-1)-dimensional space is then given by:

K nearest neighbor

The k-nearest neighbor algorithm (k-NN) is a method for classifying objects based on closest training examples in the feature space.

To see the sample in feature space only three Eigen vector are taken in FLD process. So we can see the sample in 3D space. Plot is given below. In the plot ten objects are shown. Five sample of every object is very close to each other and make a cluster. These objects can easily be differentiated by taking the Euclidian distance from the testing image.

Figure 2 (3 dimensional feature space)

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