Introduction

The idea behind eigenfaces is similar (to a certain extent) to the one behind the periodic signal representation as a sum of simple oscillating functions in a Fourier decomposition. The technique described in this tutorial, as well as in the original papers, also aims to represent a face as a linear composition of the base images (called the eigenfaces).

The recognition/detection process consists of initialization, during which the eigenface basis is established and face classification, during which a new image is projected onto the "face space" and the resulting image is categorized by the weight patterns as a known-face, an unknown-face or a non-face image.

Demonstration

To download the software shown in video for 32-bit x86 platform, click here. It was compiled using Microsoft Visual C++ 2008 and uses GSL for Windows.

Establishing the Eigenface Basis

First of all, we have to obtain a training set of $M$ grayscale face images $I_1, I_2, ..., I_M$. They should be:

1. face-wise aligned, with eyes in the same level and faces of the same scale,
2. normalized so that every pixel has a value between 0 and 255 (i.e. one byte per pixel encoding), and
3. of the same $N \times N$ size.

So just capturing everything formally, we want to obtain a set $\{ I_1, I_2, ..., I_M \}$, where \begin{align} I_k = \begin{bmatrix} p_{1,1}^k & p_{1,2}^k & ... & p_{1,N}^k \\ p_{2,1}^k & p_{2,2}^k & ... & p_{2,N}^k \\ \vdots \\ p_{N,1}^k & p_{N,2}^k & ... & p_{N,N}^k \end{bmatrix}_{N \times N} \end{align} and $0 \leq p_{i,j}^k \leq 255.$

Once we have that, we should change the representation of a face image $I_k$ from a $N \times N$ matrix, to a $\Gamma_k$ point in $N^2$-dimensional space. Now here is how we do it: we concatenate all the rows of the matrix $I_k$ into one big vector of dimension $N^2$. Can it get any more simpler than that?

This is how it looks formally:

Since we are much more interested in the characteristic features of those faces, let's subtract everything what is common between them, i.e. the average face.
The average face of the previous mean-adjusted images can be defined as $\Psi = {{1}\over{M}} \sum_{i=1}^{M} \Gamma_i$, then each face differs from the average by the vector $\Phi_i = \Gamma_i - \Psi$.

Now we should attempt to find a set of orthonormal vectors which best describe the distribution of our data. The necessary steps in this at a first glance daunting task would seem to be:

1. Obtain a covariance matrix
   $C = {{1}\over{M}} \sum_{i=1}^{M} \Phi_i \Phi_i^T = AA^T$, where $A = \left[ \Phi_1 \Phi_2 ... \Phi_M \right]$.
2. Find the eigenvectors $u_k$ and eigenvalues $\lambda_k$ of $C$.

However, note two things here: $A$ is of the size $N^2 \times M$ and hence the matrix $C$ is of the size $N^2 \times N^2$. To put things into perspective - if your image size is $128 \times 128$, then the size of the matrix $C$ would be $16384 \times 16384$. Determining eigenvectors and eigenvalues for a matrix this size would be an absolutely intractable task!

So how do we go about it? A simple mathematical trick: first let's calculate the inner product matrix $L = A^T A$, of the size $M \times M$. Then let's find it's eigenvectors $v_i, i = 1, ..., M$ of $L$ (of the $M$-th dimension). Now observe, that if $L v_i = \lambda_i v_i$, then

and hence $u_i = A v_i$ and $\lambda_i$ are respectively the $M$ eigenvectors (of $N^2$-th dimension) and eigenvalues of $C$. Make sure to normalize $u_i$, such that $\left\| u_i \right\| = 1$.

We will call these eigenvectors $u_i$ the eigenfaces. Scale them to 255 and render on the screen, to see why.

It turns out that quite a few eigenfaces with the smallest eigenvalues can be discarded, so leave only the $R \leq M$ ones with the largest eigenvalues (i.e. only the ones making the greatest contribution to the variance of the original image set) and chuck them into the matrix $U = \left[ u_1 u_2 ... u_R \right]_{N^2 \times R}$

After you have done that - congratulations! We won't need anything else, but the matrix $U$ for the next steps - face detection and classification.

Face Classification Using Eigenfaces

Once the eigenfaces are created, a new face image $\Gamma$ can be transformed into it's eigenface components by a simple operation:

The weights $\omega_i \in \Omega$ describe the contribution of each eigenface in representing the input face image. We can use this vector for face recognition by finding the smallest Euclidean distance $\epsilon_{rec}$ between the input face and training faces weight vectors, i.e. by calculating $\epsilon_{rec} = min \left\| \Omega - \Omega_i \right\|$. If $\epsilon_{rec} < \Theta_{rec}$, where $\Theta_{rec}$ is a treshold chosen heuristically, then we can say that the input image is recognized as the image with which it gives the lowest score.

The weights vector can also be used for an unknown face detection, exploiting the fact that the images of faces do not change radically when projected into the face space, while the projection of non-face images appear quite different. To do so, we can calculate the distance $\epsilon_{det}$ from the mean-adjusted input image $\Phi = \Gamma - \Psi$ and its projection onto face space $\Phi_f = \sum_{i=1}^R \omega_i u_i$, i.e. $\epsilon_{det} = \left\| \Phi - \Phi_f \right\|$. Again, if $\epsilon_{det} < \Theta_{det}$ for some treshold $\Theta_{det}$ (also obtained heuristically, for example, by observing $\epsilon_{det}$ for an input set consisting only of face images and a set of non-face images) we can conclude that the input image is a face.

References

1. Face Recognition Using Eigenfaces, Matthew A. Turk and Alex P. Pentland, MIT Vision and Modeling Lab, CVPR ‘91.
2. Eigenfaces for Recognition, Matthew A. Turk and Alex P. Pentland, Journal of Cognitive Neuroscience ‘91.
3. Eigenfaces. Sheng Zhang and Matthew Turk (2008), Scholarpedia, 3(9):4244.