Manfredas Zabarauskas' Blog » eigenfaces http://blog.zabarauskas.com We are what we repeatedly do; excellence, then, is not an act but a habit. -- Aristotle Fri, 04 Jun 2010 12:26:10 +0000 http://wordpress.org/?v=2.8 en hourly 1 Eigenfaces Tutorial http://blog.zabarauskas.com/eigenfaces-tutorial/ http://blog.zabarauskas.com/eigenfaces-tutorial/#comments Fri, 02 Oct 2009 16:43:22 +0000 Manfredas Zabarauskas http://blog.zabarauskas.com/?p=286 The main purpose behind writing this tutorial was to provide a more detailed set of instructions for someone who is trying to implement an eigenface based face detection or recognition systems. It is assumed that the reader is familiar (at least to some extent) with the eigenface technique as described in the original M. Turk and A. Pentland papers (see “References” for more details).

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 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} 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:

\Gamma_k = \begin{bmatrix} p_{1,1}^k \\ p_{1,2}^k \\ \vdots \\ p_{1,N}^k \\ p_{2,1}^k \\ p_{2,2}^k \\ \vdots \\ p_{2,N}^k \\ \vdots \\ p_{N,1}^k \\ p_{N,2}^k \\ \vdots \\ p_{N,N}^k \end{bmatrix}_{N \times 1}, where k = 1, ..., M and p_{i,j}^k \in I_k

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

\begin{array} {rcl} A L v_i &=& \lambda_i A v_i \Rightarrow \\ A A^T A v_i &=& \lambda_i A v_i \Rightarrow \\ C A v_i &=& \lambda_i A v_i \end{array},

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 \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:

\Omega = U^T (\Gamma - \Psi) = \begin{bmatrix} \omega_1 \\ \omega_2 \\ \vdots \\ \omega_R \end{bmatrix}_{R \times 1}.

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.

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