Summer Science Institute 1995
Weizmann Institue of Science, Israel
Ian Schechter, USA & Michael Vorburger, Switzerland
Mentor: Uri Pryadkin
Previous mathematicians [Turk and Pentland], have
theorized that certain cognitive processes, such as face recognition, can be emulated
through the use of principal component analysis. In the project which we have worked on,
at the International Summer Science Institute, under the supervision of our mentor Uri
Pryadkin, we have attempted to use techniques of principal component analysis, more
specifically, eigen-vector analysis, to develop a computer program capable of face
recognition. Specifically, the goal of our project was to investigate a mathematical basis
and model for face recognition using principal component analysis with eigenvectors, and
then implement a working model in the form of a computer program.
The fundamental idea behind principal component analysis with
eigenvectors has its basis in linear algebra. Put simply, if there are a series of
multi-dimensional vectors representing objects which have similarities, it is possible to
use a transformation matrix and eigenvectors to orient a space which requires fewer
dimensions to accurately describe these multidimensional vectors. For instance, if in
three dimensional space, there was a cloud of particles that lied in a two dimensional
plane skewed from the axes (Fig 1), it would be possible to orient a new space with a new
origin and new unit vectors such that the cloud which previously required a three
dimensional representation could now easily be represented in only two dimensions.
Figure 1: Series of objects in 3D space which can be represented by 2D
The new unit vectors of the lower dimensional space, which now
sufficiently describes where the points are located, can be derived by normalizing the
eigenvectors of the transformation matrix used to create the new origin. Lower
eigenvalues, that is eigenvalues approaching zero, correspond to unit vectors in
dimensions which dont have to be represented. Consequently, the vectors
corresponding to these eigenvlaues can be disregarded thus reducing the dimensions of the
space even further.
The task of having computers capable of recognizing faces, without
using principal component analysis, is a formidable one. To a computer, a face, like any
other image, is a matrix of several hundred pixels by several hundred pixels. Dealing with
many faces , in the form of pictures, can be very time consuming and difficult. If,
however, one applies principal component analysis, the task becomes much more manageable.
If the pictures pixel matrix is turned into a vector, then that picture will have a
location vector in a hundred thousand dimensional space. If the same is done with other
pictures, their location vectors will wind up in the same area of this huge
multi-dimensional space. We call this small subsection of the multi-dimensional space the
"face space." Then using eigenvectors to create a new basis for the representing
the images, one can now represent images as coordinates in a much smaller dimensional
space. Images of the same person, will, of course, be located nearer to one another in
this space, as they are similar to one another. At this level, the task of recognition
becomes merely a matter of determining the location of the vector form of a new picture in
a lower dimensional space relative to pictures of recognized faces.
2.2 Initialization with Training Set
In order to apply the ideas of ideas of principal component analysis to
face recognition, we had to take care of preliminaries. Firstly, the faces being discussed
are nine 512*352 pixel 8-bit TIFF images, consisting of three separate images for three
separate people, which we converted into vectors of 180,224 dimensions. Our first task was
to find an average face, y. This was easily
done by taking the arithmetic mean of the face vector.
Figure 2: The "average face"y
of our nine sample faces
We then constructed a matrix A, which consisted of the differences
between each face and the average face. Then, by creating a coovariance matrix C = A *
A, we can easily find the eigenvectors of C, which by proof [Turk and Pentland,
p.6], will span the "face space." The eigenvectors whose corresponding
eigenvalues are less than one can be discarded. Then, if the eigen vectors are normalized
(ie. divided by their length), they will now, also by proof [ibid], compose the basis of
this lesser dimensional space. Also, they will be orthogonal and orthonormal. Now, instead
of being a 180,224 byte matrix, the images we used became eight numerical
co-ordinates in our new "face space"!
2.3 Recognizing Faces
Once faces can be represented as coordinate vectors in a low
dimensional space, the task of face recognition itself is rendered trivial. First we look
at the placement of different vectors representing different pictures of the same person.
We average these vectors to find the "center" image of this person and then
calculate a "radius" from this center to the different images. New pictures are
broken down into lower dimensional vectors using simular techniques as used to establish
the coordinates of the test faces. Essentially, we make a vector and multiply by the
eigenvectors to determine coordinates. Now, face recognition simply becomes the
mathematical calculation of distance, to see whether or not a new face falls into a
"sphere of recognition" of a previously recognized face.
3.1 Using the Algorithm for Face Recognition
We did ... phase1.. This took a few hours to run on a Silicon Graphics,
which was running "some" other processes at this time. Here are three of the
nine images; theses are of the same person:
Figure 3: Three of the nine images in the training set
To recognize an image, we wrote a new function...
Here is another image of that person. After running for some minutes,
our program identified him as being the same as the three above.
Figure 4: A new image that our program identified well
3.2 Using the Algorithm for Image Compression
The idea that faces can be represented with only a few
"coordinates" and the principal components, which are only stored once, inspired
us to use this method for image compression. In fact, [Turk, Pentland, p.5] refer to
[Sirovich, Kirby] and state that this algorithm was originally designed for compression!
Here are the results of three experiments that we performed with
1. We took a completely new face of a person that was not in the
training set, calculated its (in our example) eight weights describing the position of
the face in the "face space" and rebuilt the image using the linear combination
of the eight principle components. What comes out is the following:
Figure 4: A reconstructed face of a person that was not in the training
As one can see, the result is very poor. It has to be said that this
could probably be improved by using more than eight principle components (eigenfaces).
[Turk, Pentland] speaks of using about 40 eigenfaces.
2. In another decomposing/rebuilding experiment, we did the same with
an unknown face of a person of whom we have had three different images in the training
set. The following figure shows that the result is much better in this case:
Figure 5: A reconstructed face of a person that was in the training set
(with different faces)
3. As a third experiment, we decomposed/rebuilt the image of a face
that was itself part of the initial training set. As one can imagine, the quality of this
image was perfect; no difference from the original picture could be seen.
4.1 Suitability of MATLAB
We were using MATLAB for the realization of our project. Though we had
no previous knowledge and expierence with this computer algebra system, we found it very
easy to learn and useful for experimenting. Especially calculations invoking matrices and
vectors are very simple. Still, it has to be stated that we cannot think of using MATLAB
in a real application, say for a security system, because it is far too slow. We would use
a C++ program for this purpose; the matrix calculations could be simplified and optimized
by using an industry standard math library such as Matrix.h++ from RogueWave Sofware Inc.
4.2 Efficiency of the Algorithm & Suitability for Industry Use
Recognizing faces could theoretically (see previous chapter concerning
MATLAB or C++) be very fast, because it consists only of a matrix subtraction, one matrix
multiplication and a bit of comparison. On the other hand, training new faces is a
comparably complicated calculation, especially if the "face space" has to be
recalculated as soon as new persons are added. We would therefore say that this method
could be useful in a serious industry application only to recogninze a group of people
that does not often change after initial training.
We want to thank our Mentor Uri Pryadkin and the staff of the Summer
 Matthew Turk, Alex Pentland: Eigenfaces for Recognition; Vision and
Modeling Group, The Media Laboratory, Massachusetts Institute of Technology; September
 L. Sirovich and M. Kirby: Low-dimensional procedure for the characterization of
human faces; J. Opt. Soc. Am. A, Vol. 4, No. 3, March 1987, 519-524.
 M. Kirby and L. Sirovich: Application of the Karhunen-Loeve procedure for the
characterization of human faces; IEEE Trans. Pattern Analysis and Machine Intelligence,
Vol 12, No. 1, Jan. 1990.
7 Appendix: MATLAB Source Files
see MATLAB scripts (http://www.vorburger.ch/projects/faces/matlab.html)
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