An eigenface is the name given to a set of eigenvectors when used in the computer vision problem of human face recognition.[1] The approach of using eigenfaces for recognition was developed by Sirovich and Kirby and used by Matthew Turk and Alex Pentland in face classification.[2] [3] The eigenvectors are derived from the covariance matrix of the probability distribution over the high-dimensional vector space of face images. The eigenfaces themselves form a basis set of all images used to construct the covariance matrix. This produces dimension reduction by allowing the smaller set of basis images to represent the original training images. Classification can be achieved by comparing how faces are represented by the basis set.
The eigenface approach began with a search for a low-dimensional representation of face images. Sirovich and Kirby showed that principal component analysis could be used on a collection of face images to form a set of basis features. These basis images, known as eigenpictures, could be linearly combined to reconstruct images in the original training set. If the training set consists of M images, principal component analysis could form a basis set of N images, where N < M. The reconstruction error is reduced by increasing the number of eigenpictures; however, the number needed is always chosen less than M. For example, if you need to generate a number of N eigenfaces for a training set of M face images, you can say that each face image can be made up of "proportions" of all the K "features" or eigenfaces: Face image1 = (23% of E1) + (2% of E2) + (51% of E3) + ... + (1% En).
In 1991 M. Turk and A. Pentland expanded these results and presented the eigenface method of face recognition. In addition to designing a system for automated face recognition using eigenfaces, they showed a way of calculating the eigenvectors of a covariance matrix such that computers of the time could perform eigen-decomposition on a large number of face images. Face images usually occupy a high-dimensional space and conventional principal component analysis was intractable on such data sets. Turk and Pentland's paper demonstrated ways to extract the eigenvectors based on matrices sized by the number of images rather than the number of pixels.
Once established, the eigenface method was expanded to include methods of preprocessing to improve accuracy.[4] Multiple manifold approaches were also used to build sets of eigenfaces for different subjects[5] [6] and different features, such as the eyes.[7]
A set of eigenfaces can be generated by performing a mathematical process called principal component analysis (PCA) on a large set of images depicting different human faces. Informally, eigenfaces can be considered a set of "standardized face ingredients", derived from statistical analysis of many pictures of faces. Any human face can be considered to be a combination of these standard faces. For example, one's face might be composed of the average face plus 10% from eigenface 1, 55% from eigenface 2, and even −3% from eigenface 3. Remarkably, it does not take many eigenfaces combined together to achieve a fair approximation of most faces. Also, because a person's face is not recorded by a digital photograph, but instead as just a list of values (one value for each eigenface in the database used), much less space is taken for each person's face.
The eigenfaces that are created will appear as light and dark areas that are arranged in a specific pattern. This pattern is how different features of a face are singled out to be evaluated and scored. There will be a pattern to evaluate symmetry, whether there is any style of facial hair, where the hairline is, or an evaluation of the size of the nose or mouth. Other eigenfaces have patterns that are less simple to identify, and the image of the eigenface may look very little like a face.
The technique used in creating eigenfaces and using them for recognition is also used outside of face recognition: handwriting recognition, lip reading, voice recognition, sign language/hand gestures interpretation and medical imaging analysis. Therefore, some do not use the term eigenface, but prefer to use 'eigenimage'.
To create a set of eigenfaces, one must:
λ
| (λ1+λ2+...+λk) | |
| v |
>\epsilon
These eigenfaces can now be used to represent both existing and new faces: we can project a new (mean-subtracted) image on the eigenfaces and thereby record how that new face differs from the mean face. The eigenvalues associated with each eigenface represent how much the images in the training set vary from the mean image in that direction. Information is lost by projecting the image on a subset of the eigenvectors, but losses are minimized by keeping those eigenfaces with the largest eigenvalues. For instance, working with a 100 × 100 image will produce 10,000 eigenvectors. In practical applications, most faces can typically be identified using a projection on between 100 and 150 eigenfaces, so that most of the 10,000 eigenvectors can be discarded.
Here is an example of calculating eigenfaces with Extended Yale Face Database B. To evade computational and storage bottleneck, the face images are sampled down by a factor 4×4=16.
Performing PCA directly on the covariance matrix of the images is often computationally infeasible. If small images are used, say 100 × 100 pixels, each image is a point in a 10,000-dimensional space and the covariance matrix S is a matrix of 10,000 × 10,000 = 108 elements. However the rank of the covariance matrix is limited by the number of training examples: if there are N training examples, there will be at most N − 1 eigenvectors with non-zero eigenvalues. If the number of training examples is smaller than the dimensionality of the images, the principal components can be computed more easily as follows.
Let T be the matrix of preprocessed training examples, where each column contains one mean-subtracted image. The covariance matrix can then be computed as S = TTT and the eigenvector decomposition of S is given by
Svi=
| Tv | |
| TT | |
| i |
=λivi
| TTu | |
| T | |
| i |
=λiui
| TTu | |
| TT | |
| i |
=λiTui
Let denote the data matrix with column as the image vector with mean subtracted. Then,
covariance(X)=
| XXT | |
| n |
Let the singular value decomposition (SVD) of be:
X=U{\Sigma}VT
XXT
XXT=U{\Sigma}{{\Sigma}T}UT=U{Λ}UT
XXT
The eigenfaces = the first
k
k\leqn
U
The ith eigenvalue of
XXT=
| 1 | |
| n |
(
X)2
Using SVD on data matrix, it is unnecessary to calculate the actual covariance matrix to get eigenfaces.
Facial recognition was the motivation for the creation of eigenfaces. For this use, eigenfaces have advantages over other techniques available, such as the system's speed and efficiency. As eigenface is primarily a dimension reduction method, a system can represent many subjects with a relatively small set of data. As a face-recognition system it is also fairly invariant to large reductions in image sizing; however, it begins to fail considerably when the variation between the seen images and probe image is large.
To recognise faces, gallery images – those seen by the system – are saved as collections of weights describing the contribution each eigenface has to that image. When a new face is presented to the system for classification, its own weights are found by projecting the image onto the collection of eigenfaces. This provides a set of weights describing the probe face. These weights are then classified against all weights in the gallery set to find the closest match. A nearest-neighbour method is a simple approach for finding the Euclidean distance between two vectors, where the minimum can be classified as the closest subject.
Intuitively, the recognition process with the eigenface method is to project query images into the face-space spanned by eigenfaces calculated, and to find the closest match to a face class in that face-space.
U\in\Ren
M
Then form a weight vector
W=[w1,w2,...,wk,...,wn]
Wm
d<\epsilon1
\epsilon1<d<\epsilon2
d>\epsilon2,U
The weights of each gallery image only convey information describing that image, not that subject. An image of one subject under frontal lighting may have very different weights to those of the same subject under strong left lighting. This limits the application of such a system. Experiments in the original Eigenface paper presented the following results: an average of 96% with light variation, 85% with orientation variation, and 64% with size variation.
Various extensions have been made to the eigenface method. The eigenfeatures method combines facial metrics (measuring distance between facial features) with the eigenface representation. Fisherface uses linear discriminant analysis[9] and is less sensitive to variation in lighting and pose of the face. Fisherface uses labelled data to retain more of the class-specific information during the dimension reduction stage.
A further alternative to eigenfaces and Fisherfaces is the active appearance model. This approach uses an active shape model to describe the outline of a face. By collecting many face outlines, principal component analysis can be used to form a basis set of models that encapsulate the variation of different faces.
Many modern approaches still use principal component analysis as a means of dimension reduction or to form basis images for different modes of variation.
Eigenface provides an easy and cheap way to realize face recognition in that:
However, the deficiencies of the eigenface method are also obvious:
To cope with illumination distraction in practice, the eigenface method usually discards the first three eigenfaces from the dataset. Since illumination is usually the cause behind the largest variations in face images, the first three eigenfaces will mainly capture the information of 3-dimensional lighting changes, which has little contribution to face recognition. By discarding those three eigenfaces, there will be a decent amount of boost in accuracy of face recognition, but other methods such as fisherface and linear space still have the advantage.