3 Face recognition tasks

Given a database consisting of a set, , of N known people, different face recognition tasks can be envisaged. Four tasks are defined here as follows:

1. Face classification: The task is to identify the subject under the assumption that the subject is a member of .
2. Known/Unknown: The task is to decide if the subject is a member of .
3. Identity verification: The subject's identity is supplied by some other means and must be confirmed. This is equivalent to task 2 with N=1.
4. Full recognition: The task is to determine whether or not the subject is a member of , and if so to determine the subject's identity.

When considering appearance-based approaches to these tasks it is helpful to know something of the topology of sets of face images in an image space. The set of all faces forms a small number of extended, connected regions . Furthermore, a face undergoing transformations such as rotation, scaling and translation results in a connected but strongly non-convex subregion in the image space. Whilst these transformations might be approximately corrected using linear image-plane transformations, large rotations in depth, illumination changes and facial expressions cannot be so easily ``normalised''. Therefore, the set of images of a single face will form at least one and possibly several, highly non-convex, connected regions in image space.

Figure 4: Plotted in a hypothetical face space, , are example faces from 3 different people. Suitable decision boundaries are shown for the four recognition tasks.

Figure 4 illustrates the four recognition tasks defined above in a hypothetical face space , where is assumed to contain all possible face images and to exclude all other images. Plotted in are example faces for three different people. Suitable decision boundaries for performing the recognition tasks are shown. The separability of face identities in will depend upon the technique used to model . However, it is likely that each identity will form strongly non-convex regions in this subspace. In the face classification task, all N classes can be modelled. In contrast, the other three tasks all suffer from the need to consider the class of unknown faces. Each task will now be discussed in greater detail.

3.1 Face classification

The face classification task is an N-class classification problem in which all N classes can be modelled. It can be tackled by collecting representative data for each of the N classes and applying one of many possible pattern classification techniques. The probability of misclassifying a face x is minimised by assigning it to the class with the largest posterior probability , where

p(x) is the unconditional density, is the class-conditional density and is the prior probability for class . Since p(x) is the same for every class it need not be evaluated in order to maximise posterior probability [3]. Therefore, one approach to the classification task is to model the class-conditional probability densities, , for each class. This approach is explored in this work. An alternative approach is to estimate discriminant functions using e.g. Linear Discriminant Analysis (LDA) [4].

3.2 Face verification

Face verification can be treated as a 2-class classification problem. The two classes and correspond to the cases where the claimed identity is true and false respectively. In order to maximise the posterior probability, x should be assigned to if and only if

 

Density represents the distribution of faces other than the claimed identity. This is difficult to model but a simple assumption is that it is constant over the relevant region of space, falling to zero elsewhere. In this case, Inequality (7) is equivalent to thresholding . Perhaps a more accurate assumption is that the density is smaller in regions of space where is large. If is chosen to be of the form , where F is a monotonically decreasing function, then this assumption is also equivalent to thresholding . In this case, the threshold takes the form , where . Since G is monotonic, is unique. Utilising only data from class , it is therefore reasonable to perform verification by thresholding .

In order to achieve more accurate verification, negative data, i.e. data from class , would need to be used in order to better estimate the decision boundaries. Only data which are ``close'' to are relevant here. An iterative learning approach can be used in which incorrectly classified unknown faces are selected as negative data. Furthermore, the face images used to train the face detection network also provide a suitable source of negative examples for identity verification [8].

3.3 Known/Unknown

This task can also be treated as a 2-class classification problem. The two classes and correspond to the cases where the subject is and is not a member of the known group , respectively. The methods discussed above for face verification can be similarly applied to this 2-class problem.

A slightly different approach involves building an identity verifier for each person in . The known/unknown task is performed by carrying out N identity verifications. If the numerator in the threshold of Inequality (7) is the same for all verifiers then they can be combined in a straightforward manner.

3.4 Full recognition

The full recognition task can be performed by combining N identity verifiers similarly to the second approach described above for known/unknown.